# Program in Applied Mathematics

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 M1. BASIC ALGEBRA 1 No. of Credits: 3, and no. of ECTS credits: 6 Semester or Time Period of the course: Fall Semester Prerequisites: linear algebra, introductory abstract algebra Course Level: introductory Brief introduction to the course: 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. The goals of the course: 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 learning outcomes of the course: 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. More detailed display of contents (week-by-week) Groups: permutations groups, orbit-stabilizer theorem, cycle notation, conjugation, conjugacy classes of S_n, odd/even permutations, commutator subgroup, free groups, geerators and relations, Dyck’s theorem, solvable and simple groups, simplicity of A_n, classical linear groups, Polynomials: Euclidean Algorithm, uniqueness of factorisatoin, Gauss Lemma, cyclotomic polynomials, polynomials in several variables, homogeneous polynomials, symmetric polynomials, formal power series, Newton's Formulas, Sturm’s Theorem on the number of real roots of a polynomial with real coefficients. Rings and modules: simplicity of matrix rings, quaternions, Frobenius Theorem, Wedderburn’s Theorem, submodules, homomorphisms, direct sums of modules, free modules, chain conditions, composition series. Partially ordered sets and lattices: Hasse-diagram, chain conditions, Zorn Lemma, lattices as posets and as algebraic structures, modular and distributive lattices, modularity of the lattice of normal subgroups, Boolean algebras, Stone Representation Theorem. Universal algebra: subalgebras, homomorphisms, direct products, varieties, Birkhoff Theorem. Optional topics:Resultants, polynomials in non-commuting 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 I-II, WH Freeman and Co., San Francisco, 1974/1980. 2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994M2. BASIC ALGEBRA 2 No. of Credits: 3, and no. of ECTS credits: 6 Semester or Time Period of the course: Winter Semester Prerequisites: Basic Algebra 1 Course Level:  intermediate Brief introduction to the course: 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. The goals of the course: 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 learning outcomes of the course: 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. More detailed display of contents: 1. Groups: composition series, Jordan-Hölder Theorem, 2. conjugation, centralizer, normalizer, class equation, p-groups,3. nilpotent groups, Frattini subgroup, Frattini argument, 4. direct product, Krull-Schmidt 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. Artin-Schreier 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 I-II, WH Freeman and Co., San Francisco, 1974/1980.2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994M3. REAL ANALYSIS No. of Credits: 3, and no. of ECTS credits: 6 Semester or Time Period of the course: Fall Semester Prerequisites: Undergraduate calculus, Elementary Linear Algebra Course level: introductory course 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 Objective and learning outcomes of the course: Introduction to Lebesgue integration theory; measure, σ-algebra, σ-finite measures. Different notion of convergences; product spaces, signed measure, Radon-Nikodym 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. Detailed contents of the course: Outer measure, measure, σ-algebra, σ-finite measure lim inf and lim sup of sets; their measure. The Borel-Cantelli lemma. Complete measure Caratheodory outer measure on a metric space. Borel sets. Lebesgue measure. Connection between Lebesgue measurable sets and Borel sets Measurable functions. Measurable functions are closed under addition and multiplication. Continuous functions are measurable. Example where the composition of measurable functions is not measurable Limits of measurable functions, sup, inf, lim sup, lim inf Egoroff's theorem: if fi converges pointwise a.e to f then it converges uniformly with an exceptional set of measure <ε. Convergence in measure; pointwise convergence for a subsequence. Lusin's theorem: a Lebesgue measurable function is continuous with an exceptional set of measure <ε. Converging to a measurable function by simple functions. Definition of the integral; conditions on a measurable function to be integrable Fatou's lemma, Monotone Convergence Theorem; Lebesgue's Dominated Convergence Theorem. Counterexample: a sequence of functions tends to f, but the integrals do not converge to the integral of f. Hölder and Minkowsi inequalities; Lp is a normed space. Riesz-Fischer theorem: Lp is complete, conjugate spaces, basic properties Signed measure, absolute continuity, Jordan and Hahn