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
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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.
Language Greek or English
4
Course Title Functional Analysis
Course Code MAS604
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
Course Content Compact operators. Spectral theory. Self adjoint operators. Closed and orthonormal operators. Spectral theorem. Semigroups.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Functional Analysis, K. Yosida, Springer Verlag 6th edition 1980
Assessment Final exam
Language Greek or English
5
Course Title Second Order Elliptic Partial Differential Equations
Course Code MAS605
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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.
Language Greek or English
6
Course Title Function Theory of One Complex Variable
Course Code MAS606
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 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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
7
Course Title Function Theory of Several Complex Variables
Course Code MAS607
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Teaching Methodology
Lectures (4 hours per week)
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.
Language Greek or English
8
Course Title Ordinary Differential Equations
Course Code MAS613
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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.)
Language Greek or English
9
Course Title Topics in Mathematical Analysis I
Course Code MAS617
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
Course Content Topics in Real Analysis, Complex Analysis or Differential Equations, depending on the special interests of the faculty member teaching the course.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
10
Course Title Topics in Mathematical Analysis II
Course Code MAS618
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
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.
Prerequisites - Required -
Course Content Topics in Real Analysis, Complex Analysis or Differential Equations, depending on the special interests of the faculty member teaching the course.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
11
Course Title Topics in Mathematical Analysis III
Course Code MAS619
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
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.
Prerequisites - Required -
Course Content Topics in Real Analysis, Complex Analysis or Differential Equations, depending on the special interests of the faculty member teaching the course.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
12
Course Title Approximation Theory Course Code MAS620 Course Type Elective Level Graduate Year / Semester 1st or 2nd semester Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
Students will learn some basic methods to approximate sufficiently smooth functions with polynomials and wavelet-type orthonormal basis
Learning Outcomes Upon successful completion of this course, students are expected to be able to:
• 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
Prerequisites - Required -
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
Teaching Methodology
Lectures (4 hours per week)
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
13
Course Title Numerical Linear Algebra
Course Code MAS621
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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)
Language Greek or English
14
Course Title Coding Theory
Course Code MAS622
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
Course Content Finite fields. Linear codes, syndrome decoding. Cyclic codes. BCH codes and Reed – Solomon codes. MDS codes. Permutation decoding.
Teaching Methodology
Lectures (4 hours per week)
Bibliography R. Hill, A First Course In Coding Theory
J.H. van Lint, Introduction to Coding Theory
Assessment Midterm examination and final examination
Language Greek or English
15
Course Title Number Theory
Course Code MAS623
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Jarvis Algebraic Number Theory Ireland and Rosen, A Classical Introduction to Modern Number Theory. Frohlich and Taylor, Algebraic Number Theory
Assessment Midterm examination and final examination
Language Greek or English
16
Course Title Introduction to Commutative Algebra
Course Code MAS624
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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.
Language Greek or English
17
Course Title Group Theory
Course Code MAS625
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 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
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
Assessment Midterm examination and final examination
Language Greek or English
18
Course Title Field and Galois Theory
Course Code MAS626
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
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)
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
Bibliography 1. J.J. Rotman, Galois Theory. 2. P. Morandi, Fields and Galois Theory. 3. S. Roman, Field Theory.
Assessment Midterm examination and final examination
Language Greek or English
19
Course Title Group Representation Theory I
Course Code MAS627
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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.
Assessment Midterm examination and final examination
Language Greek or English
20
Course Title Group Representation Theory II
Course Code MAS628
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
Assessment Midterm examination and final examination
Language Greek or English
21
Course Title Topics in Algebra
Course Code MAS629
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
Course Content Topics from algebra
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Midterm examination and final examination
Language Greek or English
22
Course Title Algebraic Geometry
Course Code MAS630
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
Bibliography • Algebraic Geometry, Chapter 1, R. Hartshorne. • Elementary Algebraic Geometry, K. Hulek. • Basic Algebraic Geometry I, I. Shafarevich.
Assessment Homeworks and final exam.
Language Greek or English
23
Course Title Differential Topology
Course Code MAS631
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
24
Course Title Riemannian Geometry
Course Code MAS632
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
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
theoretical results.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
25
Course Title General Relativity
Course Code MAS633
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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
Prerequisites - Required -
Course Content Lorentz geometry. Special relativity. Newton spacetime, Minkowski spacetime. Lorentz transformation. Einstein equations. Special solutions (Schwarzschild).
Teaching Methodology
Lectures (4 hours per week)
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.
Language Greek or English
26
Course Title Algebraic Topology I
Course Code MAS634
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
Course Content Fundamental group, Van Kampen theorem, Covering spaces, Homology and Cohomology, The Mayer Vietoris sequence, Excision, Relation between homology and fundamental group.
Teaching Methodology
Lectures (4 hours per week)
Bibliography • Algebraic Topology, A. Hatcher. • A Basic Course in Algebraic Topology, W. Massey.
