Department of Mathematics

Departmental Courses II
Courses Offered by Mathematics Department
(Beginning with Fall 2016)
Math 401 History of Mathematics (3+2+0) 3 ECTS 6
Selected topics in the history of mathematics and related fields.
Prerequisite: Consent of instructor

Math 404 Computational Mathematics (3+2+0) 3 ECTS 6
Introduction to computational mathematics, basics of a mathematics software (Sage, Mathematica, Maple, MATLAB), solving systems of linear equations, interpolation, locating roots of equations, least squares problems, numerical integration, numerical differentiation and solution of ordinary differential equations.
Prerequisite: (Math 202 and Math 221) or consent of the instructor
Math 411 Mathematical Logic (3+2+0) 3 ECTS 6
Propositional and quantificational logic, formal grammar, semantical interpretation, formal deduction, completeness theorems, selected topics from model theory and proof theory.
Prerequisite: Math 111

Math 412 Introduction to Set Theory (3+2+0) 3 ECTS 6
Sets, relations, functions, order, set-theoretical paradoxes, axiom systems for set theory, axiom of choice and its consequences, transfinite induction, recursion, cardinal and ordinal numbers.
Prerequisite: Math 111

Math 413 Model Theory (3+2+0) 3 ECTS 6
Language and structure, theory, definable sets and interpretability, compactnees theorem, complete theories, Löwenheim-Skolem theorems, quantifier elimination, algebraic examples.
Prerequisite: Math 111

Math 425 Introduction to Algebraic Geometry (3+2+0) 3 ECTS 6
Affine varieties, Hilbert’s Nullstellensatz, projective varieties, rational functions and morphisms, smooth points, dimension of a variety.
Prerequisite: Math 323

Math 426 Introduction to Arithmetic Geometry (3+2+0) 3 ECTS 6
Introduction to algebraic number theory and algebraic curves, geometric introduction to function fields of curves, affine and projective varieties, divisors on curves, Riemann-Roch theorem, basics of elliptic curves.
Prerequisite: Math 323 or consent of the instructor

Math 427 Elementary Number Theory II (3+2+0) 3 ECTS 6
Quadratic Forms, quadratic number fields, factorization of ideals in quadratic number fields, ramification theory, ideal classes and units in quadratic number fields, elliptic curves over rationals.
Prerequisite: Math 162

Math 432 Complex Analysis II (3+2+0) 3 ECTS 6
Convergent series of meromorphic functions, entire functions, Weierstrass' product theorem, partial fraction expansion theorem of Mittag-Leffler, gamma function, normal families, theorems of Montel and Vitali, Riemann mapping theorem, conformal mapping of simply connected domains, Schwarz-Christoffel formula, applications.
Prerequisite: Math 338

Math 433 Fourier Analysis (3+2+0) 3 ECTS 6
Fourier series, Dirichlet and Poisson kernels, Cesàro and Abel summability. pointwise and mean-square convergence, Weyl's equidistribution theorem, Fourier transform on the real line and Schwartz space, inversion, Plancherel formula, application to partial differential equations, Poisson summation formula.
Prerequisite: Math 338 or consent of the instructor

Math 436 Functional Analysis (3+2+0) 3 ECTS 6
Review of vector spaces, normed vector spaces, lP and LP spaces, Banach and Hilbert spaces, duality, bounded linear operators and functionals.
Prerequisite: Math 331

Math 437 Optimization Theory (3+2+0) 3 ECTS 6
Normed linear spaces, Hilbert spaces, least-squares estimation, dual spaces, geometric form of Hahn-Banach theorem, linear operators and their adjoints, optimization in Hilbert spaces, local and global theory of optimization of functionals, constrained and unconstrained cases.
Prerequisite: Math 331

Math 451 Numerical Solutions of Differential Equations (3+2+0) 3 ECTS 6
Numerical solutions of initial value problems for ordinary differential equations (ODE), Picard-Lindelof theorem, single step methods including Runge-Kutta methods, examples and consistency, stability and convergence of multistep methods, numerical solution of boundary value problems for ODE’s, shooting, finite difference, and collocation methods, finite element methods, Riesz and Lax-Milgram lemmas, weak solutions, numerical solutions of partial differential equations, examples of finite difference methods and their consistency, stability, and convergence including Lax-Richtmeyer equivalence theorem, Courant-Friedrichs-Lewy condition, and von Neumann analysis, Galerkin methods, Galerkin orthogonality, Cea’s lemma, finite element methods for elliptic, parabolic and hyperbolic equations.
Prerequisite: (Math 102 or Math 132) and Math 202

