MATH 510 CONCEPTS OF GEOMETRY FOR MATHEMATICS TEACHERS (3+0+0) 3 (ECTS 10)
Euclidean and non-Euclidean geometries. Calculus on and submanifolds of . Differential geometry of curves and surfaces. Vectors, tensors, multivectors and differential forms. Flows, symmetries and geometries. Structure equations. Lie groups and Lie algebras. Constructions in modern differential geometry.
MATH 511 DIFFERENTIAL GEOMETRY (3+0+0) 3 (ECTS 10)
Smooth mappings. Implicit function theorem. Submanifolds of Euclidean space. , analytic and smooth manifolds. Examples: projective spaces, Grassmann manifolds, Riemann surfaces. Manifolds with boundary. Partition of unity. Mappings of manifolds, regular and singular points, immersions, submersions and embeddings. Sard’s theorem. Tangent bundle. Existence of Riemannian metric. Vector fields, flows and differentials. Algebra of vector fields. Cotangent bundle. Tensor fields, multi-vectors, exterior forms and their algebras. Applications to mechanics and Lie groups.
MATH 512 RIEMANNIAN GEOMETRY (3+0+0) 3 (ECTS 10)
Riemannian manifolds. Absolute differentiation and connection. Riemann curvature, Bianchi identities. Geometry of hypersurfaces, Riemannian immersions and submersions. Completeness. Isometries and Killing vectors. Homogeneous and symmetric spaces. Properties of curvature tensors. Variational principles.
Prerequisite: Math 511.
MATH 513 TOPICS IN GEOMETRY (3+0+0) 3 (ECTS 10)
MATH 520 MATHEMATICS FOR SOCIAL SCIENTISTS (3+0+0) 3 (ECTS 10)
Basic set theory and logic, algebra and numbers. Functions, limit and continuity, derivative and integral. Fundamental theorem of calculus. Applications. Matrix algebra and applications to systems of algebraic equations. Basic probability and statistics. The aim of this course is to give a core knowledge and meaningful understanding of mathematics rather than its specific applications to social sciences.
Prerequisites: an introductory undergraduate course on some of above topics.
MATH 521 ALGEBRA I (3+0+0) 3 (ECTS 10)
Groups; Sylow theorems. Direct sums and direct products. Free groups. Action of a group on a set. Rings; homomorphisms, commutative rings. Principal ideal domains, unique factorization domains. Noetherian rings. Rings of quotients. Localization.
MATH 522 ALGEBRA II (3+0+0) 3 (ECTS 10)
Galois theory. Categories and functors. Module categories. Tensor products. Projective and injective modules. Primitive rings. Jacobson radical. Semi-simple rings. Decomposition theorems.
MATH 523 LIE GROUPS AND LIE ALGEBRAS (3+0+0) 3 (ECTS 10)
Manifolds, Lie groups and Lie algebras. Exponential map, Baker-Campbell-Hausdorff formula. Lie’s fundamental theorems. Nilpotent and solvable Lie algebras. Cartan’s criterion. Semisimple Lie algebras. Casimirs. The theorem of Weyl. Levi decomposition. Global results.
Prerequisite: Math 511.
MATH 524 GEOMETRIC CONTROL THEORY (3+0+0) 3 (ECTS 10)
General control systems, orbits, transitivity, reachability, controllability, observability, minimal realization. Linear control systems on , rank conditions. Polynomial control systems, feedback, bounded controls, bang-bang principle. Systems on Lie groups and homogeneous spaces, controllability of affine systems, characterization of observability, normalizer, drift vector field. Applications.
Prerequisite: Math 523.
MATH 525 ALGEBRAIC NUMBER THEORY I (3+0+0) 3 (ECTS 10)
Introduction to global fields, ring of integers, decomposition of ideals, discriminant and different, ideal class group. Local methods.
MATH 526 ALGEBRAIC NUMBER THEORY II (3+0+0) 3 (ECTS 10)
Local fields, higher remification theory, local Artin reciprocity law, adeles and ideles, global Artin reciprocity law.
