MSC IN MATHEMATICS
The two-year MSc program in Mathematics provides a profound knowledge of mathematics competitive both in the academic and non-academic sectors,  covering topics of algebra and number theory, analysis, geometry, probability theory and statistics, discrete mathematics, operations research, stochastics and data science. There is a large flexibility in choosing subjects according to the personal interest of the student. Foundational courses are provided to those who need them. Theoretical courses are taught mainly by internationally recognized scientists of the university. The students may enter leading-edge research projects of the Institute of Mathematics. Completing our program the graduated students have excellent career opportunities in the research sector, universities and research institutes (by becoming PhD students at either our university or some cutting-edge universities in the US or Europe), as well as in the industrial or commercial sector (by getting well-paid jobs that require greater general mathematical knowledge or a strong ability of abstract thinking and problem solving).
CURRICULUM
Code Title Parameters* ECTS credits
per semester
Lc Pr Lb Rq Cr I II III IV
Theoretical foundations (20 ECTS credits). Earlier not completed, prescribed subjects from BSc in Math. One of the following courses related to the main block of Professional subjects is cumpolsory, others are prescribed by the intructor of the student.  Those students who – due to a solid theoretical preparation – need less than 20 credits from BSc in Math subjects will obtain the remaining credits by choosing other optional courses of professional character.  12 4 4  
BMETE91AM39 Algebra 2 4 0 0 v 4 4      
BMETE92AM45 Partial Differential Equations 2 2 0 v 4   4    
BMEVISZA026 Combinatorics and Graph Theory 2 2 2 0 v 4   4    
BMETE94AM20 Differential Geometry 2 3 1 0 v 4   4    
BMETE94AM22 Convex Geometry 2 2 0 v 4   4    
BMETE95AM41 Stochastic Processes 5 0 0 v 6 6      
Professional subjects (30 ECTS credits must be completed).  Courses must be chosen from at least 4 different blocks. Students must chose a main block from where at least 2 courses must be completed. From this main block of Professional subjects and Professional subjects of specialization at least 20 ECTS credits must be completed. 5 5 10 10
  Block of algebra and number theory          
BMETE91MM01 Commutative Algebra and Algebraic Geometry 3 1 0 f 5 5      
BMETE91MM03 Group Theory 3 1 0 v 5       5
BMETE91MM08 Algebraic and Arithmetical Algorithms 3 1 0 f 5 5      
  Block of analysis                  
BMETE93MM02 Dynamical Systems 3 1 0 v 5       5
BMETE92MM00 Fourier Analysis and Function Series 3 1 0 v 5     5  
BMETE93MM03 Partial Differential Equations 2 3 1 0 f 5       5
  Block of discrete mathematics                  
BMETE91MM00 Theoretical Computer Science 3 1 0 f 5       5
BMEVISZM020 General and Algebraic Combinatorics 3 1 0 f 5     5  
BMEVISZM029 Combinatorial Optimization 3 1 0 v 5   5    
  Block of geometry                  
BMETE94MM00 Differential Geometry and Topology 3 1 0 v 5     5  
BMETE94MM02 Combinatorial and Discrete Geometry 3 1 0 f 5     5  
BMETE94MM01 Projective Geometry 3 1 0 f 5 5      
  Block of operations research                  
BMETE93MM01 Linear Programming 3 1 0 v 5     5  
BMETE93MM04 Nonlinear Programming 3 1 0 v 5       5
BMETE93MM30 Game Theory 3 1 0 v 5     5  
  Block of stochastics                  
BMETE95MM05 Introduction to Stochastic Analysis 3 1 0 v 5       5
BMETE95MM05 Mathematical Statistics and Information Theory 3 1 0 v 5       5
BMETE95MM15 Multivariate Statistics 3 1 0 v 5     5  
Professional subjects of specialization (30 ECTS credits). From the main block (chosen by the student) of Professional subjects and Professional subjects of specialization at least 20 ECTS credits must be obtained. Besides this from other two blocks of Professional subjects of specialization and Professional subjects at  least 10-10 ECTS credits must be obtained (these courses must be different from the professional courses of 30 ECTS credits already chosen). 
5 13 5 7
  Block of algebra          
BMETE91MM04 Representations of Groups and Algebras 3 1 0 f 5   5    
BMETE91MM05 Advanced Linear Algebra 2 0 0 v 3     3  
BMETE91MM06 Homological Algebra 2 0 0 f 2     2  
BMETE91MM07 Algebraic Number Theory 2 0 0 v 3   3    
BMETE95MM13 Analytic Number Theory 2 0 0 f 2   2    
BMETE91AM19 Logical Methods in Artifical Intelligence 2 0 0 v 3       3
  Block of analysis                  
BMETE92MM03 Matrix Analysis 2 0 0 v 3 3      
BMETE92MM05 Operator Theory 3 1 0 v 5     5  
BMETE92MM39 Introduction to Quantum Information Theory 2 0 0 f 2 2      
BMETE92MM08 Inverse Scattering Problems 2 0 0 v 3 3      
BMETE92MM22 Distribution Theory and Green Functions 2 0 0 v 2       2
BMETE92MM07 Numerical Methods 2 - Partial Differential Equations 2 0 2 v 5   5    
  Block of discrete mathematics                  
BMEVISZM031 Algorithms and Complexity 3 1 0 f 5   5    
BMEVISZM032 Graphs, Hypergraphs and Their Applications 3 1 0 f 5   5    
BMETE93MM28 Selected Topics in Data Science 2 0 0 v 4   4    
BMETE91MM20 Advanced Machine Learning 2 0 0 v 4 4      
  Block of geometry                  
BMETE94MM14 Introduction to Riemannian Geometry and Morse Theory 3 1 0 v 5       4
BMETE94MM?? Non-Euclidean Geometry 2 0 0 f 3     3  
  Block of operations research                  
BMETE93MM05 Stochastic Programming 3 1 0 v 5       5
BMETE93MM06 Operations Research Software 0 0 2 f 2     2  
BMETE93MM00 Global Optimization 3 1 0 f 5   5    
  Block of stochastics                  
BMETE95MM07 Markov Processes and Martingales 3 1 0 v 5 5      
BMETE95MM10 Limit- and Large Deviation Theorems of Probability Theory 3 1 0 v 5       5
BMETE95MM11 Stochastic Models 2 0 0 f 2   2    
BMETE95MM12 Advanced Theory of Dynamical Systems 2 0 0 f 2   2    
BMETE95MM09 Statistical Program Packages 2 0 0 2 f 2     2  
BMETE95MM20 Nonparametric Statistics 2 0 0 v 3     3  
BMETE95MM26 Time Series Analysis with Applications in Finance  2 0 0 f 3 3      
BMETE95MM34 Markov Decision Processes and Reinforcement Learning 2 0 0 v 3   3    
Obligatory common subjects (30 ECTS credits) 1 4 10 15
BMETE92MM01 Individual Projects 1 0 0 4 f 4   4    
BMETE92MM02 Individual Projects 2 0 0 4 f 4     4  
BMETE95MM01 Mathematical Modelling Seminar 1 2 0 0 f 1 1      
BMETE95MM02 Mathematical Modelling Seminar 2 2 0 0 f 1     1  
BMETE90MM90 Report 0 0 0 a 0   0    
BMETE90MM98 Preparatory Course for Master's Thesis 0 2 0 f 5     5  
BMETE90MM99 Master's Thesis 0 8 0 f 15       15
Elective courses (10 ECTS credits must be completed) 5 5    
*Parameters:
 
Lc = lecture, Pr = practice, Lb = laboratory (hours per week);
 
Rq = requirement or exam type (v = examination, f = midterm exam);
 
Cr = ECTS credits.
