XIII Summer Workshop in Mathematics
XIII Summer Workshop in Mathematics
Book of Abstracts
Coordinators
Jaqueline Godoy Mesquita (Universidade de Brasilia, Brazil)
Ma To Fu (Universidade de Brasilia, Brazil)
Scientific Committee
Chang Dorea (Universidade de Brasilia, Brazil)
Evgeny Khukhro (University of Lincoln, United Kingdom)
Francesco Russo (ENSTA/Paris, France)
Jose Espinar (Universidade de Cadiz, Spain)
Keti Tenenblat (Universidade de Brasilia, Brazil)
Liliane de Almeida Maia (Universidade de Brasilia, Brazil)
Maurıcio Ayala (Universidade de Brasilia, Brazil)
Pavel Shumyatsky (Universidade de Brasilia, Brazil)
Jinyun Yuan (Universidade Federal do Parana, Brazil)
Organizing Committee
Alberto Ohashi (Universidade de Brasilia, Brazil)
Daniele Nantes (Universidade de Brasilia, Brazil)
Hemar Godinho (Universidade de Brasilia, Brazil)
Irina Sviridova (Universidade de Brasilia, Brazil)
Joao Paulo dos Santos (Universidade de Brasilia, Brazil)
Lucas Seco (Universidade de Brasilia, Brazil)
Luıs Henrique de Miranda (Universidade de Brasilia, Brazil)
Matheus Bernardini (Universidade de Brasilia, Brazil)
Raquel Dorr (Universidade de Brasilia, Brazil)
Yuri Sobral (Universidade de Brasilia, Brazil)
Local Committee
Aryel Kathleen Araujo - Web Designer (Universidade de Brasilia, Brazil)
Carol Lafeta - Logistical Support (Universidade de Brasilia, Brazil)
Claudia Queiroz - Secretary (Universidade de Brasılia, Brazil)
Felipe Oliveira Lima - Logistical Support (Universidade de Brasılia, Brazil)
Gabriel Nobrega Bufolo - Logistical Support (Universidade de Brasılia, Brazil)
Gabriella Magalhaes Valadares - Logistical Support (Universidade de Brasılia, Brazil)
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Gabrielle Maria Sousa da Silva e Silva - Design and Diagramming (Universidade de Brasılia,
Brazil)
Ingrid de Sousa Andrade - Secretary (Universidade de Brasılia, Brazil)
Jose Freitas - Technical Support (Universidade de Brasılia, Brazil)
Karla Souza - Logistical Support (Universidade de Brasılia, Brazil)
Marta Chagas - Secretary (Universidade de Brasılia, Brazil)
Mateus Fleury - Web Designer (Universidade de Brasılia, Brazil)
Mattheus Pereira - Logistical Support (Universidade de Brasılia, Brazil)
Mirelly Oliveira - Logistical Support (Universidade de Brasılia, Brazil)
Thematic sessions
Algebra
Session link:
https://us02web.zoom.us/j/83041765040?pwd=eG8yZ3J0ckUwODFoU05hWWkrVGdNUT09
Meeting ID: 830 4176 5040
Passcode: 709400
Analysis
Session link:
https://us02web.zoom.us/j/97884700822?pwd=YUtMOTI0VFBzMVNPVnArOExxRGhNUT09
Meeting ID: 978 8470 0822
Passcode: 16zgK9
Numerical Passcode: 480670
Dynamical Systems
Session link:
https://us02web.zoom.us/j/83366143071?pwd=cWdLQU1OTVVhclI1aVZ3L25pZGVZUT09
Meeting ID: 833 6614 3071
Passcode: 551854
Geometry
Session Link:
https://us02web.zoom.us/j/82448403797?pwd=UXNPb0lHRWJjbzZ2MGdTR0NsMDZoZz09
Meeting ID: 824 4840 3797
Passcode: 641815
Mechanics
Session link:
https://us02web.zoom.us/j/83565978698?pwd=VXE0TlFGeXhiTExQbit0V2ZDRHJQUT09
Meeting ID: 835 6597 8698
Passcode: 724334
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XIII Summer Workshop in Mathematics
Mathematics Education
Session Link:
https://us02web.zoom.us/j/89263942629?pwd=NnRRUW1tMFlPZkJRTnpmTjRoVkd4dz09
Meeting ID: 868 3966 5048
Passcode: 975990
Number Theory
Session link:
https://us02web.zoom.us/j/87644282878?pwd=SWt2NzNqQWhqV0xZd0tCWjBPN1UrZz09
Meeting ID: 876 4428 2878
Passcode: 2021
Plenary Talks
Each talk have a specific link. All links can be found here:
https://mat.unb.br/verao2021/plenarias en.html
Probability
Session link:
https://us02web.zoom.us/j/86839665048?pwd=NWhvaXVxdkFZRnh3c21McmdYZXErdz09
Meeting ID: 868 3966 5048
Passcode: 975990
Theory of Computation
Session Link:
https://us02web.zoom.us/j/88272156520?pwd=TFI1a1dENE8xVkYrS1dJcVBpZjVwZz09
Meeting ID: 882 7215 6520
Passcode: 223389
Address
Universidade de Brasılia
Institute of Exact Sciences
Department of Mathematics
University campus Darcy Ribeiro
70910-900 Brasilia - DF
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XIII Summer Workshop in Mathematics
Contents
General Information 14
The XIII Summer Workshop in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 15
The Department of Mathematics at the University of Brasılia . . . . . . . . . . . . . . . . 16
The Graduate Program in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Plenary Talks 17
Alf Coles
An ecological turn in mathematics education . . . . . . . . . . . . . . . . . . . . . . . 18
Carolina Araujo
Symmetries in Algebraic Geometry and Cremona transformations . . . . . . . . . . . 19
Cecılia Salgado
Arithmetic and geometry of algebraic surfaces: the place of elliptic fibrations and k3
surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Filomena Pacella
Overdetermined elliptic problems and constant mean curvature surfaces . . . . . . . . 21
Francesco Russo
Fokker-Planck equations with terminal condition and related McKean probabilistic
representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Haim Brezis
A surprising formula for Sobolev norms . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Jean Mawhin
The Cours d’analyse infinitesimale of Charles de La Vallee Poussin and the ‘ghost
volume two’ of its third edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Jose M. Espinar
Minimal Surfaces: from soap films to Black Holes . . . . . . . . . . . . . . . . . . . . 25
Maria Ekiel-Jezewska
Dynamics of elastic microfilaments in fluids . . . . . . . . . . . . . . . . . . . . . . . . 26
Martin Rasmussen
Bifurcations in Random Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 27
Rostislav Grigorchuk
Self-similar groups and problems of modern mathematics . . . . . . . . . . . . . . . . 28
Valeria de Paiva:
Categorical Models of Explicit Substitutions . . . . . . . . . . . . . . . . . . . . . . . 29
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Posters 31
Alexssandra Thais Pereira Alves de Souza
Lindelof Spaces and Michael’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Anna Carolina Lafeta
Unique solutions for functional Volterra–Stieltjes integral equations . . . . . . . . . . 33
Elizabeth Cristine Jeronimo da Silva
Mathematical book: The classroom at home . . . . . . . . . . . . . . . . . . . . . . . 34
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Ellen Peixoto de Oliveira
The Abel’s Theorem via monodromy groups . . . . . . . . . . . . . . . . . . . . . . . 35
Gabriel Andre Asmat Medina
The Banach-Mazur game and product of Baire spaces . . . . . . . . . . . . . . . . . . 37
Henrique Matsuoka Medeiros
Singular Homology Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . 38
Thematic Sessions 39
Algebra 40
Adilson A. Berlatto
Self-similar Compressible T-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Ana Cristina Vieira
Superalgebras with graded involution . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Carmine Monetta
Non-abelian tensor square and related constructions of finite p-groups . . . . . . . . . 43
Csaba Schneider
The geometry of diagonal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Daniela A. Amato
Highly arc transitive and descendant-homogeneous digraphs with finite out-valency . 45
Dessislava Kochloukova
Homology and cohomology for partial actions . . . . . . . . . . . . . . . . . . . . . . 46
Iryna Kashuba, Olivier Mathieu
On the Free Jordan Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Joao Azevedo
Conciseness of some words related to non-commutators . . . . . . . . . . . . . . . . . 48
John Freddy Moreno Lozada
σ-Anti-automorphisms on Graded Primitve Associative Algebras . . . . . . . . . . . . 49
John MacQuarrie
Block Theory for Profinite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Kostiantyn Iusenko
Han’s conjecture for finite-dimensional algebras and its analogue for pseudocompact
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Mattheus P. S. Aguiar
Infinite graphs of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Nathalia Nogueira Goncalves
The q-tensor square of potent p-groups, q ≥ 0.∗∗ . . . . . . . . . . . . . . . . . . . . . 53
Sara Raissa Silva Rodrigues
Exponent of a finite group admitting a coprime automorphism of prime order . . . . . 54
Tulio Santos:
Centralizers of intransitive self-similar abelian groups . . . . . . . . . . . . . . . . . . 55
Viviane Ribeiro Tomaz da Silva
On the minimal varieties of a given exponent and the factorability of their T -ideals . 57
Vyacheslav Futorny:
Differential operators on algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . 58
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Yuri Santos Rego
Twisted conjugation in groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Analysis 61
Agnieszka B. Malinowska
On the fractional Cucker–Smale optimal control problem . . . . . . . . . . . . . . . . 62
Antonia Passarelli di Napoli
Regularity results for a class of obstacle problems . . . . . . . . . . . . . . . . . . . . 63
Augusto C. Ponce
A topological toolbox for Sobolev maps . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Benedetta Pellacci
Optimizing the principal eigenvalue for some weighted Neumann Problems . . . . . . 65
David Arcoya
Local Landesman-Lazer Condition in Quasilinear Problems . . . . . . . . . . . . . . . 66
David Ruiz
The Nirenberg problem for the disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Everaldo de Mello Bonotto
Recursive properties on generalized ODEs . . . . . . . . . . . . . . . . . . . . . . . . 68
Everaldo Souto ode Medeiros
On a Hardy–Sobolev type inequality and applications . . . . . . . . . . . . . . . . . . 69
Francesca Faraci
On a quasi-linear elliptic equation depending on the gradient . . . . . . . . . . . . . . 70
Gabriela Planas
Cahn-Hilliard/Allen-Cahn system with degenerate mobility . . . . . . . . . . . . . . . 71
Humberto Ramos Quoirin
Uniqueness and positivity issues in a subhomogeneous indefinite elliptic problem . . . 72
Joao Vitor da Silva
Gradient estimates for fully nonlinear PDEs with non-homogeneous degeneracy . . . . 73
Liliane A. Maia
A dynamical system approach to a class of radial fully nonlinear equations . . . . . . 74
Luigi Orsina
Schrodinger-Maxwell systems with interplay between coefficients and data . . . . . . . 75
Martin Bohner
Hyers–Ulam and Hyers–Ulam–Rassias Stability
of First-Order Linear Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 76
Pietro d’Avenia
Ground states for an Hartree-Fock type system . . . . . . . . . . . . . . . . . . . . . 77
Sabrina Streipert
The Beverton–Holt Model and some of its modifications . . . . . . . . . . . . . . . . 78
Stefano Buccheri
Gradient behaviour for large solutions to semilinear elliptic problems . . . . . . . . . 79
Teresa Faria
Permanence and Stability for non-autonomous Nicholson’s blowflies systems . . . . . 80
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XIII Summer Workshop in Mathematics
Zhi-Qiang Wang
Sign-changing solutions of coupled elliptic equations . . . . . . . . . . . . . . . . . . . 81
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Dynamical Systems 83
Alexandre Santana
Controllability of affine control systems . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Daniel Cavalcante Oliveira
Characteristic Classes and Quaternionic Flags . . . . . . . . . . . . . . . . . . . . . . 86
Leonardo F. Cavenaghi
On the geometry of smooth structures:
Prescribing scalar curvature on fiber bundles . . . . . . . . . . . . . . . . . . . . . . . 87
Phillipo Lappicy
Bifurcations and Chaos in Horava-Lifshitz Cosmology . . . . . . . . . . . . . . . . . . 88
Thaıs Fernanda Mendes Monis,
On selfmaps of complex Grassmann manifolds . . . . . . . . . . . . . . . . . . . . . . 89
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Geometry 91
Carla Cederbaum
On CMC-foliations of asymptotically Euclidean manifolds . . . . . . . . . . . . . . . . 92
Giuseppe Tinaglia
The geometry of constant mean curvature
surfaces in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Isabel Fernandez
The Bernstein problem for Weingarten surfaces . . . . . . . . . . . . . . . . . . . . . 94
Jose Antonio Galvez
Classification of immersed Weingarten spheres
in homogeneous three-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Jose M. Espinar
A Schoen type theorem for elliptic special Weingarten surfaces of minimal type . . . . 96
Lucas Ambrozio
Systoles, diastoles and minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Mariel Saez Trumper
Short-time existence for the network flow . . . . . . . . . . . . . . . . . . . . . . . . . 98
Miguel Brozos Vazquez
Several aspects of critical metrics for quadratic curvature functionals . . . . . . . . . 99
Pablo Mira
Minimal 2-spheres in homogeneous 3-spheres . . . . . . . . . . . . . . . . . . . . . . . 100
Rayssa Caju
The Allen-Cahn equation in the sphere: geometric and variational properties . . . . . 101
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Mechanics 103
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Aine Byrne
Mathematical Neuroscience: An Introduction . . . . . . . . . . . . . . . . . . . . . . . 104
Alan Santos Gois, Yuri Dumaresq Sobral, Jorge Carlos Lucero, Andrea Genovese de Oliveira
A dinamica de casais lineares: condicoes matematicas para o sucesso de uma relacao. 105
Aline Souza de Paula
Chaos Control Applied to Pendula Systems . . . . . . . . . . . . . . . . . . . . . . . . 106
Caio Vinıcius Schurgelies de Sa, Andre von Borries Lopes
Modelling of aqueous humour flow in the human eye . . . . . . . . . . . . . . . . . . . 107
Carolina Barros de Oliveira, Rafael Medeiros da Cunha Oliveira, Andrea Genovese Oliveira
Analise de Estabilidade de Modelos de Proliferacao de Doencas Infecciosas . . . . . . 108
Draga Pihler-Puzovic
Flows in elastic-walled Hele-Shaw cells: from understanding fundamental interfacial
instabilities to designing a fluidic fuse . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Gabriel Silva Povoa, Andrea Genovese Oliveira
Introducao a modelagem matematica do fluxo sanguıneo . . . . . . . . . . . . . . . . 111
Gabriel Nobrega Bufolo, Yuri Dumaresq Sobral
Collapsing columns in DEM: Differences between integration schemes. . . . . . . . . 112
Guilherme Mendes Santana, Roberto Francisco Bobenrieth-Miserda
Numerical simulation of the influence of the nonlinear effects on the active noise control 113
Gustavo Carreiro Matias, Rafaela Moreira Borges, Andre von Borries Lopes
Rivulet Flow Down a Vertical Surface: Lubrication Analysis and Numerical Solution . 114
Jorge Augusto Cassis Modesto, Yuri Dumaresq Sobral
Aggregation patterns in systems composed of few magnetic particles . . . . . . . . . 115
Luiza Sampaio, Roberto Francisco Bobenrieth-Miserda
Convergence Analysis for Explicit and Implicit Interpolation Methods in Finite Vol-
umes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Sarah Hormozi
Slurries of complex fluids: Rheology, microstructure and fluid mechanics . . . . . . . . 117
Taygoara Felamingo de Oliveira
Level Set mao na massa: uma introducao pratica . . . . . . . . . . . . . . . . . . . . 118
Wildemberg Ribeiro Rocha, Andre von Borries Lopes
Poiseuille flow inside ducts with rectangular cross section: analytical and numerical
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Yves Garnard Irilan, Francisco Ricardo Cunha
A comprehensive study of locomotion of particles and microorganism at low Reynolds
number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Mathematics Education 122
Adriano Costa da Silva, Thaymara Cristina de Souza Romulo, Thamyres Ribeiro Medeiros,
Aparecida de Fatima Andrade da Silva
O uso de softwares na sala de aula durante o perıodo de pandmia: Possibilidades e
utilizacao de novas ferramentas de ensino para Matematica . . . . . . . . . . . . . . . 123
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Alessandro Jacques Ribeiro and Marcia Aguiar
Promover aprendizagens profissionais a professores utilizando-se do racicınio dos es-
tudantes ao resolver tarefas matematicas . . . . . . . . . . . . . . . . . . . . . . . . . 125
Aluska Dias Ramos de Macedo, Paula Moreira Baltar Bellemain
Lesson Study e Engenharia Didatica: Uma formacao didatico-pedagogica de futuros
professores de Matematica do estagio curricular supervisionado . . . . . . . . . . . . . 126
Brigitte Lutz-Westphal and Pauline Linke
How to Create Mathematics Lessons with Focus on Discovery, Investigatory and Dia-
logic Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Cılia Cardoso Rodrigues da Silva
Saberes e Sabores matematicos em tempo de espera: possibilidades de mediacao e
intervencao com estudantes no Ensino Remoto . . . . . . . . . . . . . . . . . . . . . . 129
Cleyton Hercules Gontijo, Mateus Giani Fonseca
Intervencao em criatividade no campo da matematica: Um estudo de caso no ensino
fundamental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Daiane Soares Veras
Uso da Criptografia como Ferramenta de Ensino . . . . . . . . . . . . . . . . . . . . . 133
Ellen Michelle Barbosa de Moura, Karla Vanessa Gomes dos Santos, Joeanne Neves Fraz
Educacao Matematica e inclusao escolar na perpectiva da educacao para todos . . . 134
Fatima aparecida kian; Ailton paulo de oliveira junior; Luıs delcides rodrigues da silva
A modelagem matematica como alumbramento a reflexao e aprendizagem cotidiana . 136
Fatima aparecida kian; Ailton paulo de oliveira junior; Luıs delcides rodrigues da silva
Historia da matematica no ensino e aprendizagem: colaboracao na formacao do professor138
Fatima aparecida kian; Ailton paulo de oliveira junior; Luıs delcides rodrigues da silva
Os elementos ludico e sensorial fora dos limites da sala de aula e o aprendizado da
matematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Fredy Enrique Gonzalez
Introducao a historia social da educacao matematica - HISOEM . . . . . . . . . . . . 142
Janaına Mendes Pereira da Silva, Natalia Lami Zanettini e Sabrina Saito
Producoes Cientıficas sobre as Relacoes entre Covid - 19 e o Ensino de Ciencias e de
Matematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Josinalva Estacio Menezes, Maria Dalvirene Braga, Rui Seimetz, Poliana Maria da Silva,
Samara Araujo da Silva
Impressoes de alunos da licenciatura em matematica sobre o ensino remoto em uma
universidade publica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Kellton de Oliveira Sabino, Keila Cecılia Goncalves, Aparecida de Fatima Andrade da Silva,
Thamyres Ribeiro Medeiros
Experiencias na residencia pedagogica . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Letıcia Pereira de Almeida, Guilherme Flaviano Pereira, Thamyres Ribeiro Medeiros, Apare-
cida de Fatima Andrade de Silva
Metodologias para o ensino remoto de matematica . . . . . . . . . . . . . . . . . . . . 149
Luciana Avila Rodrigues, Thais Regina Duarte Marcal
A survey on the PreCalculus course offer in the Midwest Higher Education Institutions151
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Luciano Goncalves Batista, Renata Dourado Roque, Thamyres Ribeiro Medeiros, Aparecida
de Fatima Andrade da Silva
Experiencias, vivencias e acoes na Escola Estadual Raul de Leoni no contexto da
pandemia da Covid-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Marcia Rodrigues Leal, Liviam Santana Fontes, Cleyton Hercules Gontijo
Percepcao de professores formadores em licenciaturas de matematica sobre a criativade
no desenvolvimento do estagio supervisionado . . . . . . . . . . . . . . . . . . . . . . 154
Mateus Gianni Fonseca, Cleyton Hercules Gontijo
Pensamento crıtico e criativo em matematica a partir de leitura de informacoes
midiaticas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Raquel Carneiro Dorr, Brendo Oliveira Damasceno, Izabella Sabino da Silva
Evasao na Licenciatura em Matematica: o que apontam as pesquisas . . . . . . . . . 158
Rosangela Maria Kowalek
Mathematical Modeling in Mathematical Literacy: The Study of the Brazilian Mon-
etary System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Thiago Araujo de Lima Matos, Beatriz Barbosa Lopes, Daniella Isadora de Assis, Marcio
Gustavo Vieira, Aparecida de Fatima Andrade da Silva
Metodologias alternativas no ensino remoto de matematica . . . . . . . . . . . . . . . 162
Vania Batista Flose Jardim, Eduardo Goedert Dona e Janaına Mendes Pereira da Silva
Analise fundamentada de uma oficina de Trigonometria . . . . . . . . . . . . . . . . . 164
Waldineia Ferreira de Oliveira, Marcia Rodrigues Leal, Cleyton Hercules Gontijo
A ludicidade na aprendizagem e no desenvolvimento da criatividade em matematica . 166
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Number Theory 169
Ana Paula Chaves
On the Arithmetic Behaviour of Liouville
Numbers Under Rational Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Christopher Skinner
Some recent progress on the
Birch–Swinnerton-Dyer conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Cicero Carvalho
On the existence of pairs of primitive elements, related by
a rational function, on finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Guilherme Tizziotti
Permutation Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Herivelto Borges
Special maps on the projective line over finite fields . . . . . . . . . . . . . . . . . . . 174
Jose Plınio de Oliveira Santos
A New recurrence relation for the partition function . . . . . . . . . . . . . . . . . . . 175
Maria Bras-Amoros
Numerical Semigroups and Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Michael P. Knapp, Hemar Godinho
Counterexamples to a Conjecture of Norton . . . . . . . . . . . . . . . . . . . . . . . 177
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Shalom Eliahou
Three problems on numerical semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 178
Victor Gonzalo Lopez Neumann
Elementos primitivos 2-normais sobre Corpos Finitos . . . . . . . . . . . . . . . . . . 179
Wanderson Tenorio
Almost symmetric generalized numerical semigroups . . . . . . . . . . . . . . . . . . . 180
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Probability 182
Alexandre Richard, Xiaolu TAN, Fan YANG
Discrete-time Simulation of Stochastic Volterra Equations . . . . . . . . . . . . . . . 183
Christian Olivera
Quantitative particle approximation of nonlinear Fokker-Planck equations with singu-
lar kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Ciprian Tudor
Random matrices in Wiener chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Daniel Yasumasa Takahashi
Phase transitions in processes with long-memory . . . . . . . . . . . . . . . . . . . . . 186
Evelina Shamarova
Hormander’s theorem for path-dependent SDEs . . . . . . . . . . . . . . . . . . . . . 187
Frederi Viens, Soukaina Douissi, Khalifa Es-Sebaiy, Fatimah Alshahrani
AR (1) processes driven by second-chaos white noise: precise speed of convergence
results for parameter estimation using analysis on Wiener space . . . . . . . . . . . . 188
Luiz Renato Fontes, Pablo Gomes, Tom Mountford, Daniel Ungaretti, Maria Eulalia Vares
Contact process with renewal cures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Marcelo Hilario, Oriane Blondel, Frank den Hollander, Daniel Kious, Renato dos Santos,
Vladas Sidoravicius
Random walks on dynamic random environments with non-uniform mixing . . . . . . 190
Michael Hoegele
Cutoff thermalization for Ornstein-Uhlenbeck systems with small Levy noise in the
Wasserstein distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Mikhail Neklyudov
Rescaling nonlinear noise for sstochastic paarabolic equations . . . . . . . . . . . . . . 192
Samy Tindel
A coupling between Sinai’s random walk and Brox diffusion . . . . . . . . . . . . . . 193
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Theory of Computation 195
Claudia Nalon
The Early Years of Automated Reasoning: The Logic Theory Machine. . . . . . . . . 196
David Cerna
Anti-unification: Applications and Recent Results . . . . . . . . . . . . . . . . . . . . 197
Deivid Vale
Tuple Interpretations for Higher-Order Term Rewriting Systems . . . . . . . . . . . . 198
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XIII Summer Workshop in Mathematics
Flavio L. C. de Moura, Leandro O. Rezende
A Formalization of the Z property in Coq . . . . . . . . . . . . . . . . . . . . . . . . 199
Gabriel Ferreira Silva
Formalising Completeness of AC-unification . . . . . . . . . . . . . . . . . . . . . . . 200
Leonardo Melo
Towards a fixed-point approach to Nominal Disunification . . . . . . . . . . . . . . . 201
Sandra Alves
Quantitative measures for pattern matching . . . . . . . . . . . . . . . . . . . . . . . 202
Serdar Erbatur
Equational Unification modulo Non-disjoint Union of Theories . . . . . . . . . . . . . 203
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
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XIII Summer Workshop in Mathematics
General Information
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XIII Summer Workshop in Mathematics
The XIII Summer Workshop in
Mathematics
The Summer Workshop in Mathematics is one of the activities of the MAT/UnB Summer School.
Its main objective is to promote the exchange and dissemination of the research work developed by
researchers from Brazil and abroad, as well as the work of undergraduate and postgraduate students.
In this 13th edition, minicourses and talks in different mathematical areas will be offered, pro-
viding participants with diverse perspectives and topics on mathematics research. Moreover, there
will be opportunities for participants to contribute with oral communication and poster.
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XIII Summer Workshop in Mathematics
The Department of Mathematics at the
University of Brasılia
The Department of Mathematics at the University of Brasılia is part of the Institute of Exact Sciences.
It offers about 35 undergraduate courses each semester. The diversity and richness of this selection
is an enormous asset for students interested in the mathematical sciences.
The Department includes over seventy tenured faculty members, several post-docs and some
visiting professors with interests ranging over all areas of mathematics. Moreover, 33% of the faculty
members comes from abroad (Italy, Russia, China, Spain, Argentina, Belarus, Peru, and others).
The Department of Mathematics also has a rigorous program for a major in Mathematics with
about three hundred and fifty students. There are two undergraduate programs in Mathematics:
the first one leads to the degree “Bachelor in Mathematics” and the second one that leads to the
“Licentiate in Mathematics” degree.
The Graduate Program in Mathematics
The Graduate Program in Mathematics started in 1962 and offers a Master (MSc) degree and
a Doctoral (Ph.D.) degree in Mathematics in the subareas of Algebra, Analysis, Geometry and
Applied Mathematics (Theory of Probability, Numbers Theory, Dynamical Systems, Mechanics,
and Theory of Computation). The faculty maintains an active research program, regularly and
actively participates prominently in scientific meetings and editorial boards of scientific journals,
and maintains scientific exchange with various institutions in the country and abroad. The number
of graduate students is about 150 and a great part has financial support during their academic
programs.
Program Coordinator: Carlos Alberto Pereira dos Santos
Head of the Department: Ricardo Ruviaro
Director of the Institute of Exact Sciences: Gladston Luiz da Silva
Rector of the University of Brasilia: Marcia Abrahao Moura
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XIII Summer Workshop in Mathematics
Plenary Talks
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XIII Summer Workshop in Mathematics
An ecological turn in mathematicseducation
Alf Coles
Email: [email protected]
University of Bristol, United Kingdom
Abstract
In this plenary talk, I make the case for the need for an “ecological turn” in mathematics
education. The Covid pandemic has exposed the fragility of the world economy and its systems,
in the face of global events. I argue that there are more concerning global challenges, such as
climate change and totalitarianism, and that mathematics education can no longer assume a
neutral position. Mathematics itself is mobilized in the formulation and communication of
challenges, such as the global pandemic. Having set out some theoretical considerations, I then
describe a research project I co-lead, taking place in Mexico, where the aim is to bring the
concerns of a disadvantaged community into dialogue with the school mathematics curriculum.
I conclude with some thoughts on possible futures for mathematics education, in school and
higher education.
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XIII Summer Workshop in Mathematics
Symmetries in Algebraic Geometry andCremona transformations
Carolina Araujo
Instituto de Matematica Pura e Aplicada, Brazil
Abstract
In this talk I will discuss symmetries of algebraic varieties. When studying a projective
variety X, one usually wants to understand its automorphisms. Conversely, the structure of
the group of automorphisms of X encodes relevant geometric properties of the variety. After
describing some examples of automorphism groups of projective varieties, I will discuss why
the notion of automorphism is too rigid in the scope of birational geometry. We are then led
to consider another class of symmetries of X, its birational self-maps. Birational self-maps of
the projective space Pn are called Cremona transformations. Describing the structure of the
group of Cremona transformations of the plane is a classical problem that goes back to the 19th
century. In higher dimensions, not so much is known, and a natural problem is to construct
interesting subgroups of the Cremona group. I will end by discussing a recent work with Alessio
Corti and Alex Massarenti, where we investigate subgroups of the Cremona group consisting of
symmetries preserving special objects.
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XIII Summer Workshop in Mathematics
Arithmetic and geometry of algebraicsurfaces: the place of elliptic fibrations
and k3 surfaces
Cecılia Salgado
Email: [email protected]
Universidade Federal do Rio de Janeiro, Brazil
Rijksuniversiteit Groningen, Holanda
Abstract
In this talk I will go over recent progress on the arithmetic of algebraic surfaces, review some
contributions of the study of elliptic fibrations to it and present open problems summarising
the state of the art. K3 surfaces have a special place in the classification of algebraic surfaces:
they lie in an intermediate spot, being not too simple (as rational surfaces) nor too complicated
(as surfaces of general type). They are ubiquitous in mathematics, having been studied by
differential and algebraic geometers, dynamicists, arithmeticians, analysts and also by physicists.
A special feature of K3 surfaces is that they are the only class of surfaces that might admit
more than one elliptic fibration, with section, that is not of product type. The classification of
such different fibrations has been the object of several papers since the 90’s. I plan to outline a
new approach to this problem and, if time allows, discuss arithmetic and geometric applications
of the method.
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XIII Summer Workshop in Mathematics
Overdetermined elliptic problems andconstant mean curvature surfaces
Filomena Pacella
La Sapienza Universita di Roma, Italy
Abstract
After recalling the classical result by J.Serrin on overdetermined elliptic problems and the
one by A.D. Aleksandrov on constant mean curvature closed surfaces, we present recent results
for both types of problems in the case of domains or surfaces inside cones. In particular we
will discuss the relationships between the PDE and the geometric problem, as well as their
connection with isoperimetric inequalities.
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XIII Summer Workshop in Mathematics
Fokker-Planck equations with terminalcondition and related McKean
probabilistic representation
Francesco Russo
http://uma.ensta-paristech.fr/∼russo
ENSTA Paris, Institut Polytechnique de Paris, France
Abstract
Stochastic differential equations (SDEs) in the sense of McKean are stochastic differential
equations, whose coefficients do not only depend on time and on the position of the solution
process, but also on its marginal laws. Often they constitute probabilistic representation of
conservative PDEs, called Fokker-Planck equations;
In general Fokker-Planck PDEs are well-posed if the initial condition is specified. Here,
alternatively, we consider the inverse problem which consists in prescribing the final data: in
particular we give sufficient conditions for existence and uniqueness.
