 # Phd program

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• SYLLABI CENTRAL EUROPEAN UNIVERSITY

Department of Mathematics and its Applications

PhD Courses

Program established in 2001

Program Accreditation

Program approved and registered by the New York State Education Department

Zrinyi u. 14, Third Floor

H-1051 Budapest

Hungary

Email: Mathematics@ceu.hu

Internet: http://mathematics.ceu.hu

MANDATORY COURSES

M1. Topics in Algebra

M2. Topics in Analysis

M3. Topics in Combinatorics

M4. Topics in Topology and Geometry

Forms of assessment for mandatory courses: weekly homework, midterm, final

### SYLLABI

#### Mandatory Courses

M1. TOPICS in ALGEBRA

Course Coordinator: Matyas Domokos

No. of Credits: 3, and no. of ECTS credits: 6

Time Period of the course: Fall Semester

Prerequisites: Basic Algebra 1-2

Course Level: introductory PhD

Brief introduction to the course:

Advanced topics in Abstract Algebra are discussed.

The goals of the course:

The main goal of the course is to introduce students to the most important advanced concepts and topics in abstract algebra.

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents:

Noncommutative Algebra:

Week 1. The concepts of simple, primitive, prime, semisimple, semi-primitive, semi-simple rings, their equivalent characterizations and logical hierarchy; the Jacobson radical of a ring.

Week 2. Completely reducible modules, Schur’s Lemma, bimodules, the Jacobson-Chevalley density theorem, nilpotency of the radical of an artinian ring, the Wedderburn-Artin theorems, module theoretic characterization of semisimple artinian rings.
Week 3. Classical groups, the notion of topological and Lie groups, Lie algebras, enveloping algebras, solvable and semisimple Lie algebras.
Week 4. Generators and relations for groups, associative and Lie algebras, Nielsen-Schreier theorem.
Group Actions and Representation Theory:

Week 5. Basic concepts of group representations, the space of matrix elements associated to a finite dimensional representation, dual representation, permutation representations, the two-sided regular representation, group algebras, Maschke’s theorem.

Week 6. Tensor products of vector spaces (and more generally of bimodules), product of representations, the irreducible representations of a direct product, induced representations.
Week 7. Unitary representations, orthogonality of unitary matrix elements of irreducible complex representations of a finite group or a compact group, characters, examples of character tables, the dimension of an irreducible representation divides the order of the group, Burnside’s theorem on solvability of groups whose order has only two prime divisors or the theorem on Frobenius kernel.
Week 8. Group actions in various areas of mathematics (e.g. Cayley graphs, actions on manifolds,automorphism groups).
Commutative and Homological Algebra:

Week 9. Integral extensions, the Noether Normalization Lemma, the existence of a common zero of a proper ideal in a multivariate polynomial ring over an algebraically closed field, the Hilbert Nullstellensatz, differential criterion of separability.

Week 10. Localization, associated primes, primary ideals, the Lasker-Noether theorem for finitely generated modules over a noetherian ring.
Week 11. Affine algebraic sets and their coordinate rings, rational functions, local rings, the Zariski topology, the prime spectrum.
Week 12. Free and projective resolutions, the Hilbert syzygy theorem.

References:

1. N Jacobson, Basic Algebra II, WH Freeman and Co., San Francisco, 1974/1980.

2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994

M2. TOPICS in ANALYSIS

Course Coordinator: András Stipsicz

No. of Credits: 3, and no. of ECTS credits: 6

Time Period of the course: Fall Semester

Prerequisites: calculus

Course Level:introductory PhD

Brief introduction to the course:

Basic concepts and fundamental theorems in functional analysis and measure theory are presented.

The goals of the course:

The main goal of the course is to introduce students tobasic concepts of analysis, with a special attention to functional analysis and measure theory.

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents:

Week 1. Metric spaces, topological properties, Bolzano-Weierstrass theorem.

