ΝΕΑ ΑΙΤΗΣΗ PHD ΑΓΓΛΙΚΑ · Integral representations of holomorphic functions of...

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COURSE DESCRIPTION

Course Title Measure theory and Integration

Course Code MAS601

Course Type Compulsory or Elective

Level Postgraduate

Year / Semester 1st or 2nd semester

Teacher’s Name

ECTS 10 Lectures / week 2 Laboratories / week

0

Course Purpose and Objectives Comprehend the mathematical concepts of measure theory and integration

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of measure theory and integration

with applications to other fields. • Use the necessary mathematical tools needed for establishing certain

theoretical results in integration theory

Prerequisites - Required -

Course Content Metric spaces. σ- algebras, measures, outer measures. Borel measures on the real line. Measurable functions. Integration. General convergence theorems. Signed measures. Product measures n-dimensional Lebesque integral. The Radon Nikodym Theorem. Lp spaces.

Teaching Methodology

Lectures (4 hours per week)

Bibliography 1. Gerald B. Folland, Real Analysis, Modern Techniques and Their Applications, Second Edition, John Wiley and Sons, Inc., 1999.

2. Robert G. Bartle, The Elements of Integration and Lebesgue Measure, John Wiley and Sons, Inc., 1995.

3. H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, 1988.

4. Elias M. Stein and Rami Shakarchi: Real Analysis, Measure Theory, Integration and Hilbert Spaces. Princeton Lectures in Analysis III. Princeton University Press 2005.

Assessment Two mid-term and one final written examination

Language Greek or English

2

Course Title Fourier Analysis Course Code MAS602 Course Type Elective Level Postgraduate Year / Semester 1st or 2nd semester Teacher’s Name


0

Course Purpose and Objectives

• Learning of the mathematical concepts of Fourier Analysis and Harmonic Analysis

Learning Outcomes Upon successful completion of this course, students are expected to be able to:

• Comprehend the mathematical concepts of Fourier Analysis and Harmonic Analysis

• Use the necessary mathematical tools needed for establishing certain theoretical results in Fourier analysis and Applications to partial differential equations.


Course Content The Schwarz space. Fourier transform. Plancherel’s formula. Convergence of Fourier series and integrals. Applications in partial differential equations. Distributions. Tempered distributions, compactly supported distributions. Sobolev spaces.



Bibliography 1. Georgi P. Tolstov, FOURIER SERIES, (translated from the Russian by Richard A. Silverman), Dover publications, 1978. 2. Allan Pinkus and Samy Zafrany, FOURIER SERIES AND INTEGRAL TRANSFORMS, Cambridge University Press, 2001. 3. Gerald B. Folland, FOURIER ANALYSIS AND ITS APPLICATIONS, Brooks/Cole Publishing company, 1992. 4. A. Zygmund, TRIGONOMETRIC SERIES, 3rd edition, Cambridge University Press, Cambridge, 2002. 5. Elias M. Stein and Rami Shakarchi, FOURIER ANALYSIS, Princeton University press, 2005.

Assessment Two mid-term and one final written examinations Language Greek or English

3

Course Title Partial Differential Equations Course Code MAS603 Course Type Compulsory or Elective Level Postgraduate Year / Semester 1st or 2nd semester Teacher’s Name


0


Partial Differential equations represent deterministic mathematical formulations of phenomena in physics and engineering as well as biological processes among many other Applied Sciences. The objective of this course is to present the main results in the context of Partial Differential Equations that allow learning about the theory of models governed by these equations, with a complete presentation of the properties of their solutions.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of Partial Differential Equations. • Use the necessary mathematical tools needed for establishing certain

theoretical results. • Analyze, synthesize, organize and plan projects in the field. • Recognize the type of the equation (elliptic, parabolic, hyperbolic) and

state and prove fundamental theorems on existence, uniqueness and stability of solutions for the Laplace, heat and wave equations.


Course Content First order quasi-linear equations, the method of characteristics. Classification and normal forms. Existence theorem of Cauchy- Kovalevskaya and uniqueness theorem of Holmgren. Distributions and weak solutions. Hyperbolic theory, characteristics, propagation of singularities. Wave equation in one, two and three space dimensions. Conservation laws and shock waves. Elliptic theory, Laplace and Poisson equations, fundamental solutions, harmonic functions. Variational formulation of elliptic boundary value problems. Parabolic theory, heat equation, parabolic initial/boundary value problems.



Bibliography L. Evans, Partial Differential Equations, 2nd edition American Math Society. F. John Partial Differential Equations, Applied Mathematical Sciences, Springer. W. Strauss, Partial Differential Equations, An Introduction, 2nd Edition Wiley.

Assessment The assessment method includes one Final Exam and one Midterm Exam. An additional project with presentation is up to the discretion of the teacher.


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Course Title Functional Analysis

Course Code MAS604

Course Type Elective

Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives To become familiar with the notions of operator theory and its applications.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: . understand the spectral theory of operators and . to apply its methods and results.


Course Content Compact operators. Spectral theory. Self adjoint operators. Closed and orthonormal operators. Spectral theorem. Semigroups.