Preliminary requirements can be seen at the subjects below.
DESCRIPTION OF SUBJECTS
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE91AM39 Algebra 2 4 0 0 v 4   4    
Course coordinator: Dr. Erzsébet Lukács
Descripton: Field extensions, construction and uniqueness of simple algebraic extensions, finite and algebraic extensions. Normal extensions, splitting field, separable extension, finite fields, Wedderburn's theorem, Galois group, irreducibility of the cyclotomic polynomials, Galois groups of radical extensions, Galois correspondence, Fundamental theorem of Galois theory. Applications of Galois theory: Fundamental theorem of algebra, ruler and compass constructions, solvability of equations by radicals, Abel–Ruffini theorem. Existence and uniqueness of algebraic closure, transcendental extensions, transcendence of e, Gelfand-Schneider theorem. - Review of the basic concepts of number theory, Euler ϕ function. Linear congruences and systems of congruences, binomial congruences of higher degree, discrete logarithm, congruences of prime power moduli. Quadratic congruences, Legendre and Jacobi symbol, quadratic reciprocity. Prime numbers: Euclid's theorem, gaps between primes, Chebyshev's theorem, harmonic series of primes, Dirichlet's theorem for (nk + 1). Arithmetic functions: d(n), σ(n), ϕ(n). Multiplicativity, convolution, Möbius function, the Möbius inversion formula. Prime number theorem, magnitude of the nth prime, prime tests, Rabin–Miller test, RSA function. Diophantine equations: linear diophantine equations, Pythagorean triples, Fermat's two squares theorem, Gaussian integers.
Literature:
– I. Stewart: Galois Theory, CRC Press, 2003.
– Niven, Zuckerman, Montgomery: An Introduction to the Theory of Numbers, John Wiley & Sons, 1960.
– M.B. Nathanson: Elementary Methods in Number Theory, Springer, 2000.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE92AM45 Partial Differential Equations 2 2 0 v 4        
Course coordinator: Dr. János Karátson
Descripton: Classification of partial differential equations (PDEs). First order linear PDEs. Convection transport processes. First order quasilinear PDEs. Parabolic Cauchy problems. Heat conduction problem, qualitative properties. Hyperbolic Cauchy problems. Wave equation in one space dimension: vibrating string, travelling and standing waves. Wave equation in two and three space dimensions using surface integral. Elliptic boundary value problems. Elliptic models: stationary heat distribution, elastic torsion. Uniqueness of the solution. The problem of the notion of solution. Theoretical background: Hilbert spaces, Fourier series, symmetric operators. Fourier series expansion for elliptic boundary value problems using eigenfuctions. Theoretical background: distributions, Sobolev spaces. Weak solution of elliptic problems. Weak eigenvalue problem. Parabolic and és hyperbolic initial-boundary value problems. Elliptic fundamental solution, mathematical description of the potential for a point source, Green’s function.
Literature:
– L.C. Evans: Partial Differential Equations, AMS, 2010.
Code Title Lc Pr Lb Rq Cr I II III IV
BMEVISZA026 Combinatorics and Graph Theory 2 2 2 0 v 4        
Course coordinator: Dr. Tamás Fleiner
Descripton: Geometric and abstract duality, weak isomorphism (2-isomorphism) and the Whitney theorems. Vertex and edge coloring, Mycielsky's construction, Brooks' theorem. 5-colour theorem, Vizing's theorem, connection of edge-colouring to matchings, Petersen's theorem. List colouring of graphs, Galvin's theorem. Perfect graphs, interval graphs and the perfect graph theorem. Ramsey's theorem, Erdős-Szekeres theorem, Erdős' lower bound and the probabilistic method. Turán's theorem, Erdős-Stone theorem, Erdős-Simonovits theorem. Hypergraphs, Erdős-Ko-Rado theorem, Sperner's theorem and the LYM inequality. De Bruijn-Erdős theorem, finite planes, construction from finite field, and from difference sets. Generating functions, Fibonacci numbers, Catalan numbers. Posets, Dilworth's theorem.
Literature:
– R. Diestel: Graph Theory, online available.