We also provide a probabilistic representation of those PDEs in the form a solution of a
McKean type equation corresponding to the time-reversal dynamics of a diffusion process.
The research is motivated by some application consisting in representing some semilinear
PDEs (typically Hamilton-Jacobi-Bellman in stochastic control) fully backwardly.
This work is based on a collaboration with L. Izydorczyk (ENSTA), N. Oudhane (EDF), G.
Tessitore (Milano Bicocca).
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XIII Summer Workshop in Mathematics
A surprising formula for Sobolev norms
Haim Brezis
Email: [email protected]
Rutgers, The State University, United States
Abstract
The Sobolev spaces, introduced in the 1930s, have become ubiquitous in Analysis and Ap-
plied Mathematics. They involve Lp norms of the gradient of a function f. We present an
alternative point of view where derivatives are replaced by appropriate finite differences and
the Lebesgue space Lp is replaced by the slightly larger Marcinkiewicz space Mp (aka weak
Lp space). Surprisingly, these new spaces coincide with the standard Sobolev spaces, a fact
which sheds a new light onto these classical objects and should have numerous applications.
In particular, it rectifies some well-known irregularities occurring in the theory of fractional
Sobolev spaces. The proof relies on original calculus inequalities which might be useful in other
situations. The lecture is based on a recent joint work with Jean Van Schaftingen and Po-Lam
Yung.
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XIII Summer Workshop in Mathematics
The Cours d’analyse infinitesimale ofCharles de La Vallee Poussin and the‘ghost volume two’ of its third edition
Jean Mawhin
Email: [email protected]
Universite Catholique de Louvain, Belgium
Abstract
The well-known Cours d’analyse infinitesimale of the Belgian mathematician Charles de La
Vallee Poussin (1866-1962) has seen twelve editions of its volume 1 between 1903 and 1959, and
nine editions of its volume 2 between 1906 and 1957. It has been translated in Russian and
reproduced by Dover. Mathematically, the most interesting editions are the second and the
third ones, being the only ones where the Lebesgue integral is developed.
Unfortunately, the third edition of the second volume, planned for the Fall of 1914, was
never published, having been destroyed in the printing house joining the main Library of the
Universite Catholique de Louvain, when this library was intentionally burned down by the
German army during the First World War.
It was discovered recently that the galley proofs of a substantial part of the destroyed volume
had been offered to a great-grandson of de La Vallee Poussin, together with pages of the second
edition annotated by him in view of the third edition. This has allowed a reconstruction of
this ‘ghost volume’, which constitutes an essential link in the development of the ideas of de La
Vallee Poussin about the Lebesgue integral and related questions.
We discuss the evolution of the Cours d’analyse infinitesimale, with special attention to the
‘ghost volume’.
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XIII Summer Workshop in Mathematics
Minimal Surfaces: from soap films toBlack Holes
Jose M. Espinar
Partially supported by Spanish MEC-FEDER (Grant MTM2016-80313-P and Grant
RyC-2016-19359)
Email: [email protected]
Department of Mathematics
University of Cadiz, Spain
Abstract
Geometric analysis is, nowadays, one of the most important fields in mathematics due to
the interplay between geometry and partial differential equations (PDE). Minimization (or least
energy) is not only a fundamental concept in geometry and analysis, but also in physics. In
general, surfaces that minimize some kind of functional energy are frequent in studies of quantum
mechanics, general relativity or string theory. In Leibniz words: Nature is thrifty in all its
actions.
In this talk we will overview the history (not all obviously, but some parts that interests to
us) of minimal surfaces, how they appear in General Relativity as Black Holes; and some of the
most relevant results in the fields, from the 19th century to nowadays.
Also, we will recall some open problems.
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XIII Summer Workshop in Mathematics
Dynamics of elastic microfilaments influids
Maria Ekiel-Jezewska
Email: [email protected]
Institute of Fundamental Technological Research, Polish Academy of Sciences
Warsaw, Poland
Abstract
Flagella of bacteria or algae, chains of diatoms, actins or articially made microfibers are just
some examples of elongated deformable microobjects moving in fluids. The basic question is
how elasticity influences dynamics of such objects. Different dynamical modes and migration
of flexible fibers entrained by the shear flow [1, 2] or sedimenting under gravity [3, 4] will be
discussed.
References
[1] P. J. Zuk, A. M. S lowicka, M. L. Ekiel-Jezewska, H. A. Stone, Universal features of the shape
of elastic fibers in shear flow, J. Fluid Mech., (2021), in press.
[2] A. M. S lowicka, H. A. Stone, M. L. Ekiel-Jezewska, Flexible fibers in shear flow approach
attracting periodic solutions, Phys. Rev. E 101 (2), (2020), 023104.
[3] M. Gruziel-S lomka, P. Kondratiuk, P. Szymczak, M. L. Ekiel-Jezewska, Stokesian dynamics of
sedimenting elastic rings, Soft Matter 15, (36), (2019), 7262-7274.
[4] M. Bukowicki, M. L. Ekiel-Jezewska, Sedimenting pairs of elastic microfilaments, Soft Matter
15 (46), (2019), 9405-9417.
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XIII Summer Workshop in Mathematics
Bifurcations in Random DynamicalSystems
Martin Rasmussen
Email: [email protected]
Partially supported by funding from the Engineering and Physical Sciences Research Council
(EPSRC) and the European Union
Imperial College London, United Kingdom
Abstract
In this talk, an overview of recent progress on bifurcations in random dynamical systems
[1] will be given, and the focus will be on a detailed analysis of the pitchfork bifurcation under
additive noise. It is well known that under addition of noise, certain properties of the pitchfork
bifurcation are destroyed [2]. I would like to present three different points of view that demon-
strate that the bifurcation is still present. The first two approaches concern a finite-time and
local analysis of the (unbounded noise) random dynamical system, revealing structural changes
at the bifurcation point that can be described by the dichotomy spectrum and conditional Lya-
punov exponents [3, 4]. In the last part of the talk, we study the situation for bounded noise,
and show that the pitchfork bifurcation can be described by means of a discontinuous bifurca-
tion of minimal invariant sets. Joint work with Mark Callaway, Thai Son Doan, Maximilian
Engel, Jeroen Lamb, and Christian Rodrigues.
References
[1] Ludwig Arnold, Random Dynamical Systems, Springer, 1998.
[2] Hans Crauel and Franco Flandoli, Additive noise destroys a pitchfork bifurcation, Journal of
Dynamics and Differential Equations 10 (1998), 259–274.
[3] Mark Callaway, Thai Son Doan, Jeroen S.W. Lamb, and Martin Rasmussen, The dichotomy
spectrum for random dynamical systems and pitchfork bifurcations with additive noise, Annales
de l’Institut Henri Poincare Probabilites et Statistiques 53, 4 (2017), 1548–1574.
[4] Maximilian Engel, Jeroen S.W. Lamb, and Martin Rasmussen, Conditioned Lyapunov exponents
for random dynamical systems, Transactions of the American Mathematical Society 372 (2019),
6343–6370.
[5] Jeroen S.W. Lamb, Martin Rasmussen, and Christian S. Rodrigues, Topological bifurcations of
set-valued dynamical systems, Proceedings of the American Mathematical Society 143, 9 (2015),
3927–3937.
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XIII Summer Workshop in Mathematics
Self-similar groups and problems ofmodern mathematics
Rostislav Grigorchuk
Email: [email protected]
Texas A&M University, United States
Abstract
Self-similar groups were introduced in the 80th in works of S.Sidki, N.Gupta and speaker.
At the beginning they were used to solve several difficult problems in group theory. Then, later
it started to be evident that they are related to the numerous branches of mathematics.: from
dynamical systems and operator algebras to graph theory and computer science.
In my talk I will give an overview of the results, connections and applications of this re-
markable class of groups.
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XIII Summer Workshop in Mathematics
Categorical Models of ExplicitSubstitutions
Valeria de Paiva
Topos Institute
Berkeley, USA
Abstract
The advantages of functional programming are well-known: programs are easier to write,
understand and verify than their imperative counterparts. However, functional languages tend
to be more memory intensive and these problems have hindered their wider use in industry. The
xSLAM project addresses these issues by using explicit substitutions to construct and implement
more efficient abstract machines. In this work we provide models for the linear calculi of explicit
substitutions we are interested in.
Indexed categories provide models of cartesian calculi of explicit substitutions. However,
these structures are inherently non-linear and hence cannot be used to model linear calculi of
explicit substitutions. This paper replaces indexed categories with pre-sheaves, thus providing
a categorical semantics covering both the linear and cartesian cases. We justify our models by
proving soundness and completeness results.
Our models of calculi of explicit subsitutions appear to be the only ones so far.
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Schedule
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XIII Summer Workshop in Mathematics
Posters
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XIII Summer Workshop in Mathematics
Lindelof Spaces and Michael’s Problem
Alexssandra Thais Pereira Alves de Souza
Supported by FAPESP, process number 19/11642-0.
Email: [email protected]
Universidade Federal do ABC (UFABC)
Santo Andre, Brasil
Abstract
Lindelof spaces are a natural generalization of compact spaces, which present several dif-
ferences compared to compactness in their basic properties. For instance, unlike compactness,
Lindelofness is not preserved by the product topology. Particularly, we can ask whether the
topological product of a Lindelof space and the set of irrational numbers must be a Lindelof
space. That question is known as Michael’s problem, and is one of the principal open problem
in general topology. A regular Lindelof space such that its product topology with the set of
irrationals is Lindelof is called a Michael space; some consistent examples of Michael spaces
exist but a final answer for Michael’s problem is yet to be obtained.
The main goal of this project is to study Lindelof spaces and some of the principal open
problems related to them. We study the Arkhangel’skiı theorem about the cardinality of first
countable Lindelof spaces and the Hajnal and Juhasz problem about the reflection of the Lindelof
property in subspaces of cardinality ℵ1.
References
[1] A. V. Arhangel’skii, On the cardinality of bicompacta satisfying the first axiom of countability,
Soviet Mathematics. Doklady 10 (1969), 951–955.
[2] P. S. Alexandroff e P. S. Urysohn, Memoire sur les espaces topologiques compacts, Proceedings
of the Koninklijke Nederlandse Akademie van Wetenschappen. Series A 14 (1929), 1–96.
[3] A. Hajnal e I. Juhasz, Remarks on the cardinality of compact spaces and their Lindelof subspaces,
Proceedings of the American Mathematical Society 59:1 (1976), 146–148.
[4] E. A. Michael, Paracompactness and the Lindelof property in finite and countable Cartesian
products, Compositio Mathematica 23:2 (1971), 199–214.
[5] E. Michael, The product of a normal space and a metric space need not be normal, Bulletin of
the American Mathematical Society 69 (1963), 375–376.
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XIII Summer Workshop in Mathematics
Unique solutions for functionalVolterra–Stieltjes integral equations
Rogelio Grau
Universidad del Norte, Barranquilla - Colombia
Anna Carolina Lafeta
Universidade de Brasılia, Brasilia - Brazil
Jaqueline Godoy Mesquita
Universidade de Brasılia, Brasilia - Brazil
Abstract
In this work, we deal with integral functional Volterra–Stieltjes integral equations: x(t) = φ(0) +
ˆ t
t0
a(t, s)f(xs, s)dg(s), t ≥ t0,
xt0 = φ,
(1)
where t0 ∈ R, φ ∈ G([−r, 0],Rn), f : G([−r, 0],Rn) × [t0,+∞) → Rn, a : [t0,+∞)2 → R,
g : [t0,+∞) → R, xs : [−r, 0] → Rn is defined by xs(θ) = x(s + θ) and the integral on the
right–hand side is in the sense of Henstock–Kurzweil–Stieltjes.
We shall present some conditions with respect to the functions a and g, as well as some
conditions with respect to the integral
ˆ τ2
τ1
b(t, s)f(xs, s)dg(s),
when b : [t0,+∞)2 → R is a regulated function and t0 ≤ τ1 ≤ τ2 ≤ t0 + σ < d, for some
0 < σ < d− t0.
With these conditions, we will be able to prove the existence and uniqueness of local and
maximal solutions for equation (1).
References
[1] M. Federson, R. Grau and J. G. Mesquita, Prolongation of solutions of measure differential
equations and dynamic equations on time scales, Mathematische Nachrichten, 292(1), 22-55,
2019.
[2] D. Frankova, Regulated functions, Mathematica Bohemica. 116 (1) (1991), 20-59.
[3] R. Grau, A. C. Lafeta and J. G. Mesquita, Existence and uniqueness of local and maximal
solutions for functional Volterra Stieltjes integral equations and applications, submitted.
[4] R. Henstock, A Riemann-type integral of Lebesgue power. Canad. J. Math. 20, (1968) 79–87.
[5] S. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Series in Real Anal.,
vol. 5, 1992.
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XIII Summer Workshop in Mathematics
Mathematical book: The classroom athome
Elizabeth Cristine Jeronimo da Silva, Tamires Castro Dias
Department of Mathematics and Pedagogy
University of Brasilia, UniCesumar
Brasilia, Brasil
Abstract
This poster aims to present the construction of a mathematical textbook, which will be
the object of study in the second half of the year, it is a dynamic, inclusive, intentional and
contextualized book, which seeks to present concepts using playful objects of teaching such
as string, geometric constructions on cardboard, gouache paint, among other items present in
the book itself or easily accessible. Thus, it will be used as a methodological tool in teaching
learning, initially directed towards the end of elementary school I and beginning of elementary
school II, during its application the points of interest of this study will be analyzed, which aim to
open a discussion about the importance and use of textbooks inside and outside the classroom,
and their strength as a tool in times of pandemic. In such a way, we bring this presentation not
as a finalized research, but as a debate, which cuts across all spheres and levels of education,
based on answering the following question, which is necessary for a mathematical book, so that
it is in fact a object of study and not a primer with summaries and fixation exercises?.
References
[1] Marcelo Silva Bastos, O livro didatico nas aulas de matematica: Um estudo a partir das con-
cepcoes dos professores,VIII Encontro Nacional de Educacao Matematica, SBEM, 2004.
[2] Andreza Regina Lopes da Silva, Daiana da Silva, Guia do professor conteudista,IFSC, 2015.
[3] Monica Johansson, Teaching mathematics with textbooks: a classroom and curricular perspec-
tive, Lulea: Lulea tekniska universitet, 2006. , p. 76
[4] A educacao em tempos de pandemia: Solucoes emergenciais pelo mundo, Rev. OEMESC. (2020).
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XIII Summer Workshop in Mathematics
The Abel’s Theorem via monodromygroups
Ellen Peixoto de Oliveira
Email: [email protected]
Department of Mathematics
Federal University of Vicosa, Brazil
Abstract
We present a topological proof of Abel’s Theorem that establishes the impossibility of quintic
equations through of radicals. Usually the demonstration this theorem is done through of Galois’
classic theory via field extensions and Galois groups. However, this work we follow the proof
effectued by the Professor V.I. Arnold [1], that show topological character of Abel’s Theorem.
We study the connection between groups theory and complex functions theory, mainly exploring
Riemann surfaces, algebraic functions and monodromy.
Monodromy are a topological concept. Given a polynomial
pz(w) = wn + an−1wn−1 + · · ·+ a1w + z
in C[w], we consider the algebraic function f : C → C defined by f(z) = w ∈ C; pz(w) = 0.This function is multivalued and determines a Riemann surface. Closed paths around their
branch points induce permutations in their sheets, generating what we call of monodromy
groups.
Theorem 1 If a complex functions h(z) is representable by radicals, its monodromy group is
soluble.
The solubility this groups determines when the function f(z) and, consequently the roots of
pz(w), are expressed by radicals.
Theorem 2 (Abel’s Theorem) For n ≥ 5 the general algebraic equation of degree n
wn + an−1wn−1 + · · ·+ a1w + z = 0
is not solvable by radicals.
The work is the result of a scientific initiation project carried out at PIVIC-UFV.
References
[1] V.B. Alekseev, Abel’s Theorem in Problems and Solutions. Based on the lectures of Professor
V. I. Arnold, Kluxer Academic Publishers, 2004.
[2] R. V. Churchill; J. W. Brown; R. G. Verhey, Complex Variables and Applications, McGraw-Hill,
1974.
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XIII Summer Workshop in Mathematics
[3] H. S. Cruz, A Survey on the Monodromy Groups of Algebraic Functions, (2016), 1-15.
[4] A. Garcia; L. Yves, Elementos de Algebra, Projeto Euclides, 2002.
[5] C. P. Milies, Grupos Nilpotentes: Uma Introducao, Rev. Matematica Universitaria, (2003),
55-100.
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XIII Summer Workshop in Mathematics
The Banach-Mazur game and product ofBaire spaces
Gabriel Andre Asmat Medina
Partially supported by CAPES
Email: andre [email protected]
Department of Mathematics
University of Sao Paulo, Sao Carlos, Brazil
Abstract
We study Baire spaces and analyze the problem of product of Baire spaces. Then we present
some conditions using the Banach-Mazur game to show that the Baire property is preserved in
the product. Then we analyze the difference of the infinite product of Baire spaces, between
the box product and Tychonoff product. We also present a multiboard version for this problem.
Finally we present some open problems regarding the product of Baire spaces.
References
[1] L. Aurichi and R. Dias, A minicourse on topological games, Topology Appl, 2019, 305-335.
[2] F. Galvin and M. Scheepers, Baire spaces and infinite games, M. Arch. Math. Logic, 2016,
85-104.
37
XIII Summer Workshop in Mathematics
Singular Homology Theory andApplications
Henrique Matsuoka Medeiros
Advisor: Dahisy Valadao de Souza Lima
Email: [email protected]
Centro de Matematica, Computacao e Cognicao
Universidade Federal do ABC, Santo Andre, Brasill
Abstract
We propose the study of one of the most important topological invariants, the singular
homology group, which has a fundamental role in the application of algebra in topology. The
singular homology is a theory of homology that associates to each topological space a sequence
of abelian groups and to each continuous application, between two given topological spaces,
a sequence of induced homomorphisms. The objective is to present the construction of this
topological invariant as well as its main properties, such as homotopy invariance and the Mayer-
Vietoris sequence. We will end by showing the power of this tool through some applications,
such as the Hairy Ball Theorem, Jordan Brouwer Separation Theorem, Borsuk-Ulam Theorem
and the definition of Euler’s Characteristic on manifolds.
References
[1] HATCHER, A. Algebraic topology, Cambridge Univ. Press, Cambridge xii (2002).
[2] ROTMAN, Joseph J. An introduction to homological algebra. Springer Science & Business Media,
2008.
[3] VICK, J.W. Homology theory An Introduction to algebraic topology. 2. ed. New York: Springer-
Verlag, 1994.
[4] VALDES, Wendy Diaz; FEMINA, Ligia Lais. Calculando detalhadamente os grupos de homologia
simplicial do toro, do plano projetivo e da garrafa de Klein. 2018.
[5] MASSEY, William S. Algebraic topology: an introduction. New York: Springer, 1967.
[6] KIRBY, R. C.; SIEBENMANN, L. C. On the triangulation of manifolds
and the Hauptvermutung. Bull. Amer. Math. Soc. 75 (1969), no. 4, 742–749.
https://projecteuclid.org/euclid.bams/1183530633.
38
XIII Summer Workshop in Mathematics
Thematic Sessions
39
XIII Summer Workshop in Mathematics
Algebra
40
XIII Summer Workshop in Mathematics
Self-similar Compressible T-groups
Adilson A. Berlatto
Email: [email protected]
Instituto de Ciencias Exatas e da Terra
Universidade Federal de Mato Grosso, Brazil
Abstract
According P. Hall’s notation, a finitely generated torsion-free nilpotent group G of nilpotency
class c is called Tc-group. G is compressible if every proper finite index subgroup of G contains
a finite index subgroup which is isomorphic to G (which implies that G is non-cohopfian). For
example, T2-groups, 2-generated T3-groups and the r-generated free nilpotent groups of class c
(denoted by F (r, c), with r, c ∈ N) are compressible. In this talk will be proved that F (r, c) acts
on a regular tree of valency m as a transitive self-similar group. Yet, this action is finite-state
and the number m is determined by the Mobius function.
References
[1] A. Berlatto; S. Sidki, Virtual Endomorphisms of Nilpotent Groups, Groups Geom. Dyn., 1
(2007) 21-46.
[2] V. V. Bludov; B. V. Gusev, Geometric Equivalence of Groups, Proceedings of the Steklov Institute
of Mathematics (2007), Suppl. 1, S61–S82.
[3] I.V. Bondarenko; R.V. Kravchenko, Finite-state self-similar actions of nilpotent groups, Geom.
Dedicata, 163 (2013), 339-348.
[4] P. Hall, Nilpotent Groups, Queen Mary College Mathematical Notes, London 1969.
[5] A.G. Makanin, On compressibility of finitely generated torsion-free nilpotent groups, Moscow
University Mathematics Bulletin, Vol. 46, No. 3, (1991), 34 - 38.
[6] G.C. Smith, Compressibility in Nilpotent Groups, Bull. London Math. Soc. 17, (1985), 453-457.
41
XIII Summer Workshop in Mathematics
Superalgebras with graded involution
Ana Cristina Vieira
Partially supported by CNPq
Email: [email protected]
Departamento de Matematica
Universidade Federal de Minas Gerais, Brazil
Abstract
A polynomial f(x1, . . . , xn) in non commuting variables is an identity of an algebra A if
it vanishes under all evaluations by elements in A and A is a PI-algebra if it satisfies a non
trivial identity. In our case, we deal with associative algebras over a field F of characteristic
zero and for an algebra A, we consider its sequence of codimensions cn(A)n≥1. This sequence
was introduced by Regev in 1972 and it has been an efficient way to measure the growth of the
identities satisfied by A.
The behavior of the codimension sequence has been extensively studied in the last years. In
fact, when A is a PI-algebra, either the sequence cn(A) grows exponentially or is polynomially
bounded, i.e, there exist constants a, k such that cn(A) ≤ ank, for all n ≥ 1 and specific
situations have been treated by several researchers in the area. In particular, we are interested
in PI-algebras having polynomial growth and endowed with additional structures, such as graded
algebras and algebras with involution.
In this talk I will present some recent results about the identities and the sequence of
codimensions of graded algebras endowed with an involution which preserves the grading.
42
XIII Summer Workshop in Mathematics
Non-abelian tensor square and relatedconstructions of finite p-groups
Carmine Monetta
Partially supported by GNSAGA
Email: [email protected]
Department of Mathematics
University of Salerno, Fisciano, Italy
Abstract
The non-abelian tensor square of a group, as defined in [2, 3], plays an important role both in
group theory and in geometry. In [4], Rocco presented a new tool to deal with the non-abelian
tensor square of a group G. Indeed, he considered a new group ν(G), whose derived subgroup
contains a copy of G⊗G. Therefore, in the context of ν(G) all the commutator calculus can be
used.
In this talk we will describe new results obtained jointly with R. Bastos, E. de Melo and
N. Goncalves [1], concerning the problem to determine bounds for the exponent of ν(G) and
G ⊗ G when G is a finite p-group. More specifically, we came out with new upper bounds for
exp(ν(G)) and exp(G⊗G), depending on exp(G) and either the nilpotency class or the coclass
of G.
References
[1] R. Bastos, E. de Melo, N. Goncalves and C. Monetta, The exponent of the non-abelian
tensor square and related constructions of p-groups, Mathematische Nachrichten, (2021),
https://doi.org/10.1002/mana.202000218.
[2] R. Brown, and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26, (1987),
311-335.
[3] C. Miller, The second homology group of a group: relations among commutators, Proc. Am.
Math. Soc. 3, (1952), 588-595.
[4] N. R. Rocco, On a construction related to the non-abelian tensor square of a group, Bol. Soc.
Brasil Mat. 22, (1991), 63-79.
43
XIII Summer Workshop in Mathematics
The geometry of diagonal groups
Csaba Schneider
Email: [email protected]
Departamento de Matematica
Universidade Federal de Minas Gerais, Brazil
Abstract
Diagonal groups are one of the classes of finite primitive permutation groups occurring in
the conclusion of the O’Nan-Scott theorem. Several of the other classes have been described
as the automorphism groups of geometric or combinatorial structures such as affine spaces or
Cartesian decompositions, but such structures for diagonal groups have not been studied. The
main purpose of this paper is to describe and characterise such structures, which we call diagonal
semilattices. Unlike the diagonal groups in the O’Nan-Scott theorem, which are defined over
finite characteristically simple groups, our construction works over any group, finite or infinite.
A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square,
the Cayley table of T, though in fact any Latin square satisfies our axioms. However, for m¿=3,
the group T emerges naturally and uniquely from the axioms.
This is a joint work with Rosemary Bailey, Peter Cameron and Cheryl Praeger.
44
XIII Summer Workshop in Mathematics
Highly arc transitive anddescendant-homogeneous digraphs with
finite out-valency
Daniela A. Amato
Email: [email protected]
Department of Mathematics
University of Brasılia, Brasil
Abstract
We investigate infinite highly arc transitive digraphs with two additional properties, descendant-
homogeneity and Property Z. A digraph D is highly arc transitive if for each s ≥ 0 the auto-
morphism group of D is transitive on the set of directed paths of length s; and D is descendant-
homogeneous if any isomorphism between finite generated subdigraphs of D extends to an
automorphism of D. A digraph is said to have property Z if it has a homomorphism onto a
directed line. We show that if D is a highly arc transitive descendant-homogeneous digraph
with Property Z and F is the subdigraph spanned by the descendant set of a directed line in D,
then F is a locally finite 2-ended digraph with equal in and out-valencies. If, moreover, D has
prime out-valency then there is only one possibility for the digraph F . This knowledge is the
used to classify the highly arc transitive descendant-homogeneous digraph of prime out-valency
which have Property Z.
References
[1] Daniela Amato and John K. Truss, Descendant-homogeneous digraphs , Combinatorica, 2019,
1203–1223.
[2] Daniela Amato, Highly arc transitive and Descendant-homogeneous digraphs with finite out-
valency, Journal of Combinatorial Theory. Series A, 2011, 403–424.
45
XIII Summer Workshop in Mathematics
Homology and cohomology for partialactions
Dessislava Kochloukova
Email: autor1@email
Department of Mathematics
State University of Campinas (UNICAMP), Brazil
Abstract
This is a joint work with Misha Dokuchaev (USP) and Marcelo Munız Alves (UFPR).
We will discuss homological properties of the partial group algebra KparG, where K is a
field. In particular we define a partial cohomological dimension cdparK (G) and conjecture that
it coincides with the ordinary cohomological dimension cdK(G) over the field K. This is a non-
trivial question as even in the case G = Z it turns out the conjecture is true but the trivial
KparG-module K has a projective resolution of the right length but it is not a free one.
References
[1] M. M. Alves, M. Dokuchaev, D. Kochloukova, Homology and cohomology via the partial group
algebra , ArXiv 2006.10173
46
XIII Summer Workshop in Mathematics
On the Free Jordan Algebras
Iryna Kashuba
Olivier Mathieu
Abstract
Let K be a field of characteristic zero. For integers n, D ≥ 1, let Jn(D) be the degree n
component of the free Jordan algebra J(D) over D generators. A conjecture for the character
(in particular for the dimension) of the GL(D)-module Jn(D) is proposed.
Let sl2 J(D) be the Tits-Allison-Gao construction of J(D) (this is a certain version of the
famous Tits-Kantor-Koecher construction of J(D)), see [1]. Two natural conjectures for the
homology of Lie algebra sl2 J(D) are stated, and each of them implies the previous conjecture.
We also provide some numeric evidence that the proposed formulae for dimensions of Jn(D)
reflects the known phenomenons: for D = 3 and n = 8 the conjecture predicts that the space
of special identities has dimension 3, which is correct: those are the famous Glennie’s Identities
[2]. Similarly for D = 4 the conjecture agrees that some tetrads are missing in J(4), as it has
been observed by Cohn [3].
References
[1] B.N. Allison and Y. Gao, Central quotients and coverings of Steinberg unitary Lie algebras,
Canad. J. Math. 48 (1996) 449-482.
[2] C.M. Glennie, Some identities valid in special Jordan algebras but not valid in all Jordan algebras,
Pacific J. Math. 16 (1966) 47-59.
[3] P. M. Cohn, On a generalization of the Euclidean algorithm, Proc. Cambridge Philos. Soc. 57
(1961) 18-30.
47
XIII Summer Workshop in Mathematics
Conciseness of some words related tonon-commutators
Joao Azevedo
Supported by CNPq
Email: [email protected]
Department of Mathematics
University of Brasılia, Brazil
Abstract
A group word w = w(x1, x2 . . . , xn) is an element of the free group F of rank n. If G is any
group, we denote by Gw the set of all elements of G obtained by replacing x1, x2, . . . , xn in w
by arbitrary elements g1, g2, . . . , gn of G, and by w(G) the subgroup generated by Gw. We say
that w is concise if w(G) is finite for all G such that Gw is finite, and we say that w is boundedly
concise if, whenever |Gw| = m, there exists a bound ν(w,m) such that |w(G)| ≤ ν. Well-known
results in the subject are that all non-commutator words and all multilinear commutator words
are concise, due to P. Hall, in an unpublished work, and Turner-Smith [3].
In [1] we provide good evidence for the following conjecture: if u1, u2, . . . , un are non-
commutator words in disjoint sets of variables, is [u1, u2, . . . , un] concise? We will prove the
validity of the conjecture for the case n = 3, as well as the case where the words ui are the
same, but in mutually disjoint sets of variables. This extends a Theorem of [2], where the result
is proved for n = 2. Moreover, it is a well-know result, due to Turner-Smith [3], that if u is a
concise word, then [u, x] will also be, where x is a variable not appearing in u. The methods
found by us give a similar result by proving that, when u is a multilinear commutator word and
v is a non-commutator, then [u, v] is concise. Some considerations about bounded conciseness
in residually finite groups are also made. This is a joint work with P. Shumyatsky.
References
[1] J. Azevedo, P. Shumyatsky, On Finiteness of verbal subgroups, arXiv:2009.10121v1
[2] C. Delizia, P. Shumyatsky, A. Tortora, M. Tota, On conciseness of some commutator words,
Arch. Math. 112 (2019) 27–32.
[3] R.F. Turner-Smith, Marginal subgroup properties for outer commutator words, Proc. Lond.
Math. Soc. (3) 14 (1964) 321–341.