Assessment Homeworks and final exam.
Language Greek or English
27
Course Title Lie Groups and Lie Algebras
Course Code MAS635
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
28
Course Title Algebraic Topology II
Course Code MAS636
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
Course Content Obstruction theory. Bundles and K- theory. Bordism. Spectral sequences. Characteristic classes.
Teaching Methodology
Lectures (4 hours per week)
Bibliography J. Milnor and J. D. Stasheff, Characteristic Classes, Princeton University Press 1974.
Assessment Presentation and Final Exam
Language Greek or English
29
Course Title Spectral Geometry
Course Code MAS637
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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
theoretical results.
Prerequisites - Required -
Course Content Laplace operator. Minimax principle. Isoparametric inequalities. Heat kernel.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Spectral Geometry of the Laplacian, H. Urakawa, World Scientific 2017
Assessment Presentation and Final Exam
Language Greek or English
30
Course Title Spin Geometry
Course Code MAS638
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives An introduction to Spin geometry and its applications.
Learning Outcomes Upon successful completion of this course, students are expected to be able to:
• 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.
Prerequisites - Required -
Course Content Clifford algebras. Spin groups and representations. Spin structures. Spin connection. Spin manifolds. Dirac operator. Bochner formula. Lichnerowicz's Theorem.
Teaching Methodology
Lectures (4 hours per week)
Bibliography H. Blaine Lawson & Marie-Louise Michelsohn, Spin Geometry, Princeton Mathematical Series, 1990.
Assessment Presentation and Final Exam
Language Greek or English
31
Course Title Topics in Geometry I
Course Code MAS640
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
Course Content Topics from Differential Geometry, Algebraic Geometry and Algebraic Topology.
Depends on the special interests of the staff member teaching it
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Homework (problem sets), presentation, final exam
Language Greek or English
32
Course Title Topics in Geometry IΙ
Course Code MAS641
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
Course Content Topics from Differential Geometry, Algebraic Geometry and Algebraic Topology.
Depends on the special interests of the staff member teaching it
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Homework (problem sets), presentation, final exam
Language Greek or English
33
Course Title Probability Theory
Course Code MAS660
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week
2 Laboratories / week 0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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).
Language Greek or English
34
Course Title Numerical Solution of Ordinary Differential Equations
Course Code MAS671
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
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
Bibliography A First Course in the Numerical Analysis of Differential Equations, A. Iserles
Assessment Final exam, one midterm exam, one computational project.
Language Greek or English
35
Course Title Numerical Solution of Partial Differential Equations
Course Code MAS672
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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.
Language Greek or English
36
Course Title Finite Element Methods
Course Code MAS673
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
37
Course Title Topics in Numerical Analysis I
Course Code MAS677
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives Study of special topics not included in other courses.
Learning Outcomes Vary with the content
Prerequisites - Required -
Course Content Topics in Computational Mathematics and Approximation Theory.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Homeworks and final exam
Language Greek or English
38
Course Title Topics in Numerical Analysis IΙ
Course Code MAS678
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives Study of special topics not included in other courses.
Learning Outcomes Vary with the content
Prerequisites - Required -
Course Content Topics in Computational Mathematics and Approximation Theory.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Homeworks and final exam
Language Greek or English
39
Course Title Topics in Numerical Analysis IΙΙ
Course Code MAS679
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives Study of special topics not included in other courses.
Learning Outcomes Vary with the content
Prerequisites - Required -
Course Content Topics in Computational Mathematics and Approximation Theory.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Homeworks and final exam
Language Greek or English
40
Course Title Classical Mechanics
Course Code MAS682
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
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.
Prerequisites - Required -
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.
Teaching Methodology
Lectures (4 hours per week)
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.
Language Greek or English
41
Course Title Fluid Dynamics
Course Code MAS683
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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
Prerequisites - Required -
Course Content Equations of motion. Steady or transient viscous flows. Stokes flows. Non- Newtonian and viscoelastic flows.
Teaching Methodology
Lectures (4 hours per week)
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
Language Greek or English
42
Course Title Scientific Computing with MATLAB
Course Code MAS684
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
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.
Prerequisites - Required -
Course Content Introduction to MATLAB. Data and function approximation. Linear Systems. Eigenvalues and Eigenvectors. Ordinary Differential Equations. Numerical Methods for boundary value problems.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Experiments with MATLAB by Cleve Moler, The MathWorks Inc.