– J.A. Bondy, U.S.R. Murty: Graph Theory with Applications.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE94AM20 Differential Geometry 2 3 1 0 v 4        
Course coordinator: Dr. Szilárd Szabó
Descripton: Differentiable manifolds, tangent space, tangent bundle. Integral curve of a vector field. Vector bundles and related algebraic constructions (direct sum, tensor product, dual, homomorphisms). Differential forms, pull-back, exterior product, exterior derivation. Integration on compact oriented manifolds, Stokes' theorem. Lie-derivative, Lie-Cartan formula. Riemannian metric, examples. Geodetics, exponential map. Lie groups and algebras. Hopf-Rinow theorem and its consequences. Connections on a vector bundle, parallel transport, integrability. Levi-Civita connection, the Riemann curvature tensor. Properties of the curvature tensor, Ricci curvature. First and second variation of arc length, Jacobi vector fields.
Literature:
– P. Petersen: Riemannian geometry, Graduate Texts in Mathematics, 171. Springer.
– S. Gallot, D. Hulin, J. Lafontaine: Riemannian geometry, Universitext, Berlin, Springer.
– M. Berger, B. Gostiaux: Differential geometry: manifolds, curves, and surfaces, Graduate Texts in Mathematics.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE94AM22 Convex Geometry 2 2 0 v 4        
Course coordinator: Dr. Zsolt Lángi
Descripton: Introduction: affine and convex sets, affine dependence, independence, affine and convex combinations, affine hull, isolation theorem, characterization of closed, convex sets as the intersection of closed half spaces. Convex hull, theorems of Radon, Helly and Carathéodory, their applications. Linear functionals and their connection with hyperplanes, Minkowski sum, separation of convex sets with hyperplanes, supporting hyperplanes, faces of a convex body, extremal and exposed points, theorems of Krein-Milman and Straszewicz. Indicator function, algebras of closed/compact convex sets, valuations, Euler characteristic and the proof of its existence. Convex polytopes and polyhedral sets, their connection, face structure of polytopes, combinatorial equivalence. The f-vector of polytopes, Euler characteristic of polytopes, theorem of Euler. Polar of a set, fundamental properties of polarity, properties of the polar of a polytope, dual polytope. Moment curve, cyclic polytopes and their face structure, Gale’s evenness condition. Hausdorff distance of convex bodies. Affine transformations, Banach-Mazur distance. Ellipsoid as an affine ball. Unique existence of largest volume inscribed, and smallest volume circumscribed ellipsoid of a convex body. The Löwner-John ellipsoid, John’s theorem for general, and centrally symmetric convex bodies.
Literature:
– B. Grünbaum: Convex Polytopes, Graduate Texts in Mathematics 221, Springer, New York, 2003.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95AM34 Stochastic Processes 5 0 0 v 6        
Course coordinator: Dr. Károly Simon
Descripton: Basic notions: finite dimensional marginals, Kolmogorov’s fundamental theorem, strongly and weakly stationary processes, processes with stationary and/or independent increments. Discrete Markov chains: linear algebra of stochastic matrices, classification of states. Finite Markov chains: stationary measures and ergodic behaviour. Reversibility, random walk on graphs. Urn models. Countable Markov chains: transience, null-recurrence, positive-recurrence. Random walks on Zd: Polya’s theorem. Random walks on countable graphs, branching processes, discrete time birth-and-death processes, queuing problems. Random walks on Z1: the reflection principle and limit distribution of the maximum, difference equations. Continuous time, discrete space Markov processes: the Poisson process, jump rates, exponential clocks. Stochastic semigroup: Kolmogorov-Chapman equations, infinitesimal generator. Complements of measure theory: filtrations, adapted processes, natural filtration. The general notion of conditional expectation (Kolmogorov’s theorem), fundamental properties. Discrete time martingales: sub/super/martingales, stopping times, stopped martingales. Optional stopping theorem, Wald identity, martingale convergence theorem, submartingale inequality, maximal inequality. Azuma-Hoffding inequality, applications. The Brownian motion: defining properties, covariances. Sketch of Paul Levy’s construction, basic analytic properties. Applications.
Literature:
– Essentials of Stochastic Processes (2nd edition), Springer, 2012.
– R. Durrett: Probability Theory with Examples, 4th edition, Cambridge U. Press, 2010.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM00 Global Optimization 3 1 0 f 5   5    
Course coordinator: Dr. Boglárka Gazdag-Tóth
Different forms of global optimization problems, their transformation to each other, and their reduction to the one-dimensional problem. Comparison of the complexity of global optimization and linear programming problems. Classifications of the global optimization methods. Lagrange function, Kuhn–Tucker theorem, convex and DC programming. Basic models and methods of stochastic programming. Multi-start and stochastic methods for global optimization, their convergence properties and stopping criteria. Methods based on Lipschitz constant, and their convergence properties. Branch and Bound schema, methods based on interval analysis, automatic differentiation. Multi-objective optimization.
Literature:
– R. Horst, P. Pardalos: Handbook of Global Optimization, Kluwer, 1995.
– R. Horst, P.M. Pardalos, N.V. Thoai: Introduction to Global Optimization, Kluwer, 1995.
– A. Törn, A. Zilinskas: Global Optimization, Springer, 1989.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM01 Linear Programming 3 1 0 v 5 5      
Course coordinator: Dr. Tibor Illés
System of linear equations: solution and solvability. Gauss-Jordan elimination method. System of linear inequalities. Alternative theorems, Farkas lemma and its variants. Solution of system of linear inequalities using pivot algorithms. Convex polyhedrons. Minkowski-, Farkas- and Weyl-theorems. Motzkin-theorem. Primal-dual linear programming problems. Feasible solution set of linear programming problems. Basic solution of linear programming problem. Simplex and criss-cross algorithms. Cycling, anti-cycling rules: Bland’s minimal index rule. Two phase simplex method. Revised simplex method. Sensitivity analysis. Decomposition methods: Dantzig-Wolfe. Special type of pivot algorithms: lexicographic and lexicographic dual simplex methods. Monotonic build-up simplex algorithms. Interior point methods of linear programming problems. Self-dual linear programming problem. Central path and its uniqueness. Computation of Newton-directions. Analytical centre, Sonnevend-theorem. Dikin-ellipsoid, affine scaling primal-dual interior point algorithm and its polynomial complexity. Tucker-model, Tucker theorem. Rounding procedure. Khachian’s ellipsoid algorithm. Karmarkar’s potential function method. Special interior point algorithms.