48
XIII Summer Workshop in Mathematics
σ-Anti-automorphisms on Graded
PrimitveAssociative Algebras
John Freddy Moreno Lozada
Partially supported by CNPq and CAPES
Email: [email protected]
Department of Mathematics
University of Brasilia, Brazil
Abstract
Let G be a group and F a field. In [1], Bahturin, Bresar and Kochetov characterized prim-
itive G-graded associative rings (F-algebras) with a minimal G-graded left ideal and used this
characterization to study graded anti-automorphisms. Afterwards, in [3], to G an abelian group
and σ : G×G −→ F× an anti-symmetric 2-cocycle, K. Sousa and I. Sviridova presented a similar
characterization to primitive G-graded associative rings (F-algebras) with a minimal G-graded
right ideal in terms of σ-adjoints related to nondegenerate graded bilinear forms.
In this talk we use the characterization of Sousa and Sviridova and present a description of
σ-anti-automorfismos on G-graded associative S-algebras with a minimal G-graded right ideal
when S is an unitary commutative associative ring with trivial G-grading, U(S) is the set of all
invertible elements of S and σ : G × G −→ U(S) is an anti-symmetric 2-cocycle. This work is
joint with I. Sviridova.
References
[1] Yu. A. Bahturin, M. Bresar, M. Kochetov, Group grandings on finitary simple Lie algebras, Int.
J. Algebra Comp., 22(2012), 125-146.
[2] A. Elduque, M. Kochetov, Gradings on simple Lie algebras, Amer. Math. Soc., Mathematical
Surveys and Monographs, vol. 189, 2013.
[3] K. SOUZA, Involucoes coloridas em aneis primitivos graduados, Universidade de Brasılia, 2016.
49
XIII Summer Workshop in Mathematics
Block Theory for Profinite Groups
John MacQuarrie
Department of Mathematics
UFMG, Brazil
Abstract
If G = lim←−G/N is a profinite group and k is a field, the representation theory of G seeks to
understand the (pseudocompact) modules for the (pseudocompact) algebra k[[G]] = lim←− k[G/N ].
As with finite dimensional algebras, we may write k[[G]] as a direct product of indecomposable
algebras, known as the blocks of G. Rather than worry about the k[[G]]-modules all at once,
it is frequently convenient to study the representation theory of these blocks one at a time, as
certain blocks may have a much easier theory. The difficulty of a block B of G is measured by
a pro-p subgroup D of G, unique up to conjugation in G, called the defect group of B: morally,
the smaller D is, the easier B is (extreme case: B is a simple algebra if, and only if, D = 1).
A fundamental theorem in the block theory of finite groups (part of the so-called “local/global
principal”) is the Brauer Correspondence, which states that the blocks of the finite group G with
a given defect group D are in natural correspondence with the blocks of the subgroup NG(D)
with defect group D. I will explain a bit about block theory of finite and profinite groups, and
extend the Brauer Correspondence to blocks of arbitrary profinite groups.
Joint work with Ricardo Joel Franquiz Flores (UFMG)
50
XIII Summer Workshop in Mathematics
Han’s conjecture for finite-dimensionalalgebras and its analogue for
pseudocompact algebras
Kostiantyn Iusenko
Partially supported by FAPESP
Email: [email protected]
Department of Mathematics
University of Sao Paulo, Brazil
Abstract
Given a finite dimensional algebra A over an algebraically closed field k, Han’s conjecture
relates two homological invariants associated to A: its global dimension (which measures the
complexity of category of A-modules) and its Hochschild homology. Namely, it states (see [3])
that global dimension of A is finite if and only if Hochschild homology of A vanishes for large
enough degrees.
Recently, using the relative version of Hochschild homology with respect to a subalgebra B,
some progress was made in showing the Han’s conjecture (see [1, 2] and references therein). It
is based on existence of Jacobi-Zariski long nearly exact sequences which relates the usual and
relative versions of Hochschild homology. Such approach allows the authors to show that the
class of algebras verifying Han’s conjecture is closed by so-called bounded extensions of algebras.
Pseudocompact algebras naturally generalize the finite-dimensional algebras. So one can
consider certain analogue of Han’s conjecture for such class of algebras and existence of Jacobi-
Zariski sequences in this context. In my talk I aim to discuss the above topics with recent
advances made.
Based on joint works in progress with John MacQuarrie (UFMG).
References
[1] Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos, and Andrea Solotar. Cibils et al., Split
bounded extension algebras and Han’s conjecture, Pacific J. Math. 307 (2020), no. 1, 63–77.
[2] Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos, and Andrea Solotar. Cibils et al., Han’s
conjecture for bounded extensions, arXiv:2101.02597 (2021), 17p.
[3] Y. Han, Hochschild (co)homology dimension, J. London Math. Soc. (2) 73 (2006), no. 3, 657–
668.
51
XIII Summer Workshop in Mathematics
Infinite graphs of groups
Mattheus P. S. Aguiar
Partially supported by CAPES
Email: [email protected]
Department of Mathematics
University of Brasılia Brasılia, Brazil
Abstract
Given an infinite graph of groups (, ) we construct a profinite graph of groups (, ) such that is
densely embedded in , the fundamental profinite group Π1(, ) is the profinite completion of π1(, )
and the standard tree S(, ) embeds densely in the standard profinite tree S(, ). This answers a
Ribes’ question [4, Question 6.7.1]. Generalising the main results of [3] and [2] we answer two
other questions of Ribes [4, Questions 15.11.10 and 15.11.11] proving that a virtually free group
G is subgroup conjugacy separable and the normalizer NG(H) of a finitely generated subgroup
H of G is dense in NG(H). We also give an entirely new description of the fundamental group
of a profinite graph of groups using the language of paths, more closely related to the abstract
case introduced in [5].
This is a joint work with Pavel Zalesski.
References
[1] Aguiar, M. P. S. and Zalesskii, P. A., The profinite completion of the fundamental group of
infinite graphs of groups (2020) arXiv:2010.12720v1 [math.GR].
[2] Chagas, S. C. and Zalesskii, P. A., Subgroup conjugacy separability of free-by-finite groups.
Arch. Math. 104, (2015), 101-109.
[3] Ribes, L. and Zalesskii, P. A., Normalizers in groups and in their profinite completions. Rev.
Mat. Iberoam. 30, (2014), 165-190.
[4] Ribes, L., Profinite graphs and groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.
Folge, A Series of Modern Surveys in Mathematics 66, Springer, Berlin, 2017.
[5] Serre, J.-P., Trees. Springer-Verlag, Berlin-New York, 1980.
52
XIII Summer Workshop in Mathematics
The q-tensor square of potent p-groups,q ≥ 0.∗∗
Nathalia Nogueira Goncalves
Partially supported by CNPq
Email: [email protected]
Department of Mathematics
University of Brasılia, Brazil
Abstract
Let p be a prime number and G be a finite p-group. We say that G is potent if γp−1(G) ≤ Gp,for p odd, or if [G,G] 6 G4, for p = 2. If N is a normal subgroup of G and satisfies [N,p−2G] 6Np, for p ≥ 3, or [N,G] 6 N4, for p = 2, then N is said to be potently embedded in G. In this
talk we consider the group νq(G), q a non-negative integer, as described for instance by Bueno
and Rocco in [2], which happens to be an extension of the q-tensor square G ⊗q G by G × G.
Our purpose is to address some results concerning νq(G), G⊗qG and some normal subgroups of
the νq(G), under the assumption that G is a potent p-group, which generalize results for q = 0
found in [1]. The results here obtained are similar to those proved in [3] for another family of
finite p-groups.
∗∗ This is a joint work with Noraı Romeu Rocco.
References
[1] R. Bastos, E. de Melo, N. Goncalves and R. Nunes, Non-abelian tensor square and related
constructions of p-groups, Arch. Math., 114, (2020) 481-490.
[2] T. P. Bueno, N. R. Rocco, On the q-tensor square of a group, J. Group Theory, 14, 2011,
785-805.
[3] N. N. Goncalves and N. R. Rocco, The q-tensor square of a powerful p-group, J. Algebra, 551,
2020, 9-22.
[4] J. Gonzalez-Sanchez, A. Jaikin-Zapirain, On the structure of normal subgroups of potent p-
groups, J. Algebra, 276, 2004, 193-209.
53
XIII Summer Workshop in Mathematics
Exponent of a finite group admitting acoprime automorphism of prime order
Sara Raissa Silva Rodrigues
Partially supported by CAPES
Email: [email protected]
Department of Mathematics
University of Brasilia, Brazil
Abstract
Let G be a finite group admitting an automorphism φ of prime order p such that (|G|, p) = 1.
Denote by Gφ the fixed-point subgroup of φ in G, by G−φ the set x−1xφ; x ∈ G and by [G,φ]
the subgroup generated by G−φ. In this talk, some results bounding the exponent of [G,φ] will
be present. Among them, a sketch of the following theorem will be given:
Theorem: Let e and r be positive integers. If Gφ has rank r and xe = 1 for each x ∈ G−φ,
then the exponent of [G,φ] is (e, p, r)-bounded.
This is a joint work with Pavel Shumyatsky. It is important to point out that the results
mentioned in this talk are published in [1] and [2].
References
[1] S. R. S. Rodrigues, P. Shumyatsky, Exponent of a finite group of odd order with an involutory
automorphism, Archiv der Mathematik, 113 (2019), 113-118.
[2] S. R. S. Rodrigues, P. Shumyatsky, Exponent of a finite group admitting a coprime automorphism
of prime order, J. Group Theory, (2020), DOI 10.1515/jgth-2020-0141.
54
XIII Summer Workshop in Mathematics
Centralizers of intransitive self-similarabelian groups
Tulio Santos
Partially supported by CAPES
Email: [email protected]
Department of Mathematics
University of Brasılia, Brazil
Abstract
A self-similar group is a group G acting on a one-rooted m-regular tree Tm in such a way that
the states of its elements are themselves elements of G. Famous examples of self-similar groups
include the infinite torsion 2-group of Grigorchuk and the Gupta-Sidki p-groups. A successful
method for constructing self-similar groups is based on the notion of virtual endomorphisms of
groups, introduced by Nekrashevych and Sidki [4]; however this produced groups which act tran-
sitively on the first level of the tree. We extended the notion of a single virtual endomorphism
to a set of virtual endomorphisms by which all self-similar groups could be constructed.
Nekrashevych and Sidki characterized all self-similar free abelian subgroups of finite rank of
the group of automorphisms of T2 [4]. Later Brunner and Sidki [2] conducted the most complete
study of transitive abelian self-similar groups showing for example that the closure of such
groups under the full diagonal operations, α 7→ (α, α, · · · , α), of the group of automorphisms
of Tm is again abelian. This lead to an important translation of transitive self-similar abelian
groups to modules of the m-adic algebra Zm[[x]]. The generalization to the intransitive case
requires a careful study of a free monoid ∆ of partial diagonal operations acting on the group of
automorphisms of Tm. We show that in this setting, the closure of a self-similar abelian group
A under ∆ continues to be self-similar abelian.
In [3] Berlatto and Sidki showed that the centralizer of an abelian recurrent group is its
own topological closure, in particular is abelian. We have studied the centralizer structure of
an intransitive abelian self-similar group and its closure by ∆ and described recursively the
centralizer of an intransitive cyclic state-closed subgroup of Aut(T4). The analysis is based on
the orbit-type of the group. In particular we show that the centralizer of the double addition
machine is not abelian, but the centralizer of its closure by ∆ is abelian.∗∗Joint work with Said Sidki and Alex Dantas
References
[1] A. C. Dantas, T M. G. Santos; S. N. Sidki, Intransitive self-similar groups, Journal of Algebra
567, 2021, 564-581.
[2] A. M. Brunner; S. N. Sidki, Abelian state-closed subgroups of automorphisms of m-ary trees,
Groups, Geometry, and Dynamics 4, 2010, 455-471.
[3] A. Berlatto; S. N. Sidki, Virtual endomorphisms of nilpotent groups, Groups, Geometry, and
Dynamics 1, 2007, 21-46.
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XIII Summer Workshop in Mathematics
[4] V. Nekrashevych; S. N. Sidki, Automorphisms of the binary tree: state-closed subgroups and dy-
namics of 1/2-endomorphisms, Groups: topological, combinatorial and aritmetic aspects, London
Mathematical Society Lecture Note Series 311. Cambridge University Press, 2004, 375-404.
56
XIII Summer Workshop in Mathematics
On the minimal varieties of a givenexponent and the factorability of their
T -ideals
Viviane Ribeiro Tomaz da Silva
Partially supported by Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico (CNPq) -
grant 306534/2016-9 and Fundacao de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG)
- grant APQ-01149-18.
Email: [email protected]
Department of Mathematics
Federal University of Minas Gerais, Brazil
Abstract
Let F be a field of characteristic zero. In 2003, Giambruno and Zaicev established some
interesting results relating minimal varieties of a given exponent and the factorability of their
T -ideals (see [6, 7]). In this talk, we deal with varieties generated by PI-algebras with some
additional structures and we present some recent results [1, 2, 3, 4, 5] concerning the minimality
of these varieties and the factorability of their polynomial identities.
References
[1] D. V. Avelar, O. M. Di Vincenzo, V. R. T. da Silva, The factorability of T2-ideals of minimal
supervarieties, Comm. Algebra 47 (2019), 1595-1607.
[2] O. M. Di Vincenzo, M. A. S. Pinto, V. R. T. da Silva, On the factorability of polynomial
identities of upper block triangular matrix algebras graded by cyclic groups, Linear Algebra
Appl. 601 (2020), 311-337.
[3] O. M. Di Vincenzo, V. R. T. da Silva, E. Spinelli, A characterization of minimal varieties of
Zp-graded PI algebras, J. Algebra 539 (2019), 397-418.
[4] O. M. Di Vincenzo, V. R. T. da Silva, E. Spinelli, Minimal varieties of PI-superalgebras with
graded involution, Israel J. Math., in press.
[5] O. M. Di Vincenzo, V. R. T. da Silva, E. Spinelli, On the factorability of the ideal of ∗-graded
polynomial identities of minimal varieties of PI ∗-superalgebras, submitted.
[6] A. Giambruno, M. Zaicev, Minimal varieties of algebras of exponential growth, Adv. Math. 174
(2003), 310-323.
[7] A. Giambruno, M. Zaicev, Codimension growth and minimal superalgebras, Trans. Amer. Math.
Soc. 355 (2003), 5091-5117.
57
XIII Summer Workshop in Mathematics
Differential operators on algebraicvarieties
Vyacheslav Futorny
Email: [email protected]
Department of Mathematics
University of Sao Paulo, Brazil
Abstract
Lie algebras of polynomial vector fields on algebraic varieties play an important role in
different areas of Mathematics and Physics. Classical examples include Virasoro and Witt
algebras which correspond to the case of the torus and whose representation theory is well
understood. Recently the case of arbitrary varieties was developed in the cooperation with
Y.Billig, J.Nilsen and A.Zaidan. After discussing these results we will consider algebras of
invariant differential operators on algebraic varieties. In particular, the relation between the
Noncommutative Noether’s Problem and the classical Noether’s Problem will be explained based
on joint results with J.Schwarz.
58
XIII Summer Workshop in Mathematics
Twisted conjugation in groups
Yuri Santos Rego
Email: [email protected]
Institute of Algebra and Geometry
Otto von Guericke University of Magdeburg, Germany
Abstract
Twisted conjugation emerges in a variety of contexts throughout maths, e.g. topology,
functional analysis and theoretical computer science. Specifically in group theory, the study of
Reidemeister numbers (which count twisted conjugacy classes) was initiated nearly a century
ago and became a very active area over the past three decades. See e.g. [1, 3] for background.
In the first part of the talk we will give a ‘crash course’ on twisted conjugation for groups
in general. In the second part we concentrate on recent developments regarding Reidemeister
numbers of arithmetic groups, with particular focus on the (still mysterious) relationship be-
tween the so-called property R∞ and the structure of our groups of interest [2, 4].
Based on joint work with Paula Macedo Lins de Araujo (KU Leuven Kulak, Belgium)
References
[1] B. Jiang, A primer of Nielsen fixed point theory, in: Handbook of topological fixed point theory,
ed. by Brown, Furi, Gorniewicz and Hiang. Springer, Dordrecht, 2005, pp. 617–645.
[2] T. Mubeena and P. Sankaran, Twisted conjugacy classes in lattices in semisimple Lie groups,
Transform. Groups 19, (2014), 159–169.
[3] Y. Santos Rego, Arithmetic groups and Reidemeister classes, in: Geometric Structures in Group
Theory, Oberwolfach Reports, no. 16/2020, org. by Bridson, Drutu, Kramer, Remy and Schwer.
EMS Pub. House, Zurich, 2020, pp. 24–27.
[4] P. M. Lins de Araujo and Y. Santos Rego, Twisted conjugacy in soluble arithmetic groups,
Preprint, arXiv e-prints, (2020), pp. 47, arXiv:2007.02988.
59
XIII Summer Workshop in Mathematics
Schedule
60
XIII Summer Workshop in Mathematics
Analysis
61
XIII Summer Workshop in Mathematics
On the fractional Cucker–Smale optimalcontrol problem
Agnieszka B. Malinowska
Partially supported by the Bialystok University of Technology grant W/WI-IIT/1/2020 and funded
by the resources for research by Ministry of Science and Higher Education, email:
Faculty of Mathematics and Computer Science
Bialystok University of Technology, Poland
Ricardo Almeida
Faculty of Mathematics and Computer Science
University of Lodz, Poland
Tatiana Odzijewicz
Department of Mathematics and Mathematical Economics
SGH Warsaw School of Economics, Poland
Abstract
This work develops a sparse flocking control for the fractional Cucker–Smale multi-agent
model. The Caputo fractional derivative, in the equations describing the dynamics of a consen-
sus parameter, makes it possible to take into account in the self-organization of group its history
and memory dependency. External control is designed based on necessary conditions for a local
solution to the appropriate optimal control problem. Numerical simulations demonstrate the
effectiveness of the control scheme.
References
[1] Almeida Ricardo, Kamocki Rafa l, Malinowska Agnieszka B., Odzijewicz Tatiana, On the neces-
sary optimality conditions for the fractional Cucker–Smale optimal control problem, Communi-
cations in Nonlinear Science and Numerical Simulation, 96, (2021), 1–22.
[2] Almeida Ricardo, Kamocki Rafa l, Malinowska Agnieszka B., Odzijewicz Tatiana, On the exis-
tence of optimal consensus control for the fractional Cucker–Smale model, Archives of Control
Sciences, 30(4), (2020), 625–651.
62
XIII Summer Workshop in Mathematics
Regularity results for a class of obstacleproblems
Antonia Passarelli di Napoli
Email: [email protected]
Department of Mathematics ”R. Caccioppoli”
University of Naples ”Federico II”
Naples, Italy
Abstract
We present some higher differentiability results of integer and fractional order of the gradient
of solutions to variational obstacle problems of the form
min
ˆΩF (x, u,Du)dx : u ∈ u0 +Kψ(Ω)
,
where Ω ⊂ Rn is a bounded set, n ≥ 2. The boundary datum u0 and the obstacle ψ belong to
the Sobolev class W 1,p(Ω) and the admissible class Kψ(Ω) is defined as follows
Kψ(Ω) = v ∈ u0 +W 1,p(Ω) : v ≥ ψ.
The energy density F is assumed to be convex and of class C2 and satisfies p- growth condition,
p ≥ 2, with respect to the gradient variable.
We show that a Besov regularity assumption on the gradient of the obstacle ψ transfer to the
gradient of the solution.
The results are contained in joint works with Michela Eleuteri ([1],[2]).
References
[1] M. Eleuteri; A.Passarelli di Napoli, Higher differentiability for solutions to a class of obstacle
problems, Calc. Var. and PDE’s, 2018, 57:115.
[2] M. Eleuteri; A.Passarelli di Napoli, Regularity results for a class of non-differentiable obstacle
problems, Nonlinear Anal. 194, (2020), 111434.
63
XIII Summer Workshop in Mathematics
A topological toolbox for Sobolev maps
Augusto C. Ponce
Partially supported by FNRS-Belgium
Email: [email protected]
Institut de recherche en mathematique et physique UCLouvain
Louvain-la-Neuve, Belgium
Abstract
Classical works by F. Bethuel and by F. Hang and F-H. Lin have identified the local and
global topological obstructions that prevent smooth maps from being dense in the Sobolev
space W 1,p(Mm;Nn) between two Riemannian manifolds when p < m. They are related to the
extension of continuous maps from subsets of Mm to Nn. In this talk I will present some work
in progress with P. Bousquet (Toulouse) and J. Van Schaftingen (UCLouvain), inspired from
the notions of modulus introduced by B. Fuglede and degree for VMO maps by H. Brezis and
L. Nirenberg. I shall explain how one can decide whether a specific Sobolev map u : Mm → Nn
can be approximated or not by smooth ones, even in the presence of topological obstructions
from Mm or Nn.
64
XIII Summer Workshop in Mathematics
Optimizing the principal eigenvalue forsome weighted Neumann Problems
Benedetta Pellacci
Partially supported by ERC project Advanced Grant 2013 n. 339958: “Complex Patterns for
Strongly Interacting Dynamical Systems - COMPAT” PRIN-2015KB9WPT Grant: “Variational
methods, with applications to problems in mathematical physics and geometry”, by the
INdAM-GNAMPA group.
Email: [email protected]
Department of Mathematics and Physics
Universita della Campania “Luigi Vanvitelli”
Caserta, Italy
Abstract
We study the positive principal eigenvalue of a weighted problem associated with the Neu-
mann Laplacian. This analysis is related to the investigation of the survival threshold in popula-
tion dynamics. When trying to minimize such eigenvalue with respect to the weight, one is lead
to consider a shape optimization problem, which is known to admit spherical optimal shapes
only in very specific cases. We investigate whether spherical shapes can be recovered in general
situations, in some singular perturbation limit. In the case of planar polygons domains, we show
quantitative estimates of the optimal level convergence, as well as of the involved eigenvalues.
We will also study the optimization analysis with respect to diffusion. These are joint works
with Dario Mazzoleni (Universita di Pavia) and Gianmaria Verzini (Politecnico di Milano).
References
[1] Dario Mazzoleni, Benedetta Pellacci, Gianmaria Verzini, Asymptotic spherical shapes in some
spectral optimization problems. J. Math. Pures Appl., 135, 2020, 256-283.
[2] Dario Mazzoleni, Benedetta Pellacci, Gianmaria Verzini, Quantitative analysis of a singularly
perturbed shape optimization problem in a polygon. 2018 MATRIX Annals, MATRIX Book
Series 3; Springer International Pub- lishing AG, part of Springer Nature 2018D.R. Wood et al.
(eds.)
[3] Benedetta Pellacci, Gianmaria Verzini,“Best dispersal strategies in spatially heterogeneous envi-
ronments: optimization of the principal eigenvalue for indefinite fractional Neumann problems”.
Journal of Mathematical Biology, 2018, 76, 1357-1386.
65
XIII Summer Workshop in Mathematics
Local Landesman-Lazer Condition inQuasilinear Problems
David Arcoya
Partially supported by FEDER-MINECO (Spain) grant PGC2018-096422-B-I00 and Junta de
Andalucıa FQM-116
Email: [email protected]
Departamento de Analisis Matematico
Universidad de Granada, Spain
Abstract
In this talk I present a joint work with Manuela C. M. Rezende and Elves A. B. Silva
[1] on the existence and multiplicity of solutions for quasilinear problems in bounded domains
involving the p-Laplacian operator under local versions of the Landesman-Lazer condition. The
main results do not require any growth restriction at infinity on the nonlinear term which may
change sign. Variational methods, truncation arguments and approximation techniques based
on a compactness result for the inverse of the p-Laplacian operator are the main tools.
References
[1] D. Arcoya, M. C. M. Rezende and E. A. B. Silva, Quasilinear problems under local Landesman-
Lazer condition, Calc. Var., (2019), 58:210 https://doi.org/10.1007/s00526-019-1650-9.
66
XIII Summer Workshop in Mathematics
The Nirenberg problem for the disk
David Ruiz
Email: [email protected]
Departamento de Analisis Matematico
Universidad de Granada
Spain
Abstract
The problem of prescribing the Gaussian curvature on compact surfaces is a classic one, and
dates back to the works of Berger, Moser, Kazdan and Warner, etc. The case of the sphere
receives the name of Nirenberg problem and has deserved a lot of attention in the literature. In
the first part of the talk we will review the known results about compactness and existence of
solutions to that problem.
If the domain is the disk D, a boundary condition is in order. The most natural one is to
prescribe also the geodesic curvature h(x) of the boundary. This problem reduces to solve a
semilinear elliptic PDE under a nonlinear Neumann boundary condition.
First we perform a blow-up analysis of the solutions. We will show that, if a sequence of
solutions blow-up, it tends to concentrate around a unique point p ∈ ∂D. We are able to give
conditions on such point that, quite interestingly, involve the derivatives of K(x) but depend on
h(x) in a nonlocal way. This is joint work with A. Jevnikar, R. Lopez-Soriano and M. Medina.
Secondly, we will give existence results. We will show how the blow-up analysis developed
before can be used to compute the Leray-Schauder degree of the problem.
67
XIII Summer Workshop in Mathematics
Recursive properties on generalizedODEs
Everaldo de Mello Bonotto
Partially supported by FAPESP grant 2019/03188-7 and CNPq grant 310497/2016-7
Email: [email protected]
Department of Applied Mathematics and Statistics
University of Sao Paulo
Sao Carlos, Brazil
Abstract
In this talk, we present some results on topological properties of semiflows in the framework
of generalized ordinary differential equations. We define the concept of generalized semiflow
and we present some recursive properties as minimality and recurrence.
References
[1] E. M. Bonotto; M. Federson; M. C. Gadotti, Recursive properties on generalized ordinary dif-
ferential equations and applications, Submitted.
68
XIII Summer Workshop in Mathematics
On a Hardy–Sobolev type inequality andapplications
Everaldo Souto ode Medeiros
Partially supported by CNPq
Email: [email protected]
Department of Mathematics
Universidade Federal da Paraıba
Joao Pessoa, Brazil
Abstract
In this talk, we will prove a Hardy-Sobolev type inequality in the borderline case. We will
use this inequality to address the existence of solutions for a class of quasilinear elliptic problems
in an exterior domain with Robin boundary condition. 1
1This is a joint work with Jonison Lucas Carvalho (UFPB) and Marcerlo Fernandes Furtado (UnB)
69
XIII Summer Workshop in Mathematics
On a quasi-linear elliptic equationdepending on the gradient
Francesca Faraci
Email: [email protected]
Department of Mathematics and Computer Sciences
University of Catania, Italy
Abstract
In this talk we show an existence theorem for a quasilinear elliptic problem with dependence
on the gradient of the following type−∆pu = f(x, u,∇u) in Ω
u = 0 on ∂Ω,
under general growth conditions. Our aim is to combine variational techniques with fixed point
methods in order to prove the existence of a solution. More precisely our approach is based
on sub-supersolution techniques, extremal solutions for an auxiliary parametric problem and
Schaefer’s fixed point theorem. A multiplicity theorem is also obtained. The existence of sign
changing solutions will also be addressed.
Based on [1] and [2].
References
[1] F. Faraci, D.Motreanu, D. Puglisi, Quasi-linear elliptic equations with dependence on the gra-
dient, Calc. Var. Partial Differential Equations (2015) 525–538.
[2] F. Faraci, D. Puglisi, Nodal solutions for semilinear elliptic equations with dependence on the
gradient, work in progress.
70
XIII Summer Workshop in Mathematics
Cahn-Hilliard/Allen-Cahn system withdegenerate mobility
Gabriela Planas
Email: [email protected]
Universidade Estadual de Campinas
Campinas, Brazil
Abstract
We present a mathematical analysis to a Cahn-Hilliard/Allen-Cahn system with degenerate
mobility that models an isothermal process of solidification of a binary alloy. This model
is able to predict an observable phenomenon called solute trapping. The existence of global
weak solutions for the system is proved. We approximate the degenerate system and show
the convergence of solutions to the approximated non-degenerate problem to a solution of the
degenerate one. We also investigated deeply the non-degenerate system by showing the existence
of global weak solutions, the existence of global strong solutions in the two-dimensional case,
and local strong solutions in the three-dimensional case, as well as, providing conditions to the
uniqueness be satisfied.
Joint work with Andre Ferreira e Pereira (CEFET - MG, Belo Horizonte, Brazil)
71
XIII Summer Workshop in Mathematics
Uniqueness and positivity issues in asubhomogeneous indefinite elliptic
problem
Humberto Ramos Quoirin
Email: [email protected]
CIEM-FaMAF, Universidad Nacional de Cordoba, Argentina.
Abstract
We deal with uniqueness and positivity properties for the problem
(Pλ)
−∆pu = λup−1 + a(x)uq−1 in Ω,
u ≥ 0 in Ω,
u = 0 on ∂Ω,
where Ω is a bounded domain of RN , ∆p is the p-Laplacian operator, λ ∈ R, a ∈ C(Ω) changes
sign and 1 < q < p (the so-called subhomogeneous case). These conditions on a and q enable
the existence of dead core solutions and, consequently, multiplicity of nontrivial solutions may
occur. We discuss uniqueness results in accordance with the sign of λ. Some extensions to more
general right-hand sides will be considerded as well. Finally, the existence of positive solutions
is analyzed with respect to a and q.
72
XIII Summer Workshop in Mathematics
Gradient estimates for fully nonlinearPDEs with non-homogeneous degeneracy
Joao Vitor da Silva
Partially supported by CNPq-Brazil under Grant No. 310303/2019-2
Email: [email protected]
Departmento de Matematica
Universidade Estadual de Campinas, Brasil
Abstract
In this Lecture we present C1,βloc regularity estimates for bounded solutions of a class of fully
nonlinear elliptic equations with non-homogeneous degeneracy, whose simplest model case is
given by [|Du|p + a(x)|Du|q] ∆u = f(x, u) in Ω, for a bounded and open set Ω ⊂ RN , and
appropriate data p, q ∈ (0,∞), a and f . Such regularity estimates simplify and generalize, to
some extent, earlier ones via different modus operandi (cf. [1], [3] and [5]). Our approach is based
on geometric tangential methods and makes use of a refined oscillation mechanism combined
with compactness and scaling techniques. In the end, we also present some connections of our
results with geometric free boundary problems (see, [4]) and relevant nonlinear models in the
theory of elliptic PDEs (see, [2]).
References
[1] D.J. Araujo; G.C. Ricarte; E.V. Teixeira, Geometric gradient estimates for solutions to degen-
erate elliptic equations, Calc. Var. Partial Differential Equations 53 (2015), 605-625.
[2] M. Colombo; G. Mingione, Regularity for double phase variational problems, Arch. Rational
Mech. Anal. 215 (2015) 443-496.
[3] J.V. da Silva; G.C. Ricarte, Geometric gradient estimates for fully nonlinear models with non-
homogeneous degeneracy and applications, Calc. Var. Partial Differential Equations, 59, 161
(2020).
[4] J.V. da Silva; H. Vivas, The obstacle problem for a class of degenerate fully nonlinear operators,
To appear in Revista Matematica Iberoamericana.
