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
H1051 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 12
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, semiprimitive, semisimple rings, their equivalent characterizations and logical hierarchy; the Jacobson radical of a ring.
Week 2. Completely reducible modules, Schur’s Lemma, bimodules, the JacobsonChevalley density theorem, nilpotency of the radical of an artinian ring, the WedderburnArtin 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, NielsenSchreier 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 twosided 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 LaskerNoether 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, BolzanoWeierstrass theorem.
Week 2. Normed linear spaces. Banach spaces. A characterization of finite dimensional normed spaces.
Week 3. ArzelaAscoli 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. HahnBanach and BanachSteinhaus 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., 2^{nd} ed. 1991.
2. W. Rudin: Real and complex analysis, 3^{rd} 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. Trianglefree 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 4cycles
Week 8. BondySimonovits theorem on graphs with no 2kcycle, regularity lemma and its applications
Week 9. Extremal set family problems (basic problems, Sperner theorem, ErdosKoRado theorem)
Week 10. More advanced probabilistic methods, Lovasz Local Lemma
Week 11. The dimension bound (Fisher’s inequality, 2distance 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 MayerVietoris 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 LeviCivita 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: MayerVietoris long exact sequence, axioms for singular homology
Week 6: Applications of homology. Definition of cohomology
Week 7: CWhomology
Week 8: Manifolds, bundles, vector bundles, examples
Week 9: Connections on bundles, parallel transport, holonomy
Week 10: Curvature
Week 11: Riemannian manifolds, LeviCivita connection, Riemanncurvature tensor
Week 12: Basic theorems in Riemannian geometry
References:

E. Spanier: Algebraic Topology, 1981.

S. MacLane: Homology, 1995.

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

APPLIED FUNCTIONAL ANALYSIS

EVOLUTION EQUATIONS AND APPLICATIONS

FUNCTIONAL METHODS IN DIFFERENTIAL EQUATIONS

OPTIMAL CONTROL

PARTIAL DIFFERENTIAL EQUATIONS

APPROXIMATION THEORY

NONLINEAR FUNCTIONAL ANALYSIS

SPECIAL FUNCTIONS AND RIEMANN SURFACES

COMPLEX MANIFOLDS

INTRODUCTION TO CCR ALGEBRAS

ENUMERATION

EXTREMAL COMBINATORICS

RANDOM METHODS IN COMBINATORICS

INTRODUCTION TO THE THEORY OF COMPUTING

COMPLEXITY THEORY

BLOCK DESIGNS

HYPERGRAPHS, SET SYSTEMS, INTERSECTION THEOREMS

LARGE SPARSE GRAPHS, GRAPH CONVERGENCE AND GROUPS

SELECTED TOPICS IN GRAPH THEORY

COMPUTATIONAL GEOMETRY

COMBINATORIAL OPTIMIZATION

THEORY OF ALGORITHMS

QUANTUM COMPUTING

RANDOM COMPUTATION

HOMOLOGICAL ALGEBRA

HIGHER LINEAR ALGEBRA

REPRESENTATION THEORY I.

REPRESENTATION THEORY II.