Literature:
– K.G. Murty: Linear and combinatorial programming. John Wiley & Sons Inc., New York, 1976.
– C. Roos, T. Terlaky,  J.P. Vial: Interior Point Methods for Linear Optimization. Springer US, New York, 2005.
– A. Schrijver: Theory of Linear and Integer Programming, John Wiley, New York, 1986.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE91MM00 Theoretical Computer Science 3 1 0 f 5       5
Course coordinator: Dr. Miklós Ferenczi
Foundations of logic programming and automated theorem proving. Finite models and complexity. Non classical logics in Computer Science: temporal dynamic and programming logics. Recursive functions and lambda calculus. Boole algebras, relational algebras and their applications. Some important models of computation. Basic notions of complexity theory, some important time and spaces classes. NP completeness. Randomised computation. Algorithm design techniques. Advanced data structures, amortised costs. Pattern matching in text. Data compression.
Literature:
– A. Galton: Logic for Information Technology, Wiley, 1990.
Code Title Lc Pr Lb Rq Cr I II III IV
BMEVISZM020 General and Algebraic Combinatorics 3 1 0 f 5     5  
Course coordinator: Dr. Katalin Friedl
Combinatorics of the Young tableaux, tableau rings. Pieri formulas, Schur polynomials, Kostka numbers. Robinson-Schensted-Knuth correspondence. Littlewood-Richardson numbers, Littlewood-Richardson theorem. Important symmetric polynomials, their generating functions. Cauchy-Littlewood formulas. Garsia's generalization of the fundamental theorem on symmetric polynomials. Bases of the ring of symmetric functions. Topics from combinatorial optimization: greedy algorithm, augmenting methods. Matroids, their basic properties, matroid intersection algorithm. Approximation algorithms (set cover, travelling salesman, Steiner trees). Scheduling algorithms (single machine scheduling, scheduling for parallel machines, bin packing).
Literature:
– W. Fulton, Y. Tableaux: With Applications to Representation Theory and Geometry, London Math. Soc. Student Texts, Paperback, Cambridge Univ. Press, 1996.
– R.P. Stanley: Enumerative Combinatorics I.- II., Cambridge University Press, 2001.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM02 Dynamical Systems 3 1 0 v 5       5
Course coordinator: Dr. Péter Bálint
Continuous-time and discrete-time dynamical systems, continuous versus descrete: first return map, discretization. Local theory of equilibria: Grobman–Hartman lemma, stable-unstable-center manifold, Poincaré's normal form. Attractors, Liapunov functions, LaSalle principle, phase portrait. Structural stability, elementary bifurcations of equilibria, of fixed points, and of periodic orbits, bifurcation curves in biological models. Tent and logistic curves, Smale horseshoe, solenoid: properties from topological, combinatorial, and measure theoretic viewpoints. Chaos in the Lorenz model.
Literature:
– P. Glendinning: Stability, Instability and Chaos, Cambridge University Press, Cambridge, 1994.
– C. Robinson: Dynamical Systems, CRC Press, Boca Raton, 1995.
– S. Wiggins: Introduction to Applied Nonlinear Analysis and Chaos, Springer, Berlin, 1988.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE92MM00 Fourier Analysis and Function Series 3 1 0 v 5     5  
Course coordinator: Dr. Miklós Horváth
Completeness of the trigonometric system. Fourier series, Parseval identity. Systems of orthogonal functions, Legendre polynomials, Haar and Rademacher systems. Introduction to wavelets, wavelet orthonormal systems. Fourier transform, Laplace transform, applications. Convergence of Fourier series: Dirichlet kernel, Dini and Lischitz convergence tests. Fejer’s example of divergent Fourier series. Fejer and Abel-Poisson summation. Weierstrass-Stone theorem, applications. Best approximation in Hilbert spaces. Müntz theorem on the density of lacunary polynomials. Approximations by linear operators, Lagrange interpolation, Lozinski-Harshiladze theorem. Approximation by polynomials, theorems of Jackson. Positive linear operators Korovkin theorem, Bernstein polynomials, Hermite-Fejer operator. Spline approximation, convergence, B-splines.
Literature:
– G. Lorentz, M.V. Makovoz: Constructive Approximation, Springer, 1996.
– M.J.D. Powell: Approximation Theory and Methods, Cambridge University Press, 1981.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM03 Partial Differential Equations 2 3 1 0 f 5       5
Course coordinator: Dr. Márton Kiss
The Laplacian in Sobolev space (revision). Weak and strong solutions to second order linear parabolic equations. Ritz-Galerkin approximation. Linear operator semigroups (According to Evans and Robinson). Weak and strong solutions to reaction-diffusion (quasilinear parabolic) equations. Ritz–Galerkin approximation. Nonlinear operator semigroups (According to Evans and Robinson). Only in examples: monotonicity, maximum principles, invariant regions, stability investigations for equilibria by linearization, travelling waves (According to Smoller). Global attractor. Inertial manifold (According to Robinson).
Literature:
– L.C. Evans: Partial Differential Equations, AMS, Providence R.I., 1998.
– J. Smoller: Shock Waves and Reaction-Diffusion Equations, Springer, Berlin, 1983.