[5] C. De Filippis, Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous
degeneracy, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, (2020) 1-23.
doi:10.1017/prm.2020.5.
73
XIII Summer Workshop in Mathematics
A dynamical system approach to a classof radial fully nonlinear equations
Liliane A. Maia
Partially supported by CNPq, Capes and FAPDF
Email: [email protected]
Department of Mathematics
University of Brasılia
Brasılia, Brazil
Abstract
We will present some recent results obtained on the existence, nonexistence and classification
of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal
operators. Our study is entirely based on the analysis of the dynamics induced by an autonomous
quadratic system which is obtained after a suitable transformation. This method allows to treat
both regular and singular solutions in a unified way, without using energy arguments. This is
a work in collaboration with Gabrielle Nornberg (ICMC/USP, Brazil) and Filomena Pacella
(Sapienza Universita di Roma, Italy).
References
[1] Liliane Maia, Gabrielle Nornberg and Filomena Pacella, A dynamical system approach to a class
of radial weighted fully nonlinear equations, Communications in Partial Differential Equations,
2020, 1-30. https://doi.org/10.1080/03605302.2020.1849281
74
XIII Summer Workshop in Mathematics
Schrodinger-Maxwell systems withinterplay between coefficients and data
Luigi Orsina
Email: [email protected]
Dipartimento di Matematica
“Sapienza” Universita di Roma
Roma, Italy
Abstract
I will present some results, obtained in collaboration with David Arcoya and Lucio Boc-
cardo, concerning existence and summability of solutions for a Schrodinger-Maxwell system of
equations: −div(M(x)u) + |u|r−2u = f(x) , in Ω,
−div(N(x)) = |u|r , in Ω,
u == 0 , on ∂Ω.
Here M and N are uniformly elliptic, bounded matrices, r > 1 and Ω is a bounded open subset
of RN . The main feature of the system is the fact that the datum f(x) is “controlled” by the
coefficient a(x) (which only belongs to L1(Ω)) in the sense that there exists a positive constant
Q such that |f(x)| ≤ Qa(x).
75
XIII Summer Workshop in Mathematics
Hyers–Ulam and Hyers–Ulam–RassiasStability of First-Order Linear Dynamic
Equations
Martin Bohner
E-mail: [email protected]
Department of Mathematics and Statistics
Missouri S&T
Rolla, Missouri, USA
Abstract
We present several new sufficient conditions for Hyers–Ulam and Hyers–Ulam-Rassias sta-
bility of first-order linear dynamic equations for functions defined on a time scale with values
in a Banach space.
References
[1] Maryam A. Alghamdi, Alaa Aljehani, Martin Bohner, and Alaa E. Hamza. Hyers–Ulam
and Hyers–Ulam–Rassias stability of first-order linear dynamic equations. Publ. Inst. Math.
(Beograd) (N.S.), 2021. To appear.
[2] Martin Bohner and Allan Peterson. Dynamic equations on time scales. Birkhauser Boston, Inc.,
Boston, MA, 2001. An introduction with applications.
76
XIII Summer Workshop in Mathematics
Ground states for an Hartree-Fock typesystem
Pietro d’Avenia
Partially supported by PRIN 2017JPCAPN
Qualitative and quantitative aspects of nonlinear PDEs
Email: [email protected]
Dipartimento di Meccanica, Matematica e Management
Politecnico di Bari
Bari, Italy
Abstract
We present some existence and nonexistence results of an Hartree-Fock type system made
by two Schrodinger equations in presence of a Coulomb interacting term and a cooperative pure
power and subcritical nonlinearity.
Joint work in collaboration with Liliane Maia and Gaetano Siciliano.
77
XIII Summer Workshop in Mathematics
The Beverton–Holt Model and some ofits modifications
Sabrina Streipert
Email: [email protected]
Department of Mathematics & Statistics
McMaster University
Hamilton, Canada
Abstract
This presentation gives an overview of models in the continuous and discrete time domain,
as well as on arbitrary isolated time scales, that are related to the Beverton–Holt recurrence.
While the classical Beverton–Holt equation describes populations under the assumption of con-
stant growth and constant environmental conditions, it is reasonable to consider time-dependent
model parameters. This extension leads to the discussion of the effect of a seasonally forced
environment, originally formulated as Cushing–Henson Conjectures. For the introduced discrete
modifications of the Beverton–Holt model, a periodic environment is deleterious for the popu-
lation under the assumption of a constant growth rate. However, applying a new formulation
of periodicity on arbitrary isolated time scales [1] reveals that a periodic environment is only
deleterious for a population with constant growth rate and certain underlying time structures.
References
[1] M. Bohner; J. Mesquita; S. Streipert, Periodic functions on isolated Time Scales, Mathematische
Nachrichten, 2020, To appear.
78
XIII Summer Workshop in Mathematics
Gradient behaviour for large solutions tosemilinear elliptic problems
Stefano Buccheri
Partially supported by the Austrian Science Fund (FWF) project F65
Email: [email protected]
Faculty of Mathematics
University of Vienna
Vienna, Austria
Abstract
For any Ω ⊂ RN (N ≥ 1) smooth bounded domain, p > 1 and f Lipschitz function, it is
well known that there exists a unique u ∈ C2(Ω) that solves−∆u+ |u|p−1u = f in Ω,
u = +∞ on ∂Ω.(2)
Moreover the first and the second order terms of the asymptotic expansion of u near the bound-
ary are explicitly described, as well as the first order asymptotic expansion of the normal
derivative of u.
The main result that we present provides not only the second order behaviour of the gradient
of the large solution of (2), but also the complete asymptotic expansion of all the singular terms
of u and ∇u, for every arbitrary smooth domain, every p > 1 and every f ∈ W 1,∞(Ω). In
particular we show that there exists an explicit corrector function S, finite sum of singular
terms, such that
z := u− S ∈W 1,∞(Ω).
Moreover we prove that
∀ x ∈ ∂Ω z(x) = 0 and limδ→0
z(x− δν(x))
δ= 0,
where ν is the outward unit normal to ∂Ω. The previous result is contained in the following
paper.
References
[1] S. Buccheri, Gradient behaviour for large solutions to semilinear elliptic problems, Annali di
Matematica Pura ed Applicata 198, 1013-1040 (2019).
79
XIII Summer Workshop in Mathematics
Permanence and Stability fornon-autonomous Nicholson’s blowflies
systems
Teresa Faria
Email: [email protected]
Universidade de Lisboa
Portugal
Abstract
We study the global asymptotic behaviour of solutions for a Nicholson’s blowflies system
with patch structure and multiple discrete delays:
x′i(t) = −di(t)xi(t) +n∑
j=1,j 6=iaij(t)xj(t) +
m∑k=1
βik(t)xi(t− τik(t))e−cik(t)xi(t−τik(t)), i = 1, . . . , n,
(3)
where all the coefficients and delay functions are continuous and nonnegative. Sufficient condi-
tions for both the extinction of all the populations and the permanence of the system are given.
By refining the assumptions for permanence, criteria for the global stability of (3) are provided.
Results can be found in [1, 2].
References
[1] T. Faria, Permanence and exponential stability for generalised nonautonomous Nicholson sys-
tems, submitted (2020).
[2] T. Faria, R. Obaya, A. M. Sanz, Asymptotic behaviour for a class of non-monotone delay
differential systems with applications, J. Dynam. Differential Equations, 30 (2018), 911–935.
80
XIII Summer Workshop in Mathematics
Sign-changing solutions of coupled ellipticequations
Zhi-Qiang Wang
Email: [email protected]
Department of Mathematics
Utah State University
Logan, USA
Abstract
We report results on existence of sign-changing solutions for coupled nonlinear Schrodinger
equations. Various techniques will be discussed for constructing sign-changing solutions for this
class of systems for which it is important to distinguish different solutions due to the existence
of a large number of different type of solutions.
References
[1] Name and surname, title of reference, name of the journal, year, pages.
[2] R.P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Phys.
20(2), (1948), 367-387.
[3] R.P. Feynman; R.B. Leighton; M. Sands, The Feynman Lectures on Physics, Addison-Wesley,
1964.
81
XIII Summer Workshop in Mathematics
Schedule
82
XIII Summer Workshop in Mathematics
Dynamical Systems
83
XIII Summer Workshop in Mathematics
Controllability of affine control systems
Alexandre Santana
Email: [email protected]
Department of Mathematics
State University of Maringa
Maringa, Brazil
Abstract
In the early of 1980’s B. Bonnard, V. Jurdjevic, I. Kupka and G. Sallet (see [1], [2] and [3])
studied affine controllability in connection with linear controllability. They considered an affine
control system of the form
x = Ax+ a+
m∑i=1
ui(t)(Bix+ bi), x ∈ Rn,
whereA,B1, . . . , Bm are n×nmatrices, a, b1, . . . , bm are vectors in Rn and u(t) = (u1(t), . . . , um(t))
is a control function with values in some control constraint set Ω ⊂ Rn. Hence, the study of
such systems was made via analysis of a family of vector fields in the semidirect product of a
Lie subgroup (generated by the exponential of A,B1, . . . , Bm) and Rn.
In the context of semigroups of Lie groups consider V be an n−dimensional real vector
space. Denote by EndV the set of all linear endomorphism on V and by Gl(V ) the set of all
automorphisms on V . Consider H a subgroup of Gl(V ) with transitive action on V r 0 and
take the group G = HoV given by semidirect product of H and V . Recall that the affine group
operation is defined by (g, v) · (h,w) = (gh, v + gw) for all (g, v), (h,w) ∈ G. Let π : G → H
be the canonical projection of the affine group on the Lie group H. We call affine action the
natural action of G on V , (g, v) · w = gw + v with (g, v) ∈ G and w ∈ V . The natural action
of π(G) = H on V is called linear action. Given a semigroup S ⊂ G, the affine action of S on
V is said transitive if Sx = V for all x ∈ V . Moreover, v ∈ V is called fixed point under S if
Sv = v. Hence in this context the above result is
Theorem Consider G = H o V an affine group. Let S ⊂ G be a connected semigroup with
non-empty interior. Suppose that the linear action of π(S) is transitive on V r 0 and that S
has no fixed point. Then the affine action of S on V r 0 is transitive.
Also in the early eighties, Jurdjevic and Kupka considered an invariant control system on
Sl(n,R) given by
g = Ag + uBg,
with unrestricted controls and matrices on sl(n,R) and then presented conditions on the ma-
trices A and B ensuring that this system is controllable. The main idea was to show that the
Lie saturate of such a system is the whole Lie algebra sl(n,R).
Several years later, in [6] Rocio, O., Santana, A. and Verdi, M. considered the affine control
systems of type
x = Ax+ a+ uBx+ ub (4)
where A,B ∈ sl(2,R), a, b ∈ R2 and u ∈ R and give conditions on (A, a) and (B, b) ensuring
that the above system is controllable in R2.
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The main idea of this talk is present an initial study of controllability of invariant affine
control system via Lie saturated.
References
[1] B. Bonnard, V. Jurdjevic, I. Kupka and G. Sallet, Transitivity of families of invariant vector
fields on the semidirect products of Lie groups. Trans. Amer. Math. Soc., 271, (1982), 525-535.
[2] V. Jurdjevic and G. Sallet, Controllability of affine systems. Differential Geometric Control
Theory, R. Brockett, R. Millman and H. Sussman, Eds. Boston: Birkauser, (1982), 299-309.
[3] V. Jurdjevic and G. Sallet, Controllability properties of affine systems. SIAM J.CONTROL
OPTIM, 22(3), (1984), 501-508.
[4] V. Jurdjevic, and I. Kupka, Control systems subordinate to a group action: accessibility, J. of
Diff. Eq., 39, (1981), 186-211.
[5] V.Jurdjevic, and I. Kupka, Control systems on semisimple Lie groups and their homogeneous
spaces, Ann.Inst. Fourier , 31, (1981), 151-179.
[6] O. Rocio, A. J. Santana, and M. A. Verdi, Semigroups of affine groups, controllability of affine
systems and affine bilinear systens in sl(2,R)oR2, SIAM J. CONTROL OPTIM. 48(2), (2009),
1080–1088.
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Characteristic Classes and QuaternionicFlags
Daniel Cavalcante Oliveira
PhD supported by Capes
Email: [email protected]
IMECC
UNICAMP
Campinas, Brasil
Abstract
Given a fiber bundle over a base manifold, one often wonders if the fibers are glued in a
trivial manner. To answer this question, we’re going to take a look at the characteristic classes
of these bundles. These are topological invariants that can be calculated in many ways and
they measure how far from being trivial the bundle is. For references, see [3], [4] and [6] as they
provide a solid ground about this topic.
After a mathematical excursion in the world of algebraic topology and fiber bundles, we’re
going to focus our attention to quaternionic flag varieties and use many tools of Lie theory to
reduce the topological problem of calculating these classes to an algebraic problem using the
root system associated to a Lie algebra (see [1] and [2] for a classical overview on this matter.
The basics are covered in [4], 2nd volume). This is a joint work with professor Luiz A. B. San
Martin.
References
[1] Bernstein, I.N., Gelfand, I.M. and Gelfand, S.I., Schubert Cells and Cohomology of the Spaces
G/P . Russian Math. Surveys 28 (1973), 1-26.
[2] Borel, A.: Topics in the Homology Theory of Fibre Bundles. Lecture Notes in Mathematics
(Springer) 36 (1967).
[3] Hatcher, A.: Vector Bundles and K-Theory.
[4] Kobayashi and Nomizu: Foundations of Differential Geometry.
Real Flag Manifolds.
[5] L. A. B. San Martin: Algebras de Lie.
[6] Milnor and Stasheff: Characteristic Classes.
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On the geometry of smooth structures:Prescribing scalar curvature on fiber
bundles
Leonardo F. Cavenaghi
Partially supported by FAPESP 2017/24680-1
Email: [email protected]
Department of Mathematics
Federal University of Paraıba
Joao Pessoa, Brazil
Abstract
Since the discovery of exotic spheres in the seminal work of John Milnor (see [?]), i.e,
manifolds that are homeomorphic but no diffeomorphic to standard spheres, a lot has been
questioned about the admissible geometries of these manifolds. It is known, for instance, that
every exotic sphere of dimension 7 carries a metric of non-negative sectional curvature (see
[?] and [?]) and a lot has been done concerning metrics of positive Ricci curvature on these
manifolds (see [?, ?, ?, ?, ?, ?, ?]). On the other hand, it its not known if there is an exotic
sphere with a metric of positive sectional curvature and Hitchin proved that there are exotic
spheres that do not even admit metrics of positive scalar curvature. This raises the question: at
which extent do the smooth structures determine geometry? In this presentation we discuss the
problem of prescribing scalar curvature on the total space of general fiber bundles with compact
structure group. As applications we discuss which smooth functions are realizable as scalar
curvature functions on some exotic spheres and bundles over exotic spheres, as well as on tori
bundles. We also sketch some results and future works concerning the possibility of realizing
G-invariant functions (where G is a compact and connected Lie group) as the scalar curvature
of G-invariant Riemannian metrics. This is a joint work with Llohann Speranca.
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Bifurcations and Chaos inHorava-Lifshitz Cosmology
Phillipo Lappicy
Supported by FAPESP
Email: [email protected]
Instituto de Ciencias Matematicas e de Computacao (ICMC)
Universidade de Sao Paulo
Sao Carlos, Brasil
Abstract
The nature of generic spacelike singularities in general relativity (such as the Big Bang) is
connected with first principles, notably Lorentzian causal structure, scale invariance and general
covariance. To bring a new perspective on how these principles affect generic spacelike singulari-
ties, we consider the initial singularity modeled by an ordinary differential equation (ODE) that
describes spatially homogeneous Bianchi type VIII and IX vacuum models in Horava-Lifshitz
gravity, where relativistic first principles are replaced with anisotropic scalings of Lifshitz type
and thereby the aforementioned ODE is perturbed by a parameter. Within this class of models,
General Relativity is shown to be a bifurcation where chaos becomes generic. To describe the
chaotic features of generic singularities in Horava-Lifshitz cosmology, we introduce symbolic
dynamics within Cantor sets and iterated function systems in a billiard-like game.
References
[1] J. Hell, P. Lappicy and C. Uggla, Bifurcations and Chaos in Horava-Lifshitz Cosmology, Sub-
mitted, https://arxiv.org/abs/2012.07614, 91 pages, (2020).
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On selfmaps of complex Grassmannmanifolds
Thaıs Fernanda Mendes Monis
Partially supported by CAPES
Email: [email protected]
Department of Mathematics
Sao Paulo State University (Unesp)
Institute of Natural Sciences and Technology
Rio Claro-SP, Brazil
Abstract
Let G(k, n) be the complex Grassmann manifold of k-planes in Ck+n. A matter of interest
is to know for which values of k and n the manifold G(k, n) has the fixed point property. In
this talk, we will discuss this problem, as well as related topics.
References
[1] P. Chakraborty and P. Sankaran, Maps between certain complex Grassmann manifolds, Topology
Appl., 2014, 119–123.
[2] H. Duan, Self-maps of the Grassmannian of complex structures, Compositio Math., 2002, 159–
175.
[3] T. F. M. Monis, N. C. L. Penteado, S. T. Ura and P. Wong, A note on nontrivial intersection for
selfmaps of complex Grassmann manifolds, Bull. Belg. Math. Soc. Simon Stevin, 2017, 665–672.
[4] A. Taghavi, An alternative proof for the f.p.p. of CP 2n, Expo. Math., 2015, 105–107.
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Schedule
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Geometry
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On CMC-foliations of asymptoticallyEuclidean manifolds
Carla Cederbaum
Email: [email protected]
Mathematics Department
University of Tubingen
Germany
Abstract
Three-dimensional Riemannian manifolds are called asymptotically Euclidean if, outside a
compact set, they are diffeomorphic to the exterior region of a ball in Euclidean space, and if
the Riemannian metric converges to the Euclidean metric as the Euclidean radial coordinate r
tends to infinity. In 1996, Huisken and Yau proved existence of a foliation by constant mean
curvature (CMC) surfaces in the asymptotic end of an asymptotically Euclidean Riemannian
three-manifold. Their work has inspired the study of various other foliations in asymptotic ends,
most notably the foliations by constrained Willmore surfaces (Lamm-Metzger-Schulze) and by
constant expansion/null mean curvature surfaces in the context of asymptotically Euclidean
initial data sets in General Relativity (Metzger, Nerz).
After a rather extensive introduction of the central concepts and ideas, I will present a
new foliation by constant spacetime mean curvature surfaces (STCMC), also in the context
of asymptotically Euclidean initial data sets in General Relativity (joint work with Sakovich).
This STCMC-foliation is well-suited to define the center of mass of an isolated system in Gen-
eral Relativity and thereby answers some previously open questions of relevance in General
Relativity.
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XIII Summer Workshop in Mathematics
The geometry of constant mean curvaturesurfaces in Euclidean space
Giuseppe Tinaglia
Email: [email protected]
Mathematics Department
King’s College London, United Kingdon
Abstract
I will begin by reviewing classical geometric properties of constant mean curvature surfaces,
H > 0, in R3. I will then talk about several more recent results for surfaces embedded in R3
with constant mean curvature, such as curvature and radius estimates for simply-connected
surfaces embedded in R3 with constant mean curvature. Finally I will show applications of
such estimates including a characterisation of the round sphere as the only simply-connected
surface embedded in R3 with constant mean curvature and area estimates for compact surfaces
embedded in a flat torus with constant mean curvature and finite genus. This is joint work with
Meeks.
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XIII Summer Workshop in Mathematics
The Bernstein problem for Weingartensurfaces
Isabel Fernandez
Email: [email protected]
Departamento de Matematica Aplicada I
Universidad de Sevilla, Spain
Abstract
A surface in Euclidean 3-space is an elliptic Weingarten surface if its mean curvature H and
Gaussian curvature K are related by a smooth, elliptic equation W (H,K) = 0. A well known
open problem, proposed for instance by Rosenberg and Sa Earp in 1994, is to solve the Bernstein
problem for this class of surfaces, that is: are planes the only entire elliptic Weingarten graphs?
Up to now, it is only known that the answer is positive if the Weingarten equation is uniformly
elliptic; this follows from a deep theorem by L. Simon on entire graphs with quasiconformal
Gauss map. In this talk we present two theorems. In the first one, we extend the solution to
the Bernstein problem in the uniformly elliptic case to multigraphs, proving that planes are
the only complete uniformly elliptic Weingarten surfaces whose Gauss map image lies in an
open hemisphere. In the second one, we will solve in the affirmative the Bernstein problem for
Weingarten graphs for a large class of non-uniformly elliptic Weingarten equations. This is a
joint work with P. Mira and Jose A. Galvez.
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Classification of immersed Weingarten
spheresin homogeneous three-manifolds
Jose Antonio Galvez
Email: [email protected]
Departamento de Geometrıa y Topologıa
Universidad de Granada, Spain
Abstract
We study immersed spheres in a homogeneous three-manifold M whose mean, extrinsic and
Gauss curvatures satisfy an elliptic Weingarten relation W (H,Ke,K) = 0. When the isometry
group of M has dimension 4, we show that if the unique inextendible rotational surface S in
M that satisfies this equation and touches its rotation axis orthogonally has bounded second
fundamental form, then every immersed sphere satisfying W (H,Ke,K) = 0 must be rotational.
This proves, among other results, that any elliptic Weingarten sphere immersed in H2 × R is a
rotational sphere and also, as a particular case, recovers the Abresch-Rosenberg classification
of constant mean curvature spheres in M .
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XIII Summer Workshop in Mathematics
A Schoen type theorem for elliptic specialWeingarten surfaces of minimal type
Jose M. Espinar
Partially supported by Spanish MEC-FEDER (Grant MTM2016-80313-P and Grant
RyC-2016-19359
Email: [email protected]
Department of Mathematics
University of Cadiz
Cadiz, Spain
Abstract
In this talk, we extend the theory of complete minimal surfaces in R3 of finite total curvature
to the wider class of elliptic special Weingarten surfaces of finite total curvature; in particular,
we extend the seminal works of L. Jorge and W. Meeks and R. Schoen. Specifically, we extend
the Jorge-Meeks formula relating the total curvature and the topology of the surface and we
use it to classify planes as the only elliptic special Weingarten surfaces whose total curvature
is less than 4π. Moreover, we show that a complete (connected), embedded outside a compact
set, elliptic special Weingarten surface of minimal type in R3 of finite total curvature and two
ends is rotationally symmetric; in particular, it must be one of the rotational special catenoids
described by R. Sa Earp and E. Toubiana. This answers in the positive a question posed in
1993 by R. Sa Earp. We also prove that the special catenoids are the only connected non-flat
special Weingarten surfaces whose total curvature is less than 8π.
References
[1] Elliptic special Weingarten surfaces of minimal type in R3 of finite total curvature. Preprint;
arXiv:1907.09122.
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XIII Summer Workshop in Mathematics
Systoles, diastoles and minimal surfaces
Lucas Ambrozio
Email: [email protected]
IMPA
Rio de Janeiro, Brazil
Abstract
We will discuss projective planes embedded in a Riemannian projective space of dimension
three, and minimal two-dimensional spheres embedded in a Riemannian sphere of dimension
three. In particular, we will discuss how big the infimum of the area of these surfaces can be
when compared to the ambient volume, and what happens when the Riemannian metric varies.
This is part of a joint project with R. Montezuma (UFC).
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XIII Summer Workshop in Mathematics
Short-time existence for the network flow
Mariel Saez Trumper
Partially supported by Fondecyt Regular 1190388
Email: [email protected].
Department of Mathematics
Pontificia Universidad Catolica de Chile
Santiago, Chile
Abstract
This paper contains a new proof of the short-time existence for the flow by curvature of
a network of curves in the plane. Appearing initially in metallurgy and as a model for the
evolution of grain boundaries, this flow was later treated by Brakke [4] using varifold methods.
There is good reason to treat this problem by a direct PDE approach, but doing so requires one
to deal with the singular nature of the PDE at the vertices of the network. This was handled
in cases of increasing generality by Bronsard-Reitich [5], Mantegazza-Novaga- Tortorelli [18]
and eventually, in the most general case of irregular networks by Ilmanen- Neves-Schulze [11].
Although the present paper proves a result similar to the one in [11], the method here provides
substantially more detailed information about how an irregular network “resolves” into a regular
one. Either approach relies on the existence of self-similar expanding solutions found in [20]. As
a precursor to and illustration of the main theorem, we also prove an unexpected regularity result
for the mixed Cauchy-Dirichlet boundary problem for the linear heat equation on a manifold
with boundary. This is joint work with Jorge Lira, Rafe Mazzeo and Alessandra Pluda.
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Several aspects of critical metrics forquadratic curvature functionals
Miguel Brozos Vazquez
Email: [email protected]
Department of Mathematics
Universidade da Coruna
Ferrol, Campus de Esteiro
Abstract
It is well known that, on a closed oriented manifold M , critical metrics for the total scalar
curvature functional g 7→´M τdvolg are Einstein. Generalizing this functional, one can consider
the scalar invariants τ2, ‖ρ‖2, ‖R‖2, ∆τ , that form a basis of the space of second order curvature
invariants and give rise to the quadratic curvature functionals
Φa,b,c : g 7→ˆM
(aτ2 + b‖ρ‖2 + c‖R‖2
)dvolg, for a, b, c ∈ R.
In this talk we will first consider cone metrics and analyze the conditions for being critical to
show examples of cone metrics that are critical for all quadratic curvature invariants. After-
wards, we will concentrate on 3-dimensional homogeneous manifolds and show that there is a
critical metric for any Φa,b,c. Special attention will be paid to functionals with zero energy, that
are closely related to Ricci solitons in this context.
References
[1] M. Brozos-Vazquez, S. Caeiro-Oliveira, E. Garcıa-Rıo, Three-dimensional homogeneous
critical metrics for quadratic curvature functionals, Annali di Matematica., to appear,
https://doi.org/10.1007/s10231-020-00999-y.
[2] M. Brozos-Vazquez, S. Caeiro-Oliveira, E. Garcıa-Rıo, Critical metrics for all quadratic curva-
ture functionals, Bull. London Math. Soc., to appear, doi:10.1112/blms.12448.
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XIII Summer Workshop in Mathematics
Minimal 2-spheres in homogeneous3-spheres
Pablo Mira
Partially supported by MICINN-FEDER, MTM2016-80313-P
Email: [email protected]
Department of Applied Mathematics and Statistics
Polytechnic University of Cartagena, Spain
Abstract
We prove that, in the 3-sphere endowed with an arbitrary homogeneous metric, there exists
exactly one immersed minimal 2-sphere, up to ambient isometry. This is a result contained
in [2], for which we will give an alternative proof based on [1]. We will also describe the
basic geometric properties of such minimal 2-sphere and discuss related results and problems
regarding the classification of minimal 2-spheres in Riemannian 3-spheres.
References
[1] J.A. Galvez, P. Mira, Uniqueness of immersed spheres in three-manifolds, J. Diff. Geom. 116
(2020), 459–480.
[2] W.H. Meeks, P. Mira, J. Perez, A. Ros, Constant mean curvature spheres in homogeneous
3-spheres, J. Diff. Geom., to appear.
[3] W.H. Meeks, P. Mira, J. Perez, A. Ros, Constant mean curvature spheres in homogeneous
3-manifolds, Invent. Math., to appear.
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XIII Summer Workshop in Mathematics
The Allen-Cahn equation in the sphere:geometric and variational properties
Rayssa Caju
Email: [email protected]
Department of Mathematics
UFPB/ University of Chicago
Joao Pessoa, Brazil/ Chicago, USA
Abstract
The Allen-Cahn equation establishes an important connection between the theory of minimal
surfaces and PDEs. In particular, the characterization of solutions of this equation has been
a subject of intense interest in the past few decades. One of the most well known problems
related to this subject is the De Giorgi’s conjecture which states that the only bounded monotone
solutions to Allen-Cahn equation in Rn are one-dimensional. Our main purpose in this talk is to
characterize unstable solutions of least energy of the Allen-Cahn equation in the entire sphere
Sn. We prove that the ground state solutions are radially symmetric. Moreover they are unique
up to rotations and correspond to the equator as a minimal hypersurface. Such information is
useful in the study of the first critical values of a min-max sequence for the associated energy
and a bifurcation problem.
References
[1] Gaspar, P., and Guaraco, M. A. M., The Allen-Cahn equation on closed manifolds, Calc. Var.
Partial Differential Equations 57, 4 (2018), 101.
[2] Van Schaftingen, J., and Willem, M. Symmetry of solutions of semilinear elliptic problems. J.
Eur. Math. Soc. (JEMS) 10, 2 (2008), 439?456.
[3] Guaraco, M. A. M., Marques, F. C., and Neves, A. Multiplicity one and strictly stable allen-cahn
minimal hypersurfaces, arXiv:1912.08997 [math.DG], 2019.
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Schedule
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Mechanics
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Mathematical Neuroscience: AnIntroduction
Aine Byrne
Email: [email protected]
School of Mathematics and Statistics
University College Dublin, Ireland
Abstract
The use of mathematics has many historical successes, especially in the fields of physics and
engineering, where mathematical concepts have been put to good use to address challenges far
beyond the context in which they were originally developed. More recently, mathematics has
been employed to further our understanding of biological systems, such as the human brain.
Despite the immense complexity of the brain, mathematical modelling has allowed for major
advances to be made towards understanding behaviour, conciousness and disease. Assuming no
specific neuroscience knowledge, this talk introduces the general ideas behind mathematically
modelling the human brain. I will briefly review seminal work in the field, such as the Hodgkin-
Huxley [1] and Wilson-Cowan [2] models, before discussing more recent work investigating the
role of neural synchronisation in diseases such a epilepsy and Parkinson’s disease [3].
References
[1] A. L. Hodgkin; A. F. Huxley, A Quantitative Description of Membrane Current and its Ap-
plication to Conduction and Excitation in Nerve, The Journal of Physiology, 117(4):500–544,
1952
[2] H. R. Wilson; J. D. Cowan, Excitatory and inhibitory interactions in localized populations of
model neurons, Biophysical Journal, 12(1):1–24, 1972.
[3] A Byrne; R.O. Dea; M. Forrester; J. Ross; S. Coombes, Next generation neural mass and field
modelling, Journal of Neurophysiology. 123, 726–742, 2020.
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XIII Summer Workshop in Mathematics
A dinamica de casais lineares: condicoesmatematicas para o sucesso de uma
relacao.