Assessment Weekly assignments, final project
Language Greek or English
43
Course Title Topics in Applied Mathematics I
Course Code MAS687
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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
Prerequisites - Required -
Course Content Topics from Applied Mathematics.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Projects and final exam
Language Greek or English
44
Course Title Topics in Applied Mathematics II
Course Code MAS688
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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
Prerequisites - Required -
Course Content Topics from Applied Mathematics.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Projects and final exam
Language Greek or English
45
Course Title Topics in Applied Mathematics III
Course Code MAS689
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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
Prerequisites - Required -
Course Content Topics from Applied Mathematics.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Depends on the special interests of the staff member teaching it
Assessment Projects and final exam
Language Greek or English
46
Course Title Topics in Differential Equations I
Course Code MAS697
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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
defined. • Use the necessary mathematical tools needed for establishing certain
theoretical results. • Analyze, synthesize, organize and plan projects in the field.
Prerequisites - Required -
Course Content Topics in differential equations.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Selection of topics from bibliography and research papers in general areas of Partial Differential Equations or Ordinary Differential Equations, Calculus of Variations.
Assessment Presentation and Final Exam
Language Greek or English
47
Course Title Topics in Differential Equations II
Course Code MAS698
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of Differential Equations broadly
defined. • Use the necessary mathematical tools needed for establishing certain
theoretical results. • Analyze, synthesize, organize and plan projects in the field.
Prerequisites - Required -
Course Content Topics in differential equations.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Selection of topics from bibliography and research papers in general areas of Partial Differential Equations or Ordinary Differential Equations and Calculus of Variations.
Assessment Presentation and Final Exam
Language Greek or English
48
Course Title Topics in Differential Equations III
Course Code MAS699
Course Type Elective
Level Postgraduate
Year / Semester 1st or 2nd semester
Teacher’s Name
ECTS 10 Lectures / week 2 Laboratories / week
0
Course Purpose and Objectives
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.
Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of Differential Equations broadly
defined. • Use the necessary mathematical tools needed for establishing certain
theoretical results. • Analyze, synthesize, organize and plan projects in the field.
Prerequisites - Required -
Course Content Topics in differential equations.
Teaching Methodology
Lectures (4 hours per week)
Bibliography Selection of topics from bibliography and research papers in general areas of Partial Differential Equations or Ordinary Differential Equations and Calculus of Variations.
Assessment Presentation and Final Exam
Language Greek or English
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
Course Purpose and Objectives
Successful completion of the CE is required in order to continue with the dissertation.
Learning Outcomes -
Prerequisites - Required -
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.
Teaching Methodology
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
ECTS 0 Lectures/week 0 Laboratories / week
0
Course Purpose and Objectives
Successful completion of the CE is required in order to continue with the dissertation.
Learning Outcomes -
Prerequisites - Required -
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.
Teaching Methodology
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
Course Type Comprehensive Examination (CE)
Level Postgraduate
Year / Semester 1st semester
Teacher’s Name
ECTS 0 Lectures/week 0 Laboratories / week
0
Course Purpose and Objectives
Successful completion of the CE is required in order to continue with the dissertation.
Learning Outcomes -
Prerequisites - Required -
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.
Teaching Methodology
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
Assessment Written Examination
Language Greek
52
Course Title Comprehensive examination in Applied Mathematics
Course Code MAS783
Course Type Comprehensive Examination (CE)
Level Postgraduate
Year / Semester 1st semester
Teacher’s Name
ECTS 0 Lectures/week 0 Laboratories / week
0
Course Purpose and Objectives
Successful completion of the CE is required in order to continue with the dissertation.
Learning Outcomes -
Prerequisites - Required -
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.
Teaching Methodology
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
Assessment Written Examination
Language Greek
53
Course Title Comprehensive examination in Partial Differential Equations
Course Code MAS784
Course Type Comprehensive Examination (CE)
Level Postgraduate
Year / Semester 1st semester
Teacher’s Name
ECTS 0 Lectures/week 0 Laboratories / week
0
Course Purpose and Objectives
Successful completion of the CE is required in order to continue with the dissertation.
Learning Outcomes -
Prerequisites - Required -
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.
Teaching Methodology
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).
Assessment Written Examination
Language Greek
54
Course Title Comprehensive examination in Numerical Analysis
Course Code MAS785
Course Type Comprehensive Examination (CE)
Level Postgraduate
Year / Semester 1st semester
Teacher’s Name
ECTS 0 Lectures/week 0 Laboratories / week
0
Course Purpose and Objectives
Successful completion of the CE is required in order to continue with the dissertation.
Learning Outcomes -
Prerequisites - Required -
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).
Teaching Methodology
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.
Assessment Written Examination
Language Greek
55
Course Title Comprehensive examination in Numerical Solution of Ordinary Differential
Equations
Course Code MAS786
Course Type Comprehensive Examination (CE)
Level Postgraduate
Year / Semester 1st semester
Teacher’s Name
ECTS 0 Lectures/week 0 Laboratories / week
0
Course Purpose and Objectives
Successful completion of the CE is required in order to continue with the dissertation.
Learning Outcomes -
Prerequisites - Required -
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.
Teaching Methodology
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.
Assessment Written Examination
Language Greek
56