– J.C. Robinson: Infinite-dimensional Dynamical Systems, CUP, Cambridge, 2001.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95MM04 Stochastic Analysis and its Applications 3 1 0 v 5 5      
Course coordinator: Dr. Károly Simon
Introduction. Markov processes, stochastic semi-groups, infinitesimal generators, martingales, stopping times. Brownian motion. Brownian motion in nature. Finite dimensional distributions and continuity of Brownian motion. Constructions of the Wiener process. Strong Markov property. Self-similarity and recurrence of Brownian motion, time reversal. Reflection principle and its applications. Local properties of Brownian path: continuity, Hölder continuity, non-differenciability. Quadratic variations. Continuous martingales. Definition and basic properties. Dubbins-Schwartz theorem. Exponential martingale. Lévy processes. Processes with independent and stationary increments, Lévy-Hintchin formula. Decomposition of Lévy processes. Construction by means of Poisson processes. Subordinators, and stable processes. Examples and applications. Stochastic integration I. Discrete stochastic integrals with respect to random walks and discrete martingales. Applications, discrete Balck-Scholes formula. Stochastic integrals with respect to Poisson process. Martingales of finite state space Markov processes. Quadratic variations. Doob-Meyer decomposition. Stochastic integration II. Predictable processes. Itô integral with respect to the Wiener process, quadratic variation process. Doob-Meyer decomposition. Itô formula and its applications.
Literature:
– K.L. Chung, R. Williams: Introduction to  stochastic integration. Second edition. Birkauser, 1989.
– R. Durrett: Probability: theory and examples. Second edition. Duxbury, 1996.
– B. Oksendal: Stochastic Differential equations. Sixth edition. Springer, 2003.
– D. Revuz, M. Yor: Continuous martingales and Brownian motion. Third edition. Springer, 1999.
– G. Samorodnitsky, M.S. Taqqu: Stable Non-Gaussian Random Processes: Stochastic  Models with Infinite Variance, Chapman and Hall, New York, 1994.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95MM05 Mathematical Statistics and Information Theory 3 1 0 v 5       5
Course coordinator: Dr. Marianna Bolla
Multivariate statistical inference in multidimensional parameter spaces: Fisher’s information matrix, likelihood ratio test. Testing hypotheses in multivariate Gauss model: Mahalanobis’ distance, Wishart’s, Hotelling’s, Wilks’ distributions. Linear statistical inference, Gauss–Markov theorem. Regression analysis, one- and two-way analysis of variance as a special case of the linear model. ANOVA tables, Fisher-Cochran theorem. Principal component and factor analysis. Estimation and rotation of factors, testing hypotheses for the effective number of factors. Hypothesis testing and  I-divergence (the discrete case). I-projections,  maximum likelihood estimate as I-projection in exponential families. The limit distribution of the I-divergence statistic. Analysis of contingency tables by information theoretical methods, loglinear models. Statistical algorithms  based on  information geometry: iterative scaling,  EM algorithm. Method of maximum entropy.
Literature:
– M. Bolla, A. Krámli: Theory of statistical inference, Typotex, Budapest, 2005.
– I. Csiszár, P.C. Shields: Information Theory and Statistics. A tutorial. In: Found. and Trends in Comm. and Info. Theory, 420-525. Now Publ. Inc., The Netherlands, 2004.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE91MM01 Commutative Algebra and Algebraic Geometry 3 1 0 f 5 5      
Course coordinator: Dr. Alex Küronya
Closed algebraic sets and their coordinate rings, morphisms, irreducibility and dimension, Hilbert Nullstellensatz, the correspondence between radical ideals and subvarieties of affine space. Monomial orders, Gröbner bases, Buchberger algorithms, computations in polynomial rings. From regular functions to rational maps, local rings, fundamentals of sheaf theory, ringed spaces. Projective space and its subvarieties, homogeneous coordinate ring, morphisms, the image of a projective variety is closed. Geometric constructions: Segre and Veronese embeddings, Grassmann varieties, projection from a point, blow-up. Dimension of affine and projective varieties, hypersurfaces. Smooth varieties, Zariski tangent space, the Jacobian condition. Hilbert function and Hilbert polynomial, examples, computer experiments. Basic notions of rings and modules, chain conditions, free modules. Finitely generated modules, Cayley-Hamilton theorem, Nakayama lemma. Localization and tensor product. Free resolutions of modules, Gröbner theory of modules, computations, Hilbert syzygy theorem.
Literature:
– A. Gathmann: Algebraic geometry, 2003, www.mathematik.uni-kl.de/~gathmann/en/pub.html
– I.R. Shafarevich: Basic Algebraic Geometry I.-II., Springer Verlag, 1995.
– M. Reid: Undergraduate Commutative Algebra, Cambridge University Press, 1996.
– R. Hartshorne: Algebraic Geometry, Springer Verlag, 1977.
– M.F. Atiyah, I.G. Macdonald: Introduction to commutative algebra, Addison Wesley Publishing, 1994.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE94MM00 Differential Geometry and Topology 3 1 0 v 5     5  
Course coordinator: Dr. Szilárd Szabó
Smooth manifolds, differential forms, exterior derivation, Lie-derivation.  Stokes' theorem, de Rham cohomology, Mayer–Vietoris exact sequence, Poincaré-duality. Riemannian manifolds, Levi–Civitá connection, curvature tensor, spaces of constant curvature. Geodesics, exponential map, geodesic completeness, the Hopf–Rinow theorem, Jacobi fields, the Cartan–Hadamard theorem, Bonnet's theorem.
Literature:
– J.M. Lee: Riemannian Manifolds: an Introduction to Curvature, Graduate Texts in Mathematics 176, Springer Verlag.
– P. Petersen: Riemannian Geometry, Graduate Texts in Mathematics 171, Springer Verlag.
– J. Cheeger, D. Ebin: Comparison Theorems in Riemannian Geometry, North-Holland Publishing Company, Vol. 9, 1975.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95MM15 Multivariate Statistics 3 1 0 v 5     5  
Course coordinator: Dr. Marianna Bolla
Multivariate central limit theorem and its applications.  Density, spectra and asymptotic distribution of random matrices in multivariate statistics (Wishart-, Wigner-matrices). How to use separation theorems for eigenvalues and singular values in the principal component, factor, and correspondence analysis. Factor analysis  as low rank representation, relatios between  representations and metric clustering algorithms. Methods of classification: discriminatory analysis, hierarchical, k-means, and graph theoretical methods of cluster analysis. Spectra and testable parameters of graphs. Algorithmic models,  statistical  learning. EM algorithm, ACE algorithm, Kaplan–Meier estimates. Resampling methods: bootstrap and jackknife.  Applications in data mining, randomized methods for large matrices. Mastering the multivariate statistical methods and their nomenclature  by means of a program package (SPSS or S+),  application oriented  interpretation  of the output data.
