# Course Listings

### MATHEMATICS GRADUATE COURSE DESCRIPTION

**0410 – 501 : ALGEBRA I**

**CR : 3**

Sylow theorems. Direct Sums and free abelian groups. The dual groups and Jordan holder theorem. Rings and homomorphism, commutative rings. Modules, direct products and sums of modules. Finite algebraic extension, separable extensions. Galois theory. Finite fields.

**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, sequences and series of functions, functions of several variables, Lebesgue measure and integration on the real line.

**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**

Normed linear spaces, Hilbert spaces, Hahn-Banach extension theorems, Banach-Steinhaus theorem, closed graph and open mapping theorem, topics selected from spectral theory.

**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**

Linear operators in Hilbert space, Generalized functions, eigenfunction expansions, the Raleigh-Ritz method, the Galerkin method, Methods of least squares, eigenvalue problems, lower and upper bounds, the Weinstein method, applications.

**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 – 535 : 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 **