Week 2. Normed linear spaces. Banach spaces. A characterization of finite dimensional normed spaces.

Week 3. Arzela-Ascoli theorem. Peano theorem. Banach fixed point theorem. Applications to differential and integral equations.

Week 4. Linear operators. The dual space. Weak topologies. Hilbert spaces.

Week 5. Hahn-Banach and Banach-Steinhaus Theorems, open mapping and closed graph theorems

Week 6. The Riesz representation theorem.

Week 7. Orthonormal systems in Hilbert spaces. Fourier series.

Week 8. Distributions, Sobolev spaces.

Week 9: Fourier transforms, applications to differential equations
Week 10: The spectral theorem
Week 11: Measures, the Lebesgue measure,  Measurable functions, integration
Week 12: Abstract measure spaces, Fatou's lemma, dominated convergence theorem

References:

Handouts+

1. W.Rudin: Functional Analysis, 1973., 2nd ed. 1991.

2. W. Rudin: Real and complex analysis, 3rd ed 1987.

3. T. Tao: An introduction to measure theory, AMS, 2011

M3. TOPICS IN COMBINATORICS

Course Coordinator: Ervin Gyori

No. of Credits: 3 and no. of ECTS credits: 6

Time Period of the course: Winter Semester

Prerequisites: linear algebra

Course Level: introductory PhD

Brief introduction to the course:

More advanced concepts, methods and results of combinatorics and graph theory. Main topics: (linear) algebraic, probabilistic methods in discrete mathematics; relation of graphs and hypergraphs; special constructions of graphs and hypergraphs; extremal set families; Ramsey type problems in different structures; Regularity Lemma.

The goals of the course:

The main goal is to study advanced methods of discrete mathematics, and advanced methods applied to discrete mathematics. Problem solving is more important than in other courses!

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents:

Week 1. Colorings of graphs, Brooks’ theorem

Week 2. Triangle-free graphs with high chromatic number  (constructions of Zykov, Myczielski, shift graph)

Week 3. Famous graphs with high chromatic number (Kneser, Tutte), Erdos’ probabilistic proof for existenceof graphs with large girth and large chromatic number

Week 4. Perfect graphs, important examples, weak perfect graph theorem (linear algebraic proof), strong perfect graph theorem without proof

Week 5. Probabilistic and constructive lower bounds on Ramsey numbers

Week 6. Van der Waerden theorem, Hales Jewett theorem, threshold numbers

Week 7. Extremal graphs, Turan’s theorem, graphs with no 4-cycles

Week 8. Bondy-Simonovits theorem on graphs with no 2k-cycle, regularity lemma and its applications

Week 9. Extremal set family problems (basic problems, Sperner theorem, Erdos-Ko-Rado theorem)

Week 10. More advanced probabilistic methods, Lovasz Local Lemma

Week 11. The dimension bound (Fisher’s inequality, 2-distance sets, etc.)

Week 12. Eigenvalues, minimal size regular graphs of girth 5

References:

Reinhard Diestel, Graph Theory, Springer, 1997 or later editions +

Handouts

M4. TOPICS IN TOPOLOGY AND GEOMETRY

Course Coordinator:András Stipsicz

No. of Credits: 3, and no. of ECTS credits: 6

Time Period of the course: Winter Semester

Prerequisites: real analysis, linear algebra

Course Level: introductory PhD

Brief introduction to the course:

We introduce basic concepts of algebraic topology, such as the fundamental group (together with the Van Kampen theorem)and singular homology (together with the Mayer-Vietoris long exact sequence). We also review basic notions of homological algebra. Fiber bundles and connections on them are discussed, and we define the concept of curvature. As a starting point of Riemannian geometry, we define the Levi-Civita connection and the Riemannian curvature tensor.

The goals of the course:

The main goal of the course is to provide a quick introduction to main techniques and results of topology and geometry. In particular, singular homology and the concepts of connections and curvature are discussed in detail.