Bibliography Functional Analysis, K. Yosida, Springer Verlag 6th edition 1980

Assessment Final exam


5

Course Title Second Order Elliptic Partial Differential Equations

Course Code MAS605


Level Postgraduate


Teacher’s Name


0


Partial Differential equations allow deterministic mathematical formulations of phenomena in physics and engineering as well as biological processes among many other scenarios. The objective of this course is to present the main results in the context of elliptic partial differential equations that allow learning about the theory of models governed by this type of equations, with a complete presentation of the properties of solutions

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of Elliptic Partial Differential

Equations. • Use the necessary mathematical tools needed for establishing certain

theoretical results in the elliptic theory. • Analyze, synthesize, organize and plan projects in the field.


Course Content Laplace equation, fundamental solutions, Green's function, maximum principle, Poisson kernel, Harmonic functions and their properties, Harnack inequalities, equations with variable coefficients, Dirichlet problem, existence and regularity of solutions.



Bibliography D. Gilbarg, N. Trundiger Elliptic Partial Differential Equations of 2nd Order, Springer

O. Ladyzhenskaya, N. Uraltseva, Linera & Quasilinear Elliptic Equations, Academic Press

Assessment The assessment method includes one Final Exam and one Midterm Exam. An additional project with presentation is up to the discretion of the teacher.


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Course Title Function Theory of One Complex Variable

Course Code MAS606


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives Advanced knowledge of Complex Analysis

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of theory of functions of one

complex variable. • Use the necessary mathematical concepts and ideas needed for establishing

concrete and theoretical results in complex analysis on the plane.


Course Content Basic facts about complex numbers and complex functions of one complex variable. Differentiation. Cauchy-Riemann equations and holomorphic functions. Elementary holomorphic functions and power series. Complex integration and the Cauchy Theorem. Applications of Cauchy Theorem. Meromorphic functions and residues. Laurent series. Geometric Theory and Analytic Continuation. Conformal mappings.



Bibliography 1) Complex Variables, by S. Fisher 2) Function Theory of one Complex Variable, by R.Greene and G.Krantz 3) Complex Analysis, by L. Ahlfors. 4) Functions of One Complex Variable, by J. Conway, Springer

Assessment Homework, midterm and final exams


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Course Title Function Theory of Several Complex Variables

Course Code MAS607


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives Introductory class to Several Complex Variables

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the fundamentals of holomorphic functions of several

complex variables. • Use the necessary mathematical tools needed for concrete computations of

multidimensional complex integrals. • Understands the basic properties of orthogonal polynomials is several

variables.

Prerequisites - prerequisites -

Course Content Basic facts about holomorphic functions of several complex variables. Power series and multi-circular domains. Laurent and Hartogs series. Division Theorem and different type of convexities. Integral representations of holomorphic functions of several complex variables (Bochner-Martinelli, Cauchy -Fantappie and Bergman-Weil formulas). Chirstoffel- Darboux kernels and applications to several complex variables.



Bibliography 1) Introduction to complex variables, vol. 2 by B.V. Shabat, 2) Function theory of several complex variables, by G.Krantz 3) Integral representations and residues in Several Complex Variables, by L.

Aizenberg.

Assessment Homework, in class presentation.


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Course Title Ordinary Differential Equations

Course Code MAS613


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives In depth study of the theory of Ordinary Differential Equations (ODEs)

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the theory of ODEs, as outlined in the course content section

below. • Continue their studies at a higher level


Course Content Existence theorems: Picard- Lindelöf and Cauchy-Peano. Uniqueness theorem when Lipschitz condition is satisfied. Smooth dependence of solutions on parameters. Extensibility of solutions. Linear systems, fundamental solution matrix, systems with periodic coefficients. Stability of nonlinear systems. Poincaré-Bendixson theory.



Bibliography Gerald Teschl, “Ordinary Differential Equations and Dynamical Systems”, volume 140 of Graduate Studies in Mathematics. AMS, Providence, Rhose Island, 2012.

Assessment 1 midterm and 1 final exam. (Optional homework.)


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Course Title Topics in Mathematical Analysis I

Course Code MAS617


Level Postgraduate


Teacher’s Name


0


The objective of this course is to introduce students to main topics in Mathematical Analysis including the areas of Real Analysis, Complex Analysis and/or Differential Equations. The student will receive an in-depth presentation of the corresponding theories through study of classical texts and research papers with the purpose to improve their ability to solve theoretical problems in the field.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of Mathematical Analysis broadly

defined. • Use the necessary mathematical tools needed for establishing certain

theoretical results. • Analyze, synthesize, organize and plan projects in the field.


Course Content Topics in Real Analysis, Complex Analysis or Differential Equations, depending on the special interests of the faculty member teaching the course.



Bibliography Selection of topics from bibliography and research papers in general areas of Real Analysis, Complex Analysis and Differential Equations depending on the special interests of the faculty member teaching the course.

Assessment Presentation and Final Exam


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Course Title Topics in Mathematical Analysis II

Course Code MAS618


Level Postgraduate


Teacher’s Name


0


The objective of this course is to introduce students to main topics in Mathematical Analysis, not included in MAS617, that fall in the areas of Real Analysis, Complex Analysis and/or Differential Equations. The student will receive an in-depth presentation of the corresponding theories through study of classical texts and research papers with the purpose to improve their ability to solve theoretical problems in the field.