MATH 527 THEORY OF SCHEMES I (3+0+0) 3 (ECTS 10)
Affine algebraic varieties, Projective varieties and general algebraic varieties, Classical examples of algebraic varieties. Introduction to theory of schemes.
MATH 528 THEORY OF SCHEMES I (3+0+0) 3 (ECTS 10)
Sheaves and sheaf cohomology .Prerequisite: MATH 527.
MATH 529 TOPICS IN ALGEBRA (3+0+0) 3 (ECTS 10)
MATH 531 TOPOLOGY (3+0+0) 3 (ECTS 10)
Topological spaces and continuous mappings. Metric topology. Topology of and . Factor spaces and quotient topology. Classification of surfaces. Orbit spaces, projective and lens spaces. Operations on sets, completeness. Connectedness, countability and separation axioms. Normal spaces. Compactness and compactifications. Metrization.
MATH 532 ALGEBRAIC TOPOLOGY (3+0+0) 3 (ECTS 10)
Topology of space of continuous mappings. Homotopy. Extension. Retraction and deformation. Algebraization of topological problems. Homotopy groups. Fundamental group. Computations of the fundamental and homotopy groups of closed surfaces, topological invariance of the Euler characteristics. Homology groups of simplicial complexes and polyhedra. Barycentric subdivision, simplicial mappings. Singular homology. Homology groups of spheres, cell complexes and projective spaces. Degree of a mapping. Lefschetz number of simplicial and continuous mappings.
Prerequisite: Math 531.
MATH 533 INTRODUCTION TO DIFFERENTIAL TOPOLOGY (3+0+0) 3 (ECTS 10)
Locally trivial fiber spaces. Lifts of mappings and covering homotopy. Vector bundles and morphisms. Homotopy groups, universal covering. Monodromy. Cellular structure. Index of critical points. Morse lemma. Gradient fields. Homotopy type and change. de Rham cohomology, homotopy operator and Poincaré lemma. Stokes’ theorem. de Rham’s isomorphism theorem. Applications.
Prerequsite: Math 531, Math 511.
MATH 537 PROBABILITY AND STATISTICS FOR ENGINEERS (3+0+0) 3 (ECTS 10)
Random variables and probability distributions. Methods of estimating a parameter, hypothesis testing, confidence intervals. Linear and multiple regression. Introduction to the design of experiments. Applications.
MATH 541 ORDINARY DIFFERENTIAL EQUATIONS (3+0+0) 3 (ECTS 10)
First order equation. Cauchy-Euler method. Continuation of solution. Systems of equations. Lipschitz conditions. Linear systems. Green’s function. Singularities of linear autonomous systems. Nonlinear equations. Poincare-Bendixson theorem. Poincare index. Limit cycles.
MATH 542 PARTIAL DIFFERENTIAL EQUATIONS I (3+0+0) 3 (ECTS 10)
Review of first order equations. Hyperbolic equations in two independent variables: Characteristics, propagation of singularities, linear equations, one dimensional wave equations, systems of first order equations. Cauchy –Kowalewski theorem. Laplace equation. Hyperbolic equations in higher dimensions: spherical means, method of descent, Duhamel principle. Parabolic equations.
MATH 543 BOUNDARY-VALUE PROBLEMS AND SPECIAL FUNCTIONS (3+0+0) 3 (ECTS 10)
Standard equations and boundary conditions of the classical theoretical physics. General solution techniques. Separation of variables and Sturm-Liouville problems. Regular problems and expansions in eigen-function series. Singular problems and integral transforms. Special functions ( trigonometric, Bessel, Mathieu, Legendre etc) and integral transforms (Laplace, Fourier, Mellin, Weber, Hankel, Kontorovich-Lebedev, Mehler-Fock etc) arising in cartesian, circular cylindrical, elliptic cylindrical, spherical, toroidal etc coordinates. Some illustrative examples.
Prerequisites Math 245, Math 253
MATH 544 TOPICS IN APPLIED MATHEMATICS (3+0+0) 3 (ECTS 10)
MATH 547 PROBABILITY THEORY (3+0+0) 3 (ECTS 10)
Axioms of probability, Discrete random variables and probability functions, special discrete distributions. Independence. Continuous random variables and probability distributions, special continuous distributions. Moments, moment generating functions and characteristic functions. Chebychev inequality, Convergence concepts, laws of large numbers, central limit theorem.