Literature:
– K.V. Mardia, J.T. Kent, J.M. Bibby: Multivariate Analysis, Academic Press, Elsevier Science, 1979.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95MM20 Nonparametric Statistics 2 0 0 v 3     3  
Course coordinator: Dr. László Györfi
Density function estimation. Distribution estimation, L1 error. Histogram. Estimates by kernel function. Regression function estimation. Least square error. Regression function. Partition, kernel function, nearest neighbour estimates. Empirical error minimization. Pattern recognition. Error probability. Bayes decision rule. Partition, kernel function, nearest neighbour methods. Empirical error minimization. Portfolio strategies. Log-optimal, empirical portfolio strategies. Transaction cost.
Literature:
– L. Devroye, L. Györfi: Nonparametric Density Estimation, Wiley, 1985.
– L. Devroye, L. Györfi, G. Lugosi: Probability Theory of Pattern Recognition, Springer, New York, 1996.
– L. Györfi, M. Kohler, A. Krzyzak, H. Walk: A Distribution-Free Theory of Nonparametric Regression, Springer, New York, 2002.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95MM09 Statistical Program Packages 2 0 0 2 f 2     2  
Course coordinator: Dr. Csaba Sándor
The goal of the course is to provide an overview of contemporary computer-based methods of statistics with a review of the necessary theoretical background. How to use  the  SPSS (Statistical Package for Social Sciences) in program mode. Writing user’s  macros. Interpretation of the output  data  and setting the parameter values accordingly. Definition and English nomenclature of the  dispalyed statistics. Introduction to the S+ and R Program Packages and surveying the novel algorithmic models not available in the SPSS (bootstrap, jackknife, ACE). Practical application. Detailed analysis of a concrete  data set in S+.
Literature:
– K.V. Mardia, J.T. Kent, M. Bibby: Multivariate analysis, Academic Press, New York, 1979.
– L. Ketskeméty, L. Izsó: Introduction to the SPSS Program Package,  in Hungarian, ELTE Publishers, Budapest, 2005.
– S+ or R User's Guide (together with the program package).
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BMETE95MM07 Markov Processes and Martingales 3 1 0 v 5 5      
Course coordinator: Dr. Károly Simon
Martingales: Review (conditional expectations and tower rule, types of probabilistic convergences and their connections, martingales, stopped martingales, Doob decomposition, quadratic variation, maximal inequalities, martingale convergence theorems, optional stopping theorem, local martingales). Sets of convergence of martingales, the quadratic integrable case. Applications (e.g. Gambler's ruin, urn models, gambling, Wald identities, exponential martingales). Martingale CLT. Azuma-Höffding inequality and applications (e.g. travelling salesman problem). Markov chains: Review (definitions, characterization of states, stationary distribution, reversibility, transience-(null-)recurrence). Absorbtion probabilites. Applications of martingales, Markov chain CLT.  Markov chains and dynamical systems; ergodic theorems for Markov chains. Random walks and electric networks. Renewal processes: Laplace transform, convolution. Renewal processes, renewal equation. Renewal theorems, regenerative processes. Stationary renewal processes, renewal paradox. Examples: Poisson process, applications in queueing. Point processes: Definition of point processes. The Poisson point process in one and more dimensions. Transformations of the Poisson point process (marking and thinning, transforming by a function, applications). Point processes derived from the Poisson point process. Discrete state Markov processes: Review (infinitesimal generator, connection to Markov chains, Kolmogorov forward and backward equations, characterization of states,  transience-(null-)recurrence, stationary distribution). Reversibility, MCMC. Absorption probabilities and hitting times. Applications of martingales (e.g. compensators of jump processes). Markov processes and dynamical systems; ergodic theorems for Markov processes. Markov chains with locally discrete state space: infinitesimal generator on test functions.
Literature:
– T. Lindvall: Lectures on the Coupling Method, Dover Publications, Inc., Mineola, NY, 2002.
– J.R. Norris: Markov chains. Cambridge University Press, Cambridge, 1998.
– S. Resnick: Adventures in Stochastic Processes, Birkhäuser Boston, 1992.
– M. Rosenblatt: Markov processes: Structure and Asymptotic Behavior. Springer-Verlag, New York-Heidelberg, 1971.
– D. Williams: Probability with Martingales. Cambridge University Press, 1991.
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BMETE95MM08 Stochastic Differential Equations 3 1 0 v 5   5    
Course coordinator: Dr. Bálint Tóth
Introduction. Itô integral with respect to the Wiener process and continuous martingale, multi-dimensional stochastic integral. Local time. Local time of random walks on the line. Inverse local time, discrete Ray-Knight theorem. Local time of Brownian motion and Ray-Knight theorem. Tanaka formula and its applications. Skorohod reflection, reflected Brownian motion, a theorem by P. Lévy. Stochastic differential equations. SDEs of diffusions: Ornstein-Uhlenbeck, Bessel, Bessel-squared, exponential Brownian motion. SDE of transformed diffusions. Weak and strong solutions, existence and uniqueness. SDE with boundary conditions. Interpretation of the infinitesimal generator. Applications to physics, population dynamics, and finance. Duffusions. Basic examples:  Ornstein-Uhlenbeck, Bessel, Bessel-squared, geometrical Brownian motion. Interpretation as stochastic integrals, and Markov processes. Infinitesimal generator, stochastic semi-groups. Martingale problem. Connection with parabolic and elliptic  partial differential equations. Feyman-Kac formula. Time-change. Cameron-Martin-Girsanov formula. One-dimensional diffusions. Scale function and speed measure. Boundary conditions. Time-inversion. Application to special processes. Special selected topics. Brownian excursion. Two-dimensional Brownian motion, Brownian sheet. SLE. Additive functionals of Markov processes.
