Скачать 238.92 Kb.

M1. BASIC ALGEBRA 1
Basic concepts and theorems are presented. Emphasis is put on familiarising with the aims and methods of abstract algebra. Interconnectedness is underlined throughout. Applications are presented.
One of the main goals of the course is to introduce students to the most important concepts and fundamental results in abstract algebra. A second goal is to let them move confidently between abstract and concrete phenomena.
The students will learn some basic notions and results of abstract algebra. More importantly, they will gain expertise in using it in various areas of mathematics.
Optional topics: Resultants, polynomials in noncommuting variables, twisted polynomials, subdirect products, subdirectly irreducible algebras, subdirect representation. Categorical approach: products, coproducts, pullback, pushout, functor categories, natural transformations, Yoneda lemma, adjoint functors. Books: 0. P J Cameron, Introduction to Algebra, Oxford University Press, Oxford, 2008. 1. N Jacobson, Basic Algebra III, WH Freeman and Co., San Francisco, 1974/1980. 2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994 M2. BASIC ALGEBRA 2
Further concepts and theorems are presented. Emphasis is put on difference of questions at different areas of abstract algebra and interconnectedness is underlined throughout. Applications are presented.
One of the main goals of the course is to introduce the main distinct areas of abstract algebra and the fundamental results therein. A second goal is to let them move confidently between abstract and concrete phenomena.
The students will learn in some depth the theories in the three main areas of abstract algebra: groups, rings and fields. More importantly, they will gain some expertise in using them in various areas of mathematics.
1. Groups: composition series, JordanHölder Theorem, 2. conjugation, centralizer, normalizer, class equation, pgroups, 3. nilpotent groups, Frattini subgroup, Frattini argument, 4. direct product, KrullSchmidt Theorem, semidirect product, groups of small order. 5. Commutative rings: unique factorization, principal ideal domains, Euclidean domains, 6. finitely generated modules over principal ideal domains, Fundamental Theorem of finite abelian groups, Jordan normal form of matrices, 7. Noetherian rings, Hilbert Basis Theorem, operations with ideals. 8. Fields: algebraic and transcendental extensions, transcendence degree, 9. splitting field, algebraic closure, the Fundamental Theorem of Algebra, normal extensions, finite fields, separable extensions, 10. Galois group, Fundamental Theorem of Galois Theory, cyclotomic fields, 11. radical expressions, insolvability of the quintic equation, traces and norms: Hilbert’s Theorem, 12. ArtinSchreier theorems, ordered and formally real fields. Optional topics: Categorical approach: products, coproducts, pullback, pushout, functor categories, natural transformations, Yoneda lemma, adjoint functors. Books: 1. N Jacobson, Basic Algebra III, WH Freeman and Co., San Francisco, 1974/1980. 2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994 M3. REAL ANALYSIS
Textbooks: Online material is available at the following sites: http://www.indiana.edu/~mathwz/PRbook.pdf, http://compwiki.ceu.hu/mediawiki/index.php/Real_analysis
Introduction to Lebesgue integration theory; measure, σalgebra, σfinite measures. Different notion of convergences; product spaces, signed measure, RadonNikodym derivative, Fubini and Riesz theorems; Weierstrass approximation theorem. Solid foundation in the Lebesgue integration theory, basic techniques in analysis. It also enhances student’s ability to make their own notes. At the end of the course students are expected to understand the difference between ”naive” and rigorous modern analysis. Should have a glimpse into the topics of functional analysis as well. They must know and recall the main results, proofs, definition. As a conclusion of the course, students take an oral exam where all acquired knowledge is checked and graded.
M4. COMPLEX FUNCTION THEORY
Fundamental concepts and themes of classic function theory in one complex variable are presented: complex derivative of complex valued functions, contour integration, Cauchy's integral theorem, Taylor and Laurent series, residues, applications, conformal maps.
The goal of the course is to acquaint the students with the fundamental concepts and results of classic complex function theory.
The students will learn some basic notion and theorems (with some applications) of classic complex function theory.