Alan Santos Gois
Yuri Dumaresq Sobral
Jorge Carlos Lucero
Andrea Genovese de Oliveira
Abstract
O primeiro modelo matematico que descreve a dinamica de relacoes amorosas foi proposto
por Steven Strogatz (Mathematics Maganzine, 61, p.35, 1988) e tinha como base um sistema de
equacoes diferenciais ordinarias lineares autonomas. De fato, considerando-se um casal tıpico
(Eduardo e Monica, por exemplo), podemos definir E(t) como o sentimento de Eduardo por
Monica no instante t e M(t) o sentimento de Monica por Eduardo em t. Se E ¿ 0, Eduardo
esta apaixonado por Monica. Se E ¡ 0, Eduardo odeia Monica e se E = 0, Eduardo e indiferente
a Monica. Definicoes similares se aplicam para M (t). O sistema proposto e linear e pode ser
escrito da seguinte forma: dE/dt = aE + bM; dM/dt = cE + dM, em que a, b, c e d sao
constantes. O unico ponto fixo deste modelo esta associado a indiferenca mutua do casal, e a
ausencia de pontos fixos (estaveis) no primeiro quadrante do espaco de fase atrela o sucesso de
uma relacao a que E, M tende a infinito quando t tender a infinito. Neste trabalho, mostramos
quais sao as condicoes para as constantes do modelo de tal modo que tenhamos esse sucesso.
Neste trabalho, classificamos os perfis de casais em funcao das constantes e mostramos quais
sao as condicoes necessarias para obtermos o sucesso, bem como os casais que satisfazem tais
condicoes.
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XIII Summer Workshop in Mathematics
Chaos Control Applied to PendulaSystems
Aline Souza de Paula
Email: [email protected]
Department of Mechanical Engineering
University of Brasılia, Brazil
Abstract
Nonlinearities are responsible for a great variety of possibilities in natural systems. Chaos
is one of these possibilities being related to an intrinsic richness due to the existence of an
infinity number of unstable periodic orbits (UPOs). Chaos control may be understood as the
use of tiny perturbations for the stabilization of these UPOs. Chaos control methods may be
classified as discrete or continuous techniques. The first chaos control method was proposed
by [1], nowadays known as the OGY method. This is a discrete technique that considers small
perturbations promoted in the neighborhood of the desired orbit when the trajectory crosses
a specific surface. On the other hand, continuous methods are exemplified by the so called
delayed feedback control, proposed by Pyragas [2], which states that chaotic systems can be
stabilized by a feedback perturbation proportional to the difference between the present and
a delayed state of the system. In this work, two pendula systems are considered. In the first
dynamical system, discrete and continuous chaos control techniques are numerically applied to
a pendulum system, whose parameters have been experimentally identified, in order to stabilize
different UPOs [3]. In the second system, continuous chaos control methods are employed in
order to maintain the rotating solution of a pendulum-shaker system [4]. In this case, the
main goal is to avoid bifurcations that destabilize the rotating motion, being useful for energy
harvesting purposes. The control is carried out numerically and experimentally.
References
[1] E. Ott; C. Grebogi; J.A. Yorke, Controlling chaos, Physical Review Letters, 1990, 1196-1199.
[2] K. Pyragas, Delayed feedback control of chaos, Philosophical Transactions of the Royal Society
A, 2006, 2309-2334.
[3] A.S. de Paula; M.A. Savi, Chaos Control Applied to Mechanical Systems, Chaotic Modeling and
Simulation, 2012, 17-24.
[4] A.S. de Paula; M.A. Savi; V. Vaziri; E. Pavlovskaia; M. Wiercigroch, Experimental bifurcation
control of a parametric pendulum, Journal of Vibration and Control, 2017, 2256-2268.
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XIII Summer Workshop in Mathematics
Modelling of aqueous humour flow in thehuman eye
Caio Vinıcius Schurgelies de Sa
Andre von Borries Lopes
Abstract
We present the fluid flow of aqueous humour in the anterior chamber of the human eye,
discussing how mathematical and physical modelling can help to understand its behavior. The
relative importance of buoyancy driven flow of aqueous humor is of particular interest. After
a brief review of basic eye physiology, we introduce the anterior chamber model of Canning
et al. (2002). Using classical fluid dynamics tools, such as the Boussinesq approximation, we
present the governing equations of the flow, with proper boundary conditions. We employ the
so-called lubrication theory to simplify the governing equations and solve them analytically.
The velocity and temperature fields are analyzed for physical insight and understanding of the
model limitations. We then introduce a numerical methodology, using the open-source software
tools of SU2. The numerical solution is discussed and compared with the analytical approach.
The two methodologies are in good agreement considering the general behavior of the flow, but
the analytical solution doesn’t capture all its features. We also conclude that the buoyancy
driven flow is the dominant mechanism in the anterior chamber.
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XIII Summer Workshop in Mathematics
Analise de Estabilidade de Modelos deProliferacao de Doencas Infecciosas
Carolina Barros de Oliveira
Rafael Medeiros da Cunha Oliveira
Andrea Genovese Oliveira
Abstract
Em meio ao cenario de contaminacao por doencas infecciosas, o estudo de modelos matematicos
epidemicos e endemicos mostrou-se muito importante na descricao da forma de transmissao de
doencas por agentes infecciosos. Por meio de simulacoes numericas, analises quantitativas e
qualitativas, o estudo dos modelos pode facilitar o entendimento de como as doencas infecciosas
se espalham, ajudar a identificar parametros importantes e assim, tem-se a possibilidade de
estipular o comportamento da doenca e entao providenciar polıticas de prevencao e controle.
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XIII Summer Workshop in Mathematics
Flows in elastic-walled Hele-Shaw cells:from understanding fundamental
interfacial instabilities to designing afluidic fuse
Draga Pihler-Puzovic
Manchester Centre for Nonlinear Dynamics
University of Manchester, United Kingdom
Abstract
Flow between two rigid parallel plates separated by a small gap, or a Hele-Shaw cell, was
first studied experimentally in 1898. It has been since widely used, for example, as a labora-
tory model for porous media and for design of microfluidics, and is equally appealing to both
experimentalists and theoreticians. The former phrase its simplicity of design and amenability
to experimental measurements. The later rejoice in the quasi-two-dimensionality of the fluid
flows caused by the large aspect ratio in the cell geometry, which significantly simplifies the
mathematics of the problem. In our talk we will study flows in variants of this classical set up
- elastic-walled Hele-Shaw cells, in which one of the rigid plates is replaced by an elastic wall.
In the first half of the talk, we will discuss fingering instability at the interface of air and
a viscous fluid, which readily develops in the classical Here-Shaw cells and is an archetypical
problem of pattern formation in fluid mechanics. We find a surprisingly effective means of
suppressing this instability by replacing one of the rigid plates with a thin elastic membrane2.
The resulting fluid-structure interaction fundamentally alters the interfacial patterns that de-
velop and considerably delays the onset of fingering. The novel setup provides a missing link
between two classical interfacial instabilities, the viscous fingering and the printer’s instability
(see Figure 1 (a, c)).
In the second half of our talk, we focus on a compliant Hele-Shaw cell in which the bottom
wall has been replaced by a soft confined elastomer3. In the presence of a flow, the cell behaves
in a manner analogous to the electrical fuse; above a critical flux, the flow-induced deformation
of the cell blocks the outflow, interrupting (choking) the flow. In particular, the pressure
distribution within the fluid applies a spatially variant normal force to the soft boundary, which
causes nonuniform deformation. As a consequence of lateral confinement and incompressibility
of the soft material, this flow-induced elastic deformation manifests as bulging near the cell
outflow; bulges that come into contact with the rigid cell roof interrupt the flow (see Figure 1
(b, d)). This paves the way for the integration of passive flow limiters (which remove the need
for external actuation) into microfluidic devices.
2D. Pihler-puzovic et al., “Viscous fingering in a radial elastic-walled Hele-Shaw cell.”, J. Fluid Mech., 849, 163–191, 2018.
3F. Box et al., “Flow-induced choking of a compliant Hele-Shaw cell.” Proc. Nat. Ac. Sc., 117, 48, 30228–3023,2020.
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Figure 1: Schematic diagrams of a compliant Hele-Shaw cell (a) with a thin elastic membrane as thetop boundary, and (b) a slab of soft confined solid as the bottom boundary. Injected fluid (air ora viscous liquid) spreads outwards, deforming the elastic boundary and displacing fluid resident inthe cell. (c) The top view of the superimposed fingering patterns at the interface between two fluidswhen a viscous fluid is being displaced by an air bubble in a Hele-Shaw cell with rigid walls (left)and with an elastic wall from (a) (right). (d) The measured outflux Qout as a function of the imposedvolume flux Q of water injected into a small-scale Hele-Shaw cell from (b). For sufficiently high Q,the bulges near the cell rim make contact with the cell roof, blocking the outflow and interruptingthe flow in a manner analogous to an electrical fuse.
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Introducao a modelagem matematica dofluxo sanguıneo
Gabriel Silva Povoa
Andrea Genovese Oliveira
Abstract
O fluxo sanguıneo no sistema arterial humano pode ser considerado um problema matematico
de dinamica de fluidos. A simulacao do fluxo sanguıneo no sistema da rede arterial fornece uma
melhor compreensao da fisiologia do corpo humano. A partir de uma revisao literaria de modelos
hemodinamicos, este estudo apresenta uma modelagem matematica do fluxo sanguıneo arterial
derivada das equacoes de Navier-Stokes e algumas suposicoes para simplificacao do problema.
A partir do sistema de equacoes diferenciais parciais nao lineares para o fluxo sanguıneo e a
area da secao transversal da arteria, foi adotado um metodo numerico para resolver as equacoes
e analisar os resultados por meio de graficos.
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Collapsing columns in DEM: Differencesbetween integration schemes.
Gabriel Nobrega Bufolo
Yuri Dumaresq Sobral
Abstract
Choosing an integration method for a DEM simulation is a hard task, made harder by the
fact that there is no correct choice. Each method has its advantages and knowing these tradeoffs
and how they will alter what your software is capable of simulating as well as how it affects the
results of the collapse. In this talk we present some of these tradeofs, such as: computational
performance, path convergence of individual particles, energy conservation, greatest time step
for good collision resolution and impact on global measures (such as runout distance).
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Numerical simulation of the influence ofthe nonlinear effects on the active noise
control
Guilherme Mendes Santana
Roberto Francisco Bobenrieth-Miserda
Abstract
High bypass ratio turbofans are the most used propulsion system in modern aircraft and are
responsible for a significant part of the aircraft noise. The broadband noise is usually associated
with the turbulent effects of the boundary layers and vorticity wakes. However, the results of
research from the Computational Aeroacoustics Laboratory in the University of Brasilia show
that there is at least a part of the broadband noise that cannot be explained by the boundary
layer effects. Thus, the objective of this work is to investigate if the nonlinear effects associated
with the large pressure fluctuations cause the appearance of a broadband component in the
noise that results from the interaction of finite-amplitude sound waves. This work is composed
of three parts, which consist of the study of different types of interactions between sound waves.
The first part consists of the simulation of sound waves that interact inside an acoustic resonator
and form a standing wave. The second consists of the simulation of the noise cancelation due
to the interaction of two standing waves with opposing phases. The third part consists of the
simulation of the continuous emission of sound waves into the resonator. These simulations
use a moving-body immersed boundary method to solve the time-dependent, compressible, one-
dimensional Euler equations. The domain discretization uses a finite volumes method, with
fluxes calculated with fourth-order precision in space, and the time marching process consists of
a third-order Runge-Kutta scheme. To properly simulate the waves generated by an oscillating
piston, it was necessary to implement a change in the immersed boundary method to capture
the movement of the boundary inside the control volumes. The results showed that there is no
change from tonal to broadband noise due to the nonlinear effects for frequencies within the
human hearing range.
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Rivulet Flow Down a Vertical Surface:Lubrication Analysis and Numerical
Solution
Gustavo Carreiro Matias
Rafaela Moreira Borges
Andre von Borries Lopes
Abstract
In this work, the flow of a rivulet of incompressible Newtonian fluid down a vertical plane is
investigated. This problem has been extensively studied due to the large number of applications
in which it can be observed. In most works found in the literature, the lubrication approximation
is used to obtain a reduced model. We show that the widely used model is not the most
appropriate for the solution of this problem. In addition, a new reduced model is developed with
an appropriate perturbation parameter and the effect of the free surface curvature conservation
in the lubrication analysis is also evaluated. It is shown that this new model considerably
extends the application range of the lubrication analysis. To verify this statement, the full
problem is solved using the finite element method. It is possible to obtain the velocity field
of the flow for contact angles between 0° and 180°. Comparing the models presented with the
complete solution, it becomes evident that incorporating the exact expression of the free surface
shape in the reduced model further increases the application of the lubrication theory.
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Aggregation patterns in systemscomposed of few magnetic particles
Jorge Augusto Cassis Modesto
Yuri Dumaresq Sobral
Abstract
In this work, we study the aggregation patterns that we observe in systems composed of
two and three magnetic particles interacting magnetically and via solid-solid contact in two-
dimensions. We use a discrete element method to take into account solid-solid interactions
between pairs of particles. The particles are initially separated and, in the case of three particles,
placed in the vertex of an equilateral triangle, and then are allowed to evolve for a given period
of time. The final configurations are identified and associated to different levels of magnetic
potential energy. We construct bifurcation diagrams for both cases by considering several initial
conditions of the orientation of the magnetic dipole and identifying the final state configurations.
In both cases, the majority of the final configurations is the aggregation of the particles in chains,
but a non-negligible number of initial configurations leads to complete dispersion of the particles.
Finally, we explore in more detail the surprising fractal nature of the bifurcation diagram.
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Convergence Analysis for Explicit andImplicit Interpolation Methods in Finite
Volumes
Luiza Sampaio
Roberto Francisco Bobenrieth-Miserda
Abstract
In this work, grid-convergence analyzes are performed for different interpolation methods,
explicit and implicit, used in numerical simulations based on finite volume discretizations. The
main differential of this work is in the approach given to the convergence analysis, which is
performed in the domains of space and time together. The objective is to verify, for each of
the interpolation methods, whether the convergence-order corresponds to the convergence-order
of design and, with this information, determine the computational cost/benefit ratio for each
of the explicit and implicit methods tested. For this, the shock-tube problem will be used,
which will be simulated using the VAT code (Virtual Aeroacoustic Tunnel) developed by the
Computational Aeroacoustics Laboratory of the University of Brasılia (Caalab-UnB).
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Slurries of complex fluids: Rheology,microstructure and fluid mechanics
Sarah Hormozi
Email: [email protected]
Cornell University
Ithaca, NY, United States of America
Abstract
Slurries of complex fluids are suspensions of non-Brownian particles in complex fluids (e.g.
shear thinning, shear thickening, yield stress and viscoelastic fluids). These materials exist all
around us. Examples include the following: food products, hygiene products, mine tailings,
concrete, fracturing fluids, shock absorbing materials, drilling muds and muds in mudslides,
lava, and numerous others. As expected from their ubiquity, non-Newtonian slurries contribute
significantly to the economy both positively and negatively. Therefore, even small increases in
efficiency when processing slurries of complex fluids could make significant positive economic
and environmental impacts.
Obviously, a thorough understanding of the rheology and fluid mechanics of these materials
in natural and industrial settings is essential to improving the efficiency of production. However,
this is extremely challenging due to the complex rheology of the suspending fluids, the interaction
of fluid and particle phases, and multiple-body and short-range interactions of particles. My
presentation will introduce an array of experimental and modeling techniques that my research
team uses to investigate rheological properties and fluid dynamical behavior of the slurries
of complex fluids. The goal is to establish a continuum framework and refine it through a
series of microstructure investigations. I will discuss how our recent results can be used to
address and resolve mixing and pumping problems in concrete industries and to maximizing
petroleum reservoir production using hydraulic fracturing processes. Finally, open questions will
be disclosed, which must be answered to build a firm foundation for a long-term contribution
to the area of complex suspensions.
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Level Set mao na massa: uma introducaopratica
Taygoara Felamingo de Oliveira
Abstract
O objetivo desse mini curso e fornecer uma introducao ao metodo de Level Set para a cap-
tura de interfaces. O metodo de Level Set pode ser considerado uma “tecnologia numerica” bem
estabelecida, util na simulacao do movimento de interfaces como os observados no escoamentos
de gotas e bolhas, ondas, capsulas e do sangue (na escala das hemacias). Tambem e empre-
gado no caso de interfaces que sao impelidas a moverem-se em razao de outros mecanismos.
Por isso, encontra aplicacoes em processamento de imagens, computacao grafica, simulacao de
processos biologicos e ate em astrofısica. Nesse curso, vamos estudar os elementos do metodo:
equacoes basicas, discretizacao numerica, suavizacao de interfaces, tecnicas de reinicializacao
e acoplamento com um escoamento externo. Vamos realizar implementacoes durante o curso
para demonstrar as caracterısticas da tecnica. Havera atividades praticas, desenvolvidas em Oc-
tave. Para melhor aproveitamento do curso, sao desejaveis conhecimentos previos de equacoes
diferenciais, metodos numericos basicos e de programacao (so um pouquinho ja e suficiente!).
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Poiseuille flow inside ducts withrectangular cross section: analytical and
numerical approach
Wildemberg Ribeiro Rocha
Andre von Borries Lopes
Abstract
The modeling and study of how flows are developed inside rectangular ducts has a wide
range in the mechanical industry as well as in the modeling of complex systems, so that the
results obtained here can be implemented in the optimization of products and processes. The
present study investigates a classic solution of the Navier-Stokes equations: the laminar flow
of an incompressible Newtonian fluid in a rectangular duct. In order to model and solve the
problem analytically, we combine the solution for the plane Poiseuille flow (1844) and a Fourier
series method. In addition, a numerical investigation of the problem is carried out through
the Finite Differences Method (FDM), which is based on the solution of differential equations
through the approximation of its derivatives by finite differences taken over an appropriate
grid. Finally, a comparative study between the results obtained in the present work and those
available in the literature is presented.
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A comprehensive study of locomotion ofparticles and microorganism at low
Reynolds number
Yves Garnard Irilan
Francisco Ricardo Cunha
Abstract
This work presents an investigation on the locomotion of particles and organisms at low
Reynolds numbers using mathematical, numerical models as well as experimental studies. First,
we simulate an artificial swimmer to captures the physics involved, we carried out some experi-
ments with prototype microorganisms in creeping flow motion for this end a macroscopic swim-
mer which propels itself by mimicking the helical flagella is developed. Three swimming models
propelled by a helical tail with different wavelengths are tested and the movement is examined
for the ambient solvent being Newtonian or non-Newtonian. Secondly, we apply the Slender
Body Theory (SBT) in order to describe the dynamics of swimming micro-organism with mo-
tion produced by helical flagellum propulsion. Also the problem was modelled and simulated
using computational fluid dynamics (CFD). The propulsive velocity, the trajectory, propulsive
force, and torque are measured in the experiments and they are also compared with those
predicted using numerical simulation as function nondimensional parameters Strouhal number
and Deborah number. A very good agreement and compatibility were registered between the
experimental and numerical results. Finally, we study active matter magnetic organisms. For
this end, we erforme simulation with a model of a kind of magnetotactic bacteria composed of a
cluster of spherical magnetic nanoparticles aligned in chain. This enable this magnetic bacteria
to respond even to weak magnetic fields such as the one of the Earth (about 0,7 Gauss). In
our simulation the motion of the magnetic organisms is investigated in the presence of steady
and oscillatory fields. Since the suspended magnetic organisms are easily manipulated and con-
trolled under an applied field, the system studied here behaves like an active suspension. We
verify collective response of the magnetic organisms in the presence of an external applied fields.
In order to explore collective effects in the magnetic suspension investigated, hydrodynamic and
dipolar particle interactions under condition of creeping flow are incorporated to the computer
simulations by using appropriated Green’s functions.
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Schedule
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Mathematics Education
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O uso de softwares na sala de auladurante o perıodo de pandmia:
Possibilidades e utilizacao de novasferramentas de ensino para Matematica
Adriano Costa da Silva
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Thaymara Cristina de Souza Romulo
Email: [email protected]
E. E. Raul de Leoni, Brasil
Thamyres Ribeiro Medeiros
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Aparecida de Fatima Andrade da Silva
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Abstract
Diante das condicoes impostas pela pandemia do novo coronavırus, diversos setores tiveram
e ainda estao tendo que se adaptar e atualizar para enfrentar tal doenca. Essas mudancas
nao foram diferentes no setor de ensino, toda comunidade escolar precisou se organizar de
modo diferenciado para cumprir com as exigencias do ensino de acordo com recomendacoes da
Organizacao Mundial da Saude. Sendo assim, se fez necessario buscar alternativas de novos
metodos de ensino eficazes que substituıssem com qualidade os metodos utilizados no ensino
presencial. Devido a tais motivacoes, propomos apresentar softwares utilizados no programa de
Residencia Pedagogica da Universidade Federal de Vicosa, na Escola Estadual Raul de Leoni na
cidade de Vicosa-MG, ferramentas tais como, jogos no Geogebra, quiz no programa Socrative,
utilizacao de folder digital e a exposicao de conteudos no Word e PowerPoint, que se fizeram
presentes nas salas de aula virtuais da escola, no Regime Especial de Atividades nao Presencial
– REANP, adotado pelo sistema de ensino do estado de Minas Gerais e que se mostraram
um grande aliado para o desenvolvimento dos conteudos que foram trabalhados no perıodo
de pandemia pelo Programa de Ensino Tutorado-PET. Desta forma, o presente trabalho visa
relatar atraves dos residentes do Programa de Residencia Pedagogica da CAPES a experiencia
da utilizacao dos softwares no sistema de ensino remoto imposto pela pandemia, tais como
suas dificuldades e qualidades, assim como oferecer instrumentos que possam ser utilizados ate
mesmo no ensino presencial.
References
[1] A. Bakker; D. Wagner, Pandemic: lessons for today and tomorrow? Educational Studies in
Mathematics, 2020, 104, 1-4.
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[2] Geogebra. Disponıvel em: https://www.geogebra.org/?lang=pt Acesso em: 15 de dez. de 2020.
[3] Portal do Professor. Disponıvel em:http://portaldoprofessor.mec.gov.br/index.html Acesso em:
15 de dez. de 2020.
[4] J. Engelbrecht; M. C. Borba; S. Llinares; G. Kaise, Will 2020 be remembered as the year in
which education was changed? FIZ Karlsruhe, 2020.
[5] Socrative. Disponıvel em: https://www.socrative.com/ Acesso em: 15 de dez. de 2020.
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Promover aprendizagens profissionais aprofessores utilizando-se do racicınio dos
estudantes ao resolver tarefasmatematicas
Alessandro Jacques Ribeiro
Email: [email protected]
Universidade Federal do ABC, Brasil
Marcia Aguiar
Email: [email protected]
Universidade Federal do ABC, Brasil
Abstract
Conduzir discussoes coletivas com todos os estudantes em uma aula de matematica, nao
e uma atividade usual e nem tao pouco trivial para os professores. Em nosso workshop, pre-
tendemos propiciar o compartilhamento de resultados de pesquisas e de experiencias formativas
nas quais os professores, por meio do desenvolvimento de tarefas de aprendizagem profissional,
vivenciam o uso do raciocınio e diferentes estrategias de solucao dos estudantes quando orques-
tram discussoes coletivas em aulas de Matematica dos anos finais do ensino fundamental e do
ensino medio.
References
[1] BALL, D. L.; COHEN, D. K. , Developing practice, developing practitioners: Toward a practice-
based theory of professional education. In: Sykes, G.; Darling-Hammond, L. (Eds.). Teaching as
the learning profession: Handbook of policy and practice.San Francisco, CA: Jossey Bass, 1999.
p. 3-32.
[2] PONTE, J. P., Didacticas especıficas e construcao do conhecimento profissional. In: Tavares,
J. (Ed.). Investigar e formar em educacao: Actas do IV congresso da SPCE, Porto: SPCE p.
59-72, 1999.
[3] PONTE, J. P. , Discussoes coletivas no ensino-aprendizagem em Matematica. In: GTI (Ed.). A
pratica dos professores: planificacao e discussao coletiva na sala de aula.. Lisboa: APM, 2017.
p. 33-56.
[4] RIBEIRO, A. J.; AGUIAR, M.; TREVISAN, A. L. , Oportunidades de aprendizagem vivenci-
adas por professores ao discutir coletivamente uma aula sobre padroes e regularidades. Revista
Quadrante, v. 29, p. 52-73, 2020.
[5] RIBEIRO, A. J.; PONTE, J.P, Professional learning opportunities in a practice-based teacher ed-
ucation programme about the concept of function. Acta Scientiae.v. 21, n. 2, p. 49-74, mar./abr.
2019.
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Lesson Study e Engenharia Didatica:Uma formacao didatico-pedagogica defuturos professores de Matematica do
estagio curricular supervisionado
Aluska Dias Ramos de Macedo
Email: [email protected]
Universidade Federal de Campina Grande, Brasil
Paula Moreira Baltar Bellemain
Email: [email protected]
Universidade Federal de Pernambuco, Brasil
Abstract
Este e um recorte de uma pesquisa de doutorado que teve como objetivo analisar con-
tribuicoes de elementos da JK e da ED para fomentar o desenvolvimento profissional de licen-
ciandos em Matematica matriculados no Estagio Curricular Supervisionado. Como argumenta
Dauanny (2015), o Estagio nao tem sido vivenciado de maneira satisfatoria, pois existem muitos
limites sociais, polıticos, afetivos e cognitivos. Buscando ultrapassar esses limites e romper com
a dicotomia entre teoria e pratica existente em todo o curso de graduacao, especialmente, no
Estagio, apresentamos um processo de formacao didatico-pedagogica com a juncao da Les-
son Study (STIGLER; HIEBERT, 1999) e da Engenharia Didatica (ARTIGUE, 1988). Sendo
uma metodologia japonesa e outra francesa com proximidades e distanciamentos (MIYAKAWA;
WINSLØW, 2009a) que contribuem para o desenvolvimento profissional dos futuros professores
participantes do Estagio. O campo das Grandezas e Medidas, especificamente, o ensino de com-
primento e de area (LIMA; BELLEMAIN, 2010) estavam foi escolhido baseado no calendario
da escola que recebeu os estagiarios. O formador e o supervisor do Estagio participaram do
processo contribuindo com seus conhecimentos e suas experiencias para a formacao dos futuros
professores. Conseguimos ver que houve ganhos em todas as etapas do processo formativo, em-
bora algumas limitacoes tenham surgido. As dimensoes teorica e pratica estavam interligadas
em varios momentos como no estudo dos documentos oficiais, na analise dos livros didaticos,
na leitura e discussao de textos, nas aulas e, por fim, nas trocas de experiencias entre os partic-
ipantes desde o inıcio do processo.
References
[1] ARTIGUE, M. Ingenierie Didactique. Recherches en Didactique des Mathematiques. Grenoble:
La Pensee Sauvage-Editions, v. 9, n. 3, 1988, p. 281-308.
[2] DAUANNY, E. B. O Estagio no contexto dos processos formativos dos professores de
Matematica para a Educacao Basica: entre o proposto e o vivido. Tese (Doutorado em Ed-
ucacao). Faculdade de Educacao, Universidade de Sao Paulo, Sao Paulo, 2015, 375 p.
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[3] LIMA, P. F.; BELLEMAIN, P.M.B. Colecao Explorando o Ensino: Grandezas e Medidas. Vol-
ume 17. Brasılia, 2010, p. 167-200.
[4] MIYAKAWA, T.; WINSLØW, C. Didactical designs for students’ proportional reasoning: an
“open approach” lesson and a “fundamental situation”. Educational Studies in Mathematics,
vol. 72, 2009a, p. 199-218.
[5] STIGLER, J.W.; HIEBERT, J. The Teaching Gap: Best Ideas from the World’s Teachers for
Improving Education. The Free Press, 1999.
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How to Create Mathematics Lessons withFocus on Discovery, Investigatory and
Dialogic Learning
Brigitte Lutz-Westphal
Freie Universitat Berlin, Germany
Pauline Linke
Secondary teacher and teacher trainer, Germany
Abstract
In this interactive talk we demonstrate how to create alternative ideas for teaching a certain
topic in the mathematics classroom. This means to encourage teachers and future teachers to
widen their horizon of educational ideas and goals. With help of different didactical approaches
different shapes“ of the topics can be seen, different access-points will be highlighted and come
into action. We will talk about geometry, the notion of circle“. How can we initiate an ex-
ploratory approach? What part of the notion circle“ can be discovered? How can we encourage
to investigate this certain mathematical object? How to get into a real dialogue about what
students imagine, when they start thinking about circles? Classroom examples will be given
and also theoretical background.
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Saberes e Sabores matematicos em tempode espera: possibilidades de mediacao eintervencao com estudantes no Ensino
Remoto
Cılia Cardoso Rodrigues da Silva
Email: [email protected]
Secretaria de Estado de Educacao do Distrito Federal
Instituto de Educacao, ULisboa
Abstract
Saberes e Sabores matematicos e uma experiencia de ensino no espaco virtual do ensino
remoto de uma Escola Publica do Distrito Federal com estudantes do 4º e 5º ano do Ensino
Fundamental. Apresento as possibilidades de mediacao e intervencao na resolucao de problemas
envolvendo as operacoes de multiplicacao e divisao no ensino remoto com uso de aplicativos
Google Meet e Jamboard. E um estudo empırico, apoiado no paradigma interpretativo, com
abordagem qualitativa, na modalidade de estudo de caso. A mediacao, intervencao e coleta de
dados foi feita com 12 estudantes, sendo 5 criancas tıpicas e 7 atıpicas. Os encontros aconteceram
pelo google meet, simultaneo com o aplicativo jamboard. O acesso dos estudantes aconteceu
via celular, tablet ou computador. Os estudantes foram atendidos em pequenos grupos e de
modo individual, tendo duracao de 30 a 60 minutos. Durante os encontros eles utilizaram as
ferramentas do jamboard para resolver os problemas e escrever suas estrategias de calculo a
partir do seu pensamento matematico. O jamboard possibilitou que utilizassem as imagens
do google como material concreto de apoio para seus calculos. O uso simultaneo do meet e
jamboard permitiu a interacao entre os pares e a investigadora. No ensino remoto foi essencial
usar aplicativos que permitissem uma interacao em tempo real com os estudantes. O meet
e o jamboard possibilitaram essa interacao, ao mesmo tempo que os estudantes resolviam as
situacoes problemas e registravam suas estrategias de calculo foi possıvel discutir com eles os
procedimentos usados e suas respostas.
References
[1] Distrito Federal. Secretaria de Educacao. Guia Anos Iniciais: Orientacoes para Ativi-
dades de Ensino Remoto. 2020. Disponıvel em: http://www.educacao.df.gov.br/wp-
conteudo/uploads/2020/06/Guia-Anos-Iniciais-orienta%C3%A7%C3%B5es-para-atividades-de-
ensino-remoto.pdf , acesso em 17/09/2020.
[2] Greer, B. Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook
of research on mathematics teaching and learning (pp. 276-295). NewYork: Macmillan, 1992.
[3] Mendes, F. A aprendizagem da multiplicacao numa perspetiva de desenvolvimento do sentido
de numero: um estudo com alunos do 1° ciclo. Tese de doutoramento. Lisboa: Universidade de
Lisboa, 2012.