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Course Title Topics in Mathematical Analysis III

Course Code MAS619


Level Postgraduate


Teacher’s Name


0


The objective of this course is to introduce students to main topics in Mathematical Analysis, not included in MAS617 and MAS618, that fall in the areas of Real Analysis, Complex Analysis and/or Differential Equations. The student will receive an in-depth presentation of the corresponding theories through s study of classical texts and research papers with the purpose to improve their ability to solve theoretical problems in the field.











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Course Title Approximation Theory Course Code MAS620 Course Type Elective Level Graduate Year / Semester 1st or 2nd semester Teacher’s Name


0


Students will learn some basic methods to approximate sufficiently smooth functions with polynomials and wavelet-type orthonormal basis


• Comprehend the basic mathematical concepts of approximation theory • Will be able to study the existence of best polynomial approximation to

continuous functions on closed intervals • Will be able to represent function in Lebesgue, Sobolev and Besov spaces in

terms of wavelet orthonormal basis • Will be able to estimate the approximation order of smooth functions by

means of linear and nonlinear approximation method based on wavelet representations


Course Content 1. Introduction to Normed spaces and Linear Operators 2. Theorems of Stone-Weierstrass 3. Spaces of Functions 4. Best Approximation 5. Chebyshev Theorem 6. Degree of Approximation by Trigonometric and Algebraic Polynomials 7. Wavelet orthonormal basis 8. Non linear Wavelet Approximation



Bibliography R. A. DeVore, G. G. Lorentz, Constructive Approximation, Springer 1993 Y. Meyer, Wavelets and Operators, Cambridge University Press, 1991

Assessment Homework assignments, Midterm exam and Final exam. Language Greek or English

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Course Title Numerical Linear Algebra

Course Code MAS621


Level Postgraduate


Teacher’s Name


0


To learn about the theory and practice of linear algebra and to gain an understanding of the pros and cons of different methods in numerical linear algebra

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand the theory of matrix analysis • Know which modern method to use for linear systems, least-squares

problems and computing eigenvalues/eigenvectors.


Course Content Elements of matrix analysis, vector and matrix norms. Factorization and least - squares methods. Stability. Direct and iterative methods for the solution of linear systems. Methods for calculating eigenvectors and eigenvalues.



Bibliography Matrix Computations by G. Golub and C. Van Loan, Johns Hopkins Univ. Press, 1996.

Numerical Linear Algebra by L. N. Trefethen and D. Bau III, SIAM publications, 1997.

Assessment 1 midterm and 1 final exam. (Optional homework)


14

Course Title Coding Theory

Course Code MAS622


Level Postgraduate


Teacher’s Name


0


To introduce students to a subject of convincing practical relevance that relies heavily on results and techniques from Pure Mathematics.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend basic mathematical concepts of algebraic coding theory. • Use the necessary mathematical tools needed for establishing certain

theoretical results in algebraic coding theory.


Course Content Finite fields. Linear codes, syndrome decoding. Cyclic codes. BCH codes and Reed – Solomon codes. MDS codes. Permutation decoding.



Bibliography R. Hill, A First Course In Coding Theory

J.H. van Lint, Introduction to Coding Theory

Assessment Midterm examination and final examination


15

Course Title Number Theory

Course Code MAS623


Level Postgraduate


Teacher’s Name


0


To show how tools from algebra can be used to solve

problems in number theory

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • define the basic notions of algebraic number theory, such as algebraic numbers and algebraic integers, number fields, rings of algebraic integers • describe the additive and multiplicative structure the ring of integers of a number field using the proper algebraic terminology, • identify the ring of integers and the discriminant of simple examples • solve simple Diophantine equations using factorizations of algebraic integers and ideals.


Course Content Introduction to algebraic number theory. Quadratic reciprocity, Gauss and Jacobi sums. Field extensions, finite fields, ideal classes. Quadratic and cyclotomic fields. Applications to Diophantine equations.



Bibliography Jarvis Algebraic Number Theory Ireland and Rosen, A Classical Introduction to Modern Number Theory. Frohlich and Taylor, Algebraic Number Theory



16

Course Title Introduction to Commutative Algebra

Course Code MAS624


Level Postgraduate


Teacher’s Name


0


To obtain a solid understanding of basic notions of commutative algebra with a view to apply them in algebraic geometry.

Learning Outcomes • A solid understanding of basic notions of commutative algebra, like rings, modules, localization, Noetherian and Artin rings, completion and dimension.

• To develop the ability to solve problems in the field. • To apply the theory of commutative algebra in algebraic geometry


Course Content Rings and Ideals. Modules. Localization of rings and modules. Primary decomposition. Integral extensions of rings. Noetherian and Artinian rings. Completion of a ring. Dimension.



Bibliography • Introduction to commutative algebra, M. Atiyah, I. MacDonald. • Commutative ring theory, H. Matsumura. • Commutative algebra with a view towards algebraic geometry, D.

Eisenbud.

Assessment Homeworks and final exam.


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Course Title Group Theory

Course Code MAS625


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives To obtain a solid understanding of basic notions of the theory of groups

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend basic concepts of group theory like group action, Sylow

subgroup, semi-direct product of groups. • Develop the ability to solve problems in the field using the necessary

mathematical tools


Course Content Examples of groups, subgroups. Generators and relations. Direct and semi-direct products. Group actions. Sylow theorems and p-groups. Composition series and Jordan-Hölder theorem. Solvable and nilpotent groups.



