Courses
Courses
We offer courses in several fields, such as algebra, approximation theory, combinatorics, computer science, evolution equations, financial mathematics, functional analysis, geometry, information theory, mathematical logic and foundations, number theory, numerical analysis, ordinary and partial differential equations, probability theory and stochastic processes, set theory, statistics, etc. (see below). We assume flexibility in choosing elective courses. Our offer exceeds the demand and not all the courses are taken by a given student. On the other hand, further elective courses may be added to the list, depending on the specific interests of the students. On occasion we offer special courses on contemporary applications of mathematics.
Mandatory Courses
M1. Topics in Algebra
M2. Topics is Combinatorics
M3. Applied Functional Analysis
Elective Courses
01. Enumeration
02. Extremal Combinatorics
03. Random Methods in Combinatorics
04. Convex Geometry
05. Non-Euclidean Geometries
06. Differential Geometry
07. Homological Algebra
08. Smooth Manifolds and Differential Topology
09. Algebraic Topology
10. Function Spaces and Distributions
11. Nonlinear Functional Analysis
12. Introduction to Mathematical Logic
13. Modern Set Theory
14. Algebraic Logic and Model Theory
15. Elementary Prime Number Theory
16. Combinatorial Number Theory
17. Probabilistic Methods in Number Theory
18. Probability
19. Mathematical Statistics
20. Information Theory
21. Introduction to the Theory of Computing
22. Algorithms
23. Complexity Theory
24. Ergodic Theory
25. Mathematical Methods of Statistical Physics
26. Fractals and Dynamical Systems
27. Higher Linear Algebra
28. Representation Theory I
29. Representation Theory II
30. Universal Algebra and Category Theory
31. Topics in Group Theory
32. Topics in Ring Theory I
33. Topics in Ring Theory II
34. Permutation Groups
35. Lie Groups and Lie Algebras
36. Commutative Algebra
37. Algebraic Number Theory
38. Geometric Group Theory
39. Residually Finite Groups
40. Invariant Theory
41. Semigroup Theory
42. Basic Algebraic Geometry
43. The Language of Schemes
44. Galois Groups in Geometry
45. Algebraic Curves and Jacobian Varieties
46. The Arithmetic of Elliptic Curves
47. Hodge Theory
48. Introduction to Classification Theory
49. Toric Varieties
50. Dynamical Systems
51. Approximation Theory
52. Partial Differential Equations
53. Nonlinear Evolution Equations and Applications
54. Functional Methods in Differential Equations
55. Complex Manifolds
56. Geometric Analysis
57. Block Designs
58. Hypergraphs, Set Systems, Intersection Theorems
59. Selected Topics in Graph Theory
60. Finite Packing and Covering
61. Packing and Covering
62. Convex Polytopes
63. Combinatorial Geometry
64. Geometry of Numbers
65. Stochastic Geometry
66. Brunn-Minkowski Theory
67. Hyperbolic Manifolds
68. Characteristic Classes
69. Singularities of Differentable Maps: Local and Global Theory
70. Four Manifolds and Kirby Calculus
71. Symplectic Manifolds, Lefschetz Fibration
72. Advanced Intersection Theory
73. Descriptive Set Theory
74. Advanced Set Theory
75. Logical Systems
76. Set-Theoretic Topology
77. Logic and Relativity
78. Frontiers of Algebraic Logic
79. Classical Analytic Number Theory
80. Probabilistic Number Theory
81. Probabilistic Number Theory, Level II
82. Modern Prime Number Theory
83. Exponential Sums in Combinatorial Number Theory
84. Information Theoretical Methods in Mathematics
85. Selected Topics in Probability
86. Invariance Principles in Probability and Statistics
87. Stochastic Processes
88. Stochastic Analysis
89. Path Properties of Stochastic Processes
90. Nonparametric Statistics
91. Multivariate Statistics
92. Information Theoretical Methods in Statistics
93. Numerical Methods in Statistics
94. Ergodic Theory and Dynamical Systems
95. Ergodic Theory and Combinatorics
96. Data Compression
97. Cryptology
98. Combinatorial Optimization
99. Quantum Computing
100. Computational Geometry
101. Random Computation
102. Logic of Programs
103. Topics in Financial Mathematics
104. Introduction to CCR Algebras
105. Extremal Combinatorics,Graph Theory and Finite Geometry
106. Invariant theory
Syllabi
Mandatory Courses
M1. TOPICS IN ALGEBRA
Lecturer: Matyas Domokos
No. of Credits: 3, and no. of ECTS credits: 6
Semester or Time Period of the course: Fall Semester
Prerequisites: Basic Algebra 1-2 or equivalen
Course Level: advanced
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:
The students will learn some advanced topics of abstract algebra, that is, they will be enabled to do research in fields touching on algebra. Also, they will develop some special expertise in these topics.
More detailed display of contents:
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, its equivalent characterizations, description in tems of left quasi-regular elements, left-right symmetry;
2. basic properties of completely reducible modules, Schur’s Lemma, uniqueness of decomposition into the direct sum of irreducible (i.e. simple) submodules;
3. nilpotency of the radical of an artinian ring, the Wedderburn-Artin theorems, uniqueness of the Wedderburn-Artin decomposition, module theoretic characterization of semisimple artinian rings;
4. basic concept of group representations: representations as group actions on a vector space via linear transformations, invariant subspaces, irreducible representations, subrepresentations, factor representations, sums of representations, the trivial representation, homomorphisms of G-modules, isomorphism of representations, the matrix representation associated to a finite dimensional representation;
5. group algebras, correspondence between modules over the group algebra and representations of the group; Maschke’s Theorem;
6. consequences of the Wedderburn-Artin Theorem on representations of a finite group: the structure of the complex group algebra of a finite group, the number of irreducible complex representations of a finite group, the center of the group agebra, complete reducibility;
7. irreducible representations of abelian groups, Schur’s Lemma revisited, invariant scalar products, unitary representations, orthogonality of unitary matrix elements of irreducible complex representations of a finite group; character of a representation, orthonormality of irreducible characters, the isomorphism type of a representation is determined by its character, the character as a computational tool, the character tables of cyclic groups and their direct products, the character tables of the dihedral group D4, the quaternion group of order 8, the symmetric groups S3, S4, the alternating group A4, second orthogonality of characters;
8. Burnside’s p_q_ Theorem: characterization of algebraic integers as elements contained in a ring that is a finitely generated abelian group, consequences of the fact that character values are sums of roots of 1, the dimension of an irreducible representation divides the order of the group, groups of size p-power times q-power are solvable; Induced representations, Frobenius reciprocity.
9. Product of representations (tensor products of modules in a general setting and the Kronecker product in the representation setting), symmetric and alternating powers of representations and their characters; The character table of the direct product of two finite groups is the Kronecker product of the character tables;
10. Examples of commutative rings from Number Theory and Algebraic Geometry; Transcendence degree of an integral domain; Krull’s Theorem (without proof): the Krull dimension of an integral domain (which is a finitely generated algebra over a field K) equals its transcendence degree over K.
11. 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;
12. Basic concepts of affine algebraic geometry: affine n-space, algebraic sets and varieties, consequences of the Hilbert Basis Theorem, the Zariski topology, affine algebraic varieties, the correspondence between ideals and algebraic sets (radical ideals, the Nullstellensatz restated), the ideal of a variety is prime, irreducible components of an algebraic set, the coordinate ring of an affine algebraic set, the comorphism of a polynomial map between affine algebraic sets.
Books:
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 COMBINATORICS
Lecturer: Ervin Gyori
No. of Credits: 3 and no. of ECTS credits: 6
Semester: Fall Semester
Prerequisites:-
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.
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:
Knowledge of combinatorial techniques that can be applied not just in discrete mathematics but in many other areas of mathematics. Skills in solving combinatorial type problems.
More detailed display of contents:
Week 1. Definitions, examples of hypergraphs
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)
Week 4. Nesetril-Rodl hypergraphs and its properties
Week 5. Probabilistic and constructive lower bounds on Ramsey numbers
Week 6. Van der Waerden theorem, threshold numbers
Week 7. Tic-tac-toe and Hales-Jewett theorem
Week 8. Basic extremal set family problems
Week 9. Proofs by counting (Sperner theorem, Erdos-Ko-Rado theorem)
Week 10. More advanced probabilistic method, Lovasz Local Lemma
Week 11. The dimension bound (Fisher inequality, 2-distance sets, etc.)
Week 12. Eigenvalues, minimal size regular graphs of girth 5
References:
Bela Bollobas, Modern Graph Theory, Springer, 1998
Handouts
M3. APPLIED FUNCTIONAL ANALYSIS
Lecturer: Gheorghe Morosanu
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: real analysis, complex function theory
Course Level: introductory
Brief introduction to the course:
Basic concepts and fundamental theorems are presented. Some significant applications are analyzed to illustrate the power of functional analysis. Special attention is paid to linear and nonlinear evolution equations in Banach spaces.
The goals of the course:
The main goal of the course is to introduce students to some of the most important aspects of functional analysis, including ties with other fields of pure and applied mathematics.
The learning outcomes of the course:
Students will learn some important methods of functional analysis. Even more, they will learn how to use these methods to solve specific problems.
More detailed display of contents:
1. Metric spaces, topological properties, Bolzano-Weierstrass theorem.
2. Normed linear spaces. Banach spaces. A characterization of finite dimensional normed spaces.
3. Arzela-Ascoli theorem. Peano theorem. Banach fixed point theorem. Applications to differential and integral equations.
4. Linear operators. The dual space. Weak topologies. Hilbert spaces. Projections in Hilbert spaces. Orthogonal decomposition.
5. The Riesz representation theorem. The Lax-Milgram theorem.
6. Orthonormal systems in Hilbert spaces. Fourier series.
7. Distributions, Sobolev spaces.
8. Eigenvalue problems for linear compact operators. The Hilbert-Schmidt theory.
9. Semigroups of linear operators. The Hille-Yosida theorem.
10. Linear evolution equations in Banach spaces.
11. Monotone operators and nonlinear evolution equations.
12. Applications to partial differential equations.
Books:
1. H. Brezis, Analyse fonctionnelle. Theorie et applications, Masson, Paris, 1983.
2. G. Morosanu, Nonlinear Evolution Equations and Applications, D. Reidel, Dordrecht, 1988.
3. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.
4. E. Zeidler, Applied Functional Analysis, Appl. Math. Sci. 108,109, Springer-Verlag, 1995.
Elective Courses
1) ENUMERATION
Course Coordinator: Ervin Gyori
Prerequisites: Binomial coefficients, binomial theorem, permutations, variations, Fibonacci numbers
Books:
1. D.E. Knuth, The Art of Computer programming, Third Edition (Reading, Massachusetts: Addison-Wesley, 1997.
2. R.L. Graham, D.E. Knuth, O. Patashnik: Concrete Mathematics: a Foundation for Computer Science, Addison-Wesley, Reading, U.S.A., 1989.
3. H.S. Wilf: Generatingfunctionology, Academic Press, 1990.
4. G.E. Andrews, The theory of partitions, Addison-Wesley, 1976.
Commitment: 3 hours/week, 3 credits
Contents:
1. Binomial Theorem, Polynomial Theorem. Stirling Formula. Partitions of an integer. Fibonacci numbers. Counting examples from geometry and Information Theory. Generating Functions Linear congruences. Fibonnacci numbers. Recurrences. Inversion formulas. Partitions of sets and numbers Catalan numbers. Young Tableaux Cayley Theorem.
2. Rényi's examples to count trees. Asymptotic series. Watson Lemma. Saddle point method. Stirling formula.
3. Inclusion-Exclusion formulas: Sieve Method. Möbius function, Möbius inversion formula Applications in number theory. Pólya Method.
4. Using computers. Wilf-Zeilberger theory.
2) EXTREMAL COMBINATORICS
Course Coordinator: Ervin Gyori
Prerequisites: Some introductory course to discrete mathematics
Books:
1. B. Bollobás: Extremal Graph Theory, Academic Press, London, 1978.
2. R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey theory, Wiley, New
York, 1980.
Commitment: 3 hours/week, 3 credits
Contents:
1. Ramsey Theory. The Erdıs-Szekeres estimate. Hypergraph Ramsey Theorems. Van der Waerden theorem. Hales-Jewett theorem. Amalgamation method (Nesetril-Rödl)
2. Extremal Graph Theory. Turán's theorem. Erdıs-Stone-Simonovits theorem on the limit density. Nondegenerate extremal graph problems. Asymptotic structure of extremal graphs. Kıvári-T. Sós-Turán theorem. Constructions. Füredi's theorem on fourgous. Degenerate extremal graph problems. Erdıs-Gallai Theorem.
3. Supersaturated graphs
4. Szemerédi Regularity Lemma
5. Extremal graph problems for uniform hypergraphs Ruzsa-Szemerédi theorem. The Szemerédi theorem on arithmetic progressions.
3) RANDOM METHODS IN COMBINATORICS
Lecturer: Gyula Katona
No. of Credits: 3, and no. of ECTS credits: 6.
Prerequisites: -
Course Level: advanced.
Brief introduction to the course:
Introducing the random method in combinatorics. Proving the existence of certain combinatorial structures, or proving lower estimates on the number of such structures. Enumeration method, expectation method. Second momentum method, Lovász Local Lemma. Random and pseudorandom structures. The course is suggested to students oriented in combinatorics and computer science.
The goals of the course:
To show the main results and methods of the theory.
The learning outcomes of the course:
The students will know the most important results of the theory, they will be able follow the literature, apply these results in practical cases and create new results of similar nature.
More detailed display of contents:
Week 1: The basic method, applications in graph theory and combinatorics.
Week 2: Applications in combinatorial number theory.
Week 3: Probabilistic proof of the Erdős-Ko-Rado theorem and the Bollobás theorem.
Week 4: Application in Ramsey theory.
Week 5: The second moment method. The Rödl Nibble.
Week 6: The Lovász Local Lemma.
Week 7: Applications of LLL in Porperty B, Ramsey theory and geometry.
Week 8: Correlation inequalities: Ahlswede-Daykin and FKG inequalities.
Week 9: Martingales and tight concentration.
Week 10: Talagrand’s inequality and Kim-Vu’s polynomial concentration.
Week 11: Random graphs.
Week 12: Pseudorandom graphs.
Books:
1. N. Alon, J.H. Sepncer: The Probabilitstic Method, John Wiley & Sons, 1992.
2. B. Bollobás: Random Graphs, Academic Press, 1985.
3. P. Erdős: The Art of Counting, Cambridge, MIT Press, 1973.
4. P. Erdős: Joel Spencer: Probabilitstic Methods in Combinatorics, Academic Press, 1974.
4) CONVEX GEOMETRY
Course Coordinator: Imre Barany
Prerequisites: abstract and linear algebra, general topology, analysis
Books:
1. M. Berger, Geometry I-II, Springer-Verlag, New York, 1987.
2. K.W. Gruenberg and A.J. Weir, Linear Geometry, Springer, 1977.
Commitment: 3 hours/week, 3 credits
Contents:
1. Affine spaces.
2. Euclidean space, structure of the isometry group, canonical form of isometries, Cartan's theorem.
3. Spherical trigonometry.
4. Fundamental theorems on convex sets (Caratheodory, Radon, Helly, Krein-Milman, Straszewicz etc.).
5. Convex polytopes, Euler's formula, classification of regular polytopes, Cauchy's rigidity theorem, flexible polytopes.
6. Hausdorff metric, Blaschke's selection theorem, Cauchy's formula, the Steiner-Minkowski formula, symmetrizations, isoperimetric and isodiametral inequalities.
5) NON-EUCLIDEAN GEOMETRIES
Course Coordinator: Balázs Csikós
Prerequisites: abstract and linear algebra, general topology, analysis
Books:
1. M. Berger, Geometry I-II, Springer-Verlag, New York, 1987.
2. K.W. Gruenberg and A.J. Weir, Linear Geometry, Springer, 1977.
Commitment: 3 hours/week, 3 credits
Contents:
• Projective spaces over division rings, Desargues' and Pappus' theorem, axiomatic foundations, the duality principle.
• Collineations, correlations, cross-ratio preserving transformations.
• Quadrics, classification of quadrics, Pascal's and Brianchon's theorems, polarity induced by a quadric, pencils of quadrics, Poncelet's theorem.
• Hyperbolic geometry: models of the hyperbolic space and the transition between them, isometries, hyperbolic trigonometry, constructions.
6) DIFFERENTIAL GEOMETRY
Course Coordinator: Balázs Csikós
Prerequisites: abstract and linear algebra, general topology, analysis, ordinary
differential equations
Books:
1. M.P. do Carmo: Differential Geometry of Curves and Surfaces Prentice-Hall, Englewood Cliffs, NJ, 1976.
2. W. Klingenberg: A course in differential geometry, Springer, 1978.
3. W.M. Boothby: An introduction to differentiable manifolds and Riemannian geometry, Second Edition, Academic Press, 1986.
Commitment: 3 hours/week, 3 credits
Contents:
• Curves in R2.
• Hypersurfaces in R2. Theorema Egregium. Special surfaces.
• Differentiable manifolds, tangent budle, tensor bundles; Lie algebra of vector fields, distributions and Frobenius' theorem; Covariant derivation, the Levi-Civita connection of a Riemannian manifold, parallel transport, holonomy groups; Curvature tensor, symmetries of the curvature tensor, decomposition of the curvature tensor; Geodesic curves, the exponential map, Gauss Lemma, Jacobi fields, the Gauss-Bonnet theorem; Differential forms, de Rham cohomology, integration on manifolds, Stokes' theorem.
7) HOMOLOGICAL ALGEBRA
Lecturer: Pham Ngoc Anh
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Basic algebra 1, 2
Course Level: advanced
Brief introduction to the course:
An introduction to homological algebra. A description of projective and injective modules. Torsion and extension product with application to the theory of homological dimension and extensions of groups.