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents:

Week 1: Definition and basic properties of fundamental groups, Van Kampen theorem
Week 2: Applications, CW complexes, covering spaces, universal cover
Week 3: Simplicial complexes, simplicial homology
Week 4: Singular homology, basic homological algebra
Week 5: Mayer-Vietoris long exact sequence, axioms for singular homology
Week 6: Applications of homology. Definition of cohomology
Week 7: CW-homology
Week 8: Manifolds, bundles, vector bundles, examples
Week 9: Connections on bundles, parallel transport, holonomy
Week 10: Curvature
Week 11: Riemannian manifolds, Levi-Civita connection, Riemanncurvature tensor
Week 12: Basic theorems in Riemannian geometry

References:

1. E. Spanier: Algebraic Topology, 1981.

1. S. MacLane: Homology, 1995.

1. W. Boothby: An introduction to differentiable manifolds and Riemanniangeometry, 1986.

ELECTIVE COURSES

Suggested form of assessment for

• elective live courses: regular homework, and presentation or final

• elective reading courses: regular homework

LIST OF ELECTIVE PhD COURSES

1. APPLIED FUNCTIONAL ANALYSIS

2. EVOLUTION EQUATIONS AND APPLICATIONS

3. FUNCTIONAL METHODS IN DIFFERENTIAL EQUATIONS

4. OPTIMAL CONTROL

5. PARTIAL DIFFERENTIAL EQUATIONS

6. APPROXIMATION THEORY

7. NONLINEAR FUNCTIONAL ANALYSIS

8. SPECIAL FUNCTIONS AND RIEMANN SURFACES

9. COMPLEX MANIFOLDS

10. INTRODUCTION TO CCR ALGEBRAS

11. ENUMERATION

12. EXTREMAL COMBINATORICS

13. RANDOM METHODS IN COMBINATORICS

14. INTRODUCTION TO THE THEORY OF COMPUTING

15. COMPLEXITY THEORY

16. BLOCK DESIGNS

17. HYPERGRAPHS, SET SYSTEMS, INTERSECTION THEOREMS

18. LARGE SPARSE GRAPHS, GRAPH CONVERGENCE AND GROUPS

19. SELECTED TOPICS IN GRAPH THEORY

20. COMPUTATIONAL GEOMETRY

21. COMBINATORIAL OPTIMIZATION

22. THEORY OF ALGORITHMS

23. QUANTUM COMPUTING

24. RANDOM COMPUTATION

25. HOMOLOGICAL ALGEBRA

26. HIGHER LINEAR ALGEBRA

27. REPRESENTATION THEORY I.

28. REPRESENTATION THEORY II.

29. UNIVERSAL ALGEBRA AND CATEGORY THEORY

30. TOPICS IN GROUP THEORY

31. TOPICS IN RING THEORY. I

32. TOPICS IN RING THEORY. II

33. PERMUTATION GROUPS

34. LIE GROUPS AND LIE ALGEBRAS

35. INTRODUCTION TO COMMUTATIVE ALGEBRA

36. TOPICS IN COMMUTATIVE ALGEBRA

37. LINEAR ALGEBRAIC GROUPS

38. ALGEBRAIC NUMBER THEORY

39. TOPICS IN ALGEBRAIC NUMBER THEORY

40. GEOMETRIC GROUP THEORY

41. RESIDUALLY FINITE GROUPS

42. INVARIANT THEORY

43. SEMIGROUP THEORY

44. PRO-P GROUPS AND P-ADIC ANALYTIC GROUPS

45. CENTRAL SIMPLE ALGEBRAS AND GALOIS COHOMOLOGY

46. BASIC ALGEBRAIC GEOMETRY