Bibliography W. Ledermann, A. Weir, Introduction to Group Theory J.J. Rotman, An Introduction to the Theory of Groups J.S. Rose, A Course on Group Theory



18

Course Title Field and Galois Theory

Course Code MAS626


Level Postgraduate


Teacher’s Name


0


To introduce students to an important mathematical subject where elements of different branches of mathematics are brought together for the purpose of solving important classical problems.

Learning Outcomes • show familiarity with the concepts of ring and field, and their main algebraic properties • understanding of basic concepts of Galois theory like field extension, splitting field, Galois group • ability to solve problems in the area (for example computation of the Galois group in simple cases with conclusions regarding the corresponding field extension)


Course Content Polynomial rings. Field extensions, splitting fields. Separable extensions, normal extensions. The fundamental theorem of Galois theory. Roots of unity and cyclotomic polynomials. Solution by radicals and the Abel-Ruffini theorem.



Bibliography 1. J.J. Rotman, Galois Theory. 2. P. Morandi, Fields and Galois Theory. 3. S. Roman, Field Theory.



19

Course Title Group Representation Theory I

Course Code MAS627


Level Postgraduate


Teacher’s Name


0


To obtain a solid understanding of basic notions of the representation theory of finite groups with applications.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend basic concepts of representation theory like group algebra,

FG-module, the character of a representation. • Use the necessary mathematical tools needed for establishing certain

theoretical results in representation theory.


Course Content Representations. FG-modules, FG-submodules and FG-homomorphisms. Maschke’s Theorem and Schur’s Lemma. Irreducible module. The group algebra, the centre of the group algebra. Characters, relation between characters and representations. Character tables. Frobenius reciprocity theorem.



Bibliography 1. G. James, M. Liebeck, Representations and Characters of Groups. 2. L. Dornhoff, Group Representation Theory. 3. M. Burrow, Representation Theory of Finite Groups. 4. M. Collins, Representations and Characters of Finite Groups. 5. J. Alperin, R. Bell, Groups and Representations.



20

Course Title Group Representation Theory II

Course Code MAS628


Level Postgraduate


Teacher’s Name


0


To obtain a solid understanding of basic notions of the representation theory of finite and compact groups with applications.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend basic concepts of group representation theory like irreducible

module, splitting field of a representation and induced representation • Use the necessary mathematical tools needed for establishing certain

theoretical results in group representation theory.


Course Content Semi-simple rings, construction of irreducible R – modules. Splitting fields. Clifford’s theorem. Mackey Decomposition Theorem. Representations and characters of finite groups. Representations of compact groups.



Bibliography 1. L. Dornhoff, Group Representation Theory. 2. M. Collins, Representations and Characters of Finite Groups. 3. C.B. Thomas, Representations of Finite and Lie Groups



21

Course Title Topics in Algebra

Course Code MAS629


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives To obtain a solid understanding of basic notions from various areas of algebra

Learning Outcomes Upon successful completion of this course, students are expected to be able to: Comprehend basic algebraic concepts with their applications and develop the ability to solve problems in the field.


Course Content Topics from algebra



Bibliography Depends on the special interests of the staff member teaching it



22

Course Title Algebraic Geometry

Course Code MAS630


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives To obtain a solid understanding of basic notions of algebraic geometry.

Learning Outcomes • To obtain a solid understanding of basic notions of algebraic geometry. • To develop the ability to solve problems in the field.


Course Content Algebraic sets and the Hilbert-Nullstellensatz theorem. Affine, projective and quasi-projective varieties, morphisms, products. Local properties. Smooth and singular points. Tangent space. Dimension. Divisors on algebraic curves, Riemann-Roch theorem. Bezout's theorem. Elliptic curves. The group structure of an elliptic curve.



Bibliography • Algebraic Geometry, Chapter 1, R. Hartshorne. • Elementary Algebraic Geometry, K. Hulek. • Basic Algebraic Geometry I, I. Shafarevich.



23

Course Title Differential Topology

Course Code MAS631


Level Postgraduate


Teacher’s Name


0


The main objective of the course is to introduce students to central topics of Differentiable Manifolds. The student will receive an in-depth presentation of the fundamental theorems in this area, as well as their proofs through, the study of classical with the purpose to improve their ability to solve theoretical problems in the field.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts and definitions of Differentiable

Manifolds. • Use the necessary mathematical tools needed for establishing certain

theoretical results.


Course Content Differentiable manifolds. Tangent space. Partition of unity. Regular points. Sard's theorem. Vector fields and flows. Frobenius Theorem. Differential forms. Stokes’ Theorem. De Rham's Theorem.



Bibliography Morita, Geometry of Differential Forms, AMS 2000. W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1986. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Vol. 2, Publish or Perish, 1999. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983. V. Guillemin, A. Pollack, Differential Toplogy, Pearson 1975.

Assessment Homework and Final Exam


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Course Title Riemannian Geometry

Course Code MAS632


Level Postgraduate


Teacher’s Name


0


The main objective of the course is to introduce students to central topics of Riemannian Geometry. The student will receive an in-depth presentation of the fundamental theorems in this area, as well as their proofs through, the study of classical with the purpose to improve their ability to solve theoretical problems in the field.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts and definitions of Riemannian

Geometry. • Use the necessary mathematical tools needed for establishing certain



Course Content Riemannian manifolds. Geodesics, exponential map, normal coordinates. Gauss lemma. Theorem of Hopf- Rinow. Curvature. Jacobi fields. Theorems of Bonnet- Myers, Synge-Weinstein and Hadamard - Cartan. Homogeneous and symmetric spaces.