The goals of the course:
The main goal of the course is to introduce students to the most important, basic notions and techniques of homological algebra.
The learning outcomes of the course:The students will learn the most important notions of homological algebra, mainly the functors Ext and Tor.
More detailed display of contents:
1. Differential graded groups, modules. Examples from simplicial homology theory. Homology of complexes. Basic properties.
2. Exact sequence of homology. Short description of the singular homology theory.
3. Hom functor and tensor products. Projective and injective resolutions.
4. Ext
5. Ext (continued)
6. Tor
7. Tor (continued)
8. Homological dimensions.
9. Rings of low dimensions.
10. Cohomology of groups
11.Cohomology of algebras.
12. Application to theory of extensions.
Books:
1. J. Rotman: Introduction to homological algebra, Springer 2009.
2. J. P. Jains: Rings and homology, Holt, Rinehart and Winston, New York 1964.
8) SMOOTH MANIFOLDS AND DIFFERENTIAL TOPOLOGY
Lecturer: Andras Nemethi
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites:-
Course Level: advanced
Brief introduction to the course:
Basic principles and methods concerning differentiable manifolds and differentaible maps are discussed. The main concepts (submersions, transversality, smooth manifolds and manifolds with boundary, orientation, degree and intersection theory, etc.) are addressed, with special emphasis on different connections with algebraic topology (coverings, homological invariants). Many applications are discussed in detail (winding number, Borsuk-Ulam theorem, Lefschetz fixed point theory, and different connections with algebraic geometry).
The course is designed for students oriented to (algebraic) topology or algebraic geometry.
The goals of the course:
The main goal of the course is to introduce students to the theory of smooth manifolds and their invariants. We also intend to discuss different connections with algebraic topology, (co)homology theory and complex/real algebraic geometry.
The learning outcomes of the course:
The students will learn important notions and results in theory of smooth manifolds and smooth maps. They will meet the first non-trivial invariants in the classification of maps and manifolds. They will gain crucial skills and knowledge in several parts of modern mathematics. Via the exercises, they will learn how to use these tools in solving specific topological problems.
More detailed display of contents:
Week 1: Derivatives and tangents (definitions, inverse function theorem, immersions).
Week 2: Submersions (definitions, examples, fibrations, Sard's theorem, Morse functions).
Week 3: Transversality (definitions, examples, homotopy and stability).
Week 4: Manifolds and manifolds with boundary (definition, examples, one-manifolds and consequences).
Week 5: Vector bundles (definition, examples, tangent bundles, normal bundles, compex line bundles).
Week 6: Intersection theory mod 2 (definition, examples, winding number, Borsuk-Ulam theorem).
Week 7: Orientation of manifolds (definition, relation with coverings, orientation of vector bundles, applications).
Week 8: The degree (definition, examples, applications, the fundamental theorem of algebra, Hopf degree theorem).
Week 9: Oriented intersection theory (definitions, examples, applications, connection with homology theory).
Week 10: Lefschetz fixed-point theorem (the statement, examples).
Week 11: Vector fields (definition, examples, the index of singular points).
Week 12: Poincare-Hopf theorem (the Euler characteristic, discussion, examples).
Books:
1. John W. Milnor, Topology from the Differentiable Viewpoint, Princeton Landmarks in Mathematics, Princeton University Press.
2. Victor Guillemin and Alan Pollack, Differential Topology.
9) ALGEBRAIC TOPOLOGY
Lecturer: Andras Nemethi or András Stipsicz
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites:-
Course Level: advanced
Brief introduction to the course:
Basic principles and methods concerning algebraic topology are discussed. The main concepts (elementary homotopy theory, homotopy of paths, fundamental group, covering spaces, lifting criterions, singular homology theory, exact homology sequences, euler characteristic, orientation and dualities, etc.) are addressed, with special emphasis on different classical problems (regarding coverings, Jordan -Brouwer separation theorem, Brouwer fixed point theorem, Lefschetz number and fixed point theorem). Many applications and examples are discussed with different connections with differential topology and algebraic geometry). The course is designed for students oriented to (algebraic) topology or algebraic geometry.
The goals of the course:
The main goal of the course is to introduce students to algebraic topology and standard topological invariants. We also intend to discuss different connections with differentiable topology, (co)homology theory and complex/real algebraic geometry.
The learning outcomes of the course:
The students will learn important notions and results in algebraic topology, homological algebra and related invariants associated with topological spaces and continuous maps. They will gain crucial skills and knowledge in several parts of modern mathematics. Via the exercises, they will learn how to use these tools in solving specific topological problems.
More detailed display of contents:
Week 1: Homotopy theory (definitions, homotopy of paths, homotopy of maps).
Week 2: Fundamental group (definition, examples).
Week 3: van Kampen theorem (group theoretical preliminaries, examples, applications).
Week 4: Covering spaces (definitions, examples, universal covering, classification, Galois correspondence).
Week 5: Singular homology theory (definition, examples).
Week 6: Relation with the fundamental group (examples, abelian coverings).
Week 7: Excision, exact sequences (definitions, examples, applications).
Week 8: The Mayer-Vietoris exact sequence (examples, applications).
Week 9: Kunneth formula (examples, applications).
Week 10: Singular cohomology (definition, cup and cap products).
Week 11: Duality theorems for manifolds (Poincaré, Alexander, Lefschetz dualities).
Week 12: Thom class, Euler class (applications)
Optional material: Thom Isomorphism Theorem and related topics.
Books:
1. M.J. Greenberg: Lectures in Algebraic Topology
2. E. Spanier: Algebraic Topology
3. S. MacLane: Homology
10) FUNCTION SPACES AND DISTRIBUTIONS
Course Coordinator: Gheorghe Morosanu
Prerequisites: Undergraduate Calculus, Linear Algebra, Linear Functional Analysis
Books:
1. R.A. Adams, Sobolev Spaces, Academic Press, 1975.
2. H. Brezis, Analyse fonctionelle,. Theorie et applications, Masson, Paris, 1983.
3. I.M. Gel’fand and G.E. Shilov, Generalized Functions, Vols 1 and 2, Academic Press, 1968.
4. L. Schwartz, Theorie de distributions, Herman, Paris, 1967.
Commitment: 3 hours/week, 3 credits
Contents:
• Spaces of continuous functions
• Functions of bounded variation
• The spaces Lp(D), Lp(a,b; X)
• Test functions and distributions
• Density theorems
• Sobolev spaces
• The spaces Wk,p (a,b; X)
• Applications
11) NONLINEAR FUNCTIONAL ANALYSIS
Course Coordinator: Denes Petz
Prerequisites: Real and Complex Analysis, Linear Functional Analysis
Commitment: 3 hours/week, 3 credits
Contents:
• Fixed point theorems. Applications
• Variational principles and weak convergence. The n-th variation. Necessary and sufficient conditions for local extrema. Weak convergence. The generalized Weierstrass existence theorem. Applications to calculus of variations. Applications to nonlinear eigenvalue problems. Applications to convex minimum problems and variational inequalities. Applications to obstacle problems in Elasticity. Saddle points. Applications to duality theory. The von Neumann Minimax theorem on the existence of sadle points. Applications to game theory.
• Nonlinear monotone operators. Applications.
12) INTRODUCTION TO MATHEMATICAL LOGIC
Lecturer: Ildiko Sain
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites:-
Course Level: Introductory
Brief introduction to the course:
Basic concepts and basic properties of logical systems, in particular of sentential (propositional) logic and first order logic: syntax, semantics, truth, drivability; soundness and completeness, compactness, Lovenheim-Skolem theorems, some elements of model theory.
The goals of the course:
The main goal is to make the student familiar with the basic concepts and methods of mathematical logic. There will be more emphasis on the semantic aspects than on the syntactical ones.
The learning outcomes of the course:
Knowledge of the basic logical concepts and methods in such an extent that the student can apply them in other areas of mathematics or science.
More detailed display of contents:
Week 1. Sentential (propositional) logic: syntax and semantics.
Week 2. Completeness, compactness, definability in sentential logic. Connections with Boolean algebras.
Week 3. First order logic: syntax and semantics.
Week 4. A deductive calculus. Soundness and completeness theorems. Preservation theorems.
Week 5. Ultraproducts and Los lemma.
Week 6. Compactness theorem, Lovenheim-Skolem theorems, preservation theorems.
Week 7. Complete theories, decidable theories.
Week 8. Applications of the model theoretic results of the previous two weeks.
Week 9. Elementary classes. Elementarily equivalent structures.
Week 10. Godel’s incompleteness theorem.
Week 11. Definability.
Week 12. Logic in algebraic form (algebraisation of logical systems, Boolean algebras, cylindric algebras).
References:
Enderton, H.B.: A Mathematical Introduction to Logic. Academic Press, New York and London, 1972.
Ebbinghaus, H.D., Flum, J. and Thomas, W.: Mathematical Logic. Springer Verlag, Berlin, 1964, vi+216 pp.
More advanced readings:
Andreka, H., Nemeti, I. and Sain, I.: Algebraic Logic. In: Handbook of Philosophical Logic Vol.II, Second Edition, D.M. Gabbay and F. Guenthner eds., Kluwer Academic Publishers, 2001, 133-247.
Monk, J.D.: An Introduction to Cylindric Set Algebras. Logic Journal of the IGPL, Vol.8, No.4, July 2000, 449-494.
13) MODERN SET THEORY
Lecturer: István Juhász
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Basic (naive) set theory and logic
Course level: intermediate
Brief introduction to the course:
The basics of both axiomatic and combinatorial set theory will be presented.
The goals of the course:
One goal is to present the most important results in set theory. Another goal is to get the students aquainted with consistency and independence results.
The learning outcomes of the course:
The students will learn not just the axiomatic development of set theory but the more general significance of the axiomatic method in mathemematics.
More detailed display of contents:
The cumulative hierarchy and the ZFC axiom system
Axiomatic exposition of set-theory
Absoluteness and reflection
Models of set-theory, relative consistency
Constructible sets, consistency of AC and GCH
Combinatorial set-theory and combinatorial principles
Large cardinals
Basic forcing
Books:
1. András Hajnal, Peter Hamburger: Set Theory, Cambridge University Press,1999.
2. Thomas Jech: Set Theory, Spinger-Verlag, 1997.
3. Kenneth Kunen: Set theory. An introduction to Independence Proofs, Elsevier,1999
14) ALGEBRAIC LOGIC AND MODEL THEORY
Lecturer: Gábor Sági
No. of Credits: 3 and no. of ECTS credits: 6.
Prerequisites: not mandatory course.
Course level: introductory
Brief introduction to the course:
Ultraproducts, constructing universal and saturated models, the Keisler-Shelah theorem, definability, countable categoricity, basics of representation theory of cylindric algebras.
The goals of the course:
The main goal is to study some methods of mathematical logic and to learn how to apply them.
The learning outcomes of the course:
Knowledge of certain model theoretic techniques. Skills how to apply these techniques not just in logic but in many other areas of mathematics.
More detailed display of contents:
Week 1. Languages, structures, isomorphisms, elementary equivalence and some preservation theorems.
Week 2. Ultrafilters and their elementary properties. A combinatiorial application.
Week 3. Regular ultrafilters and universal models.
Week 4. Good ultrafilters and saturated models.
Week 5. Existence of good ultrafilters, 1.
Week 6. Existence of good ultrafilters, 2. The Keisler-Shelah theorem (with the generalized continuum hypothesis).
Week 7. Definability theorems.
Week 8. Omitting types and basic properties of countable catehoricity.
Week 9. Characterizations of countable categoricity.
Week 10. An example: the countable random graph. 0-1 laws.
Week 11. Cylindric and cylindric set algebras. Representations.
Week 12. An algebraic proof for the completeness theorem.
References:
1. C.C. Chang, H.J. Keisler, Model Theory, Elsevier, 1996.
2. L. Henkin, J.D. Monk, A. Tarski, Cylindric Algebras, Part II, Elsevier, 1987.
3. W. Hodges, Model Theory, Oxford Univ. Pres., 1997.
15) ELEMENTARY PRIME NUMBER THEORY
Course Coordinator: Imre Ruzsa
Prerequisites: Basic analysis, Basic group theory
Book: M. B. Nathanson, Elementary methods in number theory, Springer, Graduate
texts in mathematics 195, New York - Berling 2000, chapters 4, 6, 8, 9, 10.
Commitment: 3 hours/week, 3 credits
Contents:
• Elementary but more difficult methods in mutiplicative number theory. The elementary proof of the prime number theorem via Selberg's formula. Characters. The elementary proof of Dirichlet's
theorem.
• Sieves: Brun's sieve (and Hooley's refinement), Selberg's sieve, the large sieve. Applications: upper estimate for twin primes.
16) COMBINATORIAL NUMBER THEORY
Course Coordinator: Gergely Harcos or Imre Ruzsa
Prerequisites: Undergraduate number theory
Books:
1. H. Halberstam, K. F. Roth, K. F, Sequences, Clarendon, London, 1966; 2nd ed. Springer, 1983.
2. M. B. Nathanson, Additive number theory: Inverse problems and the geometry of sumsets, Springer, Graduate texts in mathematics 165, New York - Berlin 1996.
Commitment: 3 hours/week, 3 credits
Contents:
• Concepts of density. Estimates for the density of sumsets: the theorems of Schnirel-mann, Mann, Kneser, Erdıs, Plünnecke. Inequalities for the cardinality of sumsets of finite sets of integers, residues, lattice points. Structure of sets with small sums.
• Sidon sets.
17) PROBABILISTIC METHODS IN NUMBER THEORY
Course Coordinator: Imre Ruzsa
Prerequisites: Basic probability theory
Book: N. Alon, J. H. Spencer, P. Erdıs, The probabilistic method, Wiley, New York, 1992.
Commitment: 3 hours/week, 3 credits
Contents:
• Tools from probability theory: Bernstein's inequality, the Esary-Proschan-Walkup (or Fortuin-Kasteleyn-Ginibre) inequality, Janson's inequality.
• Properties of random sets. Thresholds for properties: being a basis, being a Sidon set.
• Random constructions: regular bases, dense or regular Sidon sets, pseudo-squares with a dense sumset, essential components, sum-intersective sets.
18) PROBABILITY
Course Coordinator: Endre Csáki
Prerequisites: Introduction to Probability and Statistics I-II; Measure and integral.
Books:
1. L. Breiman: Probability. Addison-Wesley, Reading, Massachusetts (1968).
2. W. Feller: An Introduction to Probability Theory and its Applications, Vol. II. Second edition. Wiley, New York (1971).
3. A. Rényi: Probability Theory. North-Holland, Amsterdam (1970).
4. Y.S. Chow-H. Teicher: Probability Theory. Independence, Interchangeability, Martingales. Third edition. Springer, New York (1997).
5. P. Major: Series of Problems in Probability Theory. http://www.renyi.hu~major
Commitment: 3 hours/week, 3 credits
Contents:
Random variables, distributions, expectations, moments; Independence. Conditional distributions; Weak convergence of probability distributions, equivalent formulations; The method of characteristic functions in proving weak convergence: the central limit theorem; Laws of large numbers, large deviations, law of the iterated; logarithm. Elements of the theory of stochastic processes. Poisson process, Wiener process, Markov processes; Martingales.
19) MATHEMATICAL STATISTICS
Course Coordinator: Endre Csáki
Prerequisites: Stochastics, Probability, Measure and integral.
Books:
1. M.G. Kendall & A. Stuart: The Advanced Theory of Statistics. Vol. I., II. Griffin, London 1958, 1961.
2. C.R.Rao: Linear Statistical Inference and its Applications. Wiley, 1965.
Commitment: 3 hours/week, 3 credits
Contents:
General Concepts:
Statistical space, statistical sample. Important statistics. Empirical distribution.
Histogram. Multidimensional normal distribution. Fisher information. Sufficiency. Completeness. Exponential family.
Theory of estimation:
Unbiased estimators. Efficiency. Cramer-Rao inequality. Admissibility. Asymptotic properties of estimators: consistency, asymptotic normality. Method of moments, least squares, maximum likelihood.
Theory of hypothesis testing.
Elements of Bayesian statistics.
20) INFORMATION THEORY
Course Coordinator: Imre Csiszár
Prerequisites: Introduction to probability and statistics, vector spaces over finite fields.
Books:
1. T.M. Cover & J.A. Thomas: Elements of Information Theory. Wiley, 1991.
2. I. Csiszar & J. Korner: Information Theory. Academic Press, 1981.
Commitment: 3 hours/week, 3 credits
Contents:
• Definition and formal properties of Shannon's information measures
• Source and channel models. Source coding, block and variable length codes, entropy rate. Arithmetic codes. The concept of universal coding.
• Channel coding (error correction), operational definition of channel capacity. The coding theorem for discrete memoryless channels. Shannon's source-channel transmission theorem.
• Outlook to multiuser and secrecy problems.
• Exponential error bounds for source and channel coding. Compound and arbitrary varying channels. Channels with continuous alphabets; capacity of the channel with additive Gaussian noise.
• Elements of algebraic coding theory; Hamming and Reed-Solomon codes.
21) INTRODUCTION TO THE THEORY OF COMPUTING
Course Coordinator: Gyula Katona
Prerequisites: -
Book: T. H. Cormen, C. L. Leiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, Cambridge, MA, 1990.
Commitment: 3 hours/week, 3 credits
Contents:
Communication games, examples. Dynamic programming: maximal interval-sum, largest all-one square submatrix, the optimal bracketing of matrix-products. The knapsack problem. The scaling method of Ibarra and Kim: approximating the optimum solution of the knapsack problem. Recursive functions. Halting problem.
The domino-problem. Deterministic time- and space complexity classes. For any recursive f(x), there exists a recursive language, which is not in DTIME(f(x)).
Non-deterministic Turing-machines.
Other NP-complete problems: Hypergraph hitting-set, edge-cover, hypergraph 2-colorability. 3-chromatic graphs, Independent set is NP-complete. Subset-sum,
Knapsack is NP-complete. Non-approximability results: graph-coloring. Parallel computing.
