Department of Mathematics

Departmental Courses I
Courses Offered by Mathematics Department
(Beginning with Fall 2016)
Math 101 Calculus I (4+2+0) 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

Math 102 Calculus II (4+2+0) 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

Math 105 Introduction to Finite Mathematics (4+2+0) 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

Math 106 Introduction to Calculus for Social Sciences (4+2+0) 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

Math 111 Introduction to Mathematical Structures (4+2+0) 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

Math 131 Calculus of a single variable (4+2+0) 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

Math 132 Calculus of several variables (4+2+0) 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

Math 162 Discrete Mathematics (4+2+0) 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

Math 201 Matrix Theory (4+2+0) 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

Math 202 Differential Equations (4+2+0) 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)

Math 221 Linear Algebra (4+2+0) 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

Math 222 Group Theory (4+2+0) 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

Math 231 Advanced Calculus I (4+2+0) 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

Math 234 Advanced Calculus II (4+2+0) 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

Math 323 Rings, Fields and Galois Theory (4+2+0) 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

Math 324 Representation Theory of Finite Groups (3+2+0) 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

Math 325 Matrix Groups (3+0+2) 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

Math 327 Number Theory (3+2+0) 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

Math 331 Metric Spaces (4+2+0) 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

Math 332 Lebesgue Integration (3+2+0) 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

Math 334 Analysis on Manifolds (3+2+0) 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

Math 336 Numerical Analysis (3+2+0) 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)

Math 338 Complex Analysis I (4+2+0) 4 ECTS 6
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

Math 344 Introduction to Probability and Statistics (3+2+0) 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

Math 345 Probability (3+2+0) 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

Math 351 Qualitative Theory of Ordinary Differential Equations (3+2+0) 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

Math 352 Partial Differential Equations (3+2+0) 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)

Math 361 Combinatorics (3+2+0) 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

Math 363 Graph Theory (3+2+0) 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