Bibliography M. Do Carmo, Riemannian Geometry, Birkhauser, 1992. J. M. Lee, Riemannian Geometry, Springer 1997 W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1986 M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Vol. 2, Publish or Perish, 1999.

Assessment Homework and Final exam


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Course Title General Relativity

Course Code MAS633


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives An introduction to General Relativity for mathematicians.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: . Comprehend the main concepts of General Relativity. . Use the necessary mathematical tools needed for establishing certain results in General Relativity


Course Content Lorentz geometry. Special relativity. Newton spacetime, Minkowski spacetime. Lorentz transformation. Einstein equations. Special solutions (Schwarzschild).



Bibliography Yvonne Choquet-Bruhat , Introduction to General Relativity, Black Holes and Cosmology, Oxford University Press, 2015.

J. Foster and J.D. Nightingale, A short course in General Relativity, Springer, 2006.

Assessment Presentation and Final Exam.


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Course Title Algebraic Topology I

Course Code MAS634


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives To obtain a solid understanding of basic notions of algebraic topology.

Learning Outcomes • To obtain a solid understanding of basic notions of algebraic topology. • To develop the ability to solve problems of the field.


Course Content Fundamental group, Van Kampen theorem, Covering spaces, Homology and Cohomology, The Mayer Vietoris sequence, Excision, Relation between homology and fundamental group.



Bibliography • Algebraic Topology, A. Hatcher. • A Basic Course in Algebraic Topology, W. Massey.



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Course Title Lie Groups and Lie Algebras

Course Code MAS635


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives To learn the basics of Lie groups and their Lie algebras.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend basic mathematical concepts of Lie groups and Lie algebras. • Understand the importance of the classification of simple Lie algebras


Course Content Differentiable manifolds. Tangent spaces and vector fields. Lie Groups. Exponential function. Homogeneous spaces. The Campbell-Hausdorf formula. Ado's Theorem. Lie algebras. Ideals and homomorphisms. Solvable and nilpotent Lie algebras. Semisimple Lie algebras. Root systems. Compact Lie groups.



Bibliography 1. James Humphreys, Introduction to Lie Algebras and Representation Theory, Springer.

2. Frank Warner, Foundations of differentiable manifolds and Lie groups, Springer.

3. Anthony Knapp, Lie Groups Beyond an Introduction, Birkhauser.

Assessment Presentation and final examination


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Course Title Algebraic Topology II

Course Code MAS636


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives An introduction to obstruction theory, K-theory and characteristic classes.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: . Comprehend the main concepts of obstruction theory. . Use the necessary technics needed for solving problems in topology.


Course Content Obstruction theory. Bundles and K- theory. Bordism. Spectral sequences. Characteristic classes.



Bibliography J. Milnor and J. D. Stasheff, Characteristic Classes, Princeton University Press 1974.



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Course Title Spectral Geometry

Course Code MAS637


Level Postgraduate


Teacher’s Name


0


The main objective of the course is to introduce students to the spectral theory of the Lapalcian. The student will receive an in-depth presentation of the fundamental theorems in this area, as well as their proofs through, the study of classical with the purpose to improve their ability to solve theoretical problems in the field.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts and definitions in the Spectral

theory of the Laplacian. • Use the necessary mathematical tools needed for establishing certain



Course Content Laplace operator. Minimax principle. Isoparametric inequalities. Heat kernel.



Bibliography Spectral Geometry of the Laplacian, H. Urakawa, World Scientific 2017



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Course Title Spin Geometry

Course Code MAS638


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives An introduction to Spin geometry and its applications.


• Comprehend the concepts of Clifford Algebra, Spin Structure and other basic concepts and definitions in Spin Geometry.

• Apply these concepts in order to study the Dirac operator and its applications.


Course Content Clifford algebras. Spin groups and representations. Spin structures. Spin connection. Spin manifolds. Dirac operator. Bochner formula. Lichnerowicz's Theorem.



Bibliography H. Blaine Lawson & Marie-Louise Michelsohn, Spin Geometry, Princeton Mathematical Series, 1990.



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Course Title Topics in Geometry I

Course Code MAS640


Level Postgraduate


Teacher’s Name


0


To introduce students to subjects related to research interests/topics of the academic staff of the department from the area of Geometry, such as Differential Geometry, Algebraic Geometry and Algebraic Topology.

Learning Outcomes Introduce students to mathematical research through the study of some classical papers and results, and to improve their ability to solve problems in the field.


Course Content Topics from Differential Geometry, Algebraic Geometry and Algebraic Topology.

Depends on the special interests of the staff member teaching it




Assessment Homework (problem sets), presentation, final exam


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Course Title Topics in Geometry IΙ

Course Code MAS641


Level Postgraduate


Teacher’s Name


0


To introduce students to subjects related to research interests/topics of the academic staff of the department from the area of Geometry, such as Differential Geometry, Algebraic Geometry and Algebraic Topology.

Learning Outcomes Introduce students to mathematical research through the study of some classical papers and results, and to improve their ability to solve problems in the field.


Course Content Topics from Differential Geometry, Algebraic Geometry and Algebraic Topology.

