Credits 4 ECTS 6

Functions, limits, continuity, differentiation and applications, integration, fundamental theorem of calculus, techniques and applications of integration, improper integrals and series, Taylor polynomials, power series, basic transcendental functions.

Prerequisite:None

Credits 4 ECTS 6

Vector calculus, functions of several variables, directional derivatives, gradient, Lagrange multipliers, multiple integrals and applications, change of variables, coordinate systems, line integrals, Green's theorem and its applications.

Prerequisite:Math 101

Credits 4 ECTS 6

Systems of linear equations and inequalities, matrices, determinants, inverses, Gaussian elimination, geometric approach to linear programming, basic combinatorics, binomial theorem, finite probability theory, conditional probability, Bayes' theorem, random variables, expected value, variance, decision theory.

Prerequisite:None

Credits 4 ECTS 6

Functions of one variable, properties of quadratic, cubic, exponential and logarithmic functions, compound interest and annuities, limits, continuity and differentiation, applied maximum and minimum problems, basic integration techniques, sequences and series.

Prerequisite:None

Credits 4 ECTS 8

Propositional logic, truth tables, equivalences, quantifiers, rules of inference, proof methods, sets, power sets, functions, sequences, countability, cardinality, divisibility, modular arithmetic, primes, mathematical induction, strong induction and wellordering principle, recursive definitions, axiomatic systems, Euclid's postulates and non-Euclidean geometries.

Prerequisite:None

Credits 4 ECTS 8

Sequences, limits and continuity, differentiation and its applications, integration and its applications, fundamental theorem of calculus, transcendental functions, improper integrals.

Prerequisite:None

Credits 4 ECTS 8

Vectors and geometry in space, vector-valued functions and motion in space, functions of several variables, partial derivatives, multiple integrals, vector fields.

Prerequisite:Math 131

Credits 4 ECTS 8

Counting, the pigeonhole principle, permutations, combinations, binomial coefficients, generalized permutations and combinations, discrete probability, linear recurrence relations, generating functions, inclusion-exclusion, relations, closures of relations, equivalence relations, construction of integers and rationals, partial orderings, graphs.

Prerequisite:None

Credits 4 ECTS 5

Systems of linear equations, Gaussian elimination, matrix algebra determinants, inverse of a matrix, Cramer's rule, rank and nullity, the eigenvalue problem, introduction to linear programming.

Prerequisite:None

Credits 4 ECTS 7

First-order differential equations, linear equations, homogeneous and non-homogeneous, series solutions, the Laplace transform, systems of first-order linear equations, boundary value problems, Fourier series.

Prerequisite: (Math 101 or Math 131) and (Math 201 or Math 221)

Credits 4 ECTS 8

Vector spaces, bases, linear transformations, matrices, subspaces, systems of linear equations, echelon and reduced echelon forms, dimension, fundamental subspaces, rank, change of coordinates, determinants, cofactor expansion, minors, eigenvalues, eigenvectors, diagonalization, inner product spaces, orthogonality, Gram-Schmidt orthogonalization process, adjoint, unitary and orthogonal transformations, dual spaces.

Prerequisite:Math 111

Credits 4 ECTS 8

Groups, subgroups, cyclic groups, generating sets, permutations, orbits, cycles, alternating groups, cosets, Lagrange's Theorem, direct products, finite abelian groups, homomorphisms, normal subgroups, factor groups, simple groups, group actions, isomorphism theorems, Sylow's theorems.

Prerequisite:Math 111

Credits 4 ECTS 8

Sequences and functions, compact sets, continuity, uniform continuity, limits of functions, discontinuities, differentiation, derivatives for functions of several variables, differentiation of composite functions, Taylor's Theorem, definite integrals, substitution in multiple integrals, improper integrals.

Prerequisite:Math 132

Credits 4 ECTS 8

Infinite series, conditionally convergent series, double series, uniform convergence, series and sequences of functions, power series, improper integrals with parameters, differentiation of transformations, linear functions, differentials and inverses of transformations, inverse and implicit function theorems.

Prerequisite:Math 231

Credits 4 ECTS 8

Infinite series, conditionally convergent series, double series, uniform convergence, series and sequences of functions, power series, improper integrals with parameters, differentiation of transformations, linear functions, differentials and inverses of transformations, inverse and implicit function theorems.

Prerequisite:Math 231

Credits 4 ECTS 8

Rings, integral domains, field of fractions, polynomials, factorization, ideals, factor rings, homomorphisms, prime and maximal ideals, extension fields, algebraic extensions, finite fields, unique factorization domains, Euclidean domains, Gaussian integers, field automorphisms, splitting fields, Galois theory, insolvability of the quintic equations.

Prerequisite: Math 222 or consent of the instructor

Credits 3 ECTS 6

Representations, irreducibility, Maschke's theorem, semisimplicity, characters, character tables, orthogonality relations, induction and restriction of characters, Mackey decomposition theorem, algebraic integers, Burnside's p^aq^b-theorem, Frobenius' normal complement theorem.

Prerequisite: Math 222 or consent of the instructor

Credits 3 ECTS 6

General linear groups, closed subgroups of real and complex general linear groups, their topological properties, associated tangent spaces, exponential and logarithm functions, manifolds, maximal tori, homomorphisms.

Prerequisite: (Math 102 or Math 132) and Math 222

Credits 3 ECTS 6

Divisibility theory, Euclidean algorithm, congruences, solutions of polynomial congruences, primitive roots, power residues, quadratic reciprocity law, arithmetical functions, distribution of prime numbers, Pell's equation, quadratic forms, some diophantine equations.