decomposition Radon-Nikodym derivative Product measure, Fubini's theorem. Counterexample where the order of integration cannot be exchanged Example for a continuous, nowhere differentiable function Example for a strictly increasing function which has zero derivative a.e. An increasing function has derivative a.e. Weierstrass' approximation theorem Basic properties of convolution M4. COMPLEX FUNCTION THEORY No. of Credits: 3 and no. of ECTS credits: 6 Semester or Time Period of the course: Winter Semester Prerequisites: calculus Course Level: introductory Brief introduction to the course: 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 goals of the course: The goal of the course is to acquaint the students with the fundamental concepts and results of classic complex function theory. The learning outcomes of the course : The students will learn some basic notion and theorems (with some applications) of classic complex function theory. More detailed display of contents. 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 theoremsBooks: 1. R. Remmert, Theory of complex functions, Springer-Verlag, 1991 2. J. B. Conway: Functions of one complex variable, Springer-Verlag, 1978, 3. A. I. Markushevich: Theory of functions of a complex variable, Chelsea, 1977 4. L. V. Ahlfors: Complex analysis, McGraw-Hill, 1979,M5. FUNCTIONAL ANALYSIS AND DIFFERENTIAL EQUATIONS No. of Credits : 3 and no. of ECTS credits : 6 Semester or Time Period of the course: Winter Semester Prerequisites: linear algebra, calculus, real and complex analysis Course Level: introductory Brief introduction to the course: The basic definitions and results of functional analysis will be presented closely following the classic book of Reed and Simon. The goals of the course: 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 learning outcomes of the course: The students will learn the basics of functional analysis. More detailed display of contents : Week 1. The geometry of Hilbert spaces. Week 2. Orthonormal systems, Fourier series. Week 3. Classical Banach spaces. Week 4. The Hahn-Banach Theorem. Week 5. The Banach-Steinhaus Theorem, The Open Mapping Theorem and The Closed Graph Theorem. Week 6. Various topologies on Banach spaces. Nets. Week 7. The Banach-Alaoglu 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: Reed-Simon: Functional Analysis (Methods of Modern Mathematical Physics Vol. I), Academic Press, 1980.M6. INTRODUCTION TO COMPUTER SCIENCE No. of Credits: 3 and no. of ECTS credits: 6 Semester or Time Period of the course: Fall Semester Prerequisites: - Course Level: introductory Brief introduction to the course: 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, NP-complete. Stochastic Turing machines, important stochastic complexity classes. Counting classes, stochastic approximation with Markov chains. The goals of the course: To learn dynamic programming algorithmsTo 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 learning outcomes of the course: The students will be able to read and understand moderately involved scientific papers related to the topic. More detailed display of contents. 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 algorithmsLecture 2.Theory: Dijstra’s algorithm and other algorithms for the shortest path problem.Preactice: Further dynamic programming algorithms.Lecture 3.Theory: Divide-and-conqueror and checkpoint algorithms. The Hirshberg’s algorithm for aligning sequences in linear spacePractice: Checkpoint algorithms. Reduced memory algorithms.Lecture 4.Theory: Quick sorting. Sorting algorithms.Practice: Recursive functions. Counting with inclusion-exclusion.Lecture 5.Theory: The Knuth-Morrison-Pratt 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 Chomsky-hierarchy of grammars. Parsing algorithms. Connections to the automaton theory.Practice: Regular expressions, regular grammars. Parsing of some special grammars between regular and context-free and between context-free and context-dependent classes.Lecture 8.Theory: Introduction to algebraic dynamic programming and the object-oriented programming.Practice: Algebraic dynamic programming algorithms.Lecture 9. Theory: Computers, Turing-machines, complexity and intractability, complexity of algorithms, the complexity classes P and NP. 3-satisfiability, and NP-complete problems.Practice: Algorithm complexities. Famous NP-complete problems.Lecture 10. Theory: Stochastic Turing machines. The complexity class BPP. Counting problems, #P, #P-complete, 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 Sinclair-Jerrum theorem: relationship between approximate counting and sampling. Practice: Some classical almost uniform sampling (unrooted binary trees, spanning trees).

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