# 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 Lyapunov 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 **

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 – ****542**** : ****SCIENTIFIC COMPUTING: MATHEMATICAL MODELS AND ALGORITHMS**

**CR : 3**

Mathematical modeling using systems of differential equations to model real situations, large systems of linear equations, sparse matrices, pseudo-inverse matrices, multilevel methods, factorization. Ordinary differential equations, initial value problems, one step and multi-step methods for solution, stiff equations, boundary value problems, shooting, difference and variational methods. Using Finite difference methods and multigrid techniques to solve partial differential equations.

**0410 – 54****3**** : ****ADVANCED NUMERICAL COMPUTING**

**CR : 3**

Fitting of Data: B-spline representations, calculating with B-splines, knot insertion algorithms, curve fitting with splines; Surface Fitting: meshed data methods, scattered data methods; Transforms and Filtration of Data: Fourier transforms, convolution and correlation, sampling theorem, deconvolution problem; Multi-level Projections: orthogonal decomposition, iterative image reconstruction, the Haar system.

**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 **