22) ALGORITHMS
Course Coordinator: Miklos Simonovits
Prerequisites: Undergraduate Algebra, Combinatorics, Advanced Calculus, and Theory of Computing.
Books:
1. T. H. Cormen, C. L. Leiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, Cambridge, MA, 1990.
2. W.J. Cook, W.H. Cunningham, W.R. Pulleyblank, and A. Schrijver, Combinatorial Optimization. Wiley, 1998.
Commitment: 3 hours/week, 3 credits
Contents:
• Algebraic algorithms. Polynomials, FFT, Matrix algorithms. Number theoretical algorithms: prime searching, factoring, RSA cryptosystem.
• Sorting networks. Elementary parallel algorithms: MIN, sorting, graph algorithms on PRAMs. Determinant computing in parallel.
• Dynamic programming. Standard examples. Greedy algorithms. Matroids-an introduction. Graph algorithms.
• Combinatorial optimization an polyhedra. The basics of linear programming.
• Optimal matchings in bipartite and general graphs.
• Maximum flow problems. Minimum cuts in undirected graphs. Multicommodity flows. Minimum-cost flow problems.
• Outlook: Agorithms on the Web. Genetic algorithms.
23) COMPLEXITY THEORY
Course Coordinator: Miklos Simonovits
Prerequisite: Computing, Elementary Algebra.
Book: M. Sipser, Introduction to the Theory of Computation, PWS Publishing Company Boston, 1997.
Commitment: 3 hours/week, 3 credits
Contents:
• Formal models of computation: Turing machines, RAM machines.
• Reduction, complete languages for NP, P, NL, PSPACE. Savitch’s theorem.
• Diagonal method: time- and space hierarchy.
• Randomization, randomized complexity classes, their relation to deterministic/non-deterministic classes, examples.
• Communication complexity, deterministic, non-deterministic, relation to each other and to matrix rank.
• Decision trees: deterministic, non-deterministic, randomized, sensitivity of Boolean functions.
24) ERGODIC THEORY
Lecturer: Péter Bálint or Domokos Szász
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: No particular course prerequisites, however, basic knowledge of linear algebra, probability theory, measure theory and functional analysis is needed
Course Level: PhD
Brief introduction to the course:
Basic concepts of ergodic theory: measure preserving transformations, ergodic theorems, notions of ergodicity, mixing and methods for proving such properties, topological dynamics, hyperbolic phenomena, examples: eg. rotations, expanding interval maps, Bernoulli shifts, continuous automorphisms of the torus.
The goals of the course:
The main goal of the course is to give an introduction to the central ideas of ergodic theory, and to point out its relations to other fields of mathematics.
The learning outcomes of the course:
The learning outcomes are twofold: on the one hand, obtaining a skill for combining tools from various fields of mathematics to solve problems specific to ergodic theory, and on the other hand, getting familiar with the ideas of ergodic theory and its role in other mathematical disciplines.
More detailed display of contents:
Week 1: Basic definitions and examples (measure preserving transformations, examples: rotations, interval maps etc.)
Week 2: Ergodic theorems (Poincare recurrence theorem, von Neumann and Birkhoff ergodic theorems)
Week 3: Ergodicity (different characterizations, examples: rotations)
Week 4: Further examples: stationary sequences (Bernoulli shifts, doubling map, baker’s transformation)
Week 5: Mixing (different characterizations, study of examples from this point of view)
Week 6: Continuous automorphisms of the torus (definitions, proof of ergodicity via characters)
Week 7: Hopf’s method for proving ergodicity (hyperbolicity of a continuous toral automorphism, stable and unstable manifolds, Hopf chains)
Week 8: Invariant measures for continuous maps (Krylov-Bogoljubov theorem, ergodic decomposition, examples)
Week 9: Markov maps of the interval (definitions, existence and uniqueness of the absolutely continuous invariant measure)
Weeks 10-12: Further topics based on the interest of the students (eg. attractors, basic ideas of KAM theory, entropy, systems with singularities etc.)
Books:
1. P. Walters: Introduction to Ergodic Theory, Springer, 2007
2. M. Brin- G.Stuck: Introduction to Dynamical Systems, Cambridge University Press 2002
25) MATHEMATICAL METHODS IN STATISTICAL PHYSICS
Course Coordinator: Bálint Tóth
Prerequisites: Probability, Functional Analysis, Complex Functions.
Commitment: 3 hours/week, 3 credits
Contents:
• The object of study of statistical physics, basic notions.
• Curie-Weiss mean-field theory of the critical point. Anomalous fluctuations at the critical point.
• The Ising modell on Zd.
• Analiticity I: Kirkwood-Salsburg equations.
• Analiticity II: Lee-Yang theory.
• Phase transition in the Ising model: Peierls' contour method.
• Models with continuous symmetry.
26) FRACTALS AND DYNAMICAL SYSTEMS
Course Coordinator: Károly Simon
Prerequisites: Ergodic Theory, Measure Theory
Books:
1. K. Falconer, Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Ltd., Chichester, 1990.
2. K. Falconer, Techniques in fractal geometry. John Wiley & Sons, Ltd., Chichester, 1997.
3. Y. Pesin, Dimension theory in dynamical systems. Contemporary views and applications Chicago Lectures in Mathematics. University of Chicago Press,
Chicago, IL, 1997.
Commitment: 3 hours/week, 3 credits
Contents:
• Fractal dimensions. Hausdorff and Packing measures.
• Basic examples of dynamically defined fractals. Horseshoe, solenoid.
• Young's theorem about dimension of invariant measure of a C2 hyperbolic diffeomorphism of a surface.
• Some applications of Leddrapier- Young theorem.
• Barreira, Pesin, Schmeling Theorem about the local dimension of invariant measures.
• Geometric measure theoretic properties of SBR measure of some uniformly hyperbolic attractors.
• Solomyak Theorem about the absolute continuous infinite Bernoulli convolutions.
27) HIGHER LINEAR ALGEBRA
Lecturer: Pham Ngoc Anh and Mátyás Domokos
No. of Credits: 3 and no. of ECTS credits:6
Prerequisites: Basic Algebra I-II
Course Level: Advanced
Brief introduction to the course:
Covers advanced topic in linear algebra beyond the standard undergraduate material.
The goals of the course:
Learn familiarity with the representation theory of quivers and its relevance for various areas.
The learning outcomes of the course:
Familiarity with an algebraic theory that has a large potential to be applied in various fields of pure and applied mathematics. The students increase their knowledge in representation theory, homological algebra, algebraic geometry, while focusing on elementary objects.
More detailed display of contents:
Week 1. Introduction, motivation, overview of the course.
Week 2. Matrix problems and their connection to modules over path algebras.
Week 3. The variety of representations, some basic properties of algebraic group actions.
Week 4. Dynkin and Euclidean diagrams.
Week 5. Gabriel’s Theorem, reflection functors.
Week 6. Auslander-Reiten translation.
Week 7. Kronecker’s classification of matrix pencils.
Week 8. Indecomposable representations of tame quivers.
Week 9. Kac’s Theorems for wild quivers.
Week 10. Schur roots, canonical decomposition of dimension vectors.
Week 11. Quivers with relations.
Week 12. Perspectives, relevance for some currently active research topics.
28) REPRESENTATION THEORY I.
Lecturer: Mátyás Domokos
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisite for Representation Theory II.
Course Level: Introductory
Brief introduction to the course:
The course gives an introduction to the theory of group representations, in a manner that provides useful background for students continuing in diverse mathematical disciplines such as algebra, topology, Lie theory, differential geometry, harmonic analysis, mathematical physics, combinatorics.
The goals of the course:
Develop the basic concepts and facts of the complex representation theory of finite groups, compact toplogical groups, and Lie groups.
The learning outcomes of the course:
A solid knowledge of some fundamental principles of representation theory, as they appear in their simplest form in mathematics.
More detailed display of contents:
Week 1. Definition of linear representations, irreducible representations, general constructions.
Week 2. Properties of completely reducible representations.
Week 3. Finite dimensional complex representations of compact groups are unitary.
Week 4. Products of representations, Schur Lemma and corollaries
Week 5. Spaces of matrix elements. Of representations.
Week 6. Decomposition of the regular representation of a finite group.
Week 7. Characters, orthogonality, character tables, a physical application.
Week 8. The Peter-Weyl Theorem
Week 9. Representation of the special orthogonal group of rank three.
Week 10. The Laplace spherical functions.
Week 11. Lie groups and their Lie algebras
Week 12. Repreentations of the complex special linear Lie algebra SL(2,C)
Reference: E. B. Vinberg: Linear Representations of Groups, Birkhauser Verlag, 1989.
29) REPRESENTATION THEORY II.
Lecturer: Mátyás Domokos
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Basic Algebra I, Representation Theory I.
Course Level: Advanced
Brief introduction to the course:
In Representation Theory I, the basic general principles of representation theory were laid. In the present course we discuss in detail the representation theory of the symmetric group, the general linear group and other classical groups, and semisimple Lie algebras.
The goals of the course:
Introduce the students the representations some of the most important groups.
The learning outcomes of the course:
The students get an idea about the richness of combinatorial structures showing up in the description of representations of certain groups, relevant for applications in several branches of mathematics, and serving as a model for many further (and probably future) theories.
More detailed display of contents:
Week 1. The irreducible representations of the symmetric group (Young symmetrizers).
Week 2. Partitions, the ring of symmetric functions, Schur functions.
Week 3. Pieri’s rule, Kostka numbers, Jacobi-Trudi formula.
Week 4. Cauchy formula, skew Schur functions.
Week 5. Induced representations, Frobenius reciprocity.
Week 6. Frobenius character formula, hook formula, branching rules.
Week 7. Schur-Weyl duality, double centralizing theorem.
Week 8. Polynomial representations of the general linear group, Schur functors.
Week 9. Semisimple Lie algebras, root systems, Weyl groups.
Week 10. Highest weight theory.
Week 11. Weyl character formula, the classical groups.
Week 12. Littlewood-Richardson rule, plethysms.
References:
1. I. G. Macdonald, Symmetric functions and Hall polynomials
2. W. Fulton, J. Harris: Representation Theory (A first course)
3. C. Procesi: Lie groups (An approach through invariants and representations)
30) UNIVERSAL ALGEBRA AND CATEGORY THEORY
Lecturer: László Márki
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: Basic Algebra I-III
Course Level: advanced
Brief introduction to the course:
Basic notions and some of the fundamental theorems of the two areas are presented, with examples from concrete algebraic structures.
The goals of the course:
The main goal of the course is to provide access to the most general parts of algebra, those on the highest level of abstraction. This also helps to understand connections between different kinds of algebraic structures.
The learning outcomes of the course:
The students will learn approaches and some of the most important tools of universal algebra and category theory, as well as some theorems of fundamental importance.
More detailed display of contents:
1. Algebra, many-sorted algebra, related structures (subalgebra lattice, congruence lattice, automorphism group, endomorphism monoid), factoralgebra, homomorphism theorem.
2. Direct product, subdirect product, subdirectly irreducible and simple algebras, Birkhoff’s theorem.
3. Ultraproduct, Łoś lemma, Grätzer-Schmidt theorem.
4. Variety, word algebra, free algebras, identities, Birkhoff’s variety theorem.
5. Pseudovariety, implicit operation, pseudoidentity, Reiterman’s theorem.
6. Equational implication, quasivariety, Kogalovskiĭ’s theorem, fully invariant congruence, Birkhoff’s completeness theorem.
7. Mal’cev type theorems.
8. Primality, Rosenberg’s theorem, functional completeness, generalizations of these notions.
9. Category, functor, natural transformation, speciaol morphisms, duality, contravariance, opposite, product of categories, comma categories.
10. Universal arrow, Yoneda lemma, coproducts and colimits, products and limits, complete categories, groups in categories.
11. Adjoints with examples, reflective subcategory, equivalence of categories, adjoint functor theorems.
12. Algebraic theories.
Books:
1. S. Burris – H.P. Sankappanavar: A course in universal algebra, Springer, 1981.; available online at www.math.uwaterloo.ca/~snburris
2. G. Grätzer: Universal algebra, 2nd ed., Springer, 1979.
3. J. Almeida: Finite semigroups and universal algebra, World Scientific, 1994.
4. S. Mac Lane: Categories for the working mathematician, Springer, 1971.
5. F. Borceux: Handbook of categorical algebra, 1-2, Cambridge Univ. Press, 1994.
31) TOPICS IN GROUP THEORY
Lecturer: Péter P. Pálfy
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Topics in Algebra
Course Level: advanced
Brief introduction to the course:
Group theory is the abstract mathematical theory of symmetries. It is the oldest branch of abstract algebra with its basic notions introduced by Evariste Galois around 1830. Nowadays group theory is a very rich subject encompassing many different areas with applications in various branches of mathematics (algebra, topology, number theory, combinatorics, geometry) and theoretical physics (quantum mechanics). Each week during the semester a different area of group theory will be discussed.
The goals of the course:
The main goal of the course is to introduce the students to the many facets of modern group theory.
The learning outcomes of the course:
The students will become familiar with the most important results and techniques of group theory. Their ability to work with abstract concepts and methods will be enhanced.
More detailed display of contents:
Week 1: Permutation groups. Transitivity, primitivity. Wreath products.
Week 2: The classification of primitive permutation groups: the O’Nan-Scott Theorem.
Week 3: Multiply transitive groups. The Mathieu groups.
Week 4: Simple groups. The simplicity of some matrix groups.
Week 5: Automorphism groups. Coherent configurations, strongly regular graphs.
Week 6: Free groups. The Nielsen-Schreier Theorem about subgroups of free groups.
Week 7: Groups extensions. Cohomology of groups. The Schur-Zassenhaus Theorem.
Week 8: Solvable groups. Hall’s Theorems for finite solvable groups.
Week 9: Nilpotent groups and finite p-groups. Commutator calculus.
Week 10: The transfer homomorphism. Normal p-complements.
Week 11: Frobenius groups. The structure of the Frobenius kernel and the complement.
Week 12: Subgroup lattices. Distributivity, modularity, Dedekind’s chain condition.
Textbooks:
Peter J. Cameron, Permutation Groups, London Mathematical Society Student Texts 45, Cambridge University Press, 1999
Derek J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, Springer-Verlag, 1993
32) TOPICS IN RING THEORY. I
Course Coordinator: Pham Ngoc Anh
Prerequisites: Basic Algebra
Books:
1. I. Kaplansky: Fields and Rings, The University of Chicago Press, 1972.
2. Lam: A First Course in Noncommutative Rings, Springer, 1991.
Commitment: 3 hours/week, 3 credits
Contents:
• Matrix rings, rings associated to directed graphs, skew polynomial rings, skew Laurent and power series rings, skew group rings, enveloping algebras of Lie algebras, Weyl algebras, free associative algebras, tensor
• Jacobson theory, rings of endomorphisms of vector spaces, Burnside and Kurosh problems, simple nil rings and Kıthe's problem
• Categorical module theory I: generators and cogenerators, flat modules and characterization of regular rings, Bass' theory of (semi-)perfect rings, Bj\"ork's results, examples on rings with the descending chain condition on finitely generated one-sided ideals
• Categorical module theory II: Morita theory on equivalence and duality, projective generators and injective cogenerators, Pickard groups
33) TOPICS IN RING THEORY. II
Course Coordinator: Anh Pham Ngoc
Prerequisites: Basic Algebra
Books:
1. C. Faith: Algebra II: Ring Theory, Springer-Verlag, 1991.
2. T. Y. Lam: A First Course in Noncommutative Rings, Springer, 1991.
Commitment: 3 hours/week, 3 credits
Contents:
• Goldie's theory
• Noncommutative localization, quotient constructions, Artin's problems on division rings
• Separable algebras, principal Wedderburn theorem, central simple algebras, cyclic (division) algebras, p-algebras, involution of algebras
• Auslander's treatment of first Brauer conjecture for artin algebras, results on Krull-Schmidt theorem
• Frobenius and quasi-Frobenius rings, serial rings
34) PERMUTATION GROUPS
Course Coordinator: Péter Pál Pálfy
Prerequisites: Undegraduate group theory
Books:
1. P. J. Cameron: Permutation Groups, Cambridge Univ. Press, 2001.
2. J.D. Dixon & B. Mortimer: Permutation Groups, Springer, 1996
Commitment: 3 hours/week, 3 credits
Contents:
• Orbits and transitivity
• The orbit-counting lemma and its consequences
• Extensions; Kantor’s lemma
• Blocks and primitivity
• Wreath products
• Doubly transitive groups: examples
• Burnside’s theorem on normal subgroups of doubly transitive groups
• Further construction of permutation groups
• Consequences of CFSG: Cameron’s theorem; classification of doubly transitive groups; rank 3 permutation groups
• Jordan groups
• Finitary permutation groups
• Oligomorphic groups
35) LIE GROUPS AND LIE ALGEBRAS
Lecturer: Balasz Ciskos or Péter Pál Pálfy
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: Differential geometry, linear algebra, abstract algebra, introductory algebraic topology (fundamental groups, covering spaces).
Course Level: advanced
Brief introduction to the course:
In the first part of the course we prove the fundamental teorems on the connection between Lie groups and Lie algebras, which enables us to convert problems on Lie groups to problems on Lie algebras. In the second part of the course the structure theory of Lie algebras is discussed.
The goals of the course:
Lie groups appear in mathematics and physics as symmetry groups of all kinds of systems. Lie was interested in the symmetries of differential equations. At the same time F. Klein pointed out the central role of the symmetry group of a geometry, which defines the given geometry as the study of the invariants of the group. Lie groups are indispensable for the study of symmetric spaces, which are natural generalizations of the spaces of constant curvature, introduced by Cartan. Lie groups and their representations are important tools also in quantum mechanics and other areas of theoretical physics. The goal of the course is to provide the basics of this useful theory
The learning outcomes of the course:
This course is an introduction to the theory of Lie groups. Finishing the course, the student will be able to apply Lie theory and study more advanced topics, where the theory of Lie groups and Lie algebras is a prerequisite.