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[4] Stake, R. E. A Arte da Investigacao com Estudos de Caso. Lisboa: Fundacao Calouste Gul-
benkian, 2016.
[5] Threlfall, J. Strategies and flexibility in mental calculation. ZDM Mathematics Education, 41(5),
541-555, 2009.
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Intervencao em criatividade no campo damatematica: Um estudo de caso no
ensino fundamental
Cleyton Hercules Gontijo
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Mateus Giani Fonseca
Email: [email protected]
Intituto Federal de Brasılia, Brasil
Abstract
Esta comunicacao tem por objetivo apresentar resultados de uma pesquisa que analisou os
efeitos de um programa de ensino de matematica apoiado no uso de tecnicas de criatividade
sobre a motivacao e a criatividade em matematica, com 51 estudantes do 4º ano do ensino
fundamental, de uma escola publica do Distrito Federal. Empregou-se uma abordagem mista
de pesquisa, com etapas qualitativas e etapas quantitativas. A abordagem quantitativa foi
caracterizada pela aplicacao de pre-testes e pos-testes, envolvendo os seguintes instrumentos:
Teste de Criatividade em Matematica – Versoes A e B(CARVALHO, 2019), Escala de Clima
de Sala de Aula para Criatividade em Matematica (CARVALHO,2015) e Escala de Percepcao
sobre as Oficinas de Criatividade em Matematica (GONTIJO, 2018). A abordagem qualitativa
foi utilizada para o desenvolvimento do programa de ensino baseado em tecnicas de criativi-
dade, composto por 8 oficinas realizadas uma vez por semana com duracao media de 3 horas
cada e para a analise dos dados obtidos com a aplicacao do Inventario de Atitudes em Relacao
a Matematica (GOMEZ CHACON, 2003). Os resultados dos instrumentos evidenciam que o
programa de ensino de matematica apoiado no uso de tecnicas de criatividade, desenvolvido a
partir do Roteiro para Oficinas de Criatividade em Matematica (GONTIJO, 2018), favoreceu
aos estudantes desenvolver atitudes positivas em relacao a matematica e proporcionou o desen-
volvimento da criatividade em matematica, elevando os escores dos estudantes no teste de Teste
de Criatividade em Matematica.
References
[1] Carvalho, Alexandre Tolentino, Relacoes entre criatividade, desempenho escolar e clima para
criatividade nas aulas de matematica de estudantes do 5º ano do ensino fundamental. 2015.
132f. Dissertacao (Mestrado em Educacao)–Faculdade de Educacao, Universidade de Brasılia,
Brasılia.
[2] Carvalho, Alexandre Tolentino, Criatividade compartilhada em matematica: do ato isolado
ao ato solidario. 2019. 359 f., il. Tese (Doutorado em Educacao) – Universidade de Brasılia,
Brasılia
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[3] Gomes Chacon, Ines Mª., Matematica emocional: os afetos na aprendizagem matematica Cria-
tividade compartilhada em matematica: do ato isolado ao ato solidario. Porto Alegre: Artmed,
2003.
[4] Gontijo, Cleyton Hercules, Roteiro para Oficinas de Criatividade em Matematica. Brasılia, 2018.
Nao publicado.
[5] Gontijo, Cleyton Hercules, Escala de Percepcao sobre as Oficinas de Criatividade em
Matematica. Brasılia, 2018. Nao publicado.
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Uso da Criptografia como Ferramenta deEnsino
Daiane Soares Veras
Email: [email protected]
Instituto Federal de Goias
Campus Valparaıso, Brasil
Abstract
As metodologias de ensino tradicionais ainda estao entre as tecnicas de ensino mais empre-
gadas, com o uso de referenciais teoricos apresentados em lousas/datashow e transcritos para o
caderno dos estudantes. Esse tipo de aula expositiva e um dos procedimentos didaticos individ-
ualizantes mais conhecidos e utilizados pelos professores, sem que haja a devida interacao na
efetivacao do aprendizado. No entanto, com os avancos tecnologicos e o facil acesso dos alunos a
tecnologias de informacao e comunicacao (TICs) torna-se necessario o uso de novas metodologias
de ensino e aprendizagem. Varios trabalhos apresentam o tema Criptografia como motivador
e gerador de situacoes didaticas que permitem o aprofundamento da compreensao de conceitos
matematicos, possibilitando ao aluno perceber a utilizacao do conhecimento matematico em
situacoes praticas, visando salientar a importancia da utilizacao de atividades didaticas que
possibilitem aos alunos resolver problemas, levantar hipoteses e trabalharem em grupo e coop-
erativamente (GROENWALD e OLGIN, 2011). Tudo isso vai ao encontro do que estabelece
a BNCC para o Ensino da Matematica e suas Tecnologias, que diz que novos conhecimentos
especıficos devem estimular processos mais elaborados de reflexao e de abstracao, que deem
sustentacao a modos de pensar que permitam aos estudantes formular e resolver problemas em
diversos contextos com mais autonomia e recursos matematicos (BRASIL, 2017, p.528). Desse
modo, este trabalho visa apresentar experiencias com o uso da Critpografia como tema gerador
de conteudos para o ensino basico e tambem o resultado de uma pesquisa sobre esse tema, feita
via Google Formularios, sob minha orientacao, durante a execucao de um projeto de Iniciacao
Cientıfica realizado com alunos do curso de Licenciatura em Matematica no IFG/Valparaıso.
References
[1] BRASIL. Ministerio da Educacao. Base Nacional Comum Curricular. Brasılia, 2017.
[2] GROENWALD, C. L. O. OLGIN, C. de A., Criptografia e o Currıculo de Mate- matica no
Ensino Medio , Revista de Educacao Matematica, v. 13, n. 15, p. 69 – 78, 2011.
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Educacao Matematica e inclusao escolarna perpectiva da educacao para todos
Ellen Michelle Barbosa de Moura
Email: [email protected]
Universidade de Brasılia, Brasil
Karla Vanessa Gomes dos Santos
Universidade de Brasılia, Brasil
Joeanne Neves Fraz
Universidade de Brasılia, Brasil
Abstract
O objetivo deste estudo e refletir sobre a dıade Inclusao Escolar e Educacao Matematica,
a partir do questionamento: quais as contribuicoes da Educacao Matematica para a inclusao
escolar na perspectiva de uma educacao para todos tendo como base artigos sobre Educacao
Matematica na perspectiva dos Direitos Humanos (VIEIRA; MOREIRA, 2018). De base qual-
itativa e descritiva, a analise utiliza como procedimento a pesquisa bibliografica. Segundo
os artigos analisados, muitos sao os obstaculos para a efetivacao de estrategias de Educacao
Matematica Inclusiva, das quais destacam-se: condicoes psicoemocionais dos profissionais da
Educacao, devido a enorme carga de trabalho, desvalorizacao docente e os baixos salarios; es-
trutura predial das escolas; o estado familiar dos alunos com e sem NEE, em funcao das novas
constituicoes familiares; formacao inicial dos professores, preconceitos e representacoes sociais
diversas. A reflexao demonstra a importancia, a atualidade, a necessidade de contınua producao
academica e discussao acerca da tematica, a fim de garantir a luta pela inclusao escolar na per-
spectiva dos Direitos Humanos, apontando caminhos diversos, ressignificacoes e experimentacoes
(MOURA; FRAZ; SANTOS, 2020). Constata-se que, compreender a relacao entre inclusao es-
colar e Educacao Matematica e essencial, na medida que aumenta a chance de os professores
lidarem com a realidade inclusiva de modo mais consciente e problematizado. Moreira (2012)
destaca a importancia de pensar a formacao docente, inclusao e modos diversificados para lidar
com alunos NEE. Para Slee (2011), a educacao inclusiva torna-se um campo da polıtica cultural
que tem o objetivo da reconstrucao social e, com efeito, os educadores inclusivos tornam-se
vigilantes culturais.
References
[1] MOREIRA, G. E. Representacoes sociais de professoras e professores que ensinam Matematica
sobre o fenomeno da deficiencia. 202 f, 2012. Tese (Doutorado em Educacao Matematica) - Pon-
tifıcia Universidade Catolica de Sao Paulo, Programa de Estudos Pos-Graduados em Educacao
Matematica, Sao Paulo, 2012.
[2] MOURA, E. M. B.; FRAZ, J. N.; SANTOS, K. V. G. Educacao Matematica na Educacao
Infantil: situacoes cotidianas na formacao de professores. In: MOREIRA, G. E. (org.). Praticas
de Ensino de Matematica em Cursos de Licenciatura em Pedagogia: Oficinas como instrumentos
de aprendizagem. Sao Paulo: Editora Livraria da Fısica, 2020. p. 20-35.
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[3] SLEE, R. O paradoxo da inclusao. In: APPLE, M. W; AU, W.; GANDIN, L. A. Educacao
crıtica analise internacional. Porto Alegre, RS: ArtMed, 2011, p.203-216.
[4] VIEIRA, L. B.; MOREIRA, G. E. Direitos Humanos e Educacao: o professor de matematica
como agente sociocultural e polıtico. Revista de Educacao Matematica, v. 15, p. 548-564, 2018.
Disponıvel em: https://www.revistasbemsp.com.br/REMat-SP/article/view/174/pdf. Acesso
em: 20 jan. 2021.
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A modelagem matematica comoalumbramento a reflexao e aprendizagem
cotidiana
Fatima Aparecida Kian
Email: [email protected]
Universidade Federal do ABC, Brasil
Ailton Paulo de Oliveira Junior
Email: [email protected]
Universidade Federal do ABC, Brasil
Luıs Delcides Eodrigues da Silva
Email: [email protected]
Centro Universitario das Faculdades Metropolitanas Unidas, Brasil
Abstract
Aprender matematica nao cansa. Ao usar a ciencia exata como linguagem para o apren-
dizado, esta revela as necessidades diarias e acessorias para tarefas comuns e de extrema im-
portancia. A presente pesquisa baseada em pesquisa bibliografica de artigos sobre a educacao
matematica faz uma importante mencao ao uso da modelagem como metodologia de apren-
dizado e simbiose de linguagens, tanto no aspecto reflexivo e da necessidade, especialmente ao
abordar o exemplo do excesso de peso da mochila escolar. O objetivo deste estudo e mostrar o
efeito do aprendizado dos alunos ao montarem uma tabela com a quantidade de massa corporea
e o peso dos apetrechos estudantis e, apos estudos, misturados com a aplicabilidade de conceitos
matematicos, chegou-se a provocar ressonancias com relacao ao futuro de saude desses garotos e
a possibilidade de afetar o desenvolvimento motor dos jovens, especialmente o desenvolvimento
de atrofias na regiao da coluna. Para isso houve a revisao bibliografica de varios trabalhos
referentes ao aprendizado da matematica e da modelagem no aprendizado de criancas, espe-
cialmente do ensino fundamental. As leituras e fichamentos das pesquisas, especialmente dos
bancos de dados das pesquisas aplicadas a educacao matematica foram de importancia indis-
pensavel por provocar reflexoes e descobrir trabalhos interessantes como o caso da professora
Carla Escorsin Roque, ao mencionar a experiencia do aprendizado da porcentagem, razao e
proporcao atraves de uma pesquisa de campo feita pelos jovens do ensino fundamental. Nao
basta apenas a continencia pelo prazo certo e para entregar o conteudo para os alunos, e pre-
ciso estimular a capacidade de pensamento, aprendizado e consideracao atraves da matematica,
como operacionalidade e linguagem.
References
[1] COSTA, Lucelida de Maia da. Leitura, Interpretacao e Construcao de Tabelas e
Graficos nos Anos Iniciais do Ensino Fundamental. 1ª Ed. Rio De Janeiro: So-
ciedade Brasileira de Matematica, 2018. Disponıvel em: https://www.sbm.org.br/wp-
content/uploads/2018/04/Leitura-interpretacao-e-construcao-de-tabelas-e-graficos.pdf Acesso
em 30 de dez 2020.
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[2] ROQUE, Carla Cristina Escorsin. Modelagem Matematica no En-
sino Fundamental. Revista da Gestao Escolar Parana. Venceslau
Braz: Revista da Gestao Escolar Parana , 2009. Disponıvel em:
http://www.gestaoescolar.diaadia.pr.gov.br/arquivos/File/producoes pde/artigo carla cristina
escorsin roque.pdf Acesso em 30 de dez 2020.
[3] PECA, Celia Maria Karpinski. Analise e interpretacao de Tabelas e Graficos Estatisticos
Utilizando Dados Interdisciplinares. Trabalho de Conclusao do PDE- Programa de desen-
volvimento educacional. Orientacao Prof. Msc. Simone Crocetti- UTFPR,2008.Disponıvel em:
http://www.diaadiaeducacao.pr.gov.br/portals/pde/arquivos/1663-8.pdf Acesso em 30 de dez.
2020.
[4] SILVA, Leonardo Brito; et. al. Modelagem Matematica: Reflexoes
Teoricas e Criacoes. Anais do VII Encontro mineiro de Educacao
Matematica, Juiz de Fora: 9 a 12 de outubro de 2015. Disponıvel em:
http://www.ufjf.br/emem/files/2015/10/MODELAGEM-MATEM%C3%81TICA-
REFLEX%C3%95ES-TE%C3%93RICAS-E-APLICA%C3%87%C3%95ES.pdf Acesso em
30 de dez de 2020.
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Historia da matematica no ensino eaprendizagem: colaboracao na formacao
do professor
Fatima Aparecida Kian
Email: [email protected]
Universidade Federal do ABC, Brasil
Ailton Paulo de Oliveira Junior
Email: [email protected]
Universidade Federal do ABC, Brasil
Luıs Delcides Eodrigues da Silva
Email: [email protected]
Centro Universitario das Faculdades Metropolitanas Unidas, Brasil
Abstract
O objetivo desta pesquisa e mostrar a possibilidade de utilizarmos a Historia da Matematica
na ensinanca e aprendizagem na formacao do professor do ensino basico. O professor do ensino
infantil, com a sua formacao superior em pedagogia, leciona nos primeiros anos do ensino funda-
mental e, em sua maioria, nao possui formacao em matematica. De acordo com estudos, a grande
maioria dos discentes em pedagogia, faz o curso pelo ensino a distancia, (EaD). Conforme as
informacoes coletadas por essa pesquisa, 60% concluem a licenciatura no formato remoto. Alem
do aumento dos cursos a distancia, em relacao ao das outras areas do Ensino Superior, a quali-
dade destes cursos deixa a desejar. Serao analisados alguns conceitos com intuito de demonstrar
a importancia do uso da Historia da Matematica no ensino e aprendizagem na formacao destes
professores. Para desenvolver este trabalho utilizamos pesquisas feitas em teses, dissertacoes,
sites especializados, e artigos. Portanto, trata-se de um estudo bibliografico referente ao tema
ao demonstrar os pontos positivos da Historia da Matematica na formacao docente, amparado
pelos Parametros Curriculares Nacionais (PCN), Brolezzi (1991) e D´Ambrosio (1997 e 1999).
Como consideracoes parciais verificamos a necessidade da utilizacao da Historia da Matematica
em algum contexto com tendencias a um papel positivo no ensino e aprendizado de conceitos e
conteudo da Matematica desde a tenra idade.
References
[1] Brasil. (1997). Ministerio da Educacao e do Desporto (MEC). Parametros Curriculares Na-
cionais: matematica, Brasılia, Brasil.
[2] BROLEZZI, Antonio Carlos. (1991). A arte de contar : uma introducao ao estudo do valor
didatico da historia da matematica. Dissertacao de Mestrado em Educacao, Departamento de
Metodologia do Ensino e Educacao Comparada da Faculdade de Educacao, Universidade do
Estado de Sao Paulo (USP), Sao Paulo, Brasil.
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[3] D´AMBROSIO, Ubiratan. (1997). Educacao Matematica: da teoria a pratica. 2 ed. Campinas:
Papirus, 1997. Colecao Perspectivas em Educacao Matematica.
[4] D´AMBROSIO, Ubiratan. (1999). A historia da matematica: questoes historiografica e polıticas
e reflexos na educacao matematica. In Bicudo, Maria Aparecida Viggiani (Org.), Pesquisa em
Educacao Matematica: Concepcoes Perspectivas (pp. 97-115). Sao Paulo: UNESP, Sao Paulo.
[5] LARA, Isabel Cristina Machado. (2013). O ensino da matematica por meio da historia da
matematica e suas possıveis articulacoes com a etnomatematica. Vidya, 33(2), 51-62.
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Os elementos ludico e sensorial fora doslimites da sala de aula e o aprendizado da
matematica
Fatima Aparecida Kian
Email: [email protected]
Universidade Federal do ABC, Brasil
Ailton Paulo de Oliveira Junior
Email: [email protected]
Universidade Federal do ABC, Brasil
Luıs Delcides Eodrigues da Silva
Email: [email protected]
Centro Universitario das Faculdades Metropolitanas Unidas, Brasil
Abstract
Matematica costuma ser a disciplina escolar repelida pelos alunos por causa de seu conteudo
macante e demasiadamente aplicado apenas em sala de aula. Ha formas de despertar os alunos
para o prazer do aprendizado e perceber a sua aplicabilidade no dia a dia. Contudo, e preciso
verificar a necessidade de vivenciar as experiencias na realidade, especialmente em atividades
extra aula ao levar os estudantes ate uma loja de materiais de construcao, supermercado para
experimentar novas experiencias enriquecedoras para o aprendizado. A presente pesquisa ap-
resenta a importancia da matematica no cotidiano, o olhar dos objetos comuns como trıades,
conforme proposto por Charles Sanders Pierce, um dos principais pensadores da Semiotica
Americana, ao mencionar em sua obra a experiencia de olhar para a estrada ao trabalhar o
conceito de paralelas e retas. Importante mencionar o ensinamento de Lewis Carrol, com toda
a sua inteligencia ao associar a historia de Alice no Paıs das Maravilhas com ensinamentos sutis
acerca da geometria espacial e os detalhes precisos acerca dos numeros e de suas caracterısticas
atraves de uma trama prazerosa e envolvente. O objetivo do presente estudo e apresentar a
importancia da educacao das ciencias exatas, especialmente fora dos limites da sala de aula, ao
proporcionar experiencia, mostrar a aplicabilidade para os alunos e a importancia dos calculos
no dia a dia.
References
[1] CARROL, Lewis. Alice no Paıs das Maravilhas. Ed. Sao Paulo: Editorial Arara Azul, 2002.
Disponıvel em: http://www.editora-arara-azul.com.br Acesso em 29 de dez de 2020.
[2] DELEUZE, Gilles. A Logica do Sentido. ed. Sao Paulo: Perspectiva Ed. Da Universidade de
Sao Paulo, 1974.
[3] GONCALVES, Talita da Cunha; et.al. Identificacao de lacunas no processo de aprendizagem
dos conteudos de geometria no ensino medio pelo metodo de Van Hiele. Revista Eletronica
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de Educacao Matematica REVEMAT , Florianopolis, v. 15, p. 01-20 , jan./dez., 2020 Uni-
versidade Federal de Santa Catarina. ISSN 1981- 1322. DOI: https://doi.org/10.5007/1981-
1322.2020.e74525 Acesso em 29 de dez de 2020.
[4] PIERCE, Charles Sanders. Semiotica. ed. Sao Paulo: Ed. Perspectiva editora da Universidade
de Sao Paulo, 2005.
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Introducao a historia social da educacaomatematica - HISOEM
Fredy Enrique Gonzalez
Email: [email protected]
Universidade Federal de Rio Grande do Norte, Brazil
Abstract
A questao norteadora desta palestra e: como aconteceu o processo que propiciou a emergencia
e o desenvolvimento da Educacao Matematica, ate se converter num campo disciplinar e de
pesquisa, tanto pratica quanto teorica? Sera oferecida uma resposta na perspectiva da Historia
Social da Educacao Matematica – HISOEM. que leva em consideracao as praticas socioculturais
associadas com processos de ensino, aprendizagem, estudo, avaliacao, criacao das Matematicas
- tanto academicas quanto escolares e cotidianas - que sao protagonizadas por diversos au-
tores/atores - tanto reconhecidos como autores/atores de referencia quanto anonimos: profes-
sores de aula, vendedores de rua, artistas de diversas areas, artesaos, bonequeiros, costureiras,
etc. O aspecto central desta perspectiva e examinar o desenvolvimento no tempo (Historia) das
interacoes entre os protagonistas (atores e autores de referencia) das diversas situacoes e praticas
sociais (Sociologia) nos multiplas contextos (cenarios de difusao) onde sao desenvolvidas praticas
de ensino, aprendizagem, estudo e avaliacao das diversas variedades da Matematica: academica,
escolar e cotidiana (a que e utilizada pelas pessoas nas suas variadas atividades, tanto profission-
ais quanto nao profissionais, como as dos marceneiros, pedreiros, e muitos outros operarios ou
tecnicos; como tambem assim os artesaos, pescadores, etc.). As nocoes teoricas assumidas sao:
as ideias de campo cientıfico, evolucionismo conceitual, pratica sociocultural, enfoque historico
cultural e situacao social. Metodologicamente, trata se de uma pesquisa teorica documental de
natureza reflexivo-interpretativa. Se conclui que a constituicao como disciplina da Educacao
Matematica e um processo epistemologico, sociologico e historico essas tres perspectivas sao o
fundamento da concepcao da HISOEM subscrita nesta exposicao.
References
[1] BOURDIEU, Pierre. Os usos sociais das ciencias: por uma sociologia clınica do campo cientıfico.
Sao Paulo: Unesp, 19832004.
[2] GONZALEZ, Fredy Enrique. Historia, Educacao, Matematica: relacoes virtuosas. Em DORR,
Raquel; NEVES, Regina. (Org.) Cenarios de Pesquisa em Educacao Matematica. Sao Paulo
Paco Editorial, 2020. Pp 95-122.
[3] MENDES, Iran; SILVA, Carlos Problematizacao de praticas socioculturais
na formacao de professores de Matematica. Revista Exitus, [S. l.], v. 7, n.
2, p. 100-126, 2017. DOI: 10.24065/2237-9460.2017v7n2ID303. Disponıvel em:
http://www.ufopa.edu.br/portaldeperiodicos/index.php/revistaexitus/article/view/303. Ac-
ceso em: 7 Jan. 2021.
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[4] TOULMIN, Stephen. La comprension humana, v. I: El uso colectivo y la evolucion de los con-
ceptos. Madrid: Alianza Editorial, 1997.
[5] VALENTE, Wagner A matematica escolar: epistemologia e historia. Revista Educacao em
Questao, v. 23, n. 9, p. 16-30, 15 ago. 2005.
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Producoes Cientıficas sobre as Relacoesentre Covid - 19 e o Ensino de Ciencias e
de Matematica
Janaına Mendes Pereira da Silva
Email: janaina.mendes @ufabc.edu.b
Universidade Federal do ABC, Brasil
Natalia Lami Zanettini
Email: [email protected]
Universidade Federal do ABC, Brasil
Sabrina Saito
Email: [email protected]
Faculdade de Tecnologia de Barueri, Brasil
Universidade Federal do ABC, Brasil
Abstract
Diante da realidade pandemica mundial, o estudo teve como objetivo a realizacao de uma
revisao sistematica na busca por pesquisas, sobre os impactos e aspectos que a pandemia da
Covid - 19 provocou e ocasionou no ensino de ciencias e matematica. A partir deste objetivo,
por meio da metodologia de pesquisa tipificada como revisao sistematica foi possıvel compi-
lar e sintetizar as evidencias encontradas nos estudos desenvolvidos entre marco a outubro de
2020. Por conta do volume de informacao, e das possıveis evidencias que foram identificadas
nas pesquisas e da necessidade de analises crıticas, exploratorias e/ou elementares, houve as
leituras e apos escolheu-se algumas pesquisas que versavam sobre a tematica deste estudo, que
permitiram a criacao de uma tabela, em que as pesquisas encontradas foram caracterizadas. A
partir destas caracterizacoes emergiram quatro categorias que contribuıram para elencar os re-
sultados e breves analises, tais sejam: Cenarios polıticos e Polıticas Publicas; Ensino: Impactos
de Tecnologia no Desempenho Escolar; Ensino: Formacao de Professores,e Ensino: Mudancas
de Trabalho. Como resultados e conclusoes as pesquisas encontradas e brevemente analisadas,
apresentam alguns impactos negativos e positivos diante da realidade educacional vivida du-
rante a pandemia provocada pela Covid-19. Outras pesquisas relatam os efeitos da pandemia
na sociedade, polıticas publicas, Fake News, ensino e saude LGBT, ensino, aprendizagem, e
avaliacao no contexto pandemico, gestao universitaria e as influencias destes impactos ao redor
do mundo.
References
[1] DIAS,E; PINTO,FCF, A Educacao e a Covid-19. Ensaio: Avaliacao e Polıticas Publicas em
Educacao, v. 28, n. 108, p. 545–54, 2020.
[2] Lopes, A.; Fracolli L, Revisao sistematica de literatura e metassıntese de literatura e metassıntese
qualitativa: Consideracoes sobre sua aplicacao na aplicacao na pesquisa em enfermagem. Texto
Contexto Enferm, Florianopolis, 2008 Out-Dez;17(4): 771-8.
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Impressoes de alunos da licenciatura emmatematica sobre o ensino remoto em
uma universidade publica
Josinalva Estacio Menezes
Email: [email protected]
Departamento de Matematica
Universidade Federal de Pernambuco, Brasil
Maria Dalvirene Braga
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Rui Seimetz
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Poliana Maria da Silva
Email: [email protected]
Universidade Federal de Pernambuco, Brasil
Samara Araujo da Silva
Email: [email protected]
Universidade Federal de Pernambuco, Brasil
Abstract
O ensino remoto hoje e alvo de uma das mais efervescentes discussoes no cenario academico
educacional em todos os nıveis, por causa do advento da pandemia do Corona Vırus, a COVID-
19, que pegou a sociedade como um todo de surpresa. Por causa disso, como se sabe, aconteceu
uma mudanca da chamada “vida normal”: as aulas foram suspensas, instalou-se o distancia-
mento social, o livre ir e vir, as saıdas de casa limitaram-se aos casos de extrema necessidade
e alguns setores ligados a servicos essenciais continuaram a funcionar. Assim, realizamos uma
pesquisa cujo objetivo geral foi descrever as impressoes de alunos do curso de Licenciatura em
Matematica participantes de atividades via ensino remoto em uma universidade publica. Con-
vidamos todos os alunos do curso de Licenciatura em Matematica que participam do ensino
remoto no perıodo, em qualquer atividade. Desses, tres alunos nao participavam e os demais
78 foram selecionados. Aplicamos e analisamos um questionario junto aos alunos atraves do
formulario google forms, o qual foi disponibilizado na sala de acolhimento disponıvel no campus
cujo acesso ao modelo e informado nas referencias. Analisamos as respostas dos alunos do ponto
de vista da frequencia estatıstica segundo Chizotti(1991) e tambem os fragmentos de fala nas
respostas livres e justificativas, segundo Bardin (1977 ). Os resultados alcancados mostraram
que o ensino remoto nao e ainda uma forma eficiente nem motivadora de processo de ensino;
ha muitas dificuldades relativas a lida com as TDIC de docentes e discentes. O andamento da
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vida social, o que inclui o ingresso dos alunos no mercado de trabalho apos a formacao, esta
a requerer urgentes medidas mais gerais para “normalizar” a vida academica, refletindo-se na
vida do egresso, requerendo uma adequacao entre essa nova realidade e os padroes de com-
portamento na sociedade assolada pela pandemia. Ha um desejo de retorno ao “normal”, com
o ensino em seu formato anterior; e paralelamente, o ensino na forma presencial nao deveria
vir antes da vacina. Assim, encaminhamos a necessidade do esforco para termos um resul-
tado satisfatorio do processo. Reafirmamos entao a grande tarefa do professor: usar sua ampla
bagagem de experiencia docente incluindo metodologias, materiais, conhecimentos e capacidade
de adaptacao, para contribuir com um ensino eficiente nessa modalidade que se impos e pode ser
benefica nessa epoca em que vivemos. Finalmente, precisamos lidar com os 4D (BEHAR, 2020):
desanimo, com as falhas nas TDIC, o desafio, ante a necessidade de sair da zona de conflito e
paralisia, o desespero, ao querermos realizar tudo da melhor forma possıvel e o desenvolvimento,
consequencia imediata da busca e exito em resolver os desafios.
References
[1] Bardin, Lawrence, Analise de conteudo. Lisboa: Edicoes 70, 1977.
[2] Behar, Patrıcia Alexandra, O ensino remoto emergencial e a educacao a distancia. Criado
em 06.07.2020. Disponıvel em https://www.ufrgs.br/coronavirus/base/artigo -o-ensino-remoto-
emergencial-e-a-educacao-a-distancia/. Acesso em 05.11.2020.
[3] Ferreira, Geiza, Pedagoga explica diferenca entre ensino remoto e EaD. Disponıvel em
https://www.uninassau.edu.br/noticias/pedagoga-explica-diferenca-entre-ensino-remoto-e-ead/.
Acesso em 05.06.2020.
[4] Silva, Joscimar Souza, Ensino remoto emergencial em contexto de pandemia. Disponıvel
em https://www.ica.ufmg.br/?noticias=ensino-remoto-emergencial-em-contexto-de-pandemia/.
Acesso em 05.11.2020.
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Experiencias na residencia pedagogica
Kellton de Oliveira Sabino
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Keila Cecılia Goncalves
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Aparecida de Fatima Andrade da Silva
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Thamyres Ribeiro Medeiros
Email: [email protected]
E. E. Raul de Leon, Brasil
Abstract
Como atuais residentes pedagogicos pela Universidade Federal de Vicosa na Escola Estadual
Raul de Leoni, e devido a pandemia da Covid-19, a fim de revisionar os conteudos estuda-
dos nos Planos de Ensino Tutorados fornecidos pelo estado de Minas Gerais, des envolvemos
pesquisas sobre materiais didaticos a serem trabalhados de forma remota para aperfeicoar o
aprendizado dos alunos do setimo ano. Buscamos por propostas simples como jogos, listas de
exercıcios, quizzes, atividades que propoem a observacao e o re conhecimento dos conceitos no
cotidiano, e elaboramos planos de aula com tais ferramentas. Pretendemos mostrar a construcao
de um Quiz que e feito no aplicativo PowerPoint de forma simples, o qual usamos para testar os
conhecimentos dos alunos sobre fracoes e medidas em uma das nossas co-regencias. Os conhec-
imentos previos para elaborar o Quiz devem ser: criar novos slides, colocar formas geometricas
e formatar. E feito atraves da criacao de hiperlinks que direcionam para os slides seguintes.
Observamos que os alunos tem interesse maior quando as atividades sao mais interativas, dessa
forma, o Quiz permite trabalhar qualquer conceito, matematico ou nao, de forma descontraıda,
uma vez que testa os conhecimentos em relacao ao conteudo abordado e fornece uma maior
memorizacao e aprendizagem do mesmo, bem como revisar conhecimentos ja adquiridos pelos
alunos.
