Coordinators - UnB

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)

2


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

3

https://us02web.zoom.us/j/83041765040?pwd=eG8yZ3J0ckUwODFoU05hWWkrVGdNUT09

https://us02web.zoom.us/j/97884700822?pwd=YUtMOTI0VFBzMVNPVnArOExxRGhNUT09

https://us02web.zoom.us/j/83366143071?pwd=cWdLQU1OTVVhclI1aVZ3L25pZGVZUT09

https://us02web.zoom.us/j/82448403797?pwd=UXNPb0lHRWJjbzZ2MGdTR0NsMDZoZz09

https://us02web.zoom.us/j/83565978698?pwd=VXE0TlFGeXhiTExQbit0V2ZDRHJQUT09


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

4

https://us02web.zoom.us/j/89263942629?pwd=NnRRUW1tMFlPZkJRTnpmTjRoVkd4dz09

https://us02web.zoom.us/j/87644282878?pwd=SWt2NzNqQWhqV0xZd0tCWjBPN1UrZz09

https://mat.unb.br/verao2021/plenarias_en.html

https://us02web.zoom.us/j/86839665048?pwd=NWhvaXVxdkFZRnh3c21McmdYZXErdz09

https://us02web.zoom.us/j/88272156520?pwd=TFI1a1dENE8xVkYrS1dJcVBpZjVwZz09


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

5


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

6


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

7


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

8


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

9


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


Historia da matematica no ensino e aprendizagem: colaboracao na formacao do professor138


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

10


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

11


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

12


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

13


General Information

14


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.

15


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

16


Plenary Talks

17


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.

18


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.

19


Arithmetic and geometry of algebraicsurfaces: the place of elliptic fibrations

and k3 surfaces

Cecılia Salgado


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.

20


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.

21


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).

22


A surprising formula for Sobolev norms

Haim Brezis


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.

23


The Cours d’analyse infinitesimale ofCharles de La Vallee Poussin and the‘ghost volume two’ of its third edition

Jean Mawhin


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’.

24


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)



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.

25


Dynamics of elastic microfilaments influids

Maria Ekiel-Jezewska


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.

26


Bifurcations in Random DynamicalSystems

Martin Rasmussen


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.

27


Self-similar groups and problems ofmodern mathematics

Rostislav Grigorchuk


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.

28


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.

29


Schedule

30


Posters

31


Lindelof Spaces and Michael’s Problem

Alexssandra Thais Pereira Alves de Souza

Supported by FAPESP, process number 19/11642-0.


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.

32


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.

33


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).

34


The Abel’s Theorem via monodromygroups

Ellen Peixoto de Oliveira



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.

35


[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.

36


The Banach-Mazur game and product ofBaire spaces

Gabriel Andre Asmat Medina

Partially supported by CAPES

Email: andre [email protected]


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


Singular Homology Theory andApplications

Henrique Matsuoka Medeiros

Advisor: Dahisy Valadao de Souza Lima


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


Thematic Sessions

39


Algebra

40


Self-similar Compressible T-groups

Adilson A. Berlatto


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


Superalgebras with graded involution

Ana Cristina Vieira

Partially supported by CNPq


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


Non-abelian tensor square and relatedconstructions of finite p-groups

Carmine Monetta

Partially supported by GNSAGA



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


The geometry of diagonal groups

Csaba Schneider




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


Highly arc transitive anddescendant-homogeneous digraphs with

finite out-valency

Daniela A. Amato



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


Homology and cohomology for partialactions

Dessislava Kochloukova

Email: autor1@email


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


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


Conciseness of some words related tonon-commutators

Joao Azevedo

Supported by CNPq



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


σ-Anti-automorphisms on Graded

PrimitveAssociative Algebras

John Freddy Moreno Lozada

Partially supported by CNPq and CAPES



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


Block Theory for Profinite Groups

John MacQuarrie


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


Han’s conjecture for finite-dimensionalalgebras and its analogue for

pseudocompact algebras

Kostiantyn Iusenko

Partially supported by FAPESP



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


Infinite graphs of groups

Mattheus P. S. Aguiar




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


The q-tensor square of potent p-groups,q ≥ 0.∗∗

Nathalia Nogueira Goncalves





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


Exponent of a finite group admitting acoprime automorphism of prime order

Sara Raissa Silva Rodrigues




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


Centralizers of intransitive self-similarabelian groups

Tulio Santos





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.