More detailed display of contents:
Week 1: Lie groups. (Definition. Examples. Cayley transformation as a tool to construct Lie group structure on matrix groups.)
Week 2: Topological constructions. (Direct and semidirect products, unit component, covering groups.)
Week 3: The Lie algebra of a Lie group. (Left invariant vector fields, one-parameter subgroups, the exponential map.)
Week 4: The derivative of the exponential map. (Adjoint representation, Lie group structure on the tangent bundle of a Lie group, one-parameter subgroups of the tangent bundle group. Reconstruction of the local group structure from the Lie algebra structure.)
Week 5: Universal envelopping algebra. (Definition, construction. Poincaré-Birkhoff-Witt theorem)
Week 6: Hopf algebras and primitive elements. (Definitions, Hopf-algebra structure on the universal envelopping algebra and its primitive elements. Dynkin form of the Campbell-Baker-Hausdorff series)
Week 7: Fundamental theorems of Lie theory. (The fundamental theorems of Lie and Cartan’s theorem on closed subgroups of a Lie group.)
Week 8-10: The structure of Lie algebras. (Nilpotent, solvable and semisimple Lie algebras. Radical, nilradical, Theorems of Jacobson and Engel. Irreducible linear Lie algebras, reductive Lie algebras. Killing form and Cartan’s criteria for solvability and semisimplicity)
Week 11: Cohomology of Lie algebras. (Definition, Casimir operator, Whitehead’s theorems, applications)
Week 12: Ado’s theorem.
Text: M.M Postnikov: Lectures in Geometry: Lie Groups and Lie Algebras (Semester V)
36) COMMUTATIVE ALGEBRA
Course coordinator: Tamás Szamuely
Prerequisites: Basic Algebra, Homological Algebra
Books:
1. M. F. Atiyah, I. G. MacDonald: Introduction to Commutative Algebra. Addison-Wesley, 1969.
2. H. Matsumura: Commutative Ring Theory. Cambridge University Press, 1988.
3. D. Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry. Springer, 1995.
4. J.-P. Serre: Local Algebra. Multiplicities. Springer, 1999.
Commitment: 3 hours/week, 3 credits
Contents:
• Review of basic concepts about commutative rings. Chain conditions, Noetherian and Artinian rings and modules. Associated primes and primary decomposition.
• The prime spectrum of a ring, localisation.
• Basic dimension theory. The Krull dimension of a finite dimensional algebra over a ring.
• Integral extensions, integral closure. Structure of discrete valuation rings and Dedekind domains. Example: rings of algebraic integers in number fields and their extensions.
• Completion of a local ring, the associated graded ring.
• Application of methods of homological algebra to local rings.
37) ALGEBRAIC NUMBER THEORY
Lecturer: T. Szamuely
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: basic algebra(groups, rings and fields), basic function theory.
Course Level: introductory
Brief introduction to the course:
The course covers basic material on algebraic number fields. In the first part the point of view will be algebraic, the required commutative algebra will be introduced along the way. At the end of the course elementary analytic methods will also be presente.
The goals of the course:
The main goal of the course is to introduce students to basic concepts of algebraic number theory. Some key examples will also be presented, as well as glimpses at more advanced and recent results.
The learning outcomes of the course:
Students will gain basic insight into many of the fundamental concepts of modern number theory. This can serve as a motivation for learning more advanced topics (e.g. class field theory, arithmetic geometry, automorphic forms), and also as background for those wanting to apply algebraic number theory in other branches of mathematics.
More detailed display of contents:
Week 1: Introduction. Rings of integers in number fields.
Week 2: Dedekind rings, unique factorization of ideals.
Week 3: Finiteness of the class number, beginning of Minkowski theory.
Week 4: Minkowski theory (continued). Dirichlet’s unit theorem.
Week 5: Extensions of number fields I: ramification.
Week 6: Extensions of number fields II: completion.
Week 7: Applications: cyclotomic fields, inverse Galois problem for abelian groups.
Week 8: Zeta and L-functions of number fields.
Week 9: hebotarev’s density theorem.
Weeks 10-12: Additional topics.
Literature:
1. J. Neukirch, Algebraic Number Theory, Springer, 1999.
2. J. S. Milne, Algebraic Number Theory, course notes available at http://www.jmilne.org .
38) GEOMETRIC GROUP THEORY
Course Coordinators: Gabor Elek
Prerequisites: Undergraduate group theory. Real analysis.
Book: P. de la Harpe: Topics in Geometric Group Theory, Univ. of Chicago Press, 2000.
Commitment: 3 hours/week, 3 credits
Contents:
• Free groups, free products and amalgams. Group actions on trees.
• Finitely generated groups, volume growth, Cayley graphs, quasi- isometries, ends, boundaries and their invariance.
• Finitely presented groups. Rips complexes.
• Amenable groups.
• Nilpotent groups. Gromov's theorem on polynomial growth.
• Groups with Kazhdan's property (T). The expander problem.
• Bloch-Weinberger homologies and their applications.
• Hyperbolic groups. Gromov's boundary.
• Bounded harmonic functions on graphs and groups.
39) RESIDUALLY FINITE GROUPS
Course Coordinator: Peter Pal Palfy
Prerequisites: Undergraduate Group Theory
Books: W. Magnus: Residually finite groups (survey) and related papers.
Commitment: 3 hours/week, 3 credits
Contents:
• Residual properties of free groups; the theorems of Iwasawa and Katz, Magnus, Wiegold
• The theorem of G. A. Jones on proper group-varieties, the Magnus conjecture
• Residual properties of free products
• Polycyclic groups are residually finite
• Linear groups are residually finite
• Basic properties of residually finite groups, the solvability of the word, problem, hopficity
• The automorphism group of a residually finite group is residually finite
• Conjugacy separability and LERF groups
• The restricted Burnside problem; Hall-Higman reduction
• Grigorchuk groups
• Residually finite groups of finite rank
• Profinite completions
• Every abstract subgroup of finite index in a finitely generated pro-p, group is open; Serre's problem
• Subgroup growth of free groups and nilpotent groups
• Groups of intermediate subgroup growth
40) INVARIANT THEORY
Lecturer: Mátyás Domokos
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Basic Algebra I-II.
Course Level: PhD
Brief introduction to the course:
Provides an introduction to Invariant Theory.
The goals of the course:
Acquaint students with classical techniques and possible research topics.
The learning outcomes of the course:
Student should get a clear view how invariant theory is related to other mathematical areas they study, and should be able to apply its methods.
More detailed display of contents:
Week 1. Overview of basic problems.
Week 2. Polarization and restitution, the theorem of Weyl.
Week 3. Matrix invariants.
Week 4. Multisymmetric polynomials: generators and relations.
Week 5. Honogeneous systems of parameters and the nullcone
Week 6. Affine quotients.
Week 7. The Hilbert-Mumford criterion
Week 8. Projective quotients
Week 9. Binary forms, the Cayley-Sylvester formula
Week 10. Hilbert series via the Weyl integration formula
Week 11. Degree bounds for finite and reductive groups
Week 12. Separating invariants
Bibliography:
1. Claudio Procesi, Lie Groups -An Approach through Invariants and Representations, Springer, 2007.
2. S. Mukai, An Introduction to Invariants and Moduli, (Cambridge studies in advanced mathematics 81), Cambridge University Press, 2003.
3. H. Dersksen, G. Kemper: Computational invariant theory
41) SEMIGROUP THEORY
Lecturer:László Márki
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Basic Algebra I-III
Course Level: advanced
Brief introduction to the course:
Basic notions and several fundamental theorems of semigroup theory are presented, with a connection to formal languages.
The goals of the course:
The main goal of the course is to point at the ubiquity and the versatility of semigroups, showing notions and results which link semigroups with various kinds of mathematical structures.
The learning outcomes of the course:
The students will learn some of the basic constructions and techniques used in semigroup theory, as well as several theorems of fundamental importance.
More detailed display of contents:
1. Basic notions, semigroups of transformations, semigroups of binary relations, free semigroups, Green's equivalences.
2. Regular D-classes, regular semigroups, (0-)simple semigroups, principal factors.
3. Completely (0-)simple semigroups, Rees’s Theorem.
4. Completely regular and Clifford semigroups, semilattice decompositions, bands, varieties of semigroups and of bands.
5. Languages, syntactic monoids, pseudovarieties, Eilenberg's theorem.
6. Piecewise testable languages and Simon's theorem, star-free languages and Schützenberger's theorem.
7. Inverse semigroups, elementary properties, Wagner-Preston theorem, Brandt semigroups.
8. Partial symmetries, local structures, E-unitary covers, congruences on inverse semigroups.
9. The Munn semigroup, fundamental inverse semigroups, the P-theorem.
10. Free inverse semigroups, solution of the word problem in free inverse semigroups.
11. Commutative semigroups, semigroup of fractions, archimedean decomposition.
12. Finitely generated commutative semigroups, Rédei's theorem, Grillet's theorem.
Books:
1. P. A. Grillet: Semigroups, Marcel Dekker, 1995.
2. J. M. Howie: Fundamentals of Semigroup Theory, Oxford University Press, 1995.
3. M. V. Lawson: Inverse Semigroups, World Scientific, 1998.
4. J. E. Pin: Varieties of Formal Languages, North Oxford Academic, 1986
42) BASIC ALGEBRAIC GEOMETRY
Lecturer: Károly Böröczky, Jr or Tamás Szamuely
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: basic algebra (groups, rings and fields), basic point set topology
Course Level: introductory
Brief introduction to the course:
Basic concepts and theorems are presented on varieties over an algebraically closed field. The point of view will be algebraic, the required commutative algebra will be introduced along the way.
The goals of the course:
The main goal of the course is to introduce students to the most basic concepts of algebraic geometry, and to show how algebraic and geometric properties of varieties are interrelated. Some key examples will also be presented, as well as glimpses at more advanced and recent results.
The learning outcomes of the course:
Students will gain basic insight into the fundamental concepts of algebraic geometry. This can serve as a motivation for learning more advanced techniques used in current research (as presented e.g. in the Language of Schemes course), and also as background for those wanting to apply basic algebraic geometry in other branches of mathematics.
More detailed display of contents:
Week 1: Affine varieties, Nullstellensatz, morphisms.
Week 2: Rational functions and maps, dimension.
Week 3: Quasi-projective varieties, products, separatedness.
Week 4: Morphisms of projective varieties, main theorem of elimination theory. Grassmannians.
Week 5: Tangent spaces, smooth points, relation with regularity.
Week 6: Normal varieties, normalization.
Week 7: Birational maps, blowups.
Week 8: Birational maps of surfaces.
Week 9: Elementary intersection theory on surfaces.
Weeks 10: Embedded resolution of singularities for curves on surfaces.
Weeks 11-12: Additional topics.
Books:
1. I. R. Shafarevich, Basic Algebraic Geometry I, Springer, 1994.
2. M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
3. R. Hartshorne, Chapter 1 of Algebraic Geometry, Springer, 1977.
43) THE LANGUAGE OF SCHEMES
Lecturer:Tamás Szamuely
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: basic algebraic geometry
Course Level: intermediate
Brief introduction to the course:
Basic concepts and theorems are presented on Grothendieck’s schemes and their cohomology. Applications are given to the theory of algebraic curves and surfaces, as well as to the construction of geometric objects classifying them.
The goals of the course:
The main goal of the course is to introduce students to the modern techniques currently used in algebraic geometry, and to show how abstract concepts describe geometric properties. Some key examples will also be presented, as well as glimpses at more advanced and recent results.
The learning outcomes of the course:
Students will gain basic insight into the fundamental concepts of modern algebraic geometry. This can serve as a background for studying advanced topics, or even recent results from the literature.
More detailed display of contents:
Week 1: Definition of sheaves and schemes.
Week 2: First properties of schemes and morphisms.
Week 3: Quasi-coherent sheaves on schemes.
Week 4: Special classes of morphisms.
Week 5: Cohomology of quasi-coherent sheaves.
Week 6: Serre’s vanishing theorem, finiteness theorem for proper morphisms.
Week 7: Cohomology of curves, Riemann-Roch formula.
Week 8: Cohomology of surfaces, adjunction formula, Hodge index theorem.
Week 9: Base change theorems in cohomology.
Weeks 10-12: Techniques of construction in algebraic geometry, the Hilbert scheme.
Books:
1. I. R. Shafarevich, Basic Algebraic Geometry II, Springer, 1994.
2. R. Hartshorne, Chapter 1 of Algebraic Geometry, Springer, 1977.
3. D. Mumford, The Red Book of Varieties and Schemes, Springer, 1999.
44) GALOIS GROUPS IN GEOMETRY
Course coordinator: Tamás Szamuely
Prerequisites: Commutative Algebra, Galois Theory, basic topology and complex function theory
Books: Handouts
Commitment: 3 hours/week, 3 credits
Contents:
• Galois theory of fields (review). Infinite Galois extensions, Krull topology. Reformulation of Galois theory in terms of \'etale algebras.
• Cover(ing space)s in topology. Galois covers and group actions. The main theorem of Galois theory for covers, relation to the fundamental group. Construction of the universal cover. Locally constant sheaves and their homotopy classification, application to differential equations.
• Riemann surfaces, branched covers. The fundamental group of the punctured line. Basic properties of schemes. The algebraic fundamental group of a locally Noetherian scheme, relation with the topological theory. Galois groups as fundamental groups, homotopy exact sequence. Fundamental groups of Dedekind schemes, relation with Galois theory. The fundamental group of the projective line minus three points, Belyi's theorem
45) ALGEBRAIC CURVES AND JACOBIAN VARIETIES
Course Coordinator: Tamás Szamuely
Prerequisites: Language of Schemes course, Homological Algebra course.
Books:
1. R. Hartshorne: Algebraic Geometry. Springer, 1977.
2. G. Cornell, J. H. Silverman (eds.) Arithmetic Geometry. Springer, 1986.
3. J.-P. Serre: Algebraic Groups and Class Fields, Springer, 1988.
Commitment: 3 hours/week, 3 credits
Contents:
• Basic theory of algebraic curves: divisors, differentials, Riemann-Roch formula and applications. Covers of algebraic curves, the Hurwitz formula. Relation with the transcendental theory.
• Algebraic curves over finite fields, Weil's theorem on the ``Riemann Hypothesis''. Bounds on the number of rational points.
• The construction of Jacobian varieties. Applications (partly only surveyed): abelian covers of curves, geometric class field theory, Mordell's conjecture, anabelian geometry.
46) THE ARITHMETIC OF ELLIPTIC CURVES
Course Coordinator: Tamás Szamuely
Prerequisites: Basic Algebraic Geometry, Algebraic Number Theory.
Book: J. H. Silverman: The Arithmetic of Elliptic Curves. Springer, 1986.
Commitment: 3 hours/week, 3 credits
Contents:
• Geometry of elliptic curves: Weierstrass equations, group law, j-invariant, singular curves and degenerate laws.
• The structure of the group of points of an elliptic curve over a p-adic field, good and bad reduction.
• Rudiments of Galois cohomology, weak Mordell-Weil theorem.
• Heights on projective space over a global field, strong Mordell-Weil theorem.
• Torsors over an elliptic curve, Selmer and Tate-Shafarevich groups, obstructions to the Hasse principle.
• Elliptic curves over finite fields, “Riemann Hypothesis”'.
• Advanced topics (optional): moduli of elliptic curves, semi-stable reduction, Weil curves, application to Fermat's Last Theorem (survey).
47) HODGE THEORY
Course Coordinator: Tamás Szamuely
Prerequisites: Basic Algebraic Geometry, Basic Differential Geometry, Functional Analysis, Algebraic Topology and Homological Algebra
Books:
1. P. Griffiths and J. Harris: Principles of algebraic geometry, Wiley Classic Library,1994.
2. R. O. Wells: Differential analysis on complex manifolds, Graduate texts in mathematics, Springer-Verlag, 1980.
Commitment: 3 hours/week, 3 credits
Contents:
• Review of complex manifolds, metrics, connections and curvature, respectively of Hodge * operator and Laplace operator;
• Kahler manifolds, DeRham cohomology, Dolbeault cohomology; Harmonic forms, Hodge theorem, Serre duality, Kunneth formula, Hodge decomposition, Lefschetz decomposition;
• Intersection form and polarization properties, the Hodge-Riemann bilinear relations;
• Kodaira vanishing theorem, Kodaira embedding theorem;
• Lefschetz theorem on hyperplane sections;
• Hodge conjecture, Lefschetz theorem on (1,1) classes;
• Algebraic DeRham complex, differential forms with logarithmic singularities.
48) INTRODUCTION TO CLASSIFICATION THEORY
Course Coordinator: Károly Böröczky, Jr.
Prerequisites: Basic Algebraic Geometry, The Language of Schemes.
Books:
1. H. Clemens, J. Kollar and S. Mori, Higher Dimensional Complex Geometry, Astérisque vol. 166, Soc. Math. France, 1988.
2. R. Hartshorne: Algebraic Geometry, Springer 1977.
Commitment: 3 hours/week, 3 credits
Contents:
• Classification of curves, birational maps of surfaces - an overview;
• Minimal surfaces, classification of surfaces, canonical surfaces;
• Minimal model conjecture, Abundance conjecture;
• Cone of curves, Kleiman's criterion for ampleness, Cone theorem, Contraction theorem;
• Flips, Flops, Flip conjecture, Minimal Model Program;
• Singularities, resolutions, discrepancy;
• Log Minimal Model Program;
• Classification of 3 dimensional terminal singularities;
• Minimal Model Program in 3 dimensions;
• Applications
49) TORIC VARIETIES
Course Coordinator: Károly Böröczky, Jr.
Prerequisites: Linear Algebra, Basic Algebraic Geometry
Book: W. Fulton: Introduction to toric varieties. Princeton University Press, Princeton, NJ, 1993.
