Course Listings

MATHEMATICS GRADUATE COURSE DESCRIPTION

0410 – 501 : ALGEBRA I

CR : 3

Fundamental topics on groups including semigroups; groups; subgroups; abelian groups; direct products; groups of permutations; cyclic groups; normal subgroups; quotient groups; homomorphisms and isomorphisms; group actions; orbits; Lagrange's Theorem; p-Groups; The Sylow Theorems; The Basis Theorem; The fundamental theorem of finite Abelian groups.

0410 – 508 : TOPICS IN ALGEBRA

CR : 3

Topics may differ from time to time, the course may be repeated for credits provided the topics are different.

0410 – 510 : REAL ANALYSIS I

CR : 3

Riemann-Stieltjes integral; uniform convergence of sequences and series of functions; functions of several real variables; and the Lebesgue integration.

0410 – 512 : COMPLEX ANALYSIS I

CR : 3

Analyticity; Cauchy’s integral formula; residues; Infinite products; Conformal mappings; Riemann mapping theorem.

0410 – 513 : ORDINARY DIFFERENTIAL EQUATIONS

CR : 3

Existence and uniqueness of solutions to initial value problems in n-dimensions.Continuation (extendability) of solutions and continuity with respect to initial conditions and parameters. Stability theory, linearization and Liapunov methods. Sturmian theory and self-adjoint boundary value problems.

0410 – 515 : FUNCTIONAL ANALYSIS

CR : 3

Metric spaces; Normed vector spaces; Inner product spaces; lp spaces; Topology of a metric space; Complete metric spaces; Banach spaces (lp and C[a,b]); Hilbert spaces; Completion of a metric space (Lp [a,b]); Finite dimensional normed vector spaces; Linear operators; Linear functionals; Normed spaces of operators; Dual space; Functionals on Hilbert space; Self-adjoint, unitary, and normal operators; Hahn-Banach theorem; Uniform boundedness theorem; Open mapping theorem; Closed graph theorem.

0410 – 517 : SPECIAL FUNCTIONS

CR : 3

Asymptotic expansions. Bessel functions and related functions, hypergeometric confluent, hypergeometric and generalized hypergeometric functions. Jacobi polynomials, Meijer’s G-functions.

0410 – 520 : BOUNDARY VALUE PROBLEMS

CR : 3

Partial differential equations of mathematical physics and engineering, the well posed problem, Dirichlet, Neumann and the mixed problems, methods of solution, Green’s function, integral equations, integral transforms.

0410 – 521 : VARIATIONAL METHODS AND EIGENVALUE PROBLEMS

CR : 3

Laplace’s equation: Fundamental solution, Mean-value property, properties of harmonic functions (max/min principles, harmonic estimates, analyticity, Liouville’s Theorem), Green’s functions, Dirichlet principle; Weak derivatives; Introduction to Sobolev theory; existence of weak solutions of various linear elliptic PDEs; Dirichlet principle in Sobolev space, Eigenvalue problems; Introduction to variational principles related to the heat and wave equations.

0410 – 522 : FINANCIAL MATHEMATICS – MODELING & COMPUTATION

CR : 3

Finance’ is one of the fastest developing areas in the modern banking and corporate world. This, together with the sophistication of modern financial products, provides a rapidly growing impetus for new mathematical models and modern mathematical methods. The course describes the modeling of financial derivative products, from modeling through analysis to elementary computation. Topics include: basic option theory, tree models, continuous time models and Black-Scholes, analytic approach to Black-Scholes, hedging numerical and binomial methods, bonds and interest rate derivatives models, computational methods for bonds, further theory of exotic and path-dependent options, foreign currency markets and exchange risks.

0410 – 523 : TOPICS IN APPLIED MATHEMATICS

CR :3

Topics may differ from time to time, the course may be repeated for credit provided the topics are different.

0410 – 525 : GENERAL TOPOLOGY

CR : 3

Abstract topological spaces; connectedness, compactness, continuous functions. Metric spaces, complete metric spaces and metrizable spaces.

0410 – 526 : ALGEBRAIC TOPOLOGY

CR : 3

Fundamental groups, surfaces and homology theory.

0410 – 530 : FOUNDATIONS OF GEOMETRY

CR : 3

Coordinatization, planar ternary rings and their algebraic properties, coordinatizing the dual plane, conditions for linearity, division rings with inverse properties, alternative division rings, the Artin-Zorn theorem, quasi fields and translation planes, division ring planes, some non desarguesian planes.

0410 – 531 : DIFFERENTIABLE MANIFOLDS

CR : 3

Manifolds, the topology of manifolds differentiation on a manifold, vector fields linear and affine connections, distributions, Riemannian manifolds.

0410 – 532 : TOPICS IN DIFFERENTIAL EQUATIONS

CR : 3

Special topics not covered in other courses. May be repeated for credit under different subtitles.

0410 – 535 : GRAPHS AND HYPER GRAPHS

CR : 3

The path problem, the flow problems, Vizing theorem, the Shannon theorem, chromatic number, chromatic polynomials, perfect graphs, hyper graphs.

0410 – 537 : COMBINATORICS

CR : 3

System of distinct representatives of a family of sets, Hall’s theorem and its generalizations, transversals, common transversals. Designs, Steiner Triple Systems, sufficient conditions for existence of a block design. Latin squares, orthogonal Latin squares.

0410 – 560 : NUMERICAL SOLUTION OF ODE’S

CR : 3

Concepts of discretization (initial value problems, boundary value problems, integral equations). Difference methods and Galerkin methods. Consistency, stability and convergence. Linear multistep methods, stability theory, spline collocation methods, stiff equations. Two-point boundary value problems, difference methods, shooting techniques, finite elements.

0410 – 561 : COMPUTATIONAL LINEAR ALGEBRA

CR : 3

Basic concepts, Gaussain Elimination and LU-decomposition, QR-Factorization and Lease Square problems, Eigenvalue problems and SVD, Iterative Methods.

0410 – 568: TOPICS IN NUMERICA MATHEMATICS

CR : 3

Topics may differ from time to time, the course may be repeated for credit provided the topics are different.

0410 – 593 : PROJECT

CR : 3

0410 – 597 to 599 : THESIS