47. THE LANGUAGE OF SCHEMES

48. GALOIS GROUPS AND FUNDAMENTAL GROUPS

49. TOPICS IN ALGEBRAIC GEOMETRY

50. THE ARITHMETIC OF ELLIPTIC CURVES

51. HODGE THEORY

52. TORIC VARIETIES

53. SMOOTH MANIFOLDS AND DIFFERENTIAL TOPOLOGY

54. CHARACTERISTIC CLASSES

55. SINGULARITIES OF DIFFERENTABLE MAPS: LOCAL AND GLOBAL THEORY

56. FOUR MANIFOLDS AND KIRBY CALCULUS

57. SYMPLECTIC MANIFOLDS, LEFSCHETZ FIBRATION

58. COMBINATORIAL NUMBER THEORY

59. COMBINATORIAL NUMBER THEORY II

60. CLASSICAL ANALYTIC NUMBER THEORY

61. PROBABILISTIC NUMBER THEORY

62. MODERN PRIME NUMBER THEORY I

63. MODERN PRIME NUMBER THEORY II

64. EXPONENTIAL SUMS IN COMBINATORIAL NUMBER THEORY

65. MODULAR FORMS AND L-FUNCTIONS I

66. MODULAR FORMS AND L-FUNCTIONS II

67. STOCHASTIC PROCESSES AND APPLICATIONS

68. PROBABILITY 1

69. PROBABILITY 2

70. STOCHASTIC MODELS

71. PROBABILITY AND GEOMETRY ON GRAPHS AND GROUPS

72. MATHEMATICAL STATISTICS

73. MULTIVARIATE STATISTICS

74. ERGODIC THEORY

75. MATHEMATICAL METHODS IN STATISTICAL PHYSICS

76. FRACTALS AND DYNAMICAL SYSTEMS

77. DYNAMICAL SYSTEMS

78. INVARIANCE PRINCIPLES IN PROBABILITY AND STATISTICS

79. STOCHASTIC ANALYSIS

80. PATH PROPERTIES OF STOCHASTIC PROCESSES

81. NONPARAMETRIC STATISTICS

82. TOPICS IN FINANCIAL MATHEMATICS

83. NUMERICAL METHODS IN STATISTICS

84. ERGODIC THEORY AND COMBINATORICS

85. INFORMATION THEORY

86. INFORMATION THEORETIC METHODS IN MATHEMATICS

87. INFORMATION THEORETICAL METHODS IN STATISTICS

88. DATA COMPRESSION

89. CRYPTOLOGY

90. INFORMATION DIVERGENCES IN STATISTICS

91. NONPARAMETRIC STATISTICS

92. INTRODUCTION TO MATHEMATICAL LOGIC

93. ALGEBRAIC LOGIC AND MODEL THEORY

94. ALGEBRAIC LOGIC AND MODEL THEORY 2

95. LOGICAL SYSTEMS (AND UNIVERSAL LOGIC)

96. LOGIC AND RELATIVITY 1

97. LOGIC AND RELATIVITY 2

98. FRONTIERS OF ALGEBRAIC LOGIC 1

99. FRONTIERS OF ALGEBRAIC LOGIC 2

100. LOGIC OF PROGRAMS

101. CONVEX GEOMETRY

102. FINITE PACKING AND COVERING BY CONVEX BODIES

103. PACKING AND COVERING

104. CONVEX POLYTOPES

105. COMBINATORIAL GEOMETRY

106. GEOMETRY OF NUMBERS

107. STOCHASTIC GEOMETRY

108. BRUNN-MINKOWSKI THEORY

109. NON-EUCLIDEAN GEOMETRIES

110. DIFFERENTIAL GEOMETRY

111. HYPERBOLIC MANIFOLDS

112. MODERN SET THEORY

113. INTRODUCTION TO FORCING

114. DESCRIPTIVE SET THEORY

116. SET-THEORETIC TOPOLOGY

117. INTRODUCTION TO ASYMPTOTIC EXPANSIONS

118. ALGEBRAIC LOGIC AND MODEL THEORY 3

119. HIGHER ORDER FOURIER ANALYSIS

120. SEIBERG-WITTEN INVARIANTS

121. HEEGARD-FLOER HOMOLOGIES

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