Commitment: 3 hours/week, 3 credits
Contents:
• Rational cones and Fans.
• Affine toric varieties, toric varieties, characterization of the projective and the complete toric varieties.
• The moment map.
• Resolution of singularities in toric setting.
• Invariant line bundles.
• Toric singularities.
• Intersection numbers.
• The Chow ring of a toric variety.
• Counting lattice points and the Hirzebuch-Riemann-Roch formula.
• About the coefficients of the Ehrhart formula.
• The Alexandrov-Fenchel inequality and the Hodge intersection inequality.
• Some applications of toric varieties to mirror symmetry.
50) DYNAMICAL SYSTEMS
Course Coordinator: Domokos Szász
Prerequisites: Ergodic Theory
Books:
1. I.P. Cornfeld and S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, 1982
2. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, 1995
Commitment: 3 hours/week, 3 credits
Contents:
• Kesten-Furstenberg theorem, Kingman's subadditive ergodic theorem.
• Oseledec' multiplicative ergodic theorem, Lyapunov exponents.
• Thermodynamic formalism, Markov-partitions.
• Chaotic maps of the interval, expanding maps, Markov-maps.
• Chaotic conservatice systems. The ergodic hyothesis. Billiards, hard ball systems. The standard map.
• Chaotic non-conservative (dissipative) systems. Strange attractors. Fractals. Exponenets and dimensions. Map of the solenoid.
• Stability: invariant tori and the Kolmogorov-Arnold-Moser theorem.
• Anosov-maps. Invariant manifolds. SRB-measure.
51) APPROXIMATION THEORY
Course Coordinator: András Kroó
Prerequisites: Real and Functional Analysis
Book: R. DeVore and G. Lorentz, Constructive Approximation, Springer, 1991.
Commitment: 3 hours/week, 3 credits
Contents:
Stone-Weierstrass theorem, positive linear operators, Korovkin theorem. Best Approximation (Haar theorem, Chebyshev polynomials, Best approximation in different norms). Polynomial inequalities (Bernstein, Markov, Remez inequalities). Splines (B-splines, Euler and Bernoulli splines, Kolmogorov-Landau inequality). Direct and connverse theorems of best approximation (Favard, Jackson, and Stechkin Theorems). Approximation by linear operators (Fourier series, Fejér operators, Bernstein polynomials). Müntz theorem.
52) PARTIAL DIFFERENTIAL EQUATIONS
Lecturer: Gheorghe Morosanu
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Real and Complex Analysis, Functional Analysis
Course Level: intermediate
Brief introduction to the course:
After a short introduction into the main typical problems, some of the most important methods and techniques are described, including both classical and modern aspects of the theory of partial differential equations. Some applications are included to illustrate the theoretical results.
The goals of the course:
The main goal of the course is to introduce students to the main topics and methods of the theory of partial differential equations.
The learning outcomes of the course:
The students will learn some basic tools, which are useful for applied mathematicians, engineers, physicists. Even more, they will learn how to use these tools in solving specific problems associated with various partial differential equations.
More detailed display of contents:
1. Physical models and typical examples of partial differential equations (PDEs).
2. First order linear PDEs.
3. Second order linear PDEs. Classification, canonical forms, characteristics.
4. Elliptic equations. Lax-Milgram lemma, the variational approach.
5. Eigenvalues and eigenvectors of the Laplace operator.
6. The heat equation in the whole space. Fundamental solution, the Cauchy problem.
7. The Dirichlet boundary value problem associated with the heat equation. The Fourier method.
8. The wave equation. The solution of the Dirichlet boundary value problem by the Fourier method.
9. The semigroup approach for linear parabolic and hyperbolic PDE’s. Introduction to the linear semigroup theory.
10. Strongly continuous semigroups of bounded linear operators.
11. Applications to linear PDEs.
12. Nonlinear PDE’s.
Books:
1. L.C. Evans, Partial Differential Equations, Graduate Studies in Math. 19, AMS, Providence, Rhode Island, 1998.
2. A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, 1969.
3. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.
53) NONLINEAR EVOLUTION EQUATIONS AND APPLICATIONS
Course Coordinator: Gheorghe Morosanu
Prerequisites: Real and Complex Analysis, Functional Analysis
Books:
1. H. Brezis, Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North Holland, Amsterdam, 1973.
2. G. Morosanu, Nonlinear Evolution Equations and Applications, Reidel, 1988.
Commitment: 3 hours/week, 3 credits
Contents:
• Preliminaries of linear and nonlinear functional analysis
• Existence and regularity of solutions to evolution equations in Hilbert spaces
• Boundedness of solutions
• Stability of solutions. Strong and weak convergence results. Periodic forcing. The asymptotic dosing problem
• Applications: Infinite delay equations, nonlinear parabolic problems, hyperbolic systems. Specific examples in viscoelasticity, heat conduction theory, electrical and electronic engineering, hydraulics, a.o.
54) FUNCTIONAL METHODS IN DIFFERENTIAL EQUATIONS
Lecturer: Gheorghe Morosanu
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: Real and Complex Analysis, Partial Differential Equations
Course Level: advanced course for PhD students
Brief introduction to the course:
In recent years functional methods have become central to the study of many mathematical problems, in particular of those described by differential equations. Significant progress have been made in different areas of functional analysis, including the theory of accretive and monotone operators (founded by G. Minty, F. Browder, H. Brezis) and the nonlinear semigroup theory (developed by Y. Komura, T. Kato, H. Brezis, M.G. Krandall, A. Pazy, a.o.). As a consequence there has been significant progress in the study of nonlinear differential equations associated with monotone or accretive operators. Our aim here is to emphasize the importance of functional methods in the study of a broad range of boundary value problems. Many applications will be discussed in detail.
The course is designed for students oriented to Applied Mathematics.
The goals of the course:
The main goal of the course is to introduce students to some important functional methods and to show their applicability to various boundary value problems. We intend to discuss specific models in appropriate functional frameworks. Using functional methods, we sometimes are able to create new models which are more general than the classical ones and better describe concrete physical phenomena.
The learning outcomes of the course:
The students will learn some basic functional methods, which are very useful for applied mathematicians, economists, engineers, physicists. Even more, they will learn how to use these tools in solving specific problems.
More detailed display of contents:
Week 1: Function spaces, distribution spaces (Lebesgue spaces, scalar distributions, Sobolev spaces, vectorial distributions)
Week 2: Monotone operators, convex functions, subdifferentials (definitions, basic properties, examples)
Week 3: Operator semigroups, linear and nonlinear evolution equations (linear semigroups, generation results, nonlinear semigroups, existence and uniqueness results for evolution equations)
Week 4: Elliptic boundary value problems (formulation, assumptions, existence results, applications)
Weeks 5-6: Parabolic problems with algebraic boundary conditions (formulation, assumptions, existence and uniqueness results, stability, applications)
Weeks 7-8: Parabolic problems with dynamic boundary conditions (formulation, assumptions, existence and uniqueness results, stability, applications)
Weeks 9-10: Hyperbolic problems with algebraic boundary conditions (formulation, assumptions, existence and uniqueness results, stability, applications)
Weeks 11-12: Hyperbolic problems with dynamic boundary conditions (formulation, assumptions, existence and uniqueness results, stability, applications)
Book:
1. Morosanu, G., Functional Methods in Differential Equations, Chapman&Hall/CRC, 2002 and some chapters of other books
55) COMPLEX MANIFOLDS
Course Coordinator: Róbert Szoke
Prerequisites: Basic topology, real analysis in several variables, functional analysis and complex analysis in one variable.
Books:
1. K. Kodaira: Complex manifolds, Holt, 1971.
2. R.O. Wells: Differential analysis on complex manifolds, Springer, 1979.
Commitment: 3 hours/week, 3 credits
Contents:
• Basic definitions, examples and constructions.
• Differential forms on manifolds, (p,q) forms. Tangent bundle, vector bundles, bundle valued forms and Dolbeault cohomology groups, metrics, Hodge * operator.
• Sobolev spaces of sections, differential operators between vector bundles and their adjoint, symbol. Pseudo-differential operators.
• Parametrix for elliptic differential operators, fundamental decomposition theorem for self-adjoint elliptic operators and complexes. Harmonic forms, complex Laplacian, Kahler manifolds, Hodge decomposition theorem on compact Kahler manifolds.
56) GEOMETRIC ANALYSIS
Course Coordinator: Jerry L. Kazdan
Prerequisites: Students will be assumed to know basic analysis---but all the background in PDE and geometry will be given as needed in the course.
Books:
1. Evans, Lawrence C., “Partial Differential Equations”, in the series Graduate studies in mathematics, v. 19, American Math. Society, 1998.
2. Robert Hardt and Michael Wolf (editors), "Nonlinear Partial Differential Equations in Differential Geometry" - AMS, 1996, 339 pp.
3. Kazdan, Jerry L. "Lecture Notes on Applications of Partial Differential Equations to Some Problems in Differential Geometry", (available from the Internet at: http://www.math.upenn.edu/~kazdan/)
4. Li, Peter, "Lecture Notes on Geometric Analysis" (available from the Internet at: http://www.math.uci.edu/faculty/pli.html)
Commitment: 3 hours/week, 3 credits
Contents:
Some applications of analysis, particularly partial differential equations (PDE), to some problems in geometry. For instance, we will prove the Hodge decomposition theorem, Newlander-Nirenberg theorem, the spectrum of the Laplacian, as well as some applications involving nonlinear PDE's arising in Riemannian geometry. The course will be flexible enough to include some topics that the class would like to cover.
57) BLOCK DESIGNS
Lecturer: Tamás Szınyi
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: introductory course to combinatorics and algebra
Course Level: intermediate
Brief introduction to the course:
After a quick introduction to the theory of block designs and strongly regular graphs, the main emphasis will be on the interplay between these two, and applications to other areas of mathematics like coding theory, group theory or extremal graph theory. Several techniques will be presented varying from combinatorial and geometrical methods to algebraic ones, like eigenvalues, polynomials, linear algebra and characters. Because of the quick introduction at the beginning, the lectures should be useful for both those not familiar with the subject and those who have already attended an introductory course on symmetric combinatorial structures.
The goals of the course:
Besides introducing the audience to areas with nice open problems, the main goal is to show different proof techniques in combinatorics.
The learning outcomes of the course:
The students will become familiar with different areas of combinatorics having connections to symmetric structures, like codes, difference sets, cages. Some open problems will be presented almost every week. Learning different proof techniques might also be useful even for students planning to do research in another area of mathematics.
More detailed display of contents:
1. Designs: basic definitions, existence, examples, square designs, extendability, Hadamard matrices and designs, projective planes, Latin squares, sharply two-transitive permutation sets.
2. Strongly regular graphs: definitions, examples, integrality conditions, necessary conditions for the existence.
3. The existence of non-trivial t-designs with t>5. Teirlinck’s theorem
4. Witt designs and Mathieu groups.
5. Quasi-residual designs. The Hall-Connor theorem.
6. Designs and projective geometries.
7. Difference sets. Multiplier theorems.
8. Basics of coding theory.
9. Codes and designs.
10. 1-factorizations of complete graphs and designs, Baranyai’s theorem.
11. Moore graphs. Generalized polygons, the Feit-Higman theorem.
12. Moore graphs and (k,g)-graphs. Constructions and bounds.
Books:
1. J. H. Van Lint, R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001.
2. P. J. Cameron, J. H. van Lint, Designs, Graphs, Codes and their Links, Cambridge University Press, 1991.
58) HYPERGRAPHS, SET SYSTEMS, INTERSECTION THEOREMS
Lecturer: Gyula Katona
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites:-
Course Level: advanced
Brief introduction to the course:
It gives the most important results and methods in extremal set theory. Largest inclusion-free and intersecting families, their combinations. Minimum size of the shadow. Methods: shifting, transformation, permutation, cycle, algebraic. The course is suggested to students oriented to combinatorics and computer science.
The goals of the course:
To show the main results and methods of the theory.
The learning outcomes of the course:
The students will know the most important results of the theory, they will be able follow the literature, apply these results in practical cases and create new results of similar nature.
More detailed display of contents:
Week 1: Inclusion-free families, antichains, 3 proofs of the Sperner theorem
Week 2: LYM (YBLM)-inequality, case of equality.
Week 3: Maximum size of the intersecting families.
Week 4: Erdős-Ko-Rado theorem for the uniform intersecting families. Cycle method.
Week 5: Shifting method, properties preserved by shifting. Left shifted families.
Week 5: Shifting method for the Erdős-Ko-Rado theorem.
Week 6: Minimum of the size of the shadow relative to an l-intersecting family.
Week 7: Maximum size of an l-intersecting family.
Week 8: Minimum size of the shadow.
Week 9: Discrete isoperimetric theorem.
Week 10: The algebraic method. “Even city”.
Week 11: Families with intersections of one fixed size. Erdős-DeBruijn theorem.
Week 12: Largest families with intersection of sizes in a given subset of integers. Ray-Chaudhury-Wilson theorem.
Book: Konrad Engel: Sperner Theory,
59) SELECTED TOPICS IN GRAPH THEORY
Course Coordinator: Ervin Gyori
Prerequisites: -
Commitment: 3 hours/week, 3 credits
Contents:
The subject of this course changes from time to time depending on the fields of interest of students.
60) FINITE PACKING AND COVERING
Course Coordinator: Károly Böröczky, Jr.
Prerequisites: -
Books: handouts
Commitment: 3 hours/week, 3 credits
Contents:
Planar arrangements: Translative packings of a centrally symmetric convex domain, the Oler inequality. Translative coverings by a centrally symmetric convex domain, the Fejes Tóth inequality. The optimal packing of equal Euclidean circles (G. Wegner). Density inside r-convex domains for arrangements of equal circles in the hyperbolic plane. The extremal perimeter for packings and coverings by congruent convex domains. The maximal perimeter for coverings by equal Euclidean circles.
The Hadwiger number in the plane. Clouds in the plane. Higher dimensional arrangements: Optimal arrangements of balls in the spherical
space. The Sausage Conjecture and Theorem for Euclidean ball packings. The extremal mean width for packings and coverings by congruent convex bodies. The Hadwiger number in high dimensions. Clouds in high dimensions. Parametric density for translative arrangements. The Wulff shape for translative lattice packings.
61) PACKING AND COVERING
Course Coordinator: Gabor Fejes Toth
Prerequisites: Geometry, Basic Linear Algebra
Books:
1. L. Fejes Tóth: Regular figures, Pergamon Press, 1964.
2. J. Pach and P.K. Agarwal: Combinatorial geometry, Academic Press, 1995.
3. C.A. Rogers: Packing and covering, Cambridge University Press, 1964.
Commitment: 3 hours/week, 3 credits
Contents:
Theorem of Groemer concerning the existence of densest packings and thinnest coverings. Dirichlet cells, Delone triangles. Theorems of Thue and Kershner concerning densest circle packings and thinnest circle coverings. Packing and covering of incongruent circles. Theorems of Dowker, generalized Dirichlet cells. Packing and covering of congruent convex discs: theorems of C.A. Rogers and L.
Fejes Tóth. The moment theorem. Isoperimetric problems for packings and coverings. Existence of dense packings and thin coverings in the plane: p-hexagons, extensive parallelograms, theorems of W. Kuperberg, D. Ismailescu, G. Kuperberg and W. Kuperberg. The theorem of E. Sas. Multiple packing and covering of circles. The problem of Thammes; packing and covering of caps on the 2-sphere. The moment theorem on S2, volume estimates for polytopes containing the unit ball. Theorem of Lindelöff, isoperimetric problem for polytopes. Packing and covering in the hyperbolic plane.
Packing of balls in Ed the method of Blichfeldt, Rogers' simplex bound. Packing in Sd, the linear programming bound. Theorem of Kabatjanskii and Levenstein. Covering with balls in Ed the simplex bound of Coxeter, Few and Rogers. Existence of dense lattice packings of symmetric convex bodies: the theorem of Minkowski-Hlawka.
Packing of convex bodies, difference body, the theorem of Rogers and Shephard concerning the volume of the difference body. Construction of dense packings via codes. The theorem of Rogers concerning the existence of thin coverings with convex bodies. Approximation of convex bodies by generalized cylinders, existence of thin lattice coverings with convex bodies
62) CONVEX POLYTOPES
Course Coordinator: Imre Barany
Prerequisites: Basic linear algebra
Book: G.M. Ziegler: Lectures on polytopes. Springer, 1995.
Commitment: 3 hours/week, 3 credits
Contents:
Rationale: Introducing the basic combinatorial properties of convex polytopes. Polytopes as convex hull of finite point sets or intersections of halfspaces. Faces of polytopes. Examples: Simplicial, simple, cyclic and neighbourly polytopes. Polarity for polytopes. The Balinski theorem. Discussion of the Steinitz theorem for three polytopes. Realizability using rational coordinates. Gale transform and polytopes with few vertices. The oriented matroid of a polytope Shelling Euler-Poincaré formula hvector of a simplicial polytope, Dehn-Sommerfield equations Upper bound theorem Stresses Lower bound theorem Weight algebra Sketch of the proof of the g-theorem.
63) COMBINATORIAL GEOMETRY
Lecturer: Endre Makai or Géza Tóth
No. of Credits: 3 and no. of ECTS credits 6
Prerequisites: No specific prerequisite, basic knowledge of graph theory and linear algebra is needed.
Course Level: PhD
Brief introduction to the course:
Convexity, separation, Helly, Radon, Ham-sandwich theorems, Erdős-Szekeres theorem and its relatives, incidence problems, the crossing number of graphs, intersection patterns of convex sets, Caratheodory and Tverberg theorems, order types, Same Type Lemma, the k-set problem
The goals of the course:
The goal of the course is to introduce some of the most important problems, results, and ideas of combinatorial geometry.
The learning outcomes of the course:
Getting familiar with combinatorial geometry, understanding the main problems, tools, and relations to some other parts of mathematics.