References
[1] M. F. R. C. Machado, O uso dos recursos didatico-tecnologicos como potencializadores ao
processo de ensino e aprendizagem. Relato de experiencia publicado no XIV EDUCERE-
CONGRESSO NACIONALDE EDUCACAO. Eixo: 12. Educacao, Tecnologia e Comunicacao.
2017. Disponıvel em: https://educere.bruc.com.br/arquivo/pdf2017/24989 14142.pdf. Acesso em
19 jan. 2021
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[2] Regime de estudo nao presencial. Plano De Estudo Tutorado, 7° Ano. Estude em casa, 2020.
Disponıvel em: https://estudeemcasa.educacao.mg.gov.br/pets/ens-fund-anos-finais. Acesso
em:19/01/2021.
[3] Como criar um quiz no Power Point. Drak Tutoriais, 2019. Disponıvel em:
https://www.youtube.com/watch?v=3CGdLpoe1nsfeature=youtu.be. Acesso em: 12/01/2021
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Metodologias para o ensino remoto dematematica
Letıcia Pereira de Almeida
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Guilherme Flaviano Pereira
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Thamyres Ribeiro Medeiros
Email: [email protected]
E. E. Raul de Leoni, Brasil
Aparecida de Fatima Andrade de Silva
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Abstract
Enquanto residentes do Programa Residencia Pedagogica da Universidade Federal de Vicosa
e atuando na Escola Estadual Raul de Leoni, elaboramos alguns planos de aula para comple-
mentar os estudos durante o perıodo de ensino remoto, os quais pudessem ser atrativos aos
estudantes. Desta forma, buscamos encontrar ferramentas que pudessem facilitar a aprendiza-
gem dos alunos do 9º ano do ensino fundamental em conteudos da disciplina de Matematica.
A elaboracao do material teve por base os PET’s disponibilizados pela Secretaria do Estado
de Educacao de Minas Gerais–SEE/MG. As subareas da Matematica abordadas foram Geome-
tria e Algebra, com as seguintes especificacoes: Teorema de Tales,Semelhanca de Triangulos,
Trigonometria, Resolucao de Equacao do 2º grau, Funcao Afim e Quadratica. Para explicacao
do Teorema de Tales e para Funcao Afim foi utilizado o software Geogebra, bem como foi ap-
resentado um vıdeo sobre a utilizacao do pantografo. Em Semelhanca de Triangulo utilizamos
exercıcios desafiadores da Olimpıada Brasileira de Matematica–OBMEP. Ja para a Trigonome-
tria foi proposto que os alunos encontrassem um triangulo em casa e fizessem o calculo dos
angulos utilizando seno, cosseno e tangente. Enquanto que no caso da Resolucao de Equacoes
do 2º grau, foi confeccionado um jogo organizado em rodadas, proporcionando um clima encora-
jador e competitivo, o qual favoreceu a resolucao dos exercıcios p elos alunos. Para o conteudo
de Funcao Quadratica utilizamos um jogo denominado “Parabola nas estrelas” o qual possi-
bilita aos alunos resolverem situacoes problemas sobre o tema. O presente trabalho apresenta
os planos que podem ser utilizados em aulas remotas ou presenciais.
References
[1] Assim, Cleber e Miranda, Tiago, Portal da Matematica. Instituto de Matematica
Pura e Aplicada(IMPA). OBMEP-Olimpıada Brasileira de Matematica. Disponıvel em:
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https://portaldaobmep.impa.br/index.php/modulo/ver?modulo=10. Acesso em 19 de janeiro de
2021.
[2] Geogebra, Dynamic Mathematics for Everyone. Version 6,0,624,0-wsuite. 2021. Disponıvel em:
https://www.geogebra.org. Acesso em 19 de janeiro de 2021.
[3] VENTURA, Geraldo, Teorema de Tales-Aplicacao pratica. Vıdeo. Disponıvel em:
https://www.youtube.com/watch?v=oquQ1x2js6U.Acesso em 19 de janeiro de 2021.
[4] Secretaria de Educacao de Minas Gerais-SEEMG, Programa de Estudos Tutorados –PET.
Disponıvel em: https://estudeemcasa.educacao.mg.gov.br/pets/ens-fund-anos-finais. Acesso em
19 de janeiro de 2021.
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A survey on the PreCalculus course offerin the Midwest Higher Education
Institutions
Luciana Avila Rodrigues
Partially supported by PET/MEC/FNDE, email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Thais Regina Duarte Marcal
Partially supported by PET/MEC/FNDE
Departamento de Matematica
Universidade de Brasılia, Brasil
Abstract
In this study we make a survey on the offer of the PreCalculus course in Public Institutions
of Higher Education (HEI) in the Midwest region. The transition from High School to Higher
Education has shown itself to be problematic, in the view of other scholars, because students
have had problems adapting in the early years of graduation, particularly in Exact Science
courses. The failure and dropout rate, especially in the Differential and Integral Calculus (DIC)
subjects, are very high in all universities in Brazil. The main objectives of the study are to
analyze what has been done in the HEIs of the Midwest region of Brazil to alleviate the problems
already encountered in the midst of this transition. To this end, we have chosen to analyze which
HEIs offer some basic discipline, known as PreCalculus, as a possible solution to this problem.
We analyzed the pedagogical projects of the 13 HEIs courses.
References
[1] Bellettini, M. T.; Souza, S. . A implantacao da disciplina de Pre-calculo como polıtica pedagogica
de permanencia nos cursos de graduacao do centro tecnologico da UFSC. In: XVIII Coloquio
Internacional de Gestao Universitaria, 2018, Florianopolis.
[2] Andrade, F. C.; Esquincalha, A. C. ; Oliveira, A. T. C. C. . O Pre-Calculo nas Licenciaturas
em Matematica das Instituicoes Publicas do Rio de Janeiro: O Prescrito. VIDYA (Santa Maria.
ONLINE), v. 39, p. 131-151, 2019.
[3] Rodrigues, Luciana Avila; Neves, Regina da Silva Pina. O Calculo Diferencial e Integral na
Universidade de Brasılia: Estrategia Metodologica em Estudo. REenCiMa. 2019.
[4] Andrade, F. C.; Esquincalha, A. C. ; Oliveira, A. T. C. C. . Un estado del conocimiento de
las investigaciones sobre pre-calculo en Brasil. UNION (San Cristobal De La Laguna), v. 16, p.
48-63, 2020.
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Experiencias, vivencias e acoes na EscolaEstadual Raul de Leoni no contexto da
pandemia da Covid-19
Luciano Goncalves Batista
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Renata Dourado Roque
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Aparecida de Fatima Andrade da Silva
Email: [email protected]
Universidade Federal de Vicosa, Brasil
Thamyres Ribeiro Medeiros
Email: [email protected]
E. E. Raul de Leon, Brasil
Abstract
O presente trabalho tem como objetivo relatar as experiencias, vivencias e acoes desen-
volvidas na Escola Estadual Raul de Leoni, pelo Programa Residencia Pedagogica da Univer-
sidade Federal de Vicosa, no contexto da pandemia da COVID-19.Neste trabalho, relatamos
as metodologias empregadas por residentes pedagogicos nas aulas de Matematica da turma do
8º do Ensino Fundamental II.Analisandoo Plano de Estudos Tutorados(PET), implementado
pelo governo do estado de Minas Gerais para auxiliar os estudos na modalidade de Ensino
Remoto, percebemos que algumas apostilas nao trazem o conteudo explicado de forma clara e
objetiva, para que os alunos possam estudar em casa com mais tranquilidade. Sabendo disso,
propomos atividades para complementacao do PET. Dentre essas atividades destacamos a elab-
oracao de planos de aulas sobre os conteudos: dızimas periodicas, plano cartesiano, geometria
plana, volume, equacoes do segundo grau e inequacoes. Nosso principal objetivo e fazer com
que os alunos aprendam os conteudos de matematica de forma significativa. Com isso, os planos
foram elaborados pensando no desenvolvimento do senso crıtico e do raciocınio logico.Na busca
de tor nar as aulas mais atraentes, utilizamos vıdeos, curiosidades, desafios, jogos, atividades
contextualizadas, contexto historico e o software geogebra.As acoes realizadas remotamente nos
conduziram a refletir o quanto o uso das tecnologias digitais e importante para a formacao do-
cente. Apesar de ser uma experiencia desafiadora, a qual nao estavamos acostumados, estamos
nos adaptando, e ao mesmo tempo desenvolvendo as algumas competencias sugeridas na BNCC.
References
[1] Andrini, A, Praticando matematica. 3ª Edicao.renovada. Sao Paulo: Editora do Brasil, 2012.
152
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[2] Giovani Junior, J.R, A conquista da matematica: 8º ano: ensino fundamental: anos finais.
4ªEdicao. Sao Paulo: FTD, 2018.
[3] Minas Gerais, Estude em casa. Disponıvel em: https://estudeemcasa.educacao.mg.gov.br/pets
.Acesso em: 15 de janeiro de 2021.
[4] O estudo de inequacoes no GeoGebra. isponıvel em: http://portaldoprofessor.mec.gov.br/
fichaTecnicaAula.html?aula=56415. Acesso em: 18 de dez. de 2020.
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XIII Summer Workshop in Mathematics
Percepcao de professores formadores emlicenciaturas de matematica sobre a
criativade no desenvolvimento do estagiosupervisionado
Marcia Rodrigues Leal
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Liviam Santana Fontes
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Cleyton Hercules Gontijo
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Abstract
Este trabalho apresenta dados preliminares de uma pesquisa em andamento que tem como
objetivo analisar as percepcoes de docentes formadores em relacao as praticas ocorridas no decor-
rer do Estagio Supervisionado em cursos de licenciatura em matematica, especificamente no que
se refere ao desenvolvimento do processo criativo do estudante. Considerando a necessaria ar-
ticulacao entre teoria e pratica, problematizamos nessa pesquisa a importancia de incluir nessa
relacao o estımulo a criatividade em matematica no desenvolvimento do Estagio durante a
formacao inicial e no decorrer do trabalho na pratica docente. Buscamos trazer o objeto de
estudo atraves de uma investigacao com 15 formadores desses cursos, no Distrito Federal e
em Goias, por meio de uma abordagem qualitativa, que coletou as percepcoes dos professores
por meio de um questionario. Algumas questoes motivadoras da pesquisa: os professores for-
madores compreendem o significado da criatividade no processo de ensino-aprendizagem em
matematica? Os professores formadores reconhecem o processo criativo no desenvolvimento do
Estagio Supervisionado dos cursos de licenciatura em matematica? Neste estudo, autores como
Gontijo1 e Lubart2 serao fundamentais na sustentacao teorica sobre criatividade e criatividade
em matematica. Para tratar do Estagio Supervisionado nos apoiamos teoricamente, entre out-
ros autores, em Pimenta e Lima3. Os resultados esperados pautam-se na expectativa que diante
dos relatos dos respondentes poderemos identificar comoocorre o processo de formacao docente
quanto aos estımulos a criatividade e as percepcoes dos professores formadores quanto aos pro-
cessos criativos que sao desenvolvidos ou nao, no desdobramento metodologico das atividades
em Estagio na docencia.
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References
[1] C. H. Gontijo; A. T. Carvalho; M. G. Fonseca; M. P. Faria, Criatividade em matematica:
conceitos, metodologias e avaliacao, Brasılia. Editora da UnB, 2019.
[2] T. I. Lubart, Psicologia da criatividade. Porto Alegre: Artmed, 2007.
[3] S. G. Pimenta; M. S. L. Lima, Estagio e Docencia. Sao Paulo: Cortez, 2011.
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Pensamento crıtico e criativo emmatematica a partir de leitura de
informacoes midiaticas
Cleyton Hercules Gontijo
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Mateus Giani Fonseca
Email: [email protected]
Intituto Federal de Brasılia, Brasil
Abstract
Com o avanco da disseminacao das tecnologias e a popularizacao do acesso a internet e as
redes sociais, passamos a ser expostos a uma grande quantidade de informacoes e numa ve-
locidade que muitas vezes impossibilita uma analise criteriosa de todas elas. Para lidar com
tanta informacao no processo de tomada de decis£o, as pessoas buscam elementos que podem
ser caracterizados como uma especie de prova social, isto e, dados de aceitacao de que as in-
formacoes sao confiaveis para um numero significativo de pessoas. Esse processo de prova social
costuma vir acompanhado de porcentagens, medias, estimativas, graficos, entre outras formas
de apresentacao de dados, quase sempre em uma linguagem matematica. Daı surge a pergunta:
sera que todos estamos preparados para lidar com a informacao matematica no formato que ela
e apresentada, de modo a n£o sermos induzidos a interpretacoes equivocadas? Afinal, estamos
em um paıs com cerca de 30% de analfabetos funcionais (IPM, 2018). E, mais, segundo a ultima
edicao do Programa Internacional de Avaliacao de Estudantes (Pisa), em um paıs onde cerca
de 2/3 dos brasileiros sabem menos do que o consider- ado basico de matematica; enquanto o
mesmo exame apontou que apenas 2% dos respondentes demonstraram plena capacidade de res-
olucao de problemas (OCDE, 2019). Dessa forma, somos levados a refletir se a populacao possui
condicoes de interpretar corretamente aquilo que lhe e apresentado, de forma crıtica. Consid-
eramos fundamental o estımulo ao desenvolvimento do pensamento crıtico e criativo ao longo
de todo o perıodo de escolarizacao, pois, este se caracteriza como um processo de tomada de
decisao que envolve a analise das estrategias e dos dados apresentados/construıdos durante uma
jornada de geracao de multiplas ideias para solucionar um problema, ou mesmo de multiplos
caminhos para encontrar uma resposta (FONSECA; GONTIJO, 2020), pois, e uma habilidade
fundamental para interpretar dados matematicos, evitando que a ”manipulacao criativa” de
informacoes, de forma maldosa, leve a decisoes equivocadas e, ao mesmo tempo, colabora para
impedir a circulaco de fake news.
References
[1] Instituto Paulo Montenegro, Indicador de alfabetismo funcional. Inaf Brasil 2018: resultados
preliminares, 2018.
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[2] Mateus Gianni Fonseca, Cleyton Hercules Gontijo, Pensamento crıtico e criativo em matematica
em diretrizes curriculares nacionais. In Ensino em Revista, 27(3),(2020), 956-978.
[3] Organizacao para a Cooperacao e Desenvolvimento Economico. Pisa 2018: Insights and inter-
pretations, 2019.
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Evasao na Licenciatura em Matematica:o que apontam as pesquisas
Raquel Carneiro Dorr
Email: [email protected]
Universidade de Brasılia, Brasil
Brendo Oliveira Damasceno
Email: [email protected]
Universidade de Brasılia, Brasil
Izabella Sabino da Silva
Email: [email protected]
Universidade de Brasılia, Brasil
Abstract
A evasao academica pode ocorrer quando ha uma interrupcao no ciclo de estudos do es-
tudante universitario. Em termos praticos, isso pode significar sua saıda da instituicao antes
da conclusao do curso. Muitos sao as causas para que isso aconteca. Entre elas podemos
citar a falta de identificacao com o curso ou instituicao escolhidos inicialmente, problemas so-
cioeconomicos do estudante, reprovacoes, entre outros. Em especial, estamos interessados na
investigacao sobre as circunstancias que tem levado um licenciando em Matematica a abandonar
ou mudar de curso. Pesquisas em universidades brasileiras tem revelado que a Licenciatura em
Matematica aparece entre os cursos com registros de altas taxas de desistencias. As inves-
tigacoes ainda destacam algumas caracterısticas desse sujeito como o seu perfil, o momento em
que abandona o curso, entre outras. A identificacao e entendimento desses motivos pode servir,
por exemplo, de auxılio aos gestores educacionais na preparacao de atividades de acolhimento
a estudantes iniciantes ou na elaboracao de seus currıculos. Assim, apresentamos nessa comu-
nicacao cientıfica alguns resultados coletados por uma pesquisa bibliografica feita sobre o tema
da evasao em cursos de Licenciatura em Matematica no ambito de um projeto iniciacao cientıfica
em realizacao em uma universidade publica do centro oeste brasileiro. Serao apresentadas as
principais conclusoes obtidas pelas pesquisas, bem como algumas questoes relacionadas a esse
assunto que poderao ser usadas em futuras investigacoes.
References
[1] Almeida, E; R.; Candido, J. P.; PAULA, M. A.; SANTOS, D. B. A producao teorica sobre
evasao escolar nas licenciaturas em matematica nos perıodos DE 2006 A 2016, 2016
[2] Bittar, M. ; Burigato, S. M. M. S.; Oliveira, A. B.; Santos, R.l M. A evasao em um curso de
matematica em 30 anos, 2012
[3] Bonato, G. C. ; Mello, K. B.Evasao no curso de licenciatura em matematica do IFRS campus
Caxias do Sul , 2017
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[4] CHAGAS, T. M. Evasao do curso de licenciatura em matematica (noturno) da Universidade de
Brasılia, 2016
[5] Souza,L. F. D. Analise da evasao dos alunos dos CURSOS da UnB: Um estudo no ambito da
graduacao, 2019.
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Mathematical Modeling in Mathematical
Literacy:The Study of the Brazilian Monetary
System
Rosangela Maria Kowalek
Email: [email protected]
Universidade Estadual de Londrina, Brazil
Daniel Jose Kmita
Email: [email protected]
Universidade Estadual de Londrina, Brazil
Abstract
Mathematical Modeling in Mathematical Education is presented as a methodology to be
used since the first school years (BURAK, 1994; MAAβ, 2005; LUNA, 2007; MARCONDES and
SILVA, 2019). Still, studies like the one by Kowalek and Veleda (2019) point to Mathematical
Modeling as a methodology that provides Mathematical Literacy addressing topics of interest to
students. Thus, taking into account the research already carried out and seeking to corroborate
with studies in the area, we developed the present work, in which we aim to briefly present
the teaching of the Brazilian monetary system through a Mathematical Modeling activity with
a multiserial class of the Literacy Cycle composed of six 1st year students and three 2nd year
students. To this end, an activity was developed using the Mathematical Modeling methodology
(BURAK, 2010) which had the theme ”games”, chosen by the students in the class. The
students’ interest was the prices of games and toys. As the students had no knowledge about
banknotes and coins, the values of the banknotes were initially studied, and the teacher presented
the class with a domino game with pieces of money so that the students, when playing, could get
to know the banknotes. Subsequently, false notes were used so that students could form specific
values. With students already familiar with banknotes and coins, a toy “shop” was held in the
classroom in which students brought toys from home, in which some sold and others bought,
providing opportunities to work with the values of the toys to be used. paid and the concept
of change. From the development of the activity it is possible to make some considerations
in relation to the work with Mathematical Modeling in the Literacy Cycle, highlighting that
the teaching linked with activity of themes of interest to the students actively involves them in
the learning process. It was also possible to realize that the work with the monetary system
provided students with recognition of the banknotes and coins that circulate in Brazil, their
values, exchanges and use that are objectives of Mathematical Literacy present in the Official
Brazilian Document (2012).
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References
[1] D. Burak, Modelagem Matematica sob um olhar de Educacao Matematica e suas implicacoes
para a construcao do conhecimento matematico em sala de aula, Revista de Modelagem em
Educacao Matematica Blumenau, v.1, n. 1, p. 47-60, 2010.
[2] D. Burak, Criterios norteadores para a adocao da Modelagem Matematica no Ensino Funda-
mental e Secundario, Zetetike v.2, n. 2, p. 10-27, 1994.
[3] A. V. A. Luna, Modelagem Matematica nas series iniciais do Ensino Fundamental: um estudo
de caso no 1o ciclo, CONFERENCIA INTERAMERICANA DE EDUCACION MATEMAT-
ICA, 12, Santiago de Queretaro. Anais... Santiago de Queretaro: Comita Interamericano de
Educacion Matematica, 2007. 1 CDROM.
[4] R. Kowalek; G. Veleda, Uma Pratica com Modelagem Matematica no Ciclo de Alfabetizacao:
Um Olhar para os Objetivos de Aprendizagem Modelagem Matematica, CONFERENCIA NA-
CIONAL SOBRE MODELAGEM MATEMATICA NA EDUCACAO MATEMATICA, 11.,
Belo Horizonte. Anais... Minas Gerais: UFMG, 2019. p. 1-16.
[5] K. Maaβ, Barriers and opportunities for the integration of modelling in mathematics classes:
results of an empirical study, Teaching Mathematics and Its Application v. 24, n. 2-3, p. 61-74,
2005.
[6] C. F. Marcondes; V. S. Silva, Modelagem matematica na educacao infantil: consideracoes a
partir de uma pratica educativa com criancas de 3 e 4 anos, Revista de Educacao Matematica
Sao Paulo, v. 16, n. 21, p. 71-87, jan. /abr. 2019.
[7] BRASIL, Elementos conceituais e metodolagicos para definicao dos direitos de aprendizagem e
desenvolvimentos do Ciclo de Alfabetizacao (1o, 2o e 3o anos) do Ensino Fundamental, MEC:
Brasılia, 2012.
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Metodologias alternativas no ensinoremoto de matematica
Thiago Araujo de Lima Matos
Email: [email protected]
Universidade Federal de Vicosa
Beatriz Barbosa Lopes
Email: [email protected]
Universidade Federal de Vicosa
Daniella Isadora de Assis
Email: [email protected]
Universidade Federal de Vicosa
Marcio Gustavo Vieira
Universidade Federal de Vicosa
Aparecida de Fatima Andrade da Silva
Universidade Federal de Vicosa
Abstract
Devido a Pandemia do novo coronavırus, as escolas do paıs suspenderam as aulas pres-
enciais e ingressaram em uma modalidade de ensino remoto. A Secretaria de Educacao de
Minas Gerais desenvolveu o Plano de Estudo Tutorado - PET, um material didatico para que
os estudantes resolvam atividades escolares programadas de forma nao presencial e autoinstru-
cional, visando minimizar o impacto da interrupcao das aulas. No entanto, notou-se que o
distanciamento do professor no processo de ensino e aprendizagem dificultou a compreensao
dos alunos em relacao aos conteudos, o que provocou a diminuicao do engajamento nos estu-
dos autonomos. Assim, com o objetivo de apoiar os alunos da Escola Estadual Dr.Raimundo
Torres nesse perıodo remoto, os residentes do Programa Residencia Pedagogica pela Universi-
dade Federal de Vicosa, produziram videoaulas como material complementar. A partir dessa
proposta foi possıvel utilizar metodologias alternativas para promover uma aprendizagem mais
significativa, como a contextualizacao do conteudo, a interdisciplinaridade, o uso de tecnologias
e mıdias digitais no processo de ensino, as quais os professores vem apresentando dificuldades de
introduzir em suas aulas. Foram produzidos videoaulas sobre: simetrias, em que suas aplicacoes
foram contextualizadas em diversas areas; potenciacao e radiciacao, trabalhando as definicoes
e propriedades atraves da ludicidade com o jogo Uno; e aspectos da arquitetura mineira sob a
perspectiva matematica para o auxılio no PET comemorativo dos 300 anos de Minas Gerais.
Alem disso, a proposta deste trabalho oportunizou o manuseio de ferramentas tecnologicas, bem
como experiencias didaticas inovadoras de ensino a distancia e aplicacao de teorias educacionais
estudadas na universidade.
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References
[1] Skovsmose, Ole, Educacao matematica crıtica: a questao da democracia. Campinas-SP: Papirus,
2013.
[2] Zabala, Antoni, Enfoque globalizador e pensamento complexo: uma proposta para o currıculo
escolarPorto Alegre: Artmed, 2002.
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Analise fundamentada de uma oficina deTrigonometria
Vania Batista Flose Jardim
Email: [email protected]
Universidade Federal do ABC, Brasil
Instituto Federal de Sao Paulo, Brasil
Eduardo Goedert Dona
Email: [email protected]
Universidade Federal do ABC, Brasil
Instituto Federal do Sul de Minas, Brasil
Janaına Mendes Pereira da Silva
Email: [email protected]
Universidade Federal do ABC, Brasil
Abstract
Considerando a importancia da pratica docente, este estudo propoe-se refletir sobre uma
pratica tipificada em uma oficina que visava o ensino da trigonometria aos estudantes de ensino
medio. Para tal, recorreu-se aos sujeitos deste processo, no caso as professoras responsaveis,
por meio de uma roda de conversa gravada, e dos registros do planejamento e execucao desta
oficina, na busca por evidencias da/na pratica e relaciona-las com alguns elementos teoricos
da Didatica da Matematica. Por meio de uma pesquisa qualitativa utilizou-se de estudos que
refletem a utilizacao de materiais manipulaveis e a utilizacao de recurso tecnologicos para o
ensino matematico, materiais para o ensino trigonometrico e a Teoria de Tres Mundos de David
Tall para fundamentar a analise. Com os dados em maos, organizamos os mesmos em tres
categorias: (i) planejamento, (ii) desenvolvimento e (iii) avaliacao e, de acordo com o referen-
cial estudado chamamos a atencao para a importancia do uso de metodologias diferenciadas no
ensino da matematica tanto para ensina-la na educacao basica, ao protagonismo do aluno e a
necessidade de oportunizar aos estudantes transitar entre o mundo corporificado, representado
pelo material manipulavel, o mundo simbolico ao se inserir sımbolos, bem como a compreensao
de conceitos e algumas demonstracoes que aproximam o aluno do mundo formal. Como as-
pectos conclusivos, observamos que as praticas exercidas em sala de aula podem considerar os
varios mundos da matematica a medida que os alunos apresentam sua compreensao sobre um
determinado assunto matematico e o professor disponibilize recursos (materiais ou didaticos)
que proporcionem reflexoes quanto a construcao e entendimento de conceito por meio deles.
References
[1] D. Fiorentini;M.A. MIORIM, Uma reflexao sobre o uso de materiais concretos e jogos no ensino
de matematica. Boletim da Sociedade Brasileira de Educacao Matematica ,Sao Paulo, v.4, n.7,
(1990), 5-10.
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[2] S. Lorenzat, Laboratorio de ensino de matematica e materiais didaticos manipulaveis. O labo-
ratorio de ensino de matematica na formacao de professores ,Campinas, SP. Autores Associados,
(2006), 3-37.
[3] P.C.T Souza, Materiais manipulaveis e recursos digitais no ensino de trigonometria. (Programa
de Mestrado Profissional em Matematica em Rede Nacional Matematica - PROFMAT) Setor de
Ciencias Exatas, Universidade Federal do Parana, (2018), 54p.
[4] D.Tall, The transition to formal thinking in mathematics. Mathematics Education Research
Journal, (2008), v.20, n.2, (2008),5-24.
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XIII Summer Workshop in Mathematics
A ludicidade na aprendizagem e nodesenvolvimento da criatividade em
matematica
Waldineia Ferreira de Oliveira
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Marcia Rodrigues Leal
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Cleyton Hercules Gontijo
Email: [email protected]
Departamento de Matematica
Universidade de Brasılia, Brasil
Abstract
Este trabalho tem por objetivo apresentar uma pesquisa que analisou a pratica cotidiana
de uma turma de estudantes do 4ºano do Ensino Fundamental de uma Escola Municipal em
Goias a fim de verificar o potencial de um conjunto de atividades ludicas na aprendizagem e
no desenvolvimento da criatividade em matematica dos estudantes. A metodologia utilizada
fundamentou-se na abordagem qualitativa, sendo a observacao participativa e a entrevista gru-
pal (roda de conversa) as principais estrategias de producao de informacoes. Foi observada a
participacao dos estudantes nas atividades ludicas conduzidas pela professora regente (como
amarelinha, jogo de argolas, pescaria dosnumeros voltadas para soma e subtracao e atividades
com ilustracoes diversas). A roda de conversa buscou coletar informacoes sobre os novos con-
hecimentos apreendidos pelos estudantes a partir das atividades ludicas e sobre como usaram a
criatividade nessas atividades. Estudiosos da area como Almeida (1),DaAmbrosio (2), Gontijo
(3), Gontijo et al(4), Lorenzato (5), trouxeram importantes reflexoes para esse estudo. Foi
possıvel perceber a relevancia do uso das atividades ludicas para possibilitar a aprendizagem
de novos conhecimentos matematicos e para o desenvolvimento de algumas habilidades, em
especial, da criatividade dos estudantes. Alem disso, observou-se a importancia do papel do
professor na mediacao da aprendizagem do estudante, na orientacao e direcionamento dos jogos
e manuseio de materiais didaticos em sala de aula, nacontribuicao positiva no desenvolvimento
do raciocıo logico do estudante, como tambem na concentracao e no domınio dos conceitos
matematicos, promovendo estımulos a criatividade em matematica.
References
[1] Almeida, A, Ludicidade como Instrumento Pedagogico. Cooperativa do Fitnnes, Disponıvel em:
http://www.cdof.com.br/recrea22.htm, Acesso em 10 jan. 2021.
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[2] DAmbrosio, U., Educacao Matematica: da teoria a pratica. Papirus: Sao Paulo, 2012.
[3] Gontijo, C. H, Relacoes entre Criatividade, Criatividade em Matematica e Motivacao em
Matematica de Alunos do Ensino Medio. 2007, 194f, Tese (Doutorado em Psicologia), Insti-
tuto de Psicologia, Universidade de Brasılia, Brasılia/DF, 2007.
[4] Gontijo, C. H. et al, Criatividade em Matematica: conceitos, metodologias e avaliacao. Brasılia:
Editora Universidade de Brasılia, 2019.
[5] Lorenzato, S., Para Aprender Matematica. Sao Paulo: Autores Associados, 2006.
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Number Theory
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On the Arithmetic Behaviour of LiouvilleNumbers Under Rational Maps
Ana Paula Chaves
Email: [email protected]
Instituto de Matematica e Estatıstica
Universidade Federal de Goias, Brazil
Abstract
The genesis of transcendental number theory, took place in 1844 with Liouville’s result “bad”
approximation of algebraic numbers by rationals. More precisely, if α is an algebraic number of
degree n > 1, then there exists a positive constant C, such that |α− p/q| > Cq−n, for all p/q ∈Q∗. Using this remarkable fact, he was able to build a non enumerable set of transcendental
numbers called Liouville numbers. Since then, several classifications of transcendental numbers
have been developed, one of them proposed by Kurt Mahler in 1932. He splited the set of
transcendental numbers on three disjoint sets: S−,T− and U−numbers. In a certain sense,
U−numbers generalize the concept of Liouville numbers. Yet, the set of U−numbers can be
splitted into Um−numbers, that are numbers “rapidly” approximable by algebraic numbers of
degree m. On this lecture, the following result, made in cooperation with D. Marques and P.
