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Remarks: 1. Each course has 3 CEU credits (3 hours / week), unless otherwise specified. 2. Spring Term course: Thesis Writing for Mathematics (12 hours = 6 lectures, no credit), mandatory for all MS students. 20092010 Syllabi
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 Teaching format: lecture combined with classroom discussions, regular homework assignments
Attendance is mandatory. Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments. Tests and Grading: There will be a midterm test (worth 30%), a final exam (worth 40%) and homework (worth 30%).
Final exam: within two weeks after the final lecture Office hours: by appointment Pál Hegedűs Zrinyi u. 14, Third Floor, Office # 312, hegedusp@ceu.hu Name of Course: 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).
Midterm test after the 7th week. Description of algorithms + exercises. (50%) Colloquia at the end of the course. 1 exercise + 1 theorem covering the material of one lecture. (50%)
Istvan Miklos Rényi Institute of Mathematics Reáltanoda utca 1315, 1053 Budapest, miklosi@renyi.hu Name of Course: Probability and Statistics
Students will learn basic probability models with applications. Laws of large numbers, central limit and large deviation theorems will be introduced together with the notion of conditional expectation that plays a crucial role in statistics. While probability theory describes random phenomena, mathematical statistics teaches us how to behave in the face of uncertainties, according to the famous mathematician Abraham Wald. Roughly speaking, we will learn strategies of treating randomness in everyday life.
The first part of the course gives an introduction to probability models and basic notion of conditional distributions, while the second part to the theory of estimation and hypothesis testing. The main concept is that our inference is based on a randomly selected sample from a large population, and hence, our observations are treated as random variables. Through the course we intensively use laws of large numbers and the Central Limit Theorem. On this basis, applications are also discussed, mainly on a theoretical basis, but we make the students capable of solving numerical exercises.
Students will be able to identify probability models, further to find the best possible estimator for a given parameter by investigating the bias, efficiency, sufficiency, and consistency of an estimator on the basis of theorems and theoretical facts. Students will gain familiarity with basic methods of estimation and will be able to construct statistical tests for simple and composite hypotheses. They will become familiar with applications to realworld data and will be able to choose the most convenient method for given reallife problems.
Literature: S. Ross, A First Course in Probability. Fifth Edition, PrenticeHall, 1998. A. Rényi, Probability Theory, Acad. Press, 1978. W. Feller, An Introduction to Probability. Theory and Its Applications, Wiley, 1966. C.R. Rao, Linear statistical inference and its applications. Wiley, New York, 1973. R. A. Johnson, G. K. Bhattacharyya, Statistics. Principles and methods. Wiley, New York, 1992. M.G. Kendall, A. Stuart, The Theory of Advanced Statistics IIII. Griffin, London, 1966. Handouts: tables of notable distributions and percentile values of basic test distributions.
For passing grade, student must solve correctly at least 50% of homework exercises (there will be 4 homework assignments, each of them containing 5 exercises) and 50% of the test exercises (there will be 2 tests, each of them containing 10 exercises). Grading in the function of the collected (maximum 40) points: A: 3640 A: 3335 B+: 3032 B: 2629 B: 2325 C+: 2022 However, the final grade can be contested by taking an oral exam at the end of the semester.
Schedule of the tests: 6^{th} and 12^{th} weeks. Contact details: Marianna Bolla, Budapest University of Technology and Economics, Institute of Mathematics, 1111. Budapest, Egry József u. 1. Bldg. H5.2 Office phone: 0614631111, ext. 5902. Email: marib@math.bme.hu, Homepage: www.math.bme.hu/~marib/ceu Name of Course: 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.
At the end of the course students take an oral exam they must recall notions, definitions, theorem and proofs. Their grades are awarded on their performance on the final oral exam. During the exam students got the topics as outlined above, have about 20 minutes preparation time, then must present a short view of the topics including theorems, proofs, motivation, counterexamples as indicated by the topic.
László Csirmaz Csirmaz@ceu.hu Name of Course: Introduction to Discrete Mathematics
Fundamental concepts and results of combinatorics and graph theory. Main topics: counting, recurrences, generating functions, sieve formula, pigeonhole principle, Ramsey theory, graphs, flows, trees, colorings.
The main goal is to study the basic methods of discrete mathematics via a lot of problems, to learn combinatorial approach of problems. Problem solving is more important than in other courses!
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.
Week 1. Basic counting problems, permutations, combinations, sum rule, product rule Week 2. Occupancy problems, partitions of integers Week 3. Solving recurrences, Fibonacci numbers Week 4. Generating functions, applications to recurrences Week 5. Exponential generating functions, Stirling numbers, derangements Week 6. Advanced applications of generating functions (Catalan numbers, odd partitions) Week 7. Principle of inclusion and exclusion (sieve formula), Euler function Application of sieve formula to Stirling numbers, derangements, and other involved problems Week 8. Pigeonhole principle, Ramsey theory, Erdos Szekeres theorem Week 9. Basic definitions of graph theory, trees Week 10. Special properties of trees, Cayley’s theorem on the number of labeled trees Week 11. Flows in networks, connectivity Week 12. Graph colorings, Brooks theorem, colorings of planar graphs References: Fred. S. Roberts, Applied Combinatorics, Prentice Hall, 1984 Fred. S. Roberts, Barry Tesman, Applied Combinatorics, Prentice Hall, 2004 Bela Bollobas, Modern Graph Theory, Springer, 1998 Teaching format: lectures combined with classroom exercises, problems and discussions, homework problems (not just easy exercises!), selected homework problems discussed in class
Written final exam within two weeks after the last lecture + graded homework problems.The final exam covers main results including connections between different areas, advanced problems.