Depends on the special interests of the staff member teaching it




Assessment Homework (problem sets), presentation, final exam


33

Course Title Probability Theory

Course Code MAS660


Level Postgraduate


Teacher’s Name

ECTS 10 Lectures / week

2 Laboratories / week 0


The purpose of the course is to provide the students with the necessary tools in probability to successfully complete the PhD in Statistics.

Learning Outcomes Upon successful completion of this course, students are expected to:

1. Know the measure-theoretic foundations of probability theory. 2. Work with and prove statements about convergence of random variables

and 0-1 laws. 3. Compute and work with conditional expectation. 4. Familiarize with central limit theorems.


Course Content Measure spaces and σ-algebras, independence, measurable functions and random variables, distribution functions, Lebesgue integral and expectation, convergence concepts, law of large numbers, characteristic functions, central limit theorem, conditional probability, conditional expectation, martingales, central limit theorem for martingales.



Bibliography 1. Durret Rick, Probability: Theory and Examples, 5th Edition, Cambridge University Press, 2019.

2. Shiryaev A., Probability, 2nd Edition, Spinger, 1996.

Assessment Final exam, midterm exam, homework (theoretical assignments).


34

Course Title Numerical Solution of Ordinary Differential Equations

Course Code MAS671


Level Postgraduate


Teacher’s Name


0


The purpose of this course is the study of numerical techniques for the solution of ordinary differential equations (ODEs).

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Solve analytically and numerically initial value problems for systems of first

order ordinary differential equations.. • Study the stability of one-step and multiple-step methods. • Solve analytically and numerically two-point boundary value problems for

second order ordinary differential equations with finite difference methods.


Course Content One-step methods and multistep methods for the numerical solution of initial value problems for first order systems of ordinary differential equations. Runge – Kutta methods. Finite difference methods for the numerical solution of two-point boundary value problems.



Bibliography A First Course in the Numerical Analysis of Differential Equations, A. Iserles

Assessment Final exam, one midterm exam, one computational project.


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Course Title Numerical Solution of Partial Differential Equations

Course Code MAS672


Level Postgraduate


Teacher’s Name


0


The purpose of this course is the study of finite difference methods for the numerical solution of partial differential equations (PDEs).

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Solve analytically and numerically initial/boundary value problems for the

one-dimensional heat equation and the one-dimensional wave equation. • Solve analytically and numerically boundary value problems for the two-

dimensional Poisson and biharmonic equations. • Study the convergence and stability of finite difference methods for the

numerical solution of the above problems.


Course Content Finite difference approximations. Numerical solution of the heat equation. Convergence and stability. ADI methods. Numerical solution of the wave equation. The Courant – Friedrichs – Lewy condition. Numerical solution of the Poisson and biharmonic equations.



Bibliography Numerical Partial Differential Equations: Finite Difference Methods, J. W. Thomas

Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations,, J. W. Thomas

Assessment Final exam, one midterm exam, one computational project.


36

Course Title Finite Element Methods

Course Code MAS673


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives Theory and implementation of the finite element method.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical theory of finite element methods. • Treat boundary value problems using Galerkin’s method.


Course Content Sobolov spaces. Ritz-Galrkin approximation. Variational formulation of elliptic boundary value problems. Finite element spaces. Polynomial approximation in Sobolev spaces. N-dimensional variational problems. Multigrid finite element methods.



Bibliography 1. Γ. Δ. Ακρίβης, Μέθοδοι πεπερασμένων στοιχείων, Πανεπιστημιακές Εκδόσεις, Πανεπιστήμιο Κύπρου, 2005.

2. C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, 1987.

3. D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2001.

Assessment 1 midterm exam, 1 final exam and a group project


http://www.mas.ucy.ac.cy/%7Exenophon/courses/mas473/pdf/nmpde.pdf

37

Course Title Topics in Numerical Analysis I

Course Code MAS677


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives Study of special topics not included in other courses.

Learning Outcomes Vary with the content


Course Content Topics in Computational Mathematics and Approximation Theory.




Assessment Homeworks and final exam


38

Course Title Topics in Numerical Analysis IΙ

Course Code MAS678


Level Postgraduate


Teacher’s Name


0










39

Course Title Topics in Numerical Analysis IΙΙ

Course Code MAS679


Level Postgraduate


Teacher’s Name


0










40

Course Title Classical Mechanics

Course Code MAS682


Level Postgraduate


Teacher’s Name


0


To learn the basic ideas of integrable systems and Lie symmetries of differential equations.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the basic theory of Integrablle Hamiltonian Systems. • Understand the method of Lie group analysis of differential equations.


Course Content Lie Groups and Lie Algebras. Equations of motion (Newton, Lagrange). Poisson structures, Integrable systems, Lax pairs, bi – Hamiltonian systems. Symmetries of Differential Equations, Noether Theorem.



Bibliography 1. Pantelis Damianou, Lecture notes on Classical Mechanics.

2. Olver, P. J. Applications of Lie groups to Differential Equations. Springer-Verlag, New York (1986).

3. Adler M., Van Moerbeke P., Vanhaecke P., Algebraic integrability, Painleve geometry and Lie algebra, Springer-Verlag, Berlin Heidelberg

4. Nail H. Ibragimov, Transformation Groups and Lie Algebras, Higher Education Press

Assessment Midterm and Final Exams.