Literature:
– K.L. Chung, R. Williams: Introduction to  stochastic integration, 2nd edition, Birkauser, 1989.
– N. Ikeda, S. Watanabe: Stochastic differential equations and diffusion processes, 2nd edition, North Holland, 1989.
– K. Ito, H.P. McKean: Diffusion processes and their sample paths, Springer, 1965.
– J. Jacod, S.N. Shiryaev: Limit theorems for stochastic processes, Springer, 1987.
– S. Karlin, H.M. Taylor: A second course in stochastic processes, Academic, 1981.
– D. Revuz, M. Yor: Continuous martingales and Brownian motion, 3rd edition, Springer, 1999.
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BMETE95MM26 Time Series Analysis with Applications in Finance  2 0 0 f 3 3      
Course coordinator: Dr. Károly Simon
White noise and basic ARMA models, lag operators and polynomials, auto- and crosscorrelation, autocovariance, fundamental representation, state space representation, predicting ARMA models, impulse-response function, stationary ARMA models, Wold Decomposition, vector autoregression (VAR): Sims and Blanchard-Quah orthogonalization, variance decomposition, VARs in state space notation, Granger causality, spectral representation, spectral density, filtering, spectrum of the filtered series, constructing filters, Hodrick-Prescott filter, random walks and unit root time series, cointegration, Beveridge-Nelson decomposition, Bayesian Vector Autoregression (BVAR) models, Gibbs Sampling, coding practice and application to financial and macroeconomic data.
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BMETE95MM10 Limit- and Large Deviation Theorems of Probability Theory 3 1 0 v 5       5
Course coordinator: Dr. Bálint Tóth
Part I.: Limit theorems: Weak convergence of probability measures and distributions. Tightness: Helly-Ptohorov theorem. Limit theorems proved with bare hands: Applications of the reflection principle to random walks: Paul Lévy’s arcsine laws, limit theorems for the maximum, local time and hitting times of random walks. Limit theorems for maxima of i.i.d. random variables, extremal distributions. Limit theorems for the coupon collector problem.  Proof of limit theorem with method of momenta. Limit theorem proved by the method of characteristic function. Lindeberg’s theorem and its applications: Erdős-Kac theorem: CLT for the number of prime factors. Stable distributions. Stable limit law of normed sums of i.i.d. random variables. Characterization of the characteristic function of symmetric stable laws. Weak convergence to symmetric stable laws. Applications. Characterization of characteristic function of general (non-symmetric) stable distributions, skewness. Weak convergence in non-symmetric case. Infinitely divisible distributions:. Lévy-Hinchin formula and Lévy measure. Lévy measure of stable distributions, self-similarity. Poisson point processes and infinitely divisible laws. Infinitely divisible distributions as weak limits for triangular arrays. Applications.  Introduction to Lévy processes: Lévy-Hinchin formula and decomposition of Lévy processes. Construction with Poisson point processes (a la Ito). Subordinators and Lévy processes with finite total variation, examples. Stable processes. Examples and applications.
    Part II.: Large deviation theorems: Introduction: Rare events and large deviations. Large deviation principle. Computation of large deviation probabilities with bare hands: application of Stirling’s formula. Combinatorial methods: The method of types. Sanov’s theorem for finite alphabet. Large deviations in finite dimension: Bernstein’s inequality, Chernoff’s bound, Cramer’s theorem. Elements of convex analysis, convex conjugation in finite dimension, Cramer’s theorem in R^d. Gartner-Ellis theorem. Applications: large deviation theorems for random walks, empirical distribution of the trajectories of finite state Markov chains, statistical applications. The general theory: general large deviation principles. The contraction principle and Varadhan’s lemma. large deviations in topological vector spaces and function spaces. Elements of abstract convex analysis. Applications: Schilder’s theorem, Gibbs conditional measures, elements of statistical physics.
Literature:
– A. Dembo, O. Zeitouni: Large deviation techniques and application.  Springer, 1998.
– R. Durrett: Probability: theory and examples. Second edition. Duxbury, 1996.
– B.V. Gnedenko, A.N. Kolmogorov: Limit theorems for sums of independent random variables, 1951.
– W. Feller: An introduction to probability theory and its applications. Vol.2. Wiley, 1970.
– D.W. Stroock: An introduction to the theory of large deviations. Springer, 1984.
– S.R.S. Varadhan: Large deviations and application . SIAM Publications, 1984.
– D. Williams: Probability with martingales. Cambridge UP, 1990.
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BMETE95MM11 Stochastic Models 2 0 0 f 2   2    
Course coordinator: Dr. Gábor Pete
Coupling methods (stochastic dominance, coupling random variables and stochastic processes, examples: connectivity using dual graphs, optimization problems, combinatorial probability problems)
Percolation (definitions, correlation inequalities, duality, contour methods)
Strongly dependent percolation: Winkler percolation, compatible 0-1 sequences
Basics of statistical physics (Gibbs measure, a few basic models). Card shuffling (completely shuffled deck, how many times should one shuffle?) Random graph models (Erdős–Rényi, Barabási–Albert; basic phenomena). Variants of random walks: scenery reconstruction, self-avoiding és self-repelling walks, loop-erased walks, random walk in random environment). Queueing models and basic behavior; stationary distribution and reversibility, Burke Theorem; systems of queues. Interacting particle systems (simple exclusion on the torus and on the infinite lattice, stationary distribution, Palm distributions, couplings, other models). Graphical construction of continuous time Markov processes (Yule model, Hammersley's process, particle systems). Self organized criticality: sandpile models (questions of construction, commutative dynamics, stationary distribution in finite volume, power law decay of correlations). Linear theory of stationary processes: strongly and weakly stationary processes, spectral properties, autoregressive and moving average processes. Analysis of time series, long memory processes. Models of risk processes.