Math 452 Dynamical Systems (3+2+0) 3 ECTS 6
Dynamical systems with discrete and continuous time, differential equations on torus, invariant sets, topological dynamics, topological recurrence and entropy, expansive maps, homoemorphisms and diffeomorphisms of the circle, periodic orbits, hyperbolic dynamics, Grobman-Hartman and Hadamard-Perron theorems, geodesic flows, topological Markov chains, zeta functions, invariant measures and the ergodic theorem.
Prerequisite: Math 331 or consent of the instructor

Math 455 Calculus of Variations (3+2+0) 3 ECTS 6
First variation of a functional, necessary conditions for an extremum of a functional, Euler's equation, fixed and moving endpoint problems, isoperimetric problems, problems with constraints, Legendre transformation, Noether's theorem, Jacobi's theorem, second variation of a functional, weak and strong extremum, sufficient conditions for an extremum, direct methods in calculus of variations, the principle of least action, conservation laws, Hamilton-Jacobi equation.
Prerequisite: Math 202

Math 462 Cryptography (3+2+0) 3 ECTS 6
Simple crypto-systems, public key cryptography, discrete logarithms and Diffie-Hellman key exchange, primality, factoring and RSA, elliptic curve crypto-systems, lattice based crypto-systems.
Prerequisite: Math 221 or consent of the instructor

Math 471 Topology (3+2+0) 3 ECTS 6
Topological spaces, compactness, connectedness, continuity, separation axioms, homotopy, fundamental group.
Prerequisite: Math 331

Math 472 Geometric Topology (3+2+0) 3 ECTS 6
Basics of point set topology, quotient topology, CW complexes and their homology and fundamental group, classification of surfaces, introduction to knot theory, Seifert surfaces and Seifert forms, signature, Alexander polynomial, and Arf invariant of knots, introduction to Morse theory, Heegaard splittings of three manifolds, Dehn surgery, Lickorish-Wallace theorem.
Prerequisite: Math 331 or consent of the instructor

Math 474 Mathematical Aspects of General Relativity (3+2+0) 3 ECTS 6
Review of special relativity, differentiable manifolds, tensors, Lie derivative, covariant derivative, parallel transport, geodesics, curvature, Einstein's field equations, Schwarzschild black hole, Cauchy problem, maximally symmetric spacetimes, singularity theorems.
Prerequisite: Consent of the instructor

Math 475 Differential Geometry (3+2+0) 3 ECTS 6
Fundamentals of Euclidean spaces, geometry of curves and surfaces in three-dimensional Euclidean space, the Gauss map, the first and the second fundamental forms, theorema egregium, geodesics, Gauss-Bonnet theorem, introduction to differentiable manifolds.
Prerequisite: Math 234 or consent of the instructor

Math 476 Differential Topology (3+2+0) 3 ECTS 6
Smooth functions and smooth manifolds embedded in Euclidean space, tangent spaces, immersions, submersions, transversality, applications of the implicit function theorem, Morse functions, Sard's theorem, Whitney embedding theorem, intersection theory mod 2, Brouwer fixed point theorem, Borsuk-Ulam Theorem, and other related results.
Prerequisite: Math 331

Math 477 Projective Geometry (3+2+0) 3 ECTS 6
Projective spaces, homogeneous coordinates, dual spaces, the groups of affine and projective transformations and their properties, Desargues' theorem, Pascal's theorem, and other classical results, classification of conics, projective plane curves, singular points, intersection multiplicity, Bezout's Theorem, the group law on an elliptic curve, cross-ratio.
Prerequisite: Math 201 or Math 221

Math 478 Groups and Geometries (3+2+0) 3 ECTS 6
Plane Euclidean geometry and its group of isometries, affine transformations in the Euclidean plane, fundamental theorem of affine geometry, finite group of isometries of R, Leonardo da Vinci's theorem, geometry on the sphere S, motions of S, orthogonal transformations of R, Euler's theorem, right triangles in S, projective plane, Desargues' theorem the fundamental theorem of projective geometry.
Prerequisite: Math 222 or consent of the instructor

Math 481-489 Selected Topics in Mathematics (3+0+0) 3 ECTS 6
Selected topics in pure and applied mathematics.
Prerequisite: Consent of the instructor.

Math 490 Project (1+0+4) 3 ECTS 6
Individual research supervised by a member of the department.
Prerequisite: Consent of the instructor.

Math 491-499 Selected Topics in Mathematics (3+0+0) 3 ECTS 6
Selected topics in pure and applied mathematics.
Prerequisite: Consent of the instructor.