More detailed display of contents:
week 1: convexity, linear and affine subspaces, separation
week 2: Radon' theorem, Helly's theorem, Ham-sandwich theorem
week 3: Erdős-Szekeres theorem, upper and lower bounds
week 4: Erdős-Szekeres-type theorems, Horton sets
week 5: Incidence problems
week 6: crossing numbers of graphs
week 7: Intersection patterns of convex sets, fractional Helly theorem, Caratheodory theorem
week 8: Tverberg theorem, order types, Same Type Lemma
week 9-10: The k-set problem, duality, k-level problem, upper and lower bounds
week 11-12: further topics, according to the interest of the students
Book: J. Matousek: Lectures on Discrete Geometry, Springer, 200
64) GEOMETRY OF NUMBERS
Course Coordinator: Zoltan Furedi
Prerequisites: Geometry, Linear Algebra, Analysis
Books:
1. J.W.S Cassels: An introduction to the geometry of numbers, Springer, Berlin, 1972.
2. P.M. Gruber, C.G. Lekkerkerker: Geometry of numbers, North-Holland, 1987.
3. L. Lovász: An algorithmic theory of numbers, graphs, and convexity, CBMS-NSF regional conference series, 1986.
Commitment: 3 hours/week, 3 credits
Contents:
Lattices, sublattices, bases, determinant of a lattice. Convex bodies, elements of the Brunn-Minkowski theory, duality, star bodies. Selection theorems of Blaschke and Mahler. The fundamental theorem of Minkowski, and its generalizations: theorems of Blichfeldt, van der Corput. Successive minima, Minkowski's second theorem. The Minkowski-Hlawka theorem. Reduction theory, Korkine-Zolotarev basis, LLL basis reduction. Connections to the theory of packings and coverings Diophantine approximation: simultaneous, homogeneous, and inhomogeneous, theorems of Dirichlet, Kronecker, Hermite, Khintchin Short vector problem, nearest lattice point problem Applications in combinatorial optimization. The flatness theorem. Covering minima Algorithmic questions, convex lattice polytopes.
65) STOCHASTIC GEOMETRY
Course Coordinator: Imre Bárány
Prerequisites: Geometry, linear algebra, analysis, basic notions of probability theory
Books:
1. L.A. Santalo, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Appl., Vol 1. Addison-Wiley, 1976.
2. J. Pach and P.K. Agarwal, Combinatorial geometry, Academic Press, 1995.
3. C.A. Rogers, Packing and covering Cambridge University Press, 1964.
Commitment: 3 hours/week, 3 credits
Contents:
• Space of lines, measures on the space of lines
• Spaces, groups, measures, intersection formulae
• Minkowski addition and projections
• Lines and flats through convex bodies, the Crofton formulae
• Random polytopes, approximation by random polytopes, expectation of the deviation in various measures
• Connections to floating bodies and affine surface area, extremal properties of balls and polytopes
• Random methods in geometry: the Erdos-Rogers theorem, the Johnson-Lindenstrauss theorem, Dvoretzki's theorem, etc
• Applications in computational geometry
66) BRUNN-MINKOWSKI THEORY
Course Coordinator: Endre Makai
Prerequisites: Geometry, linear algebra, general topology, analysis
Books:
1. T. Bonnesen, W. Fenchel, Theory of convex bodies, BSC Associates, Moscow, Idaho, 1987.
2. R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge Univ. Press, Cambridge, 1993.
Commitment: 3 hours/week, 3 credits
Contents:
• Isoperimetric inequality in the plane, sharpening with the inradius.
• Distance function.
• Support properties, support function.
• Minkowski sum, Blaschke-Hausdorff distance.
• Blaschke selection theorem.
• Almost everywhere differentiability of convex functions.
• Polyhedral approximation.
• Cauchy surface formula.
• Steiner symmetrization, isoperimetric inequality via it.
• Mixed volumes.
• Brunn-Minkowski inequality. Minkowski's inequality for mixed volumes, isoperimetric inequality
67) HYPERBOLIC MANIFOLDS
Course Coordinator: Balazs Csikos
Prerequisites: Basic Euclidean and projective geometry, elements of group theory and algebraic topology (fundamental groups and covering spaces), basic concepts of differential geometry.
Books:
1. R. Benedetti, C.~Petronio, Lectures on Hyperbolic Geometry, Springer, 1992
2. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, 1994
Commitment: 3 hours/week, 3 credits
Contents:
• Hyperbolic space. Overview of the projective, quadratic form, and conformal models. Isometries and groups of isometries.
• Hyperbolic manifolds. Hyperbolic structures, developing and holonomy, completeness. Discrete groups of isometries of hyperbolic space. The case of dimension two.
• Constructing hyperbolic manifolds. Fundamental polyhedra and the Poincaré theorems. Some arithmetic constructions.
• Mostow Rigidity. Extending quasi-isometries. The Gromov-Thurston proof of the rigidity theorem for closed hyperbolic manifolds.
• Structure of hyperbolic manifolds. Margulis' Lemma and the thick-thin decomposition of complete hyperbolic manifolds of finite volume.
• Thurston's hyperbolic surgery theorem. The space of hyperbolic manifolds. Properties of the volume function. Dehn surgery on three-manifolds and Thurston's theorem.
• The geometrization conjecture. Topology of three-manifolds: geometric structures and the role of hyperbolic geometry in Thurston's theory.
68) CHARACTERISTIC CLASSES
Course Coordinator: Richárd Rimányi
Prerequisites: Homology theory
Books:
1. J. Milnor, J. Stasheff: Characteristic Classes; Ann. Math. Studies 76, Princeton UP, 1974
2. D. Husemoller: Fibre Bundles; McGraw-Hill, 1966
3. R. M. Switzer: Algebraic Topology-Homotopy and Homology; Springer, 1975
Commitment: 3 hours/week, 3 credits
Contents:
• Differentiable manifolds, maps. Vector bundles. Algebraic manipulations on vector bundles. Examples. Pullback, universal bundle.
• Stiefel-Whitney classes- axiomatic approach. Computations, examples. Application to differentiable topology: embeddings, immersions. Review on Thom polynomial theory and geometric representations of Stiefel-Whitney classes.
• Orientability, Euler class, Almost complex structures, Chern classes. Pontryagin classes.
• Existence questions (review): Cohomology operations; Schubert cells, Schubert calculus; differential geometrical approach.
• Thom isomorphism, Poincaré duality. Applications.
• Characteristic numbers. Unoriented and oriented cobordism groups, computations. Signature formulas.
69) SINGULARITIES OF DIFFERENTABLE MAPS: LOCAL AND GLOBAL THEORY
Course Coordinator: Richárd Rimányi
Prerequisites: Multivariable calculus, algebra.
Books:
1. V. Arnold, S. Gusein-Zade, A. Varchenko: Singularities of Differentiable Maps,Vol. I.; Birkhauser, 1985
2. J. Martinet: Singularities of Differentiable Functions and Maps; London Math.Soc. Lecture Notes Series 58, 1982
3. C.T.C. Wall: Proceedings of Liverpool Singularities, Sypmosium I.; SLNM 192,1970
Commitment: 3 hours/week, 3 credits
Contents:
• Examples and interesting phenomena about the singularities of differentiable maps. Connection with physics, catastrophes.
• The notion of map germs and jets; the ring of germs of differentiable functions. Modules over this ring. Weierstrass-Malgrange-Mather preparation theorem. Applications.
• Σi singularities. Thom-Boardman singularities (examples: fold, cusp, swallow-tail, umbilics), Thom's transversality theorems.
• Useful group actions in singularity theory: A, K. Stability and infinitesimal stability. Finitely determined germs. Classification of stable germs by local algebra. The nice dimensions.
70) FOUR MANIFOLDS AND KIRBY CALCULUS
Lecturer: Andras I. Stipsicz
No. of Credits: 3 and no. of ECTS credits 6
Prerequisites: Introductory topology, Introductory algebraic topology
Course Level: advanced
Brief introduction to the course:
The course introduces modern techniques of differential topology through handle calculus, and pays special attention to the description of 4-dimensional manifolds. We also show how to manipulate diagrams representing 4-manifolds. Smooth invariants of 3- and 4-manifolds (Heegaard Floer invariants and Seiberg-Witten invariants) will be also discussed.
The goals of the course:
The aim is to get a working knowledge of all basic (algebraic) topologic notions such as homology, cohomology theory, the theory of knots and handlebodies and some aspects of differential geometry through the rich theory of 4-manifolds. This discussion quickly leads to some important and unsolved questions in the field.
The learning outcomes of the course:
A solid working knowledge with the basic concepts and tools of algebraic and differential topology, skills that can be applied in a wide variety of branches of mathematics.
More detailed display of contents:
Week 1: Knots in the 3-space
Week 2: Invariants of knots, the Alexander and the Jones polynomial
Week 3: Morse theory, handle decompositions
Week 4: Decompositions of 3-manifolds: Heegaard diagrams
Week 5: Decompositions of 4-manifolds: Kirby diagrams
Week 6: Knots in 3-manifolds and Heegaard diagrams
Week 7: Surfaces in 4-manifolds; Freedman’s theorem
Week 8: New invariants of knots: grid homology
Week 9: Heegaard Floer invariants of 3-manifolds (combinatorial approach)
Week 10: Further structures on Heegaard Floer groups
Week 11: Seiberg-Witten invariants
Week 12: Basic properties of Seiberg-Witten invariants
Lecture notes : Milnor: Morse theory
Milnor: The h-cobordism theorem
Gompf-Stipsicz: 4-manifolds and Kirby calculus
Lickorish: An introduction to knot theory
71) SYMPLECTIC MANIFOLDS, LEFSCHETZ FIBRATION
Lecturer: Andras I. Stipsicz
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: Introductory topology, Introductory algebraic topology
Course Level: (for those programs who have 2 year MAs, the programs may be divided into levels – ie level 1 credits in the first year etc; or MA and PhD; or PhD only year 1 etc) MA and PhD
Brief introduction to the course:
We will discuss symplectic and contact manifolds, with a special emphasis on dimensions four and three. By results of Donaldson, Gompf and Giroux, the topological counterparts of these structures are Lefschetz fibrations and open book decompositions. In the study of these structures we need to examine mapping class group of surfaces (closed and with nonempty boundary).
The goals of the course:
The aim is to get a working knowledge of basic notions of symplectic and contact topology, and introduce the concepts of Lefschetz fibrations and open book decompositions. If time permits, we will also discuss Floer homologies.
The learning outcomes of the course:
A solid working knowledge with the basic concepts and tools of symplectic and contact topology.
More detailed display of contents:
Week 1: Linear symplectic theory
Week 2: Symplectic and contact manifolds
Week 3: Almost complex and almost contact structures
Week 4: Constructions of symplectic manifolds
Week 5: Contact surgery and Stein manifolds
Week 6: Manifolds with no symplectic structure
Week 7: Mapping class groups and their presentations
Week 8: Lefschetz pencils and fibrations, elliptic fibrations
Week 9: Open book decompositions
Week 10: The Giroux correspondance
Week 11: Donaldson’s almost holomorphic technique, existence of Lefschetz fibrations
Week 12: Stein manifolds
Lecture notes : McDuff-Salamon: Introduction to Symplectic Topology
Gompf-Stipsicz: 4-manifolds and Kirby calculus
Ozbagci-Stipsicz: Surgery on contact 3-manifolds and Stein surfaces
72) ADVANCED INTERSECTION THEORY
Course Coordinator: Tamas Szamuely
Prerequisites: Basic Algebraic Geometry and Language of Schemes courses. Familiarity with Local Algebra is an advantage.
Book: W. Fulton: Intersection Theory. Springer, 1986.
Commitment: 3 hours/week, 3 credits
Contents:
• Cycles on algebraic varieties, Chow groups. Functorial properties of Chow groups.
• Construction of an intersection product on cycles via moving lemmas and/or Fulton's deformation technique.
• Vector bundles and their characteristic classes. Splitting principle.
• The Grothendieck K-group (resp. G-group) of locally free (resp. coherent) sheaves. Comparison of the two for regular projective schemes (by applying methods of local algebra: finite locally free resolutions, Koszul complexes). Topological filtration on K(X), relation to Chow groups.
• The Grothendieck-Riemann-Roch theorem for smooth varieties, with applications.
• A survey of advanced topics (optional): relation to higher K-theory, motivic cohomology, Arakelov geometry.
73) DESCRIPTIVE SET THEORY
Course Coordinator: Istvan Juhasz
Prerequisites: -
Books:
1. K. Kuratowski: Topology, Academic Press, 1968.
2. A.S. Kechris: Classical Descriptive Set Theory, Springer, 1995.
Commitment: 3 hours/week, 3 credits
Contents:
Borel, analytic, projective sets, universality, reduction, separation theorems, ranks, scales, games, axiom of determinancy, large cardinals, trees.
Forcing.
74) ADVANCED SET THEORY
Lecturer: Istvan Juhasz and /or Soukup Lajos
No. of Credits: 3 and no. of ECTS credits 6
Prerequisites:
Course Level: advanced
Brief introduction to the course:
The past decades have seen a spectacular development in set theory, both in applying it to other fields (like topology and analysis) and mainly as an independent discipline.
The goals of the course:
Our aim is to familiarize the students with the latest developments within set theory and thereby give them a chance to do independent study and research of the many open problems of set theory.
More detailed display of contents:
Part I: Iterated forcing and preservation theoreme
Weeks 1-6. Finite support iterations and Martin's Axiom. Countable support iterations and PFA.
Part II. Combinatorial set theory
Weeks 7-9. Combinatorial set-theory and applications to topology. Large cardinals. Basic pcf theory with applications to algebra and to topology.
Part III. Set theory of the reals
Weeks 10-12. ZFC results and forcing constructions. Determinacy, infinite games and combinatorics.
Books:
1. T. Bartoszynski and H. Judah, Set theory on the structure of the real line, A K Peters, 1995.
2. Thomas Jech, Set theory, Spinger-Verlag, 1997.
3. István Juhász, Cardinal functions in topology - ten years later. Amsterdam: Mathematisch Centrum, 1980.
4. Akihiro Kanamori, Higher Infinite, Springer-Verlag, 1994.
5. Kenneth Kunen: Set theory. An introduction to Independence Proofs, Elsevier, 1999.
75) LOGICAL SYSTEMS
Course Coordinator: Andreka Hajnal
Prerequisites: Introduction to Mathematical logic.
Books:
1. J. Barwise and S. Feferman, editors, Model-Theoretic Logics, Springer-Verlag, Berlin, 1985.
2. W.J. Blok and D.L. Pigozzi: Algebraizable Logics, Memoirs AMS, 77, 1989.
3. L. Henkin, J.D. Monk, and A. Tarski: Cylindric Algebras, North-Holland, Amsterdam, 1985.
Commitment: 3 hours/week, 3 credits
Contents:
Establishing a meta-theory for investigating logical systems (logics for short), the concept of a general logic, some distinguished properties of logics. Filter-property (syntactical) substitution property. Semantical substitution property. Structurality. Algebraizability. Algebraization of logics. Linden-baum-Tarski algebras. Characterization theorems for completeness, soundness and their algebraic counterparts; concepts of compactness and their algebraic counterparts; definability properties and their algebraic counterparts; properties and their algebraic counterparts; omitting types properties and their algebraic counterparts. Applications, examples; propositional logic; (multi-)modal logical systems; dynamic logics (logics of actions, logics of programs etc.)
Connections with abstract model theory; elements of Abstract Model Theory (AMT); absolute logics; Abstract Algebraic Logic (AAL); Lindstrom's theorem in AMT versus that in AAL
76) SET-THEORETIC TOPOLOGY
Course Coordinator: István Juhász
Prerequisites: Modern Set-Theory, Advanced Set-Theory
Books:
1. K. Kunen, J.E. Vaughan, Handbook of Set-Theoretic Topology, Noth-Holland,1995.
2. Miroslav Huvsek and Jan van Mill. Recent progress in general topology. North-Holland , 1992.
Commitment: 3 hours/week, 3 credits
Contents:
• Cardinal functions and their interrelationships
• Cardinal functions on special classes, in particular on compact spaces
• Independence results, consequences of CH, Diamond, MA and PFA
• Topological results in special forcing extensions, in particular in Cohen models
• S and L spaces, HFD and HFC type spaces
77) LOGIC AND RELATIVITY
Course Coordinator: Németi István
Prerequisites: Familiarity with the basics of first order logic, e.g. formulas, models, satisfaction, validity. The notion of a first order theory and its models.
Books:
1. Andréka, H., Madarász, J., Németi, I.., Andai, A. Sain, I., Sági, G., Tıke, Cs.: Logical analysis of relativity theory. Parts I-IV. Lecture Notes. www.mathinst.hu/pub/algebraic-logic.
2. d'Inverno, R.: Introducing Einstein's Relativity. Clarendon Press, Oxford, 1992.
3. Goldblatt, R.: Orthogonality and spacetime geometry. Springer-Verlag, 1987.
Commitment: 3 hours/week, 3 credits
Contents:
• Axiomatizing special relativity purely in first order logic. (Arguments from abstract model theory against using higher order logic for such an axiomatization.)
• Proving some of the main results, i.e. "paradigmatic effects", of special relativity from the above axioms. (E.g. twin paradox, time dilation, no FTL observer etc.)
• Which axiom is responsible for which "paradigmatic effect"
• Proving the paradigmatic effects in weaker/more general axiom systems (for relativity).
• Applications of definability theory of logic to the question of definability of "theoretical concepts" from "observational ones" in relativity. Duality with relativistic geometries.
• Extending the theory to accelerated observers. Acceleration and gravity. Black holes, rotating (Kerr) black holes. Schwarzschild coordinates, Eddington-Finkelstein coordinates, Penrose diagrams. Causal loops (closed time-like curves).
Connections with the Church-Turing thesis.