Week 1: Complex numbers Week 2: Complex differentiable functions Week 3: Complex line integral Week 4: Cauchy theorem on star regions and its corollaries Week 5: Winding number, the general form of Cauchy's theorem Week 6: Harmonic functions Week 7: Isolated singularities Week 8: The residue theorem and its applications Week 9: Further applications of the residue theorem Week 10: Conformal maps Week 11: Riemann mapping theorem Week 12: Picard's theorems Books: 1. R. Remmert, Theory of complex functions, SpringerVerlag, 1991 2. J. B. Conway: Functions of one complex variable, SpringerVerlag, 1978, 3. A. I. Markushevich: Theory of functions of a complex variable, Chelsea, 1977 4. L. V. Ahlfors: Complex analysis, McGrawHill, 1979, M5. FUNCTIONAL ANALYSIS AND DIFFERENTIAL EQUATIONS
The basic definitions and results of functional analysis will be presented closely following the classic book of Reed and Simon.
The main goal of the course is to provide a foundation for further studies in operator theory, mathematical physics as well as differential equations.
The students will learn the basics of functional analysis.
Week 1. The geometry of Hilbert spaces. Week 2. Orthonormal systems, Fourier series. Week 3. Classical Banach spaces. Week 4. The HahnBanach Theorem. Week 5. The BanachSteinhaus Theorem, The Open Mapping Theorem and The Closed Graph Theorem. Week 6. Various topologies on Banach spaces. Nets. Week 7. The BanachAlaoglu Theorem. Week 8. Locally convex vector spaces. Week 9. Distributions. Week 10. Bounded operators and their topologies. Week 11. Compact and trace class operators. Week 12. The spectral theorem. Book: ReedSimon: Functional Analysis (Methods of Modern Mathematical Physics Vol. I), Academic Press, 1980. M6. INTRODUCTION TO COMPUTER SCIENCE
Greedy and dynamic programming algorithms. Famous tricks in computer science. the most important data structures in computer science. The Chomsky hierarchy of grammars, parsing of grammars, relationship to automaton theory. Computers, Turing machines, complexity classes P and NP, NPcomplete. Stochastic Turing machines, important stochastic complexity classes. Counting classes, stochastic approximation with Markov chains.
To learn dynamic programming algorithms To get an overview of standard tricks in algorithm design To learn the most important data structures like chained lists, hashing, etc. To learn the theoretical background of computer science (Turing machines, complexity classes) To get an introduction in stochastic computing
The students will be able to read and understand moderately involved scientific papers related to the topic.
Lecture 1. Theory: The O, and notations. Greedy and dynamic programming algorithms. Kruskal’s algorithm for minimum spanning trees, the folklore algorithm for the longest common subsequence of two strings. Practice: The money change problem and other famous dynamic programming algorithms Lecture 2. Theory: Dijstra’s algorithm and other algorithms for the shortest path problem. Preactice: Further dynamic programming algorithms. Lecture 3. Theory: Divideandconqueror and checkpoint algorithms. The Hirshberg’s algorithm for aligning sequences in linear space Practice: Checkpoint algorithms. Reduced memory algorithms. Lecture 4. Theory: Quick sorting. Sorting algorithms. Practice: Recursive functions. Counting with inclusionexclusion. Lecture 5. Theory: The KnuthMorrisonPratt algorithm. Suffix trees. Practice: String processing algorithms. Exact matching and matching with errors. Lecture 6. Theory: Famous data structures. Chained lists, reference lists, hashing. Practice: Searching in data structures. Lecture 7. Theory: The Chomskyhierarchy of grammars. Parsing algorithms. Connections to the automaton theory. Practice: Regular expressions, regular grammars. Parsing of some special grammars between regular and contextfree and between contextfree and contextdependent classes. Lecture 8. Theory: Introduction to algebraic dynamic programming and the objectoriented programming. Practice: Algebraic dynamic programming algorithms. Lecture 9. Theory: Computers, Turingmachines, complexity and intractability, complexity of algorithms, the complexity classes P and NP. 3satisfiability, and NPcomplete problems. Practice: Algorithm complexities. Famous NPcomplete problems. Lecture 10. Theory: Stochastic Turing machines. The complexity class BPP. Counting problems, #P, #Pcomplete, FPRAS. Practice: Stochastic algorithms. Lecture 11. Theory: Discrete time Markov chains. Reversible Markov chains, Frobenious theorem. Relationship between the second largest eigenvalue modulus and convergence of Markov chains. Upper and lower bounds on the second largest eigenvalue. Practice: Upper and lower bounds on the second largest eigenvalue. Lecture 12. Theory: The SinclairJerrum theorem: relationship between approximate counting and sampling. Practice: Some classical almost uniform sampling (unrooted binary trees, spanning trees). 