MATH 548 THEORY OF STATISTICS (3+0+0) 3 (ECTS 10)
Point estimation. Maximum-likelihood estimation, method of moments estimation. Properties of estimators, minimum variance unbiased estimators, the Cramer-Rao inequality. Tests of Hypotheses. Interval estimation. Decision theory. Regression, correlation.
Prerequisite: Math 547
MATH 550 CONCEPTS OF ANALYSIS FOR MATHEMATICS TEACHERS (3+0+0) 3 (ECTS 10)
Cartesian tradition. Calculus. Algebraic conceptions of Euler and Lagrange. Functions. Analytical mechanics. Potential theory. Gauss, Green and Stokes theorems. Cauchy and Weierstrass. Complex functions. Riemann and Lebesgue integrals. Modern foundation of analysis. Topics in the theory of differential equations, variational calculus and functional analysis.
MATH 551 FUNCTIONAL ANALYSIS I (3+0+0) 3 (ECTS 10)
Linear vector spaces, subspaces, direct sum. Linearly independent sets, Hamel bases. Linear transformations, linear functionals. Eigenvalues and eigenvectors of linear operators. Introduction to topology. Numerical functions. Measures of sets, integration of numerical functions. Metric spaces, completeness. Contraction mappings. Compact metric spaces. Approximations. Normed linear spaces, norm and semi-norm topologies. Bounded linear operators.
MATH 552 FUNCTIONAL ANALYSIS II (3+0+0) 3 (ECTS 10)
Normed linear spaces, topological dual, weak and strong topologies. Compact and closed operators. Inner product spaces, orthogonal subspaces. Orthonormal sets and Fourier series. Duals of Hilbert spaces. Linear operators on Hilbert spaces and their adjoints. Spectral theory of linear operators, resolvent set and spectrum. Spectrum of bounded linear operators. Spectral analysis on Hilbert spaces. Introduction to non-linear functional analysis, Gâteaux and Fréchet derivatives of non-linear operators. Integration of operators.
MATH 553 COMPLEX ANALYSIS (3+0+0) 3 (ECTS 10)
Analytic functions. The argument principle. Conformal mappings. The Riemann mapping theorem. Infinite products. The Weierstrass factorization theorem. The Mittag–Lefler theorem. Analytic continuation. The Picard theorem.
MATH 554 APPROXIMATION THEORY (3+0+0) 3 (ECTS 10)
Preliminaries, polynomials of best approximation. Existence, characterizations and uniqueness of the best polynomial approximation. Best trigonometric approximation. Degree of approximation by trigonometric functions. Inverse theorems for periodic functions. Rational approximation.
MATH 555 MEASURE AND INTEGRATION THEORY (3+0+0) 3 (ECTS 10)
Measures, outer measures. Extension of measures. Measurable functions. Integrable functions. Sequences of integrable functions. Properties of integrals. Signed measures. Hahn and Jordan decompositions. The Radon-Nikodym theorem. Product Spaces.
MATH 556 TOPICS IN ANALYSIS (3+0+0) 3 (ECTS 10)
MATH 590 SEMINAR (0+0+0) 0 (ECTS 10)
MATH 600 MS THESIS (0+0+0) 0 (ECTS 30)
MATH 611 INTRODUCTION TO SYMPLECTIC GEOMETRY (3+0+0) 3 (ECTS 10)
Poisson brackets and Poisson manifolds. Hamiltonian vector fields. Symplectic manifolds. Darboux theorem. Lagrangian submanifolds. Special symplectic structures. Legendre transforms. Hamiltonian symmetries. Symplectic reduction. Applications.