Prerequisite: Math 111 or Math 162

Credits 4 ECTS 8

Topology, density, separability, convergence, compactness, connectedness, continuity, open and closed maps, equicontinuity, Arzela-Ascoli theorem, contractions and fixed point theorems, completeness, Cantor's theorem, Baire category theorem, completion.

Prerequisite: Math 231

Credits 3 ECTS 6

Elementary measure theory, sets of measure zero, Lebesgue measure, Lebesgue measurable sets and functions, Lebesgue integral, convergence theorems, the space L^1, absolutely continuous functions, functions of bounded variation, Hilbert space L^2, Fourier series.

Prerequisite: Math 234 or consent of the instructor

Credits 3 ECTS 6

Differentiation, inverse and implicit function theorems, integration, manifolds, differential forms, orientation, Stokes' theorem, Poincaré lemma, de Rham cohomology.

Prerequisite: Math 221 and Math 234

Credits 3 ECTS 6

Solutions of nonlinear equations, bisection, Newton, and fixed point iterations, direct solutions of linear systems, Gaussian elimination with partial pivoting, LU and Cholesky factorizations, iterative solutions of linear systems, vector and matrix norms, Neumann series, Jacobi, Gauss-Seidel and SOR iterations, projection methods, steepest descents, conjugate-gradient and GMRES methods, matrix eigenvalue problem, power method, Givens rotations, Jacobi iteration, Hessenberg form, QRiteration, polynomial interpolation, Lagrange polynomials, Newton’s divided differences, Chebyshev polynomials, least squares, spline interpolation.

Prerequisite:(Math 101 or Math 131) and (Math 201 or Math 221)

Credits 4 ECTS 8

Complex numbers, exponential forms, roots of complex numbers, functions of a complex variable, limits, continuity, derivatives, Cauchy-Reimann Equations, polar coordinates, analytic functions, reflection principle, exponential and logarithmic functions, branches, trigonometric and hyperbolic functions, linear transformations, definite integrals, contour integrals, branch cuts, Cauchy-Goursat theorem, simply connected domains, Cauchy integral formula, Liouville's Theorem, maximum modulus principle, Taylor and Laurent series, residues and poles, Cauchy's residue theorem, residue at infinity.

Prerequisite:Math 132

Credits 3 ECTS 6

Probability, conditional probability, Bayes’ theorem, independence, discrete and continuous probability distributions, expected value, estimation, confidence intervals, tests of hypothesis for one parameter, goodness of fit test, linear regression, analysis of variance.

Prerequisite:Math 102 or Math 132

Credits 3 ECTS 6

Axioms of probability, conditional probability, independence, discrete and continuous random variables, jointly distributed random variables, expectation, limit theorems.

Prerequisite:Math 344 or consent of the instructor

Credits 3 ECTS 6

Existence and uniqueness theorems, phase portraits in the plane, linear systems and canonical forms, non-linear systems, linearization, stability of fixed points, limit cycles, Poincaré-Bendixson theorem.

Prerequisite:Math 202

Credits 3 ECTS 6

Wave equation, heat equation, Laplace equation, classification of second order linear equations, initial value problems, boundary value problems, Fourier series, harmonic functions, Green's functions.

Prerequisite:(Math 132 and Math 202) or (Math 102 and Math 202)

Credits 3 ECTS 6

Sieve methods, lattices, distributive lattices, incidence algebra, Mobius inversion formula, Mobius algebras, generating functions, exponential formula, Lagrange inversion formula, matrix tree theorem.

Prerequisite:Math 201 or Math 221

Credits 3 ECTS 6

Basic definitions, trees, Cayley's formula, connectedness, Eulerian and Hamiltonian graphs, matchings, edge and vertex colouring, chromatic numbers, planar graphs, directed graphs, networks.

Prerequisite:Math 221 or consent of instructor

Credits 3 ECTS 6

Selected topics in the history of mathematics and related fields.

Prerequisite:Consent of instructor

Credits 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

Credits 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

Credits 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

Credits 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

Credits 3 ECTS 6

Affine varieties, Hilbert’s Nullstellensatz, projective varieties, rational functions and morphisms, smooth points, dimension of a variety.

Prerequisite:Math 323

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 3 ECTS 6

Topological spaces, compactness, connectedness, continuity, separation axioms, homotopy, fundamental group.

Prerequisite:Math 331

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 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

Credits 3 ECTS 6

Selected topics in pure and applied mathematics.

Prerequisite:Consent of the instructor.

Credits 3 ECTS 6

Individual research supervised by a member of the department.

Prerequisite:Consent of the instructor.

Credits 3 ECTS 6

Selected topics in pure and applied mathematics.

Prerequisite:Consent of the instructor.

Credits 4 ECTS 8

Credits 3 ECTS 3/5

Credits 3 ECTS 5

Credits 4 ECTS 8

Credits 3 ECTS 4/5

Credits 3 ECTS 5

Credits 4 ECTS 6

Credits 3 ECTS 5

Credits 2 ECTS 3

Credits 2 ECTS 3

Credits 3 ECTS 5

Credits 3 ECTS 5

Credits 2 ECTS 3

Credits 3 ECTS 5

Credits 3 ECTS 5

Credits 2 ECTS 3

Credits 3 ECTS 6

Credits 3 ECTS 6

Credits 3 ECTS 5

Credits 3 ECTS 6

Credits 3 ECTS 6

Credits 3 ECTS 6

Credits 3 ECTS 5

Credits 3 ECTS 5

Credits 3 ECTS 6

Credits 3 ECTS 6

Credits 3 ECTS 5

Credits 3 ECTS 5

Credits 3 ECTS 5