78) FRONTIERS OF ALGEBRAIC LOGIC
Course Coordinator: Andréka Hajnal
Prerequisites: Algebraic logic and model theory course.
Books:
1. Henkin, L. Monk, J. D. Tarski, A. Andréka, H. Németi, I.: Cylindric Set Algebras. Lecture Notes in Mathematics Vol 883, Springer-Verlag, Berlin, 1981.
2. Henkin, L. Monk, J. D. Tarski, A.: Cylindric Algebras Part II. North-Holland, Amsterdam, 1985.
3. Andréka, H., Németi, I., Sain, I.: Algebraic Logic. Chapter in Handbook of Philosophical Logic, second edition. Kluwer.
4. van Benthem, J.: Exploring Logical Dynamics. Studies in Logic, Language and Information, CSLI Publications, 1996.
Commitment: 3 hours/week, 3 credits
Contents:
• Re-thinking the role of algebraic logic in logic. Theories as algebras, interpretations between theories as homomorphisms.
• The finitization problem, its connections with finite model theory.
• Parallels and differences between algebraic logic and (new trends in) the modal logic tradition.
• Connections and differences between the algebraic logic based approach and abstract model theory (e.g. in connection with the Lindström type theorems).
• Tarskian representation theorems and duality theories in algebraic logic and their generalizations (e.g. in axiomatic geometry and relativity theory).
79) CLASSICAL ANALYTIC NUMBER THEORY
Lecturer: Gergely Harcos or Janos Pintz
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites:-
Course Level: advanced
Textbook: Hugh L. Montgomery and Robert C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge University Press, 2006
Objective and learning outcomes of the course: We shall prove the classical theorems on the distribution of prime numbers in strong analytic form. You will meet the basic objects and techniques of analytic number theory such as Dirichlet L-functions and the Mellin transform. You will develop a deeper understanding of numbers, complex analysis, and the Riemann Hypothesis.
Detailed contents of the course:
Week 1: Dirichlet series.
Week 2: Additive and multiplicative characters.
Week 3: Primes in arithmetic progressions I.
Week 4: Mellin Transform. Jensen’s Inequality. Borel-Carathéodory Lemma.
Week 5: The Prime Number Theorem.
Week 6: Primitive characters and Gauss sums.
Week 7: Quadratic characters. The Pólya Vinogradov Inequality.
Week 8: The Burgess bound.
Week 9: Analytic properties of Dirichlet L-functions I.
Week 10: Analytic properties of Dirichlet L-functions II.
Week 11: Analytic properties of Dirichlet L-functions III.
Week 12: Primes in arithmetic progressions II.
80) PROBABILISTIC NUMBER THEORY
Course Coordinator: Imre Ruzsa
Prerequisites: Probability theory, Complex function theory.
Books:
1. G. Tanenbaum, Introduction to analytic and probabilistic number theory, Cambridge studies in advanced mathematics 46, Cambridge University Press, 1995, part III
2. P. D. T. A. Elliott, Probabilistic number theory I-II, Springer, Grundlehren der Mathematischen Wissenschaften 239-240, 1979, 1980.
Commitment: 3 hours/week, 3 credits
Contents:
• Statistical properties of arithmetical functions: average, extremal orders, limiting distribution.
• The Erdıs-Wintner theorem. The Erdıs-Kac theorem and generalizations. Kubilius's model. The analytic method of Halász.
81) PROBABILISTIC NUMBER THEORY, LEVEL 2
Course Coordinator: Imre Ruzsa
Prerequisites: Probability theory, Complex function theory
Books: There are no textbooks for these subjects, the original papers have to be used.
Commitment: 3 hours/week, 3 credits
Contents:
This continues the study of additive and multiplicative functions, primarily with analytic methods, Halász's method and refinements. Topics: Concentration estimates for additive functions. Properties of the distribution function of additive functions: continuity, absolute continuity. Distribution of additive functions on shifted primes
82) MODERN PRIME NUMBER THEORY
Lecturer: Andras Biro, or Gergely Harcos, or János Pintz
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: -
Course level: PhD all years
Objective and learning outcomes of the curse:
We will discuss in detail the Green-Tao theorem on the existence of long arithmetic progressions among prime numbers and Linnik’s theorem on the least prime in arithmetic progressions. You will learn modern techniques of combinatorial and analytic number theory such as the Gowers uniformity norm and density results for the zeroes of Dirichlet L-functions.
Detailed contents of the course:
Week 1: Outline of the proof of the Green-Tao theorem. Pseudorandom measures.
Week 2: Gowers uniformity norms, and a generalized von Neumann theorem.
Week 3: Gowers anti-uniformity.
Week 4: Generalised Bohr sets and sigma-algebras.
Week 5: A Furstenberg tower.
Week 6: A pseudorandom measure which majorises the primes, Part 1.
Week 7: A pseudorandom measure which majorises the primes, Part 2.
Week 8: A pseudorandom measure which majorises the primes, Part 3.
Week 9: The log-free zero-density theorem.
Week 10: The exceptional zero repulsion.
Week 11: Proof of Linnik’s theorem.
Week 12: Discussion. Minilectures by students.
83) EXPONENTIAL SUMS IN COMBINATORIAL NUMBER THEORY
Course Coordinator: Imre Ruzsa
Prerequisites: Harmonic Analysis
Books: There are no textbooks for these subjects, the original papers have to be used.
Commitment: 3 hours/week, 3 credits
Contents:
We learn to use Fourier-analytic techniques to solve several problems on general sets of integers. In particular: to find estimates for sets free of arithmetic progressions; methods of Roth, Szemerédi, Bourgain and Gowers. To find arithmetic progressions and Bohr sets in sumsets: methods of Bogolyubov, Bourgain, and Ruzsa's construction. Difference sets and the van der Corput property.
84) INFORMATION THEORETIC METHODS IN MATHEMATICS
Course Coordinator: Imre Csizsar
Prerequisites: Probability and Statistics ; Measure and integral; measure preserving maps.
Books:
1. T.M. Cover & J.A. Thomas: Elements of Information Theory. Wiley, 1991.
2. I. Csiszar: Information Theoretic Methods in Probability and Statistics. IEEE Inform. Th. Soc. Newsletter, 48, March 1998.
3. G. Simonyi: Graph entropy: a survey. In: Combinatorial Optimization, DIMACS Series on Discrete Mathematics and Computer Science, Vol. 20, pp. 399-441,1995.
Commitment: 3 hours/week, 3 credits
Contents:
Applications of information theory in various fields of mathematics are discussed.
Probability:
• Dichotomy theorem for Gaussian measures.
• Sanov's large deviation theorem for empirical distributions, and Gibbs' conditioning principle.
• Measure concentration.
Statistics:
• Wald inequalities.
• Error exponents for hypothesis testing.
• Iterative scaling, generalized iterative scaling, and EM algorithms.
• Minimum description length inference principle Combinatorics:
• Using entropy to prove combinatorial results.
• Graph entropy and its applications.
• Graph capacities (Shannon, Sperner), applications.
• Ergodic theory:
• Kolmogorov--Sinai theorem. Information theoretic proofs of inequalities.
85) SELECTED TOPICS IN PROBABILITY
Course Coordinator: Péter Major
Prerequisites: Stochastics, Introduction to Probability and Statistics; Measure and integration; Fourier integral.
Books:
1. L. Breiman: Probability. Addison-Wesley, Reading, Massachusetts , 1968
2. M. Csörgı-P. Révész: Strong Approximations in Probability and Statistics. Academic Press, New York , 1981.
3. P. Major: Series of Problems in Probability Theory,
http://www.renyi.hu/\~{}major.
Commitment: 3 hours/week, 3 credits
Contents:
• The central limit theorem and Fourier analysis.
• General limit theorems for sums of independent random variables and infinitely divisible distributions.
• Large deviations.
• Wiener process.
• Markov processes.
• Poisson process.
• Invariance principles in Probability, strong approximations.
86) INVARIANCE PRINCIPLES IN PROBABILITY AND STATISTICS
Course Coordinator: Istvan Berkes
Prerequisites: Stochastics, Probability; elements of functional analysis.
Books:
1. M. Csörgı-P. Révész: Strong Approximations in Probability and Statistics. Academic Press, New York ,1981.
2. P. Révész: Random Walk in Random and Non-Random Environments. World Scientific, Singapore , 1990.
3. M. Csörgı-L. Horváth: Weighted Approximations in Probability and Statistics. Wiley, New York , 1993.
Commitment: 3 hours/week, 3 credits
Contents:
• Functional central limit theorem. Donsker's theorem via Skorokhod embedding. Weak convergence in D[0,1].
• Strassen's strong invariance theorem.
• Strong approximations of partial sums by Wiener process: Komlós-Major- Tusnády theorem and its extension (Einmahl, Sakhanenko, Zaitsev).
• Strong invariance principles for local time and additive functionals. Iterated processes.
• Strong approximation of empirical process by Brownian bridge: Komlós- Major- Tusnády theorem.
• Strong approximation of renewal process.
• Strong approximation of quantile process.
• Asymptotic results (distributions, almost sure properties) of functionals of the above processes.
87) STOCHASTIC PROCESSES
Course Coordinator: István Berkes
Prerequisites: Information Theory, Probability, Functional Analysis.
Books:
1. A.V. Skorokhod: Studies in the Theory of Random Processes. Addison-Wesley, Reading, Massachusetts , 1965.
2. W. Feller: An Introduction to Probability Theory and its Applications, Vol. II., Second edition. Wiley, New York , 1971.
3. D. Revuz-M. Yor: Continuous Martingales and Brownian Motion. Third edition. Springer, Berlin , 1999.
Commitment: 3 hours/week, 3 credits
Contents:
• Random walk., Renewal processes.
• Markov chains. Transition probabilities. Recurrence, ergodicity. Existence of stationary distribution.
• Processes with independent increments. Lévy processes. Stable processes. Stationary processes. Ergodicity. Bochner-Khintchine theorem. Markov processes. Infinitesimal generator. Chapman-Kolmogorov equations.
• Branching processes. Asymptotic results. Birth and death process.
• Martingales. Stopping times. Maximal inequalities. Martingale convergence theorems. Quadratic variation. Burkholder-Davis-Gundy inequalities.
88) STOCHASTIC ANALYSIS
Lecturer: Vilmos Prokaj or Balint Toth
No. of Credits: 3 and no. of ECTS credits 6
Prerequisites: Probability Theory
Course Level: PhD level
Brief introduction to the course:
Main topics are: Brownian motion (Wiener process), martingales, stochastic (Ito) integration, stochastic differential equations, diffusion processes. These tools are heavily used in financial mathematics, biology, physics, and engineering. Thus if someone wants to enter e.g. the flourishing field of financial mathematics, it is a must to complete such a course.
The goals of the course:
Review of some calculus and probability tools. A review of the theory of stochastic processes, including continuous time Markov processes. Introducing the student to the major topics of stochastic calculus, including stochastic integration, stochastic differential equations and diffusion processes. Introducing to some applications, in particular the Black-Scholes model of financial mathematics.
The learning outcomes of the course:
A good understanding of continuous time stochastic processes, including Wiener process and other diffusion processes (Ito diffusions). Understanding and competence in stochastic integration and stochastic differential equations (SDE’s), strong and weak solutions, and conditions for existence and uniqueness. Practice in solving linear SDE’s, understanding the Ornstein-Uhlenbeck process. Understanding the relationship between weak solutions and the Stroock-Varadhan martingale problem; the notion of generator of a diffusion, and the related backward and forward partial differential equations.
More detailed display of contents:
Week 1: A review of Calculus and Probability theory topics.
Conditional expectation, main properties, continuous time stochastic processes, martingales, stopping times.
Week 2: Definition and some properties of Brownian motion.
Covariance function, quadratic variation, martingales related to Brownian motion, Markov property.
Week 3: Further properties of Brownian motion; random walks and Poisson process .
Hitting times, reflection principle, maximum and minimum, zeros: the arcsine law, Brownian motion in higher dimensions. Martingales related to random walks, discrete stochastic integrals, optional stopping in discrete setting, properties of Poisson process.
Week 4: Definition of Ito stochastic integral.
Definition and stochastic integral of simple adapted processes. Basic properties of the stochastic integral of simple processes. Stochastic integral of left-continuous, square-integrable, adapted processes. Extension to regular, adapted processes.
Week 5: Ito integrals as processes, Ito formul.,
Ito integrals as martingales, Gaussian Ito integrals, Ito formula for Brownian motion, Ito processes, their quadratic variation and the corresponding Ito formula.
Week 6: Ito processes.
Ito processes, Ito formula for Ito processes. Ito formula in higher dimensions, integration by parts formulae.
Week 7: Stochastic Differential Equations; strong solution.
The physical model and the definition of Stochastic Differential Equations (SDE), SDE of Ornstein-Uhlenbeck (OU) process, geometric Brownian motion, stochastic exponential and logarithm, explicit solution of a linear SDE, strong solution of an SDE, existence and uniqueness theorem, Markov property of solutions.
Week 8: Weak solutions of SDE’s.
Construction of weak solutions, canonical space for diffusions, the Stroock-varadhan martingale problem, generator of a diffusion, backward and forward equations, Stratonovich calculus.
Week 9: Some properties of diffusion processes.
Dynkin formula, calculation of expectation. Feynman-Kac formula.
Week 10: Further properties of diffusion processes.
Time homogeneous diffusions and their generators. Diffusions in the line: L-harmonic functions, scale function, explosion, recurrence and transience, stationary distributions.
Week 11: Multidimensional diffusions.
Existence and uniqueness. Some properties. Bessel processes.
Week 12: An application in financial mathematics.
Derivatives, arbitrage, replicating portfolio, complete market model, self-financing portfolio, the Black-Scholes model.
Optional topics: Changing of probability measure, Girsanov theorem.
The course follows the textbook: Fima C. Klebaner, Introduction to stochastic calculus with applications, Second edition, Imperial College Press, 2006.
89) PATH PROPERTIES OF STOCHASTIC PROCESSES
Course Coordinator: Peter Major
Prerequisites: Invariance Principles in Probability and Statistics, Stochastic Processes.
Books:
1. M. Csörgı-P. Révész: Strong Approximations in Probability and Statistics. Academic Press, New York , 1981.
2. P. Révész: Random Walk in Random and Non-Random Environments. World Scientific, Singapore , 1990.
3. P. Révész: Random Walks of Infinitely Many Particles. World Scientific, Singapore , 1994.
4. D. Revuz-M. Yor: Continuous Martingales and Brownian Motion. Third edition. Springer, Berlin , 1999.
Commitment: 3 hours/week, 3 credits
Contents:
• Constructions of Wiener process. Modulus of continuity.
• Laws of the iterated logarithm. Strassen's theorem.
• Increments of Wiener process.
• Local times, additive functionals and their increments.
• Asymptotic properties, invariance principles for local time and additive functionals. Dobrushin's theorem.
• Path properties of random walks, their local times and additive functionals.
• Random walk in random environment.
• Random walk in random scenery.
• Branching random walk and branching Wiener process.
• Almost sure central limit theorems.
90) NONPARAMETRIC STATISTICS
Course Coordinator: Istvan Berkes
Prerequisites: Probability, Mathematical Statistics.
Books:
1. L. Takács: Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York , 1967.
2. J. Hájek: Nonparametric Statistics. Holden-Day, San Francisco , 1969.
Commitment: 3 hours/week, 3 credits
Contents:
• Order statistics and their distribution.
• Empirical distribution function. Glivenko-Cantelli theorem and its extensions.
• Estimation of the density function. Kernel-type estimators.
• U-statistics.
• Rank correlation. Kendall-s tau.
• Nonparametric tests: goodness of fit, homogeneity, independence.
• Empirical process, approximation by Brownian bridge.
• Komlós-Major-Tusnády theorem.
• Tests based on empirical distribution: Kolmogorov-Smirnov, von Mises tests.
• Quantile process. Bahadur-Kiefer process.
• Rank tests. Wilcoxon-Mann-Whitney test.
91) MULTIVARIATE STATISTICS
Course Coordinator: Peter Major
Prerequisites: Probability; Mathematical statistics; Linear Algebra.
Books:
1. T.W. Anderson: An Introduction to Multivariate Statistical Analysis. Wiley, New York (1958).
2. K.V. Mardia-J.T. Kent-J.M. Bibby: Multivariate Analysis. Academic Press, New York (1979).
Commitment: 3 hours/week, 3 credits
Contents:
• Multivariate normal distribution. Moments, correlation, partial correlations. Conditional and marginal distributions. Regression. Empirical moments, maximum likelihood estimation of the parameters. Wishart matrix and its distribution.
• Testing for independence.
• Least squares. Various regression methods (linear, polynomial, orthogonal, spline).
• Variance and covariance analysis.
• Linear models.
• Design of experiments.
• Analysis of contingency tables.
• Time series. ARMA processes.
• Factor analysis.
92) INFORMATION THEORETICAL METHODS IN STATISTICS
Course Coordinator: Imre Csiszár
Prerequisites: Probability; Mathematical Statistics; Information Theory.
Books:
1. I. Csiczár: Lecture Notes, University of Meryland , 1989.
2. J. Kullback: Information Theory and Statistics. Wiley, 1989.
3. J. Rissanen: Stochastic Complexity in Statistical Inquiry. World Scientific, 1989.
Commitment: 3 hours/week, 3 credits
Contents:
• Definition and basic properties of Kullback I-divergence, for discrete and for general probability distributions.
• Hypothesis testing: Stein lemma, Wald inequalities. Sanov's large deviation theorem, error exponents for testing simple and composite hypotheses (discrete case).
• Estimation: performance bounds for estimators via information-theoretic tools.
• Hájek's proof of the equivalence or orthogonality of Gaussian measures. Sanov's theorem, general case.
• Information geometry, I-projections. Exponential families, log-linear models. Iterative scaling, EM-algorithm. Gibbs conditioning principle.