MATH 612 GEOMETRY OF DIFFEOMORPHISM GROUPS (3+0+0) 3 (ECTS 10)
Mappings and flows. Manifolds of maps. Group of diffeomorphisms and its subgroups. Diffeomorphisms of circle. Lie groups of volume preserving and symplectic diffeomorphisms. Lie algebras of divergence free and symplectic vector fields. Lie-Poisson structures on duals of Lie algebras. Poisson maps. Generalizations to semi-direct products. Variational problems. Applications to integrable non-linear partial differential equations, symplectic topology, fluids and plasmas.
MATH 613 GEOMETRY OF MANIFOLDS OF MAPS (3+0+0) 3 (ECTS 10)
Space of maps between finite dimensional manifolds. Realization as space of sections of a trivial bundle. Vector fields and forms, tangent and cotangent bundles. Metric and symplectic structure. Actions of Lie groups. Applications from continuum theories.
MATH 621 APPLICATIONS OF LIE GROUPS TO DIFFERENTIAL EQUATIONS (3+0+0) 3 (ECTS 10)
Symmetries and generalized symmetries of differential equations, Lagrangians and Hamiltonian equations in finite and infinite dimensions. Group invariant solutions. Formal variational calculus. Bi-Hamiltonian systems. Integrability.
MATH 626 LIE GROUPOIDS AND LIE ALGEBROIDS (3+0+0) 3 (ECTS 10)
Fundamental theory and algebraic constructions. General semi-direct products. Double vector bundles. Double tangent bundle and its dual. Vector fields on Lie groupoids. Poisson structures. Poisson-Lie groups. Structure of cotangent bundle. Poisson and symplectic groupoids. Canonical isomorphisms. Lie bialgebroids. Poisson actions and moment maps.
MATH 627 INTRODUCTION TO ALGEBRAIC GROUPS (3+0+0) 3 (ECTS 10)
Basic algebraic geometry, classical groups, Jordan decomposition, Solvable groups and Borel subgroups, root systems and classification theory, highest weight theory, representation theory in characteristic zero, Schubert varieties and Demazure operators, BGG resolution.
MATH 628 LANGLANDS FUNCTORIALITY PRINCIPLE I (3+0+0) 3 (ECTS 10)
Construction of L-groups, automorphic L-functions, principle of functoriality.
MATH 629 LANGLANDS FUNCTORIALITY PRINCIPLE II (3+0+0) 3 (ECTS 10)
Examples of functoriality principle: Langlands correspondence. Proof of Langlands correspondence for GL(2).
MATH 630 INTRODUCTION TO ARTHUR TRACE FORMULA (3+0+0) 3 (ECTS 10)
Continuation of the series MATH628-629. The aim is to introduce one of the main tools to prove the functoriality principle, namely the trace formula of Arthur. The topics include : Abstract trace formula, Arthur trace formula, and its applications to the study of automorphic representations and the functoriality principle.
MATH 631 INTRODUCTION TO MOTIVES (3+0+0) 3 (ECTS 10)
Weil conjectures on the zeta-functions of algebraic varieties, Weil cohomology theory, definition of correspondences, standard conjectures, proof of Weil conjectures assuming the standard conjectures, motives.
MATH 641 EXTERIOR DIFFERENTIAL FORMS I (3+0+0) 3 (ECTS 10)
Alternating multilinear functionals, exterior forms and exterior algebra. Differentiable manifolds, vector fields, tangent spaces. Lie derivatives, Lie algebras. Mapping between manifolds. Cotangent spaces, exterior differential forms and exterior algebra. Interior product, dual forms. Ideals of exterior algebra, exterior derivative, closed ideals. Lie derivatives of exterior forms, Frobenius and Cartan theorems. Isovector fields.
MATH 642 EXTERIOR DIFFERENTIAL FORMS II (3+0+0) 3 (ECTS 10)
Integration of differential forms, Stokes’ theorem. Conservation laws. Homotopy operator. Decomposition of a form into exact and antiexact parts. Canonical forms of 1-forms and closed 2-forms. Caratheodory theorem. Solutions of exterior differential equations. Affine connections on manifolds. Covariant derivative, curvature and torsion forms. Exterior forms equivalent to system of partial differential equations. Symmetry groups. Variational Calculus. Various applications.