55


[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


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.



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


Differential operators on algebraicvarieties

Vyacheslav Futorny



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


Twisted conjugation in groups

Yuri Santos Rego


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


Schedule

60


Analysis

61


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:

[email protected]

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


Regularity results for a class of obstacleproblems

Antonia Passarelli di Napoli


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


A topological toolbox for Sobolev maps

Augusto C. Ponce

Partially supported by FNRS-Belgium


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


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.


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


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


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


The Nirenberg problem for the disk

David Ruiz


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


Recursive properties on generalizedODEs

Everaldo de Mello Bonotto

Partially supported by FAPESP grant 2019/03188-7 and CNPq grant 310497/2016-7


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


On a Hardy–Sobolev type inequality andapplications

Everaldo Souto ode Medeiros




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


On a quasi-linear elliptic equationdepending on the gradient

Francesca Faraci


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


Cahn-Hilliard/Allen-Cahn system withdegenerate mobility

Gabriela Planas


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


Uniqueness and positivity issues in asubhomogeneous indefinite elliptic

problem

Humberto Ramos Quoirin


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


Gradient estimates for fully nonlinearPDEs with non-homogeneous degeneracy

Joao Vitor da Silva

Partially supported by CNPq-Brazil under Grant No. 310303/2019-2


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


A dynamical system approach to a classof radial fully nonlinear equations

Liliane A. Maia

Partially supported by CNPq, Capes and FAPDF




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


Schrodinger-Maxwell systems withinterplay between coefficients and data

Luigi Orsina


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


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


Ground states for an Hartree-Fock typesystem

Pietro d’Avenia

Partially supported by PRIN 2017JPCAPN

Qualitative and quantitative aspects of nonlinear PDEs


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


The Beverton–Holt Model and some ofits modifications

Sabrina Streipert


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


Gradient behaviour for large solutions tosemilinear elliptic problems

Stefano Buccheri

Partially supported by the Austrian Science Fund (FWF) project F65


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


Permanence and Stability fornon-autonomous Nicholson’s blowflies

systems

Teresa Faria


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


Sign-changing solutions of coupled ellipticequations

Zhi-Qiang Wang



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


Schedule

82


Dynamical Systems

83


Controllability of affine control systems

Alexandre Santana



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.

84


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.

85


Characteristic Classes and QuaternionicFlags

Daniel Cavalcante Oliveira

PhD supported by Capes


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.

86


On the geometry of smooth structures:Prescribing scalar curvature on fiber

bundles

Leonardo F. Cavenaghi

Partially supported by FAPESP 2017/24680-1



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.

87


Bifurcations and Chaos inHorava-Lifshitz Cosmology

Phillipo Lappicy

Supported by FAPESP


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).

88


On selfmaps of complex Grassmannmanifolds

Thaıs Fernanda Mendes Monis




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.

89


Schedule

90


Geometry

91


On CMC-foliations of asymptoticallyEuclidean manifolds

Carla Cederbaum


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.

92


The geometry of constant mean curvaturesurfaces in Euclidean space

Giuseppe Tinaglia


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.

93


The Bernstein problem for Weingartensurfaces

Isabel Fernandez


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.

94


Classification of immersed Weingarten

spheresin homogeneous three-manifolds

Jose Antonio Galvez


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 .

95


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



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.

96


Systoles, diastoles and minimal surfaces

Lucas Ambrozio


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).

97


Short-time existence for the network flow

Mariel Saez Trumper

Partially supported by Fondecyt Regular 1190388

Email: [email protected].


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.

98


Several aspects of critical metrics forquadratic curvature functionals

Miguel Brozos Vazquez



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.