Ervin Gyori, Renyi Institute of Mathematics,1053 Budapest, Realtanoda utca 1315.Email: gyori@renyi.hu Name of Course: Optimization in Economics
In the last decades mathematical methods have become indispensable in the study of many economical problems, in particular, in the optimization of certain reallife phenomena. For instance, J. F. Nash received the Nobel Prize in Economics (1994) for his outstanding contributions in the field of Economics via mathematical tools. Our aim here is to emphasize the importance of Mathematics in the study of a broad range of economical problems. Many applications/examples will be discussed in detail.
The main goal of the present course is to introduce Students into the most important concepts and fundamental results of Economics by using various tools from Mathematics as calculus of variations, critical points, matrixalgebra, or even RiemannianFinsler geometry. Starting with basic economical problems, our final purpose is to describe some recent research directions concerning certain optimization problems in Economics.
The Students will learn how to use wellknown mathematical tools to treat both theoretical and practical economical problems.
Lecture 1. Introduction and motivation: some basic problems from Economics via optimization. Lecture 2. Economic applications of onevariable calculus (demand and marginal revenue, elasticity of price, cost functions, profitmaximizing output). Lecture 3. Economic applications of multivariate calculus (consumer choice theory, production theory, the equation of exchange in Macroeconomics, Paretoefficiency, application of the least square method). Lectures 4. Linear programming (application of the geometric, simplex and dual simplex method). Lecture 5. Linear economical problems (diet problem, Ricardian model of international trade). Lecture 6. Comparative statics I (equilibrium comparative statics in one and two dimensions; comparative statics with optimization, perfectly competitive firms, Cournot duopoly model). Lecture 7. Comparative statics II: n variables with and without optimization (equilibrium comparative statics in n dimensions, Grosssubstitute system, perfectly competitive firms). Lecture 8. Comparative statics III: Optimization under constraints (Lagrangemultipliers, specific utility functions, expenditure minimization problems). Lecture 10. Nash equilibrium points (existence, location, dynamics, and stability). Lecture 11. Optimal placement of a deposit between markets: a RiemannFinsler geometrical approach. Lecture 12. Economical problems via best approximations. References:
Attendance is mandatory. Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments. Tests and Grading: There will be two partial exams (a. In October: Lectures 16, 50%; b. In December: Lectures 712, 50%). Extra bonus points may be earned by presentations as well as by participation in class discussions. Final grade: from the two partial exams.
Alexandru Kristaly, alexandrukristaly@yahoo.com Name of Course: Stochastic Processes and Applications
The most common classes of stochastic processes are presented that are important in applications an stochastic modeling. Several real word applications are shown. Emphasis is put on learning the methods and the tricks of stochastic modeling.
The main goal of the course is to learn the basic tricks of stochastic modeling via studying many applications. It is also important to understand the theoretical background of the methods.
The students will learn the most common methods in stochastic processes and their applications.
Books: 1. S. M. Ross, Applied Probability Models with Optimization Applications, HoldenDay, San Francisco, 1970. 2. S. Asmussen, Applied Probability and Queues, Wiley, 1987.
Attendance is mandatory. Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments. Tests and Grading: There will be a final exam (worth 70%). The homeworks worth 30%.
Final exam: within two weeks after the final lecture Office hours: by appointment Balázs Székely szbalazs@math.bme.hu Name of Course: Basic Algebra 2 (MS)
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 Teaching format: lecture combined with classroom discussions, regular homework assignments
Attendance is mandatory. Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments. Tests and Grading: There will be a midterm test (worth 30%), a final exam (worth 40%) and homework (worth 30%). Final exam: within two weeks after the final lecture
Name of Course: 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, Teaching format: lecture combined with classroom discussions
Attendance is mandatory. Homework: will be assigned regularly. The final exam will be based to a significant extent on homework assignments. Tests and grading: the homeworks worth 30% and the final exam 70%. Final exam: written exam in two weeks after the final lecture.
Name of Course: Functional Analysis and Differential Equations
Name of Course: Combinatorial Optimization
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.
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 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.
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 Books: E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Courier Dover Publications, 2001 or earlier edition: Rinehart and Winston, 1976 Teaching format: lecture combined with classroom discussions and possible presentations.
Attendance is mandatory.Homework and quiz are to be assigned regularly. Worth of them in grading is 20 %.Midterm exam (worth 40 %) and Final exam (worth 40 %) are based on or related to homework and quizzes. Bonus might be earned by presentations.
Office hours on Monday at 11:30 am or by appointment. Ervin Győri Rényi Institute, Reáltanoda u.1315. 1053 Budapest gyori@renyi.hu Name of Course: Cryptology (Laszlo Csirmaz) (syllabus TBA) Name of Course: Quantitative Financial Risk Analysis (Balazs Janecsko and Imre Kondor) (syllabus TBA) Name of Course: Topics in Financial Mathematics (Alexandru Kristaly) (syllabus TBA) Name of Course: Statistics of Stochastic Processes (Balazs Szekely) (syllabus TBA) Annex Mathematics Entrance Examination The exam takes 3 hours and consists of problems in algebra and analysis. Of course, problems may involve a mixture of analysis and algebra. Some problems are computational, some ask for proofs, and some ask for examples or counterexamples. Here is a list of subjects required for the entrance exam: AlgebraLinear Algebra:
Abstract Algebra:
Analysis