Literature:
– G. Grimmett: Percolation, Springer-Verlag, Berlin, 1999.
– T. Liggett: Interacting Particle Systems, Springer-Verlag, Berlin, 2005.
– T. Lindvall: Lectures on the Coupling Method, Dover Publications, Inc., Mineola, NY, 2002.
– H. Thorisson: Coupling, Stationarity, and Regeneration, Springer-Verlag, New York, 2000.
– J. Walrand: An Introduction to Queueing Networks, Prentice Hall, 1988.
– W. Werner: Lectures on Two-dimensional Critical Percolation, http://arxiv.org/abs/0710.0856
– W. Werner: Random Planar Curves and Schramm–Loewner Evolutions, http://arxiv.org/abs/math/0303354
– O. Zeitouni: Lecture Notes on Random Walks in Random Environment, XXXI summer school in probability, St Flour, France, Volume 1837 of Springer's Lecture notes in Mathematics
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BMETE95MM12 Advanced Theory of Dynamical Systems 2 0 0 f 2   2    
Course coordinator: Dr. Domokos Szász
Subadditive and multiplicative ergodic theorems. Lyapunov exponents. Spectral properties of measure preserving transformations. Shadowing lemma. Markov partitions and their construction for uniformly hyperbolic systems. Perron-Frobenius operator and its spectrum. Doeblin-Fortet inequality.Stochastic properties of hyperbolic dynamical systems. Kolmogorov-Sinai entropy. Ornstein’s isomorphy theorem (without proof).
Literature:
– M. Pollicott: Lectures on Ergodic theory and Pesin Theory on compact manifolds, CUP, 1993.
– R. Bowen: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer LNM 470, 1975.
– M. Brin, G. Stuck: Introduction to Dynamical Systems, CUP, 2002.
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BMETE92MM01
BMETE92MM02
Individual Projects 1, 2 0 0 4 f 4   4 4  
Course coordinator: Dr. Márta Lángné Lázi
Within the framework of the subject the student is working on an application oriented research subject based on stochastic mathematics lead by an external supervisor. At the end of each semester the student writes a report about his results which will be also presented by him to the other students in a lecture. The activities to be exercised: literature research, modelling, computer aided problem solving, mathematical problem solving.
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BMETE95MM01
BMETE95MM02
Mathematical Modelling Seminar 1, 2 2 0 0 f 1 1   1  
Course coordinator: Dr. Domokos Szász
The aim of the seminar to present case studies on results, methods and problems from applied mathematics for promoting: the spreading of knowledge and culture of applied mathematics; the development of the connections and cooperation of students and professors of the Mathematical Institute, on the one hand, and of  personal, researchers of other departments of the university or of other firms, interested in the applications of mathematics. The speakers talk about problems arising in their work. They are either applied mathematicians or non-mathematicians, during whose work the mathematical problems arise. An additional aim of this course to make it possible for interested students to get involved in the works presented for also promoting their long-range carrier by building contacts that can lead for finding appropriate jobs after finishing the university.
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BMETE90MM90 Report 0 0 0 a 0   0    
Course coordinator: Dr. Attila Andai
A tárgyat akkor tekintjük teljesítettnek, ha a hallgató a felvételi során megkövetelt alapképzésbeli tárgyak elvégzésével az előírt legalább 65 kreditet teljesítette, továbbá a hallgatónak van elfogadott diplomatémája és témavezetője.
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BMETE90MM98 Preparatory Course for Master's Thesis 0 2 0 f 5     5  
Course coordinator: Dr. Attila Andai
A diplomamunka a matematikushallgatóknak a témavezető irányításával elért önálló kutatási, kutatás-fejlesztési eredményeit tartalmazó írásbeli beszámoló (dolgozat). A hallgató a dolgozatban mutassa be a vizsgált témát, fejtse ki a problémákat, és részletesen ismertesse eredményeit. A munkának a matematikus tanulmányok ismeretanyagára kell épülnie és a szerző önálló, saját munkája legyen. A diplomamunkának arról kell tanúskodnia, hogy a hallgató az egyetemi tanulmányai során szerzett matematikai ismereteit, képességeit a gyakorlati életben vagy az elméleti kutatásokban egy több hónapra kiterjedő munka folyamán önállóan tudja alkalmazni oly módon, hogy a megoldandó problémát felismeri, a megoldáshoz vezető út nehézségeivel megbirkózik, a megfelelő színvonalú megoldást megtalálja, és azt mások számára érthetően leírja. A dolgozat legyen tömör, de a témában nem járatos matematikus olvasó számára is érthető. A Diplomamunka előkészítés c. tárgy keretében a hallgató összegyüjti mindazokat az információkat és matematikai eredményeket, amelyek a diplomamunka megírásához szükségesek.
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BMETE90MM99 Master's Thesis 0 8 0 f 15       15
Course coordinator: Dr. Attila Andai
A diplomamunka a matematikushallgatóknak a témavezető irányításával elért önálló kutatási, kutatás-fejlesztési eredményeit tartalmazó írásbeli beszámoló (dolgozat). A hallgató a dolgozatban mutassa be a vizsgált témát, fejtse ki a problémákat, és részletesen ismertesse eredményeit. A munkának a matematikus tanulmányok ismeretanyagára kell épülnie és a szerző önálló, saját munkája legyen. A diplomamunkának arról kell tanúskodnia, hogy a hallgató az egyetemi tanulmányai során szerzett matematikai ismereteit, képességeit a gyakorlati életben vagy az elméleti kutatásokban egy több hónapra kiterjedő munka folyamán önállóan tudja alkalmazni oly módon, hogy a megoldandó problémát felismeri, a megoldáshoz vezető út nehézségeivel megbirkózik, a megfelelő színvonalú megoldást megtalálja, és azt mások számára érthetően leírja. A dolgozat legyen tömör, de a témában nem járatos matematikus olvasó számára is érthető. A Diplomamunka-készítés c. tárgy keretében a hallgató megírja a diplomamunkáját.