MATH 643 PARTIAL DIFFERENTIAL EQUATIONS II (3+0+0) 3 (ECTS 10)
Sobolev spaces:Weak derivatives, Approximation by smooth functions, Extensions, Traces, Sobolev Inequalities, The Space H - 1. Second Order Elliptic Equations: Weak solutions, Lax-Milgram Theorem, Energy Estimates, Fredholm Alternative, Regularity, Maximum principles. Linear Evolution Equations: Second Order Parabolic Equations, (Weak solutions, regularity, Maximum Principle), Second Order Hyperbolic Equations, (Weak solutions, Regularity, Propagation of disturbances)
MATH 644 GEOMETRICAL THEORY OF DYNAMICAL SYSTEMS (3+0+0) 3 (ECTS 10)
Review of differentiable manifolds and vector fields. Transversality. Structural stability. Tubular flows. Local stability. Invariant manifolds. Kupka-Smale theorem. Poincare map. Morse-Smale vector fields.
MATH 645 SOBOLEV SPACES AND ELLIPTIC OPERATOR (3+0+0) 3 (ECTS 10)
Review of Banach and Hilbert spaces. Hölder continuity. Sobolev spaces. Sobolev inequality and embedding theorem. Differential operators. Adjoints. Principal symbols. Elliptic operators. Linearization of nonlinear differential operators. Fredholm alternative. Regularity of solutions for elliptic equations. Survey of methods for the existence problems.
MATH 646 POTENTIAL THEORY (3,0,0) 3 (ETSC 10)
Poisson integral formula. Positive harmonic functions. Subharmonic functions. Criteria for subharmonicity. Integrability, convexity and smoothing. Potentials and their basic properties. Polar sets. Capacity and equilibrium measure. Removable singularities. The Generalized Laplacian. Thinness. Solution of the Dirichlet Problem. Criteria for Regularity. Harmonic measure. Green functions. The Poisson-Jensen formula. Computation and estimation of capacity. Transfinite diameter. Applications
MATH 647 BVP FOR COMPLEX DIFFERENTIAL EQ. (3,0,0) (ECTS 10)
Function Theoretical tools; Integral representations of complex functions; Singular integral operators (Cauchy – Pompeiu operator, its weak derivatives); Motivation for complex differential equations, boundary value problems for first order complex differential equations; Second order differential equations, ellipticity condition; Boundary value problems for higher order elliptic differential equations in simply connected bounded domains; Mixed problems; Problems in unbounded domains, multiply connected domains.
MATH 651 NONLINEAR FIELD THEORIES OF CONTINUOUS MEDIA - I (3+0+0) 3 (ECTS 10)
Curvilinear coordinates, vector fields, covariant derivative, Tensor fields, curvature tensor, weighted tensors, two-point tensor fields. Deformation and strain tensors, infinitesimal strain and rotation tensors, length and angle changes, deformation invariants, principal directions, polar decomposition and rotation, area and volume changes, compatibility conditions. Material derivative, velocity and acceleration, material systems, material derivatives of arc, area and volume elements, material derivatives of line, area and volume integrals, strain rates, spins and vorticity, general balance equations, fundamental axioms of continuous media, principle of objectivity. Stress hypothesis, equations of motion, principal stresses, equations of motion in material description, stress flux and objective time of rates. Energy balance, entropy inequality, thermodynamical potentials, dissipation inequality.
MATH 652 NONLINEAR FIELD THEORIES OF CONTINUOUS MEDIA - II (3+0+0) 3 (ECTS 10)
Axioms of the constitutive theory, thermomechanical materials. Hyperelastic bodies. Anisotropic and isotropic materials, formulation of boundary and initial value problems, isothermal and isentropic elasticities, incompressible materials, approximate theories, various exact solutions. Thermoelasticity. Constitutive theory, generalised thermoelasticity, linear theory. Thermo-viscous fluids. constitutive relations, Stokes fluids, Rivlin-Ericksen fluids, viscometric flows. Electromagnetic interactions.
MATH 690 SEMINAR (0+0+0) 0 (ECTS 10)
MATH 700 PH.D. THESIS (0+0+0) 0 (ECTS 30)