99


Minimal 2-spheres in homogeneous3-spheres

Pablo Mira

Partially supported by MICINN-FEDER, MTM2016-80313-P


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.

100


The Allen-Cahn equation in the sphere:geometric and variational properties

Rayssa Caju



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.

101


Schedule

102


Mechanics

103


Mathematical Neuroscience: AnIntroduction

Aine Byrne


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.

104


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.

105


Chaos Control Applied to PendulaSystems

Aline Souza de Paula


Department of Mechanical Engineering


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.

106


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.

107


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.

108


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.

109


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.

110


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.

111


Collapsing columns in DEM: Differencesbetween integration schemes.

Gabriel Nobrega Bufolo


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).

112


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.

113


Rivulet Flow Down a Vertical Surface:Lubrication Analysis and Numerical

Solution

Gustavo Carreiro Matias

Rafaela Moreira Borges


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.

114


Aggregation patterns in systemscomposed of few magnetic particles

Jorge Augusto Cassis Modesto


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.

115


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).

116


Slurries of complex fluids: Rheology,microstructure and fluid mechanics

Sarah Hormozi


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.

117


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!).

118


Poiseuille flow inside ducts withrectangular cross section: analytical and

numerical approach

Wildemberg Ribeiro Rocha


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.

119


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.

120


Schedule

121


Mathematics Education

122


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


Universidade Federal de Vicosa, Brasil

Thaymara Cristina de Souza Romulo


E. E. Raul de Leoni, 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.

123


[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.

124


Promover aprendizagens profissionais aprofessores utilizando-se do racicınio dos

estudantes ao resolver tarefasmatematicas

Alessandro Jacques Ribeiro


Universidade Federal do ABC, Brasil

Marcia Aguiar



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.

125


Lesson Study e Engenharia Didatica:Uma formacao didatico-pedagogica defuturos professores de Matematica do

estagio curricular supervisionado

Aluska Dias Ramos de Macedo


Universidade Federal de Campina Grande, Brasil

Paula Moreira Baltar Bellemain


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.

126


[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.

127


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.

128


Saberes e Sabores matematicos em tempode espera: possibilidades de mediacao eintervencao com estudantes no Ensino

Remoto

Cılia Cardoso Rodrigues da Silva


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.

129


[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.

130


Intervencao em criatividade no campo damatematica: Um estudo de caso no

ensino fundamental

Cleyton Hercules Gontijo



Universidade de Brasılia, Brasil

Mateus Giani Fonseca


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

131


[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.

132


Uso da Criptografia como Ferramenta deEnsino

Daiane Soares Veras


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.

133


Educacao Matematica e inclusao escolarna perpectiva da educacao para todos

Ellen Michelle Barbosa de Moura



Karla Vanessa Gomes dos Santos


Joeanne Neves Fraz


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



Ailton Paulo de Oliveira Junior



Luıs Delcides Eodrigues da Silva


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










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.

139


Os elementos ludico e sensorial fora doslimites da sala de aula e o aprendizado da

matematica










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.

141


Introducao a historia social da educacaomatematica - HISOEM

Fredy Enrique Gonzalez


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.

142


[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


Natalia Lami Zanettini



Sabrina Saito


Faculdade de Tecnologia de Barueri, 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.

144


Impressoes de alunos da licenciatura emmatematica sobre o ensino remoto em

uma universidade publica

Josinalva Estacio Menezes




Maria Dalvirene Braga




Rui Seimetz




Poliana Maria da Silva



Samara Araujo da Silva



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

145


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



Keila Cecılia Goncalves








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

148


Metodologias para o ensino remoto dematematica

Letıcia Pereira de Almeida



Guilherme Flaviano Pereira





E. E. Raul de Leoni, Brasil

Aparecida de Fatima Andrade de Silva



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:

149


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.

150


A survey on the PreCalculus course offerin the Midwest Higher Education

Institutions

Luciana Avila Rodrigues

Partially supported by PET/MEC/FNDE, email: [email protected]



Thais Regina Duarte Marcal

Partially supported by PET/MEC/FNDE



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.

