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. 
Math 412 
Introduction to Set Theory 
(3+2+0) 3 
ECTS 6 
Sets, relations, functions, order, settheoretical paradoxes, axiom systems for set theory, axiom of choice and its consequences, transfinite induction, recursion, cardinal and ordinal numbers. 
Math 413 
Model Theory 
(3+2+0) 3 
ECTS 6 
Language and structure, theory, definable sets and interpretability, compactnees theorem, complete theories, LöwenheimSkolem theorems, quantifier elimination, algebraic examples. 
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. 
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, RiemannRoch 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. 
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 MittagLeffler, gamma function, normal families, theorems of Montel and Vitali, Riemann mapping theorem, conformal mapping of simply connected domains, SchwarzChristoffel formula, applications. 
Math 433 
Fourier Analysis 
(3+2+0) 3 
ECTS 6 
Fourier series, Dirichlet and Poisson kernels, Cesàro and Abel summability. pointwise and meansquare 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. 
Math 437 
Optimization Theory 
(3+2+0) 3 
ECTS 6 
Normed linear spaces, Hilbert spaces, leastsquares estimation, dual spaces, geometric form of HahnBanach theorem, linear operators and their adjoints, optimization in Hilbert spaces, local and global theory of optimization of functionals, constrained and unconstrained cases. 
Math 451 
Numerical Solutions of Differential Equations 
(3+2+0) 3 
ECTS 6 
Numerical solutions of initial value problems for ordinary differential equations (ODE), PicardLindelof theorem, single step methods including RungeKutta 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 LaxMilgram lemmas, weak solutions, numerical solutions of partial differential equations, examples of finite difference methods and their consistency, stability, and convergence including LaxRichtmeyer equivalence theorem, CourantFriedrichsLewy 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, GrobmanHartman and HadamardPerron 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, HamiltonJacobi equation. 
Math 462 
Cryptography 
(3+2+0) 3 
ECTS 6 
Simple cryptosystems, public key cryptography, discrete logarithms and DiffieHellman key exchange, primality, factoring and RSA, elliptic curve cryptosystems, lattice based cryptosystems. 
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. 
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, LickorishWallace 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 threedimensional Euclidean space, the Gauss map, the first and the second fundamental forms, theorema egregium, geodesics, GaussBonnet 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, BorsukUlam Theorem, and other related results. 
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, crossratio. 
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 481489 
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 491499 
Selected Topics in Mathematics 
(3+0+0) 3 
ECTS 6 
Selected topics in pure and applied mathematics. 
Prerequisite: 
Consent of the instructor. 
FOR THE PART I OF THE COURSE CATALOGUE, SEE HERE.