41

Course Title Fluid Dynamics

Course Code MAS683


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives Introduction to Newtonian and non-Newtonian Fluid Mechanics.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Formulate Newtonian or non-Newtonian fluid flow problems in various

coordinate systems • Derive analytical solutions either directly in terms of the velocity

components or indirectly in terms of the streamfunction or the velocity potential


Course Content Equations of motion. Steady or transient viscous flows. Stokes flows. Non- Newtonian and viscoelastic flows.



Bibliography J. Marsden και A. Tromba, Διανυσματικός Λογισμός (Μετάφραση: Α. Γιαννόπουλος), Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο (1992). T. Papanastasiou, G. Georgiou and A. Alexandrou, Viscous Fluid Flow, CRC Press, Boca Raton (1999).

Assessment HWK problems, 2 midexams, and project with presentation


42

Course Title Scientific Computing with MATLAB

Course Code MAS684


Level Postgraduate


Teacher’s Name


0

Course Purpose and Objectives Study of problems whose solution requires the use of a computer

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Write MATLAB code to address problems from solid and fluid mechanics. • Choose the appropriate method for each problem.


Course Content Introduction to MATLAB. Data and function approximation. Linear Systems. Eigenvalues and Eigenvectors. Ordinary Differential Equations. Numerical Methods for boundary value problems.



Bibliography Experiments with MATLAB by Cleve Moler, The MathWorks Inc.

Assessment Weekly assignments, final project


43

Course Title Topics in Applied Mathematics I

Course Code MAS687


Level Postgraduate


Teacher’s Name


0


Introduction to methods of Applied Mathematics in modern Mathematical problems.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: . Comprehend the concepts of certain methods in Applied Mathematics . Use the necessary mathematical methods needed to solve modern Mathematical problems


Course Content Topics from Applied Mathematics.




Assessment Projects and final exam


44

Course Title Topics in Applied Mathematics II

Course Code MAS688


Level Postgraduate


Teacher’s Name


0











45

Course Title Topics in Applied Mathematics III

Course Code MAS689


Level Postgraduate


Teacher’s Name


0











46

Course Title Topics in Differential Equations I

Course Code MAS697


Level Postgraduate


Teacher’s Name


0


The objective of this course is to introduce students to main topics in of the area of Differential Equations. The student will receive an in-depth presentation of the corresponding theories through study of classical texts and research papers with the purpose to improve their ability to solve theoretical problems in the field.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of Differential Equations broadly




Course Content Topics in differential equations.



Bibliography Selection of topics from bibliography and research papers in general areas of Partial Differential Equations or Ordinary Differential Equations, Calculus of Variations.



47

Course Title Topics in Differential Equations II

Course Code MAS698


Level Postgraduate


Teacher’s Name


0


The objective of this course is to introduce students to main topics in of the area of Differential Equations, not included in MAS697. The student will receive an in-depth presentation of the corresponding theories through study of classical texts and research papers with the purpose to improve their ability to solve theoretical problems in the field.








Bibliography Selection of topics from bibliography and research papers in general areas of Partial Differential Equations or Ordinary Differential Equations and Calculus of Variations.



48

Course Title Topics in Differential Equations III

Course Code MAS699


Level Postgraduate


Teacher’s Name


0


The objective of this course is to present main topics of Differential Equations, not included in MAS697 and MAS698, that allow learning about the theory included in the areas of Ordinary or Partial Differential Equations, with a complete presentation of the corresponding theories.








Bibliography Selection of topics from bibliography and research papers in general areas of Partial Differential Equations or Ordinary Differential Equations and Calculus of Variations.



49

Course Title Comprehensive examination in Analysis Course Code MAS780 Course Type Comprehensive Examination (CE) Level Postgraduate Year / Semester 1st semester Teacher’s Name

ECTS 0 Lectures/week 0 Laboratories / week

0


Successful completion of the CE is required in order to continue with the dissertation.

Learning Outcomes -


Course Content Structure and properties of real numbers, continuity, differentiability, Riemann integrability. Metric spaces, compactness, connectedness, Bolzano-Weierstrass theorem, Heine-Borel theorem, Baire category theorem, uniform continuity, convergence of sequences and series of functions. σ-Algebras, outer measures, Borel and Lebesgue measures, measurable functions, Lebesgue dominated convergence theorem, monotone convergence theorem, Fatou’s lemma. Signed measures, Radon-Nikodym theorem, product measures, Fubini’s theorem. The complex plane, stereographic projection. Möbius transformations. Elementary analytic functions. Cauchy-Riemann equations, harmonic functions. Cauchy’s integral formula and theorem, Morera’s theorem. Liouville’s theorem. Fundamental theorem of algebra. Taylor and Laurent series, residues. Maximum Measure Principle. Schwarz’s lemma, the Argument Principle, Rouche’s theorem, conformal mapping, the Riemann mapping theorem.


