### 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 **