• Information theoretic inference principles: Maximum entropy, maximum entropy on the mean, minimum description length. The BIC model selection criterion; consistency of BIC order estimation.
• Generalizations of I-divergence: f-divergences, Bregman distances.
93) NUMERICAL METHODS IN STATISTICS
Course Coordinator: Istvan Berkes
Prerequisites: Probability; Mathematical Statistics.
Books:
1. W. Freiberger-U. Grenander: A Course in Computational Probability and Statistics. Springer, New York (1971).
2. J.E. Gentle: Random Number Generation and Monte Carlo Methods. Springer, New York (1998).
Commitment: 3 hours/week, 3 credits
Contents:
• Combinatorial algorithms with statistical applications.
• Numerical evaluation of distributions.
• Generating random numbers of various distribution.
• Computer evaluation of statistical procedures. Estimation methods. Robust procedures. Testing statistical hypothesis. Testing for normality.
• Sequential methods.
• EM algorithm.
• Statistical procedures for stochastic processes. Time series. Reliability. Life tests.
• Bootstrap methods.
• Monte Carlo methods
• Statistical software packages
94) ERGODIC THEORY AND DYNAMICAL SYSTEMS
Course Coordinator: Domokos Szász
Prerequisites: Probability, Measure and Integration.
Books:
1. I.P. Cornfeld, S. V. Fomin and Ya. G. Sinai: Ergodic Theory. Springer, Berlin (1982).
2. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press (1995).
Commitment: 3 hours/week, 3 credits
Contents:
• Fixed point theorems. Applications
• Variational principles and weak convergence. The n-th variation. Necessary and sufficient conditions for local extrema. Weak convergence. The generalized Weierstrass existence theorem. Applications to calculus of variations. Applications to nonlinear eigenvalue problems.
Applications to convex minimum problems and variational inequalities. Applications to obstacle problems in Elasticity. Saddle points.
Applications to duality theory. The von Neumann Minimax theorem on the existence of sadle points. Applications to game theory.
• Nonlinear monotone operators. Applications.
95) ERGODIC THEORY AND COMBINATORICS
Lecturer: Gábor Elek
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites:
Course Level:
Brief introduction to the course:
Very large graphs can be viewed as “almost” infinite objects. We survey the analytical methods related to the study of large dense and sparse graphs.
The goals of the course:
The students will be introduced to the developing theory of graph limits and its connection to ergodic theory.
The learning outcomes of the course:
The students will understand the relation of discrete and continuous objects, and have a new, fresh look on combinatorics.
More detailed display of contents:
Week 1. Graph and hypergraph homorphisms.
Week 2. Euclidean hypergraphs.
Week 3. Sampling. Limit notions.
Week 4. Ultrafilters, ultraproducts
Week 5. The correspondence principle.
Week 6. Removal and regularity lemmas.
Week 7. Basic ergodic theory of group actions.
Week 8. Benjamini-Schramm limits.
Week 9. Property testing, matchings.
Week 10. Hyperfiniteness.
Week 11. Testing hyperfinite families.
Week 12. Ergodic theory and samplings.
96) DATA COMPRESSION
Course Coordinator: Imre Csiszár
Prerequisites: Information theory
Books:
1. T.M. Cover-J.A. Thomas: Elements of Information Theory. Wiley, New York (1991).
2. I. Csiszár: Lecture Notes, University of Maryland (1989).
3. K. Sayood: Introduction to Data Compression. Morgan-Kauffmann, San Francisco (1996).
4. D. Salomon: Data Compression: the Complete Reference. Springer, New York (1998).
Commitment: 3 hours/week, 3 credits
Contents:
Lossless compression:
• Shannon-Fano, Huffman and arithmetic codes. "Ideal codelength", almost sure sense asymptotic optimality. Data compression standard for fax.
• Optimum compression achievable by block codes. Information sprectum, information stability, Shannon--McMillan theorem.
• Universal coding. Optimality criteria of mean and max redundancy; mixture and maximum likelihood coding distributions. Mathematical equivalence of minimax redundancy and channel capacity. Krichevsky-Trofimov distributions, minimax redundancy for memoryless and finite state sources.
• The context weighting method. Burrows-Wheeler transform. Dictionary-type codes. Lempel-Ziv codes, their weak-sense universality for stationary ergodic sources.
• Universal coding and complexity theory, Kolmogorov complexity, finite state complexity, Ziv complexity. Lossy compression:
• Shannon's rate distortion theorem for discrete memoryless sources, and its universal coding version. Extensions to continuous alphabets and sources with memory. Rate-distortion function of Gaussian sources.
• Uniform and non-uniform scalar quantization, Max-Lloyd algorithm. Companded quantization, Bennett's integral.
• Vector quantization: Linde-Buzo-Gray algorithm, tree structured vector quantizers, lattice vector quantizers. Transform techniques. Pyramidal coding. The JPEG image compression standard.
97) CRYPTOLOGY
Lecturer: Laszlo Csirmaz
Prerequisites: Mathematical Statistics, Information Theory.
Books:
1. S.W. Golomb: Shift Register Sequences. Holden Day, San Francisco (1960).
2. M.E. Hellman: An Extension of the Shannon Theory Approach to Cryptography, IEEE-IT 23 (1977), 289-294.
3. D. Kahn: The Codebreakers. MacMillan, New York (1967).
4. C.E. Shannon: Communication Theory of Secrecy Systems, Bell Syst. Techn. J.28 (1949), 656-715.
5. G.J. Simmons: Contemporary Cryptology: The Science of Information Security. IEEE Press, New York (1992).
Commitment: 3 hours/week, 3 credits
Contents:
First steps: From Caesar to the simple substitution. Transposition and its statistical solution. Language statistics, use of statistical hypothesis testing. Intuitive notion of the unicity distance. Definition of codes.
Poly-alphabetic codes: Vigen\'ere-algorithm. Statistical determination of the length of the key-word. Statistical solutions. The role of cryptographic keys.
The one time pad. Recognizing double use of key-stream: test of coincidence. Solving the two-times pad: The Kerkhoff method. Learning from the Venona papers. Solving by side-information.
Data compression and its utilization in cryptography. Dictionary look-up compression, and preparing dictionaries. Ref: parallel course on Data Compression. Generating random and pseudo-random sequences.
Stream chipers. Short introduction to LFSRs, Ref: Golomb (1960). Linear cryptanalysis. Encryption machines.
Shannon theory and its extension by Hellman. Notion of redundancy. Ref: Shannon (1949), Hellman (1977).
DES. Triple DES. Differential cryptanalysis. AES.
Introduction to public key cryptography and to security questions of public networks. Ref: Simmons (1992). The RSA algorithm.
Applications of public key cryptography. Trusted third party, CA.
Digital signature. Hash functions and algorithms. The SHA-1 standard. Critics of CRC-hashing.
Cryptographic protocols. Zero-knowledge protocols. Secret sharing.
Digital fingerprint, digital watermarks.
98) COMBINATORIAL OPTIMIZATION
Lecturer: Ervin Győri
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisites: discrete mathematics, graph theory, linear algebra
Course Level: introductory
Brief introduction to the course:
Basic concepts and theorems are presented. Some significant applications are analyzed to illustrate the power and the use of combinatorial optimization. Special attention is paid to algorithmic questions.
The goals of the course:
One of the main goals of the course is to introduce students to the most important results of combinatorial optimization. A further goal is to discuss the applications of these results to particular problems, including problems involving applications in other areas of mathematics and practice. Finally, computer science related problems are to be considered too.
The learning outcomes of the course:
The students will learn some basic notions and results of combinatorial optimization. They will learn how to use these tools in solving every day life problems as well as in software developing.
More detailed display of contents:
Week 1: Typical optimization problems, complexity of problems, graphs and digraphs
Week 2: Connectivity in graphs and digraphs, spanning trees, cycles and cuts, Eulerian and Hamiltonian graphs
Week 3: Planarity and duality, linear programming, simplex method and new methods
Week 4: Shortest paths, Dijkstra method, negative cycles
Week 5: Flows in networks
Week 6: Matchings in bipartite graphs, matching algorithms
Week 7: Matchings in general graphs, Edmonds’ algorithm
Week 8: Matroids, basic notions, system of axioms, special matroids
Week 9: Greedy algorithm, applications, matroid duality, versions of greedy algorithm
Week 10: Rank function, union of matroids, duality of matroids
Week 11: Intersection of matroids, algorithmic questions
Week 12: Graph theoretical applications: dedge disjoint and coverong spanning trees, directed cuts
Book: E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Courier Dover Publications, 2001 or earlier edition: Rinehart and Winston, 1976
99) QUANTUM COMPUTING
Course Coordinator: Miklos Simonovits
Prerequisites: An "Algorithms and complexity" type course, including the solid
understanding of randomized and deterministic Turing machines.
Book: J. Gruska, Quantum Computing, McGrawHill, 1999.
Commitment: 3 hours/week, 3 credits
Contents:
• The comparison of probabilistic and quantum Turing Machines. Probabilities vs. complex amplitudes. Positive interference and negative interference. Why complex amplitudes? Background in Physics: some experiments with bullets and electrons.
• Background in Mathematics. Linear algebra, Hilbert spaces, projections. Observables, measuring quantum states.
• The qubit. Tensor product of vectors and matrices. Properties of tensor product. Two qubit registers. Quantum entanglement, examples. n-qubit registers.
• The Fourier transform. Quantum parallelism. Van Dam's algorithm, Deutsch's problem. The deterministic solution of Deutsch's problem.
• The promise problem of Deutsch and Józsa.
• Simon's Problem.
• Grover's database-search algorithm. Lower bound for the database-search problem.
• Shor's integer factoring algorithm.
• Complexity theoretic results: BQP is in PSPACE, P is in
100) COMPUTATIONAL GEOMETRY
Course Coordinator: Endre Makai
Prerequisites: Geometry, Basic linear algebra, Algorithms, Discrete mathematics
Book: M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf: Computational Geometry - Algorithms and Applications, Springer, Berlin, 1997.
Commitment: 3 hours/week, 3 credits
Contents:
• Line segment intersection
• Convex hull
• Polygon triangulation, art gallery problems
• Linear programming
• Range searching
• Point location
• Voronoi diagrams, Delaunay triangulations
• Arrangements and duality
• Geometric data structures
• Motion planning
• Visibility graphs, ray shooting
• Applications in Computer Science, Robotics, Computer graphics, Geometric
Optimization
101) RANDOM COMPUTATION
Lecturer: Zoltan Kiraly or Miklos Simonovits
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: basic probability theory
Course Level: PhD
Brief introduction to the course:
The main topic covers: how randomization helps in design of algorithms, and the analysis of randomized algorithms. Also the related parts of complexity theory will be described, and a short introduction to derandomization will be given.
The goals of the course:
To learn designing and analyzing of randomized algorithms.
The learning outcomes of the course:
Stable skills to use randomized methods in algorithmic environment.
More detailed display of contents:
Examples (Schwartz lemma and applications. Karger’s min cut algorithm. Quicksort. Prime number testing and generation.)
Probabilistic tools (Markov, Chebysev and Chernoff inequalities, method of conditional probabilities).
Complexity (Computational models, randomized classes, relation between them, Neumann-Yao minimax theorem).
Random graphs, expanders, random walks in graphs, routing in hypercube.
Minimum spanning trees, VPN design, minimum cuts II.
Hashing.
Random sampling.
Lovász’ Local Lemma.
Randomized approximation schemes, approximating the volume in high dimensions.
Isolation lemma, parallel computing.
On-line algorithms.
Pseudorandom number generation, derandomization techniques.
Book: Motwani-Raghavan: Randomized Algorithms, Cambridge University Press, 1995.
102) LOGIC OF PROGRAMS
Lecturer: Laszlo Csirmaz
No. of Credits: 3 and no. of ETCS credits: 6
Prerequisities: Algebra II, Introduction to Logic
Course Level: Advanced PhD course on the theory of programming
Objective and learning outcomes of the course:
Logic of programs stemmed from the requirement of automatically proving that a computer program behaves as expected. Format methods are of great importance as they get rid of the subjective factor, checking by examples,
and still letting the chance of a computer bug somewhere. When an algorithm is proved to be correct formally, it will always perform according to its specification. The course takes a route around this fascinating topics. How programs are modeled, what does it mean that a program is correct, what are the methods to prove correctness, and when are these methods sufficient.
The course requires knowledge of mathematical methods, especially Mathematical Logic and Universal Algebra; and perspective students should have some programming experience as well.
The course discusses how programs can be proved correct, what are the methods, their limitations. We state and prove characterizations for some of the most well-known methods. Temporal logic is a useful tool for stating and proving such theorems. The limitations of present-day methods for complex programs containing recursive definitions or arrays is touched as well.
At the end of the course students will know an overview of the recent result and problems, will be able to start their own research in the topics. Students will be able to judge the usefulness and feasibility of formal methods in different areas, including formal verification of security protocols. The course concludes with an oral exam or by preparing a short research paper.
Detailed contents of the course:
Computability, computations, structures, models
Programs, program scheme, chart and straight-line programs.
Interpreting program runs in structures, Herbrandt universe, partial and total correctness. Formalizing statements
Methods proving program termination
Calculus of annotated programs
Example: an “evidently wrong” sorting program is, in fact, correct
Floyd-Hoare method, correctness and completeness
Programs with strange time scale: parallel execution, eventuality and liveness; Dijsktra conditional statementsDynamic logic: soundness and completeness
Temporal logic of programs: modalities and expressive power
Recursive program schemes, operators, fixed-point theorems
Weak higher order structures, weak program runs, characterizations of the Floyd-Hoare method.
BAN logic, compositional logic for proving security properties of protocols.
Textbooks: T. Gergely, L Ury, First-order programming theories, Springer, 1991, Z.Manna, Mathematical theory of computation, Courier Dover Publications, 2003
103) TOPICS IN FINANCIAL MATHEMATICS
Lecturer: Miklos Rasonyi
No. of Credits: 3 and no. of ECTS credits: 6
Prerequisities:-
Brief introduction to the course:
Basic concepts of stochastic calculus with respect to Brownian motion. Martingales, quadratic variation, stochastic differential equations. Fundamentals of continuous-time mathematical finance; pricing, replication, valuation using PDE methods. Exotic options, jump processes.
The goals of the course:
To obtain a solid base for applying continuous-time stochastic finance techniques; a firm knowledge of basic notions, methods. An introduction to most often used models.
The learning outcomes of the course:
Skills to price derivative products of various types, computation of the replicating strategies. Ability to handle most commonly used term-structure models. Being operational with Ito's formula and other fundamental rules of stochastic calculus.
More detailed display of contents:
Week 1. From random walk to Brownian motion. Quadratic variation.
Week 2. Ito integral, Ito processes. Ito's formula and its applications.
Week 3. Stochastic differential equations: existence and uniqueness of solutions.
Week 4. Black-Scholes model and option pricing formula.
Week 5. Replication of contingent claims. European options.
Week 6. American options and their valuation.
Week 7. The PDE approach to hedging and pricing.
Week 8. Exotic (Asian, lookback, knock-out barrier,...) options.
Week 9. The role of the numeraire. Forward measure.
Week 10. Term-structure modelling: short rate models, affine models.
Week 11. Heath-Jarrow-Morton models. Defaultable bonds.
Week 12. Asset price models involving jumps.
Suggested reading: Steven E. Shreve: Stochastic calculus for finance, vols. I and II, Springer, 2004
104) INTRODUCTION TO CCR ALGEBRAS
Lecturer: Dénes Petz
No. of Credits: 3, and no. of ECTS credits: 6
Course Level: advanced
The goals of the course:
The main goal of the course is to introduce students to the theory of unbounded operators, C*-algebras, orthogonal polynomials and mathematical foundations of certain area of quantum theory.
The learning outcomes of the course:
The students will learn some basic notions and results in mathematical foundation of quantum theory. The subject is an example of the application of functional analysis. It is useful for applied mathematicians and physicists.
More detailed display of contents
Week 1: Historical introduction to the formalism of canonical commutation relation (contribution of Heisenberg, Schrödinger and von Neuman in 1920's)
Week 2: A short introduction to C*-algebras, their states and representations (Gelfand-Naimark theorems, GNS-construction, tensor product structure)
Week 3: The C*-algebra of the canonical commutation relation, CCR (existence and uniquenes)
Week 4-5: The concept of symmetric Fock space (definition, second quantization, important examples of unbounded operators, exponential vectors)
Week 6: The Fock representation of the CCR (detailed study of the one-dimensional case, tensor product)
Week 7-9: States on CCR (Gaussian states 2-point function. relation to classical probability)
Week 10-11: Central limit theorem (statement and proof, maximization of entropy when 2-point function is fixed, an introduction to some unsolved problems)
Week 12: Schrödinger representation (introduction to Hermite polinomials, the P and Q operators, their complementary relation)
Book: D. Petz, The algebra of the canonical commutation relation, Leuven University Press, 1990.
105) EXTREMAL COMBINATORICS, GRAPH THEORY AND FINITE GEOMETRY
Lecturer: Aart Blokhuis
No. of Credits: 1 and no. of ECTS credits: 2
Intersecting Families, Erdős-Ko-Rado, Kneser graphs.
Turan's problems and generalizations.
Basics from Finite Geometry, q-analogues of some problems in extremal graph theory.
Linear algebraic methods in Combinatorics: Equiangular lines, few distance sets, strongly regular graphs.
Kahn and Kallai's disproof of Borsuk's Conjecture.
Extremal problems and projective planes: Arcs, ovals and hyperovals, Segre's theorem. Related strongly regular graphs and generalized quadrangles.
Blocking sets in finite projective planes.
106) INVARIANT THEORY
Lecturer: Friedrich Knop
No. of Credits: 1 and no. of ECTS credits: 2
1. Invariant theory: General methods
2. Invariant theory: Classifications
3. Algebraic transformation groups 1
4. Algebraic transformation groups 2
5. Spherical varieties: embeddings
6. Spherical varieties: classification
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