Bibliography 1) Royden, H. L. Real Analysis, New York, Mackmillan 2) Rudin, W. Principles of Mathematical Analysis 3) Rudin W. Real and Complex Analysis, New York, McGraw-Hill 4) John B. Conway, Functions of one complex variable, Springer Verlag 5) L. V. Alfors, Complex Analysis, McGraw-Hill 6) A.I. Markushevich, Theory of Functions of a Complex Variable 7) Ralph Boas, Invitation to Complex Analysis, McGraw Hill

Assessment Written Examination Language Greek

50

Course Title Comprehensive examination in Algebra

Course Code MAS781

Course Type Comprehensive Examination (CE)

Level Postgraduate

Year / Semester 1st semester

Teacher’s Name


0



Learning Outcomes -


Course Content Groups and homomorphisms, Lagrange’s theorem. Direct and semi-direct products. Cyclic, dihedral and symmetric groups. Free groups, generators and relations, finitely generated. Abelian groups. Group actions. Sylow’s theorem and p-groups. Simple groups, composition series. Solvable groups. Rings and homomorphisms. Ideals. Polynomial rings. Factorization in commutative rings. Modules and exact sequences. Extensions of fields, splitting field of a polynomial, separable extensions, normal extensions. Fundamental theorem of Galois theory. Roots of unity and cyclotomic polynomials. Solvability by radicals. Symmetric functions and Abel’s theorem.


Bibliography 1) I. Herstein, Topics in Algebra, N.Y. Wiley 2) T. Hungerford, Algebra, Springer-Verlag 3) J. Rotman, An Introduction to the theory of groups, Fourth Edition, Springer-Verlag 4) P. Cameron, Introduction to Algebra, Oxford University Press

Assessment Written Examination

Language Greek

51

Course Title Comprehensive examination in Geometry

Course Code MAS782


Level Postgraduate


Teacher’s Name


0



Learning Outcomes -


Course Content Topological and differentiable manifolds, basic examples and properties. Fundamental group. Tangent spaces. Partitions of unity. Normal values. Vector fields, flows. Frobenius’ theorem. Differentiable forms. Stokes’ theorem. Riemannian manifolds. The Riemannian connection and exterior differential forms. Geodesic curves, exponential mapping, normal coordinates, Gauss’ Lemma. Hopf-Rinow theorem. Curvature. Gauss-Bonnet theorem. Hadamard-Cartan theorem.


Bibliography 1) Bothby, W. An introduction to differentiable manifolds and Riemannian Geometry, Academic Press 2) M. Do Carmo, Riemannian Geometry, Birkhauser 3) J. M. Lee, Riemannian Geometry, Springer


Language Greek

52

Course Title Comprehensive examination in Applied Mathematics

Course Code MAS783


Level Postgraduate


Teacher’s Name


0



Learning Outcomes -


Course Content Lie groups and algebras, Equations of Motion (Newton, Lagrange), Poisson

structures, Integrable systems, Lax pairs, Bi-Hamiltonian systems, Symmetries,

Noether’s theorem, variational calculus, integral equations.


Bibliography 1) P. Olver Applications of Lie Groups to Differential Equations, Second Edition, Springer-Verlag, New York, 1993.

2) F.B. Hildebrand, Methods of Applied Mathematics, Dover, 1992


Language Greek

53

Course Title Comprehensive examination in Partial Differential Equations

Course Code MAS784


Level Postgraduate


Teacher’s Name


0



Learning Outcomes -


Course Content First order partial differential equations, Second order partial differential

equations: Wave Equation, Heat Equations, Harmonic functions. Initial boundary

value problems, Fourier series, Green’s functions, Maximum Principle.


Bibliography 1) G. D. Akrivis, D. Dougalis, Partial Differential Equations (University Publications).

2) W. A. Strauss, Partial Differential Equations: An Introduction (Chapters 1–7).

3) L. Evans, Partial Differential Equations (Chapter 2 and Chapter 3: Sections 3.1, 3.2).


Language Greek

54

Course Title Comprehensive examination in Numerical Analysis

Course Code MAS785


Level Postgraduate


Teacher’s Name


0



Learning Outcomes -


Course Content Numerical solution of nonlinear equations. Vector and matrix norms. Solution of

linear systems (direct and iterative methods). Calculation of eigenvalues and

eigenvectors. Interpolation (Lagrange and Hermite). Numerical integration

(Newton – Cotes, Gauss).


Bibliography 1) E. Süli and D. Mayers, An Introduction to Numerical Analysis, Cambridge Univ Press, 2003.

2) K. Atkinson: An Introduction to Numerical Analysis, Wiley, New York, 1978.

3) G. D. Akrivis, D. Dougalis: Introduction to Numerical Analysis, University Publications, Crete, 1997.


Language Greek

55

Course Title Comprehensive examination in Numerical Solution of Ordinary Differential

Equations

Course Code MAS786


Level Postgraduate


Teacher’s Name


0



Learning Outcomes -


Course Content Single and multistep methods and Runge-Kutta methods for the numerical

solution of initial value problems for ordinary differential equations. Finite

Difference Methods for ordinary differential equations. Finite Element Methods

for ordinary differential equations.


Bibliography 1) L. Fox and D. F. Mayers: Numerical Solution of Ordinary Differential Equations, Chapman and Hall (London, 1987). 2) A. Iserles: A First Course in the Numerical Analysis of Differential Equations, Cambdrige Univ Press, 1996. 3) G. D. Akrivis, D. Dougalis: Numerical Methods for Ordinary Differential Equations, University Publications, Crete, 2006. 4) C. Johnson: Numerical solution of partial differential equations by the finite element method, Cambridge Univ Press, 1994. 5) G. D. Akrivis: Finite Element Methods, University Lectures, Cyprus, 2005.


Language Greek

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