151


Experiencias, vivencias e acoes na EscolaEstadual Raul de Leoni no contexto da

pandemia da Covid-19

Luciano Goncalves Batista



Renata Dourado Roque








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


[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.

153


Percepcao de professores formadores emlicenciaturas de matematica sobre a

criativade no desenvolvimento do estagiosupervisionado

Marcia Rodrigues Leal




Liviam Santana Fontes








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.

154


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.

155


Pensamento crıtico e criativo emmatematica a partir de leitura de

informacoes midiaticas





Mateus Giani Fonseca


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 considerado 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.

156


[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.

157


Evasao na Licenciatura em Matematica:o que apontam as pesquisas

Raquel Carneiro Dorr



Brendo Oliveira Damasceno



Izabella Sabino da Silva



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

158


[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.

159


Mathematical Modeling in Mathematical

Literacy:The Study of the Brazilian Monetary

System

Rosangela Maria Kowalek


Universidade Estadual de Londrina, Brazil

Daniel Jose Kmita


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).

160


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.

161


Metodologias alternativas no ensinoremoto de matematica

Thiago Araujo de Lima Matos


Universidade Federal de Vicosa

Beatriz Barbosa Lopes



Daniella Isadora de Assis



Marcio Gustavo Vieira




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.

162


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.

163


Analise fundamentada de uma oficina deTrigonometria

Vania Batista Flose Jardim



Instituto Federal de Sao Paulo, Brasil

Eduardo Goedert Dona



Instituto Federal do Sul de Minas, Brasil

Janaına Mendes Pereira da Silva



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.

164


[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.

165


A ludicidade na aprendizagem e nodesenvolvimento da criatividade em

matematica

Waldineia Ferreira de Oliveira




Marcia Rodrigues Leal








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.

166


[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.

167


Schedule

168


Number Theory

169


On the Arithmetic Behaviour of LiouvilleNumbers Under Rational Maps

Ana Paula Chaves


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.

170


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).

171


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


Permutation Polynomials

Guilherme Tizziotti

Partially supported by CPNq 307037/2019-3



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.

173


Special maps on the projective line overfinite fields

Herivelto Borges


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.

174


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.

175


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


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.

176


Counterexamples to a Conjecture ofNorton

Michael P. Knapp


Loyola University Maryland, USA

Hemar Godinho


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.

177


Three problems on numerical semigroups

Shalom Eliahou


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.

178


Elementos primitivos 2-normais sobreCorpos Finitos

Victor Gonzalo Lopez Neumann

Partially supported by FAPEMIG



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.

179


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.

180


Schedule

181


Probability

182


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.

183


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.

184


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

185


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.

186


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.

187


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.

188


Contact process with renewal cures

Luiz Renato Fontes


Pablo Gomes


Thomas Mountford

Ecole Polytechnique Federale de Lausanne, Switzerland

Daniel Ungaretti


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

189


Random walks on dynamic randomenvironments with non-uniform mixing

Marcelo Hilario


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.

190


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.

191


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.

192


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.

193


Schedule

194


Theory of Computation

195


The Early Years of AutomatedReasoning: The Logic Theory Machine.

Claudia Nalon


Department of Computer Science


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.

196


Anti-unification: Applications andRecent Results

David Cerna


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.

197


Tuple Interpretations for Higher-OrderTerm Rewriting Systems

Deivid Vale

Supported by the ICHOR project


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.

198


A Formalization of the Z property in Coq

Flavio L. C. de Moura




Brasılia, Brazil

Leandro O. Rezende




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.

199


Formalising Completeness ofAC-unification

Gabriel Ferreira Silva


Supported by a CAPES scholarship


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.

200


Towards a fixed-point approach toNominal Disunification

Leonardo Melo

Partially supported by Capes



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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Quantitative measures for patternmatching

Sandra Alves



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.

202


Equational Unification moduloNon-disjoint Union of Theories

Serdar Erbatur



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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Schedule

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Coordinators - UnB

Documents

Transcript of Coordinators - UnB