Trojovsky, will be proved:
Theorem: Let ω : N → N, such that ωn → ∞, as n → ∞. Let ξ ∈ R be a Liouville number,
such that there exists an infinite sequence of rational numbers (pn/qn)n, satisfying∣∣∣∣ξ − pnqn
∣∣∣∣ < H
(pnqn
)−ωn
,
where H(pn+1/qn+1) ≤ H(pn/qn)O(ωn). Now, take α0, . . . , αl, β0, . . . , βr ∈ Q, with βr = 1 and
αl 6= 0, such that [Q(α0, . . . , αl, β0, . . . , βr) : Q] = m. Then, for P (z), Q(z) ∈ Q[z], given by
P (z) = α0 + α1z + · · ·+ αlzl and Q(z) = β0 + β1z + · · ·+ βrz
r, P (ξ)/Q(ξ) is a Um-number.
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XIII Summer Workshop in Mathematics
Some recent progress on theBirch–Swinnerton-Dyer conjecture
Christopher Skinner
Princeton, USA
Abstract
This talk will describe some recent work on the Birch–Swinnerton-Dyer conjecture, which
famously asserts that the rank of the Mordell-Weil group of an elliptic curve E is equal to the
order of vanishing of the Hasse-Weil L-function L(E, s) of E at s = 1. Emphasis will be on
results that pass from the rank of the Mordell-Weil group to the order of the L-function (the
opposite direction to that of the celebrated works of Gross–Zagier and Kolyvagin) and on results
about the conjectured value of the leading term of L(E, s) at s = 1 (the Birch–Swinnerton-Dyer
formula).
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XIII Summer Workshop in Mathematics
On the existence of pairs of primitive
elements, related bya rational function, on finite fields
Cicero Carvalho
Faculdade de Matematica
Universidade Federal de Uberlandia, Brazil
Abstract
In this talk we would like to present some results on the existence of pairs of elements in a
finite field, where the first element is either primitive or primitive and normal over a subfield,
and the second element is primitive and a rational function of the first one. This is based on
joint works with J.P Guardieiro, V. Neumann and G. Tizziotti.
172
XIII Summer Workshop in Mathematics
Permutation Polynomials
Guilherme Tizziotti
Partially supported by CPNq 307037/2019-3
Email: [email protected]
Faculdade de Matematica
Universidade Federal de Uberlandia, Brasil
Abstract
In this talk we present a survey of results on polynomials for which the associated polyno-
mial functions are permutations of a given finite fields Fq. Polynomials of this type are called
permutation polynomials and exist for any Fq since, more generally, every mapping of Fq int
itself can be expressed by a polynomial.
References
[1] R. Lidl; H. Niederreiter, Finite Fields, Cambridge University Press, 1997.
[2] Z. Zha; L. Hu; Z. Zhang, Permutation polynomials of the form x + γTrqn
q (h(x)), Finite Fields
and Their Applications, 60, (2019), 101573.
[3] D. Zheng; M. Yuan; M. Yu, Two types of permutation polynomials with special forms, Finite
Fields and Their Applications, 56, (2019), 1-16.
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XIII Summer Workshop in Mathematics
Special maps on the projective line overfinite fields
Herivelto Borges
Email: [email protected]
Instituto de Ciencias Matematicas e de Computacao
Universidade de Sao Paulo, Brazil
Abstract
Let q be a prime power, and let Fq be the finite field with q elements. For any polynomial
f ∈ Fq[x] of degree d > 1, the size of the value set Vf = f(α) = 0 | α ∈ Fq can be trivially
bounded as follows ⌈qd
⌉6 #Vh 6 q. (5)
The problem of characterizing polynomials attaining the lower (or upper) bound of (5) has been
investigated by many authors over the past decades.
In this talk, we consider the analogous problem where polynomials are replaced by rational
functions h(x) ∈ Fq(x), and we discuss its connection with Galois theory and algebraic curves.
In particular, we will present conditions for which the following statement makes sense
Vh ⊆ P1(Fq) is small if and only if Fq(x)/Fq(h(x)) is Galois.
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XIII Summer Workshop in Mathematics
A New recurrence relation for thepartition function
Jose Plınio de Oliveira Santos
Instituto de Matematica, Estatıstica e Computacao Cientıfica
Universidade Estadual de Campinas, Brazil
Abstract
We present a new recurrence relationship for the partition function based on a representa-
tion for partitions as two-line matrices. This recurrence is quite different from the well-known
relationship given by Euler. This result was obtained by a student of mine in his doctoral thesis.
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XIII Summer Workshop in Mathematics
Numerical Semigroups and Music
Maria Bras-Amoros
Partially supported by the Spanish government under grant TIN2016-80250-R and by the Catalan
government under grant 2014 SGR 537
Email: [email protected]
Department of Computer Science and Mathematics
Universitat Rovira i Virgili, Spain
Abstract
We will elaborate on the algebraic structure of the sequence of harmonics when combined
with equal temperaments. Fractals and the golden ratio appear surprisingly on the way. The
sequence of physical harmonics is an increasingly enumerable submonoid of (R+,+) whose pairs
of consecutive terms get arbitrarily close as they grow. These properties suggest the definition
of a new mathematical object which we denote a tempered monoid. Mapping the elements
of the tempered monoid of physical harmonics from R to N may be considered tantamount to
defining equal temperaments. The number of equal parts of the octave in an equal temperament
corresponds to the multiplicity of the related numerical semigroup. Analyzing the sequence of
musical harmonics we will derive two important properties that tempered monoids may have:
that of being product-compatible and that of being fractal. We will demonstrate that, up to
normalization, there is only one product-compatible tempered monoid, which is the logarithmic
monoid, and there is only one nonbisectional fractal monoid which is generated by the golden
ratio. The example of half-closed cylindrical pipes imposes a third property to the sequence of
musical harmonics, the so-called odd-filterability property. We will prove that the maximum
number of equal divisions of the octave such that the discretizations of the golden fractal monoid
and the logarithmic monoid coincide, and such that the discretization is odd-filterable is 12.
This is nothing else but the number of equal divisions of the octave in classical Western music.
References
[1] M. Bras-Amoros, Increasingly Enumerable Submonoids of R: Music Theory as a Unifying
Theme, The American Mathematical Monthly, Mathematical Association of America, 127(1),
(2020), 33-44.
[2] M. Bras-Amoros, Tempered Monoids of Real Numbers, the Golden Fractal Monoid, and the
Well-Tempered Harmonic Semigroup, Semigroup Forum, Springer, 99(2), (2019), 496-516.
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XIII Summer Workshop in Mathematics
Counterexamples to a Conjecture ofNorton
Michael P. Knapp
Department of Mathematics
Loyola University Maryland, USA
Hemar Godinho
Departamento de Matematica
Universidade de Brasılia, Brazil
Abstract
Let Γ∗(k) be the smallest integer s such that the equation
a1xk1 + · · ·+ asx
ks = 0
has a nontrivial solution in every p-adic field Qp, regardless of the values of the (rational integer)
coefficients. An old conjecture of Norton was that we should have Γ∗(k) ≡ 1 (mod k) for all
degrees k. This was disproved in 1974 by Bovey, who showed that Γ∗(8) = 39, but until a
few years ago this was the only known counterexample. In this talk, we show that there are
infinitely many counterexamples to Norton’s conjecture.
References
[1] M. P. Knapp and H. Godinho, Infinitely many counterexamples to a conjecture of Norton,
Michigan Math Journal, 2020, 533-543.
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XIII Summer Workshop in Mathematics
Three problems on numerical semigroups
Shalom Eliahou
Email: [email protected]
Laboratoire de Mathematiques Pures et Appliquees
Universite du Littoral Cote d’Opale, France
Abstract
A numerical semigroup is a cofinite submonoid of N. That is, a subset S ⊆ N containing 0,
stable under sum and with finite complement in N. Equivalently, it is a subset S ⊆ N of the
form S = 〈a1, . . . , an〉 = Na1 + · · ·+ Nan for some globally coprime positive integers a1, . . . , an.
In this talk, we shall visit three main problems concerning numerical semigroups: the Frobenius
problem (19th century), the Bras-Amoros conjectures (21st century) and Wilf’s conjecture (20th
century). We shall present selected recent results, together with some proof ideas, concerning
mostly the last two problems.
References
[1] M. Bras-Amoros, Fibonacci-like behavior of the number of numerical semigroups of a given
genus, Semigroup Forum 76 (2008) 379–384.
[2] M. Delgado, Conjecture of Wilf: a survey, arXiv:1902.03461 (2019).
[3] S. Eliahou; J. Fromentin, Gapsets and numerical semigroups, J. Combin. Th. Series A 169
(2020), 105129.
[4] S. Eliahou, A graph-theoretic approach to Wilf’s conjecture, Elec. J. Combin. 27 (2020), #P2.15,
31 pp.
[5] J. L. Ramırez Alfonsın, The Diophantine Frobenius problem, Oxford Lecture Series in Mathe-
matics and its Applications 30, Oxford University Press, 2005.
[6] H. Wilf, A circle-of-lights algorithm for the money-changing problem, Amer. Math. Monthly 85
(1978) 562–565.
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XIII Summer Workshop in Mathematics
Elementos primitivos 2-normais sobreCorpos Finitos
Victor Gonzalo Lopez Neumann
Partially supported by FAPEMIG
Email: [email protected]
Faculdade de Matematica
Universidade Federal de Uberlandia, Brazil
Abstract
Um elemento α ∈ Fqn e normal sobre Fq se B = α, αq, αq2 , · · · , αqn−1 e uma base de
Fqn como espaco vetorial sobre Fq. Sabe-se que α ∈ Fqn e normal sobre Fq se, e somente se,
gα(x) = αxn−1 +αqxn−2 + · · ·+αqn−2
x+αqn−1
e xn− 1 sao primos entre si em Fqn [x]. Usando
esta equivalencia, Huczynska et al. (ver [2]) introduziram a nocao de elemento k-normal: um
elemento α ∈ Fqn e k-normal sobre Fq se o maior divisor comum dos polinomios gα(x) e xn − 1
em Fqn [x] e de grau k; assim, um elemento normal no sentido usual e 0-normal.
No mesmo artigo Huczynska et al. propuseram varios problemas, um deles foi: determinar os
pares (n, k) para os quais existem elementos primitivos k-normais em Fqn sobre Fq. Eles mesmos
resolveram parcialmente o problema no caso k = 1. Alguns anos depois, Reis e Thomson (ver
[4]) resolveram esse caso completamente. O Teorema da Base Normal Primitiva (ver [3] e [1])
resolve o caso k = 0. Na presente palestra mostramos a solucao completa do caso k = 2. Este
e um trabalho em conjunto com Josimar J. R. Aguirre.
References
[1] S.D. Cohen and S. Huczynska, The primitive normal basis theorem without a computer, Journal
of London Mathematical Society 67(1) (2003), 41-56.
[2] S. Huczynska; G.L. Mullen; D. Panario and D. Thomson, Existence and properties of k-normal
elements over finite fields, Finite Fields and Their Applications 24 (2013), 170-183.
[3] H.W. Lenstra and R. Schoof, Primitive normal bases for finite fields, Mathematics of Computa-
tion 48 (1987), 217-231.
[4] L. Reis and D. Thompson, Existence of primitive 1-normal elements in finite fields, Finite Fields
and Their Applications 51 (2018), 238-269.
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XIII Summer Workshop in Mathematics
Almost symmetric generalized numericalsemigroups
Wanderson Tenorio
Email: wanderson [email protected]
Instituto de Matematica e Estatıstica
Universidade Federal de Goias, Brazil
Abstract
In this talk, we discuss the outcomes of an extension of the notion of almost-symmetry for
a special class of affine semigroups in higher dimensions, the so-called generalized numerical
semigroups. By exploring some symmetries occurring in these objects, we present the main
characterizations of this concept. We show that this property yields a new family of Frobenius
generalized numerical semigroups that extends the class of irreducible generalized numerical
semigroups. Furthermore, we discuss a method of computing all almost symmetric generalized
numerical semigroup having a fixed Frobenius element and organizing them in a rooted tree.
References
[1] C. Cisto, G. Failla, C. Peterson et al., Irreducible generalized numerical semigroups and unique-
ness of the Frobenius element, Semigroup Forum 99, (2019), 481–495.
[2] G. Failla, C. Peterson, R. Utano, Algorithms and basic asymptotics for generalized numerical
semigroups in Nd, Semigroup Forum 92 (2), (2016), 460–473.
[3] H. Nari, Symmetries on almost symmetric numerical semigroups, Semigroup Forum 86, (2013),
140–154.
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Probability
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Discrete-time Simulation of StochasticVolterra Equations
Alexandre Richard
Laboratoire MICS , CNRS
Centrale-Supelec, France
Xiaolu TAN
CUHK, China
Fan YANG
CUHK, China
Abstract
In this talk, I will present a couple of discrete-time simulation schemes for stochastic Volterra
equations (namely Euler and Milstein schemes), and the corresponding Multilevel Monte-Carlo
method. Stochastic Volterra equations form a particular class of path-dependent SDEs involving
a deterministic kernel which is allowed to be singular. For equations with smooth enough
coefficients, we obtain the convergence rates for these schemes towards the original equation,
under the supremum norm. I will then apply these schemes to approximate the expectation of
functionals of such Volterra equations by the (Multilevel) Monte-Carlo method, and compute
their complexity. Then for equations with non-Lipschitz coefficients (typically CIR-type Volterra
processes with square-root coefficient in the stochastic integral), we also obtain the convergence
of the Euler scheme. These models have been used a lot recently in the modeling of so-called
rough volatilities in mathematical finance.
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XIII Summer Workshop in Mathematics
Quantitative particle approximation ofnonlinear Fokker-Planck equations with
singular kernel
Christian Olivera
UNICAMP, Brazil
A. Richard
Centrale-Supelec, France
M. Tomasevic
Ecole Polytechnique, France
Abstract
We propose a new approach to obtain quantitative convergence of interacting particle sys-
tems to solutions of nonlinear Fokker-Planck equations with singular kernels. Our result only
requires very weak regularity on the interaction kernel, including the Biot-Savart kernel(2D-
Navier-Stoke equation), the family of Keller-Segel kernels in arbitrary dimension, and more
generally singular Riesz kernels. This seems to be the first time that such quantitative conver-
gence results are obtained in Lebesgue and Sobolev norms for the aforementioned kernels. In
particular, this convergence holds locally in time for PDEs exhibiting a blow-up in finite time.
The proof is based on a semigroup approach combined with stochastic calculus techniques, and
we also exploit the regularity of the solutions of the limiting equation.
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XIII Summer Workshop in Mathematics
Random matrices in Wiener chaos
Ciprian Tudor
Universite de Lille, France
Abstract
We consider a random matrix whose entries are elements in a Wiener chaos of fixed order.
These random entries are either independent or with a particular correlation structure. We
discuss the limit behavior in distribution, under the Wasserstein distance, of its associated
Wishart matrix. We use the techniques of the Malliavin calculus and the characterisation of
the independence on Wiener space.
References
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XIII Summer Workshop in Mathematics
Phase transitions in processes withlong-memory
Daniel Yasumasa Takahashi
Universidade Federal do Rio Grande do Norte, Brazil
Abstract
Stochastic chains of unbounded memories are a generalization of Markov chains in which the
transition kernels can depend on the entire past. I will discuss three types of phase transitions
that can happen in stochastic chains of unbounded memories: (1) equilibrium phase transition,
(2) dynamic phase transition, and (3) concentration phase transition.
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XIII Summer Workshop in Mathematics
Hormander’s theorem for path-dependentSDEs
Evelina Shamarova
Universidade Federal da Paraıba, Brazil
Abstract
We establish the existence of smooth densities for solutions to a broad class of path-
dependent SDEs under Hormander’s condition. Our coefficients at time t are allowed to depend
on the whole history of the solution path up to t arbitrarily. To formulate Hormander’s bracket
condition, which holds in the finite-dimensional state space for the given equation, we define
the Lie brackets in terms of vertical derivatives in the sense of the functional Ito calculus. Our
concept of the Lie bracket naturally extends the classical case. The approach we take relies on
an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus, and rough path
techniques.
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XIII Summer Workshop in Mathematics
AR (1) processes driven by second-chaoswhite noise: precise speed of convergence
results for parameter estimation usinganalysis on Wiener space
Frederi Viens
Michigan State University, United States
Soukaina Douissi
Africa Business School, Marrakech, Morocco
Khalifa Es-Sebaiy
University of Kuwait, Kuwait
Fatimah Alshahrani
Princess Nora bint Abdulrahman University, Riyadh, Saudi Arabia
Abstract
In the limit of high-frequency data for time series, also known as in-fill asymptotics, most
mean-reverting processes are well-approximated by a classical un-centered Ornstein-Uhlenbeck
processs with Gaussian noise. This is not a good approximation, however, for processes with
discrete observation, particularly for fixed-scale observation inter-arrival time. The asymptotics
in this case are of the so-called increasing horizon type, and any process with non-normal noise
will retain features which stem from the non-normality in any limit. In this talk, we describe the
asymptotic behavior of the quadratic variation for the class of AR(1) processes driven by white
noise in the second Wiener chaos. This class of distributions, whose tails are roughly exponential,
is far more flexible than Gaussian noise, because a free infinite-dimensional parameter remains
after standardization. We will explain this phenomenon using a series representation of the
second Wiener chaos. New tools from the analysis on Wiener space are established to provide
an efficient convergence rate in total variation distance. The now-classical optimal 4th moment
theorem of Nourdin and Peccati, and its 3rd-moment interpretation established by the presenter
and Neufcourt, cannot apply to our setting which involves a convolution of 4th and 2nd chaos
terms. We give an upper bound for the total-variation speed of convergence of the process’s
quadratic variation to the normal law, which we apply to study the estimation of the model’s
mean-reversion parameter.
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XIII Summer Workshop in Mathematics
Contact process with renewal cures
Luiz Renato Fontes
Universidade de Sao Paulo, Brazil
Pablo Gomes
Universidade de Sao Paulo, Brazil
Thomas Mountford
Ecole Polytechnique Federale de Lausanne, Switzerland
Daniel Ungaretti
Universidade de Sao Paulo, Brazil
Maria Eulalia Vares
Universidade Federal do Rio de Janeiro, Brazil
Abstract
The contact process is a model for the spread of an infection in which subjects lying on the
sites of a graph may be infected, reinfected and transmit the infection to/by their neighbors,
and also get healed. Most literature treats the case where for each subject the healing as well
as the infection mechanism, independent from other subjects, is memoryless. In a number of
recent studies, we analyse the case where the healing takes place according to a general renewal
process, thus going beyond the traditional exponential memoryless case in this aspect of the
model; the infection mechanism is kept memoryless, at rate lambda, which is a parameter of the
model. An issue of interest is the triviality of the critical parameter λc, above which the infection
started with a single infected subject survives indefinitely with positive probability, and below
which we have almost sure extinction of that infection. The tail of the renewal distribution
plays a key role. If that tail is too heavy, we have λc = 0; otherwise, λc > 0. Roughly, heavy tail
means an infinite first moment, but we require precise conditions on our analysis, nuancing this
issue. We will present our results for various particular cases, drawing up the above sketched
picture; and characterizing the asymptotic behavior of the model when λc = 0 (under more or
less restrictive regularity conditions on the tail of the renewal distribution).
References
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XIII Summer Workshop in Mathematics
Random walks on dynamic randomenvironments with non-uniform mixing
Marcelo Hilario
Universidade Federal de Minas Gerais, Brazil
Oriane Blondel
Universite Claude Bernard Lyon 1, France
Frank den Hollander
Universiteit Leiden, Netherlands
Daniel Kious
University of Bath, United Kingdom
Renato dos Santos
Universidade Federal de Minas Gerais
Vladas Sidoravicius
Abstract
In this talk, we will discuss recent results on the limiting behavior of random walks on
dynamic random environments. We will mainly discuss the case when then random walk evolves
on one-dimensional random environments given by conservative interacting particle systems such
as the simple symmetric exclusion process. Our results depend a great deal on space-time mixing
properties imposed on the underlying environment and also on other features like the dimension
and the type of allowed transitions. Conservation of particles leads to poor-mixing conditions
which complicate the applicability of available tools and to overcome this difficulty we use
renormalization to obtain the law of large numbers, large deviation estimates, and sometimes
central limit theorems.
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XIII Summer Workshop in Mathematics
Cutoff thermalization forOrnstein-Uhlenbeck systems with smallLevy noise in the Wasserstein distance
Michael Hoegele
Universidad de los Andes, Colombia
Abstract
This talk presents recent results on cutoff thermalization (also known as the cutoff phe-
nomenon) for a general class of general Ornstein-Uhlenbeck systems under ε-small additive
Levy noise. The driving noise processes include Brownian motion, α-stable Levy flights, finite
intensity compound Poisson processes and red noises and may be highly degenerate. Window
cutoff thermalization is shown under generic mild assumptions, that is, we see an asymptoti-
cally sharp ∞0 -collapse of the renormalized Wasserstein distance from the current state to the
equilibrium measure µε along a time window centered in a precise ε-dependent time scale tε.
In many interesting situations such as reversible (Levy) diffusions it is possible to prove the
existence of an explicit, universal, deterministic cutoff thermalization profile. The existence of
this limit is characterized by the absence of non-normal growth patterns in terms of an orthogo-
nality condition on a computable family of generalized eigenvectors of the matrix Q. With this
piece of theory at hand this article provides a complete discussion of the cutoff phenomenon for
the classical linear oscillator with friction subject to ε-small Brownian motion or α-stable Levy
flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a
generalized heat bath at low temperature.
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XIII Summer Workshop in Mathematics
Rescaling nonlinear noise for sstochasticpaarabolic equations
Mikhail Neklyudov
Universidade Federal do Amazonas, Brazil
Abstract
In the talk we will discuss the regularisation effect of nonlinear gradient noise to the solution
of 1D stochastic parabolic equation. We demonstrate convergence to the space homogeneous
martingale when we rescale noise at the extremum points of the process. In the end we present
open problems and further directions of research.
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XIII Summer Workshop in Mathematics
A coupling between Sinai’s random walkand Brox diffusion
Samy Tindel
Purdue University, United States
Abstract
Sinai’s random walk is a standard model of 1-dimensional random walk in random environ-
ment. Brox diffusion is its continuous counterpart, that is a Brownian diffusion in a Brownian
environment. The convergence in law of a properly rescaled version of Sinai’s walk to Brox dif-
fusion has been established 20 years ago. In this talk, I will explain a strategy which yields the
convergence of Sinai’s walk to Brox diffusion thanks to an explicit coupling. This method, based
on rough paths techniques, opens the way to rates of convergence in this demanding context.
Notice that I’ll try to give a maximum of background about the objects I’m manipulating, and
will keep technical considerations to a minimum.
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XIII Summer Workshop in Mathematics
Theory of Computation
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XIII Summer Workshop in Mathematics
The Early Years of AutomatedReasoning: The Logic Theory Machine.
Claudia Nalon
Email: [email protected]
Department of Computer Science
University of Brasılia
Brasılia, Brazil
Abstract
In the 65th anniversary of the Logic Theory Machine (LTM), we pay homage to the early
contributors of automated reasoning, by looking at some historical and technical aspects that
laid the foundations of the field. The LTM , which is based on a Hilbert-style calculus for
classical logic, was proposed and implemented by Newell and Simon in 1956, with subsequent
improvements in the next few years. Despite the technical challenges, mostly imposed by the
limited existing architectures at that time, and some methodological criticism, which we also
intend to discuss, their approach has established some techniques and heuristics that have
survived time and that are still in use today.
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XIII Summer Workshop in Mathematics
Anti-unification: Applications andRecent Results
David Cerna
Email: [email protected]
Institute of Computer Science
Czech Academy of Sciences
Prague, Czech Republic
Abstract
Unification[1] is a process by which two symbolic expressions may be identified through
variable replacement. Anti-unification (generalization)[4, 5], on the other hand, is a process
that derives from a set of symbolic expressions a new symbolic expression possessing certain
commonalities shared between its members. In this talk we will discuss the basics of anti-
unification, applications of the concept and some recent results concerning anti-unification over
equational theories and higher order anti-unification. For example, the existence of equational
theories over which anti-unification is nullary [3], and a generic framework for particular variants
of higher order anti-unification[2].
References
[1] Franz Baader, Wayne Snyder, Paliath Narendran, Manfred Schmidt-Schauß and Klaus U. Schulz.
Unification Theory. Handbook of Automated Reasoning (in 2 volumes), 445-532, Elsevier and
MIT Press, 2001.
[2] David M. Cerna and Temur Kutsia. A Generic Framework for Higher-Order Generalizations.
In 4th International Conference on Formal Structures for Computation and Deduction (FSCD
2019), Leibniz International Proceedings in Informatics (LIPIcs), volume 131, 10:1–10:19, 2019.
[3] David M. Cerna. Anti-unification and the theory of semirings. Theoretical Computer Science,
volume, 848, pages 133-139, 2020.
[4] Gordon D. Plotkin. A note on inductive generalization. Machine Intel., volume 5(1), pages
153-163, 1970.
[5] John C. Reynolds. Transformational systems and the algebraic structure of atomic formulas.
Machine Intel., volume 5(1), pages 135-151, 1970.
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XIII Summer Workshop in Mathematics
Tuple Interpretations for Higher-OrderTerm Rewriting Systems
Deivid Vale
Supported by the ICHOR project
Email: [email protected]
Department of Software Sciences
Radboud University Nijmegen
Nijmegen, The Netherlands
Abstract
We present a style of algebra interpretations for many-sorted and higher-order term rewrit-
ing based on interpretations to tuples; intuitively, a term is mapped to a sequence of values
identifying for instance its evaluation cost, size and perhaps other values of interest. This gives
a more fine-grained notion of the complexity of a term or TRS than notions such as runtime
or derivational complexity, which is in particular useful to obtain complexity results for higher-
order term rewriting systems.
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XIII Summer Workshop in Mathematics
A Formalization of the Z property in Coq
Flavio L. C. de Moura
Email: [email protected]
Department of Computer Science
University of Brasılia
Brasılia, Brazil
Leandro O. Rezende
Email: [email protected]
Department of Computer Science
University of Brasılia
Brasılia, Brazil
Abstract
Rewriting theory is a well established model of computation equivalent to the Turing ma-
chines, and the most well known rewriting system is the λ-calculus. Confluence is an important
and undecidable property related to the determinism of the computational process. Direct
proofs of confluence are, in general, difficult to be done. Therefore, alternative characteriza-
tions of confluence can circumvent this difficulty for different contexts. This is the case of the so
called Z property, which has been successfully used to prove confluence in several situations such
as the λ-calculus with βη-reduction, extensions of the λ-calculus with explicit substitutions, the
λµ-calculus, etc. In this work we present a direct and constructive proof that the Z property
implies confluence. In addition, we formalized our proof and an extension of the Z property,
known as the Compositional Z, in the Coq proof assistant.
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XIII Summer Workshop in Mathematics
Formalising Completeness ofAC-unification
Gabriel Ferreira Silva
Email: [email protected]
Supported by a CAPES scholarship
Department of Computer Science
Universidade de Brasılia (UnB)
Brasılia, Brazil
Abstract
Unification has several applications in computer science and mathematics: logic program-
ming, theorem proving, type inference and so on. In this talk we revisit the problem of AC-
unification, i.e., unification in the presence of associative-commutative function symbols. We
talk about why this problem is harder than standard unification and give an example that
illustrates Stickel and Fages’ first-order AC-unification algorithm. Then, we discuss an inter-
esting step of a structured proof of the algorithm completeness. To conclude, we discuss work
in progress in formalising AC-unification in PVS as a first step to extend our work on nominal
C-unification. This is joint work with Mauricio Ayala-Rincon, Maribel Fernandez and Daniele
Nantes-Sobrinho.
References
[1] Stickel, Mark E, A unification algorithm for associative-commutative functions, Journal of the
ACM (JACM), 1981, 423–434.
[2] Fages, Francois, Associative-commutative unification, Journal of Symbolic Computation, 1987,
257–275.
[3] Ayala-Rincon, Mauricio and Fernandez, Maribel and Silva, Gabriel and Nantes-Sobrinho,
Daniele. A Certified Functional Nominal C-Unification Algorithm, Lecture Notes in Computer
Science, 2019, 123–138.
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XIII Summer Workshop in Mathematics
Towards a fixed-point approach toNominal Disunification
Leonardo Melo
Partially supported by Capes
Department of Mathematics
University of Brasılia
Brasılia, Brazil
Abstract
We will present the nominal abstract syntax [1] following the fixed-point constraints ap-
proach [2], which is convenient for dealing with α-equivalence modulo equational theories in-
volving commutative operators. Then, we will present a nominal unification algorithm via
fixed-point constraints and introduce our current work on the extension of nominal disunifica-
tion via freshness to this fixed-point approach, the result obtained so far and the prospects of
future work.
References
[1] Murdoch James Gabbay and Andrew Pitts A New Approach to Abstract Syntax Involving
Binders, 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 2-5,
1999, pages 214-224, IEEE Computer Society, 1999.
[2] Mauricio Ayala-Rincon, Maribel Fernandez and Daniele Nantes-Sobrinho. On Nominal Syntax
and Permutation Fixed Points. Log. Methods Comput. Sci. volume 16 (1), 2020.
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XIII Summer Workshop in Mathematics
Quantitative measures for patternmatching
Sandra Alves
Email: [email protected]
Department of Computer Science
University of Porto
Porto, Portugal
Abstract
In this talk we explore recent approaches to quantitative typing systems for programming
languages with pattern matching features. Quantitative (non-idempotent intersection) types
have been used to characterise solvability for a pair pattern calculus, in which a qualitative
characterisation of head-normalisation was given by means of typability. We show that one
can go further and provide upper-bounds/exact measures for head-normalisation, by means of
two resource aware quantitative type systems (system U and system E), which take advantage
of specific technical tools. While system U provides upper bounds for the length of head-
normalisation sequences and the size of normal forms, system E goes even further and produces
exact measures for each of them, as well as discriminating between the different kind of reduction
steps performed.
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XIII Summer Workshop in Mathematics
Equational Unification moduloNon-disjoint Union of Theories
Serdar Erbatur
Email: [email protected]
Department of Computer Science
University of Texas at Dallas (UT Dallas)
Abstract
Unification and its applications to verification of cryptographic protocols have been explored
extensively. The main idea is that it is possible to reduce verification of protocols to solving
symbolic equations between terms exchanged between Alice and Bob who follow a protocol for
secure communication. This approach is further extended so that the terms are interpreted by
taking into account algebraic properties of function symbols that occur in them. For instance,
properties such as associativity (A) and commutativity (C) occur frequently and AC-unification
is used to verify protocols that employ AC function symbols. The underlying logical formalism
for this whole approach is first-order logic with equality (i.e., equational logic). In this logic,
the algebraic properties possessed by the function symbols are axioms of equational theories.
In this talk, I will present an overview of recent results of our group. In particular, I will
explain non-trivial modularity results when an equational theory is a non-disjoint union of other
theories.
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