UNIVERSAL ALGEBRA AND CATEGORY THEORY

TOPICS IN GROUP THEORY

TOPICS IN RING THEORY. I

TOPICS IN RING THEORY. II

PERMUTATION GROUPS

LIE GROUPS AND LIE ALGEBRAS

INTRODUCTION TO COMMUTATIVE ALGEBRA

TOPICS IN COMMUTATIVE ALGEBRA

LINEAR ALGEBRAIC GROUPS

ALGEBRAIC NUMBER THEORY

TOPICS IN ALGEBRAIC NUMBER THEORY

GEOMETRIC GROUP THEORY

RESIDUALLY FINITE GROUPS

INVARIANT THEORY

SEMIGROUP THEORY

PROP GROUPS AND PADIC ANALYTIC GROUPS

CENTRAL SIMPLE ALGEBRAS AND GALOIS COHOMOLOGY

BASIC ALGEBRAIC GEOMETRY

THE LANGUAGE OF SCHEMES

GALOIS GROUPS AND FUNDAMENTAL GROUPS

TOPICS IN ALGEBRAIC GEOMETRY

THE ARITHMETIC OF ELLIPTIC CURVES

HODGE THEORY

TORIC VARIETIES

SMOOTH MANIFOLDS AND DIFFERENTIAL TOPOLOGY

CHARACTERISTIC CLASSES

SINGULARITIES OF DIFFERENTABLE MAPS: LOCAL AND GLOBAL THEORY

FOUR MANIFOLDS AND KIRBY CALCULUS

SYMPLECTIC MANIFOLDS, LEFSCHETZ FIBRATION

COMBINATORIAL NUMBER THEORY

COMBINATORIAL NUMBER THEORY II

CLASSICAL ANALYTIC NUMBER THEORY

PROBABILISTIC NUMBER THEORY

MODERN PRIME NUMBER THEORY I

MODERN PRIME NUMBER THEORY II

EXPONENTIAL SUMS IN COMBINATORIAL NUMBER THEORY

MODULAR FORMS AND LFUNCTIONS I

MODULAR FORMS AND LFUNCTIONS II

STOCHASTIC PROCESSES AND APPLICATIONS

PROBABILITY 1

PROBABILITY 2

STOCHASTIC MODELS

PROBABILITY AND GEOMETRY ON GRAPHS AND GROUPS

MATHEMATICAL STATISTICS

MULTIVARIATE STATISTICS

ERGODIC THEORY

MATHEMATICAL METHODS IN STATISTICAL PHYSICS

FRACTALS AND DYNAMICAL SYSTEMS

DYNAMICAL SYSTEMS

INVARIANCE PRINCIPLES IN PROBABILITY AND STATISTICS

STOCHASTIC ANALYSIS

PATH PROPERTIES OF STOCHASTIC PROCESSES

NONPARAMETRIC STATISTICS

TOPICS IN FINANCIAL MATHEMATICS

NUMERICAL METHODS IN STATISTICS

ERGODIC THEORY AND COMBINATORICS

INFORMATION THEORY

INFORMATION THEORETIC METHODS IN MATHEMATICS

INFORMATION THEORETICAL METHODS IN STATISTICS

DATA COMPRESSION

CRYPTOLOGY

INFORMATION DIVERGENCES IN STATISTICS

NONPARAMETRIC STATISTICS

INTRODUCTION TO MATHEMATICAL LOGIC

ALGEBRAIC LOGIC AND MODEL THEORY

ALGEBRAIC LOGIC AND MODEL THEORY 2

LOGICAL SYSTEMS (AND UNIVERSAL LOGIC)

LOGIC AND RELATIVITY 1

LOGIC AND RELATIVITY 2

FRONTIERS OF ALGEBRAIC LOGIC 1

FRONTIERS OF ALGEBRAIC LOGIC 2

LOGIC OF PROGRAMS

CONVEX GEOMETRY

FINITE PACKING AND COVERING BY CONVEX BODIES

PACKING AND COVERING

CONVEX POLYTOPES

COMBINATORIAL GEOMETRY

GEOMETRY OF NUMBERS

STOCHASTIC GEOMETRY

BRUNNMINKOWSKI THEORY

NONEUCLIDEAN GEOMETRIES

DIFFERENTIAL GEOMETRY

HYPERBOLIC MANIFOLDS

MODERN SET THEORY

INTRODUCTION TO FORCING

DESCRIPTIVE SET THEORY

ADVANCED SET THEORY

SETTHEORETIC TOPOLOGY

INTRODUCTION TO ASYMPTOTIC EXPANSIONS

ALGEBRAIC LOGIC AND MODEL THEORY 3

HIGHER ORDER FOURIER ANALYSIS

SEIBERGWITTEN INVARIANTS

HEEGARDFLOER HOMOLOGIES
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