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Course Listings

0410 - 601 : Algebra

CR

:

3

Monoids, groups and subgroups; normal and quotient groups. Direct products, direct sums and free groups. Free products, generators and relation. Rings, factorization of commutative rings, rings of quotients and localization. Rings of polynomials and formal power series. Factorization in polynomial rings. Modules and exact sequences, injective and projective modules. Matrices, rank and equivalence.

0410 - 603 : Ring Theory

CR

:

3

Primitive rings, simple rings with minimal ideals, Jacobson  radical, semi-simple rings, and Artinian semi-simple rings. Completely reducible modules. Tensor product (elementary properties). Inductive and projective modules and semi perfect rings.

0410 - 606 : Group Theory

CR

:

3

Automorphisms of group. The holomorph, complete groups and semi-direct product of groups. Free groups and Schreier method. Solvabale groups and extended Sylow theorems and Frattini subgroups. Abelian groups:torsion, torsion free, divisible and pure subgroups.

0410 - 608 : Advanced topics in Algebra

CR

:

3

Selected Advanced topics in algebra that will reflect recent advances.

0410 - 610 : Measure and Integration Theory

CR

:

3

Abstract measure and integration, the Lebesgue integral, Dini derivatives and the fundamental theorem of Calculus.

0410 - 612 : Complex Analysis

CR

:

3

Analytic continuation, Harmonic functions, Mapping theorems, The modular function, Entire functions.

0410 - 614 : Operator Theory

CR

:

3

Hilbert spaces and  Banach spaces. Duality, linear operators, compact operators, and Fredholm operators.

0410 - 615 : Functional Analysis

CR

:

3

Function spaces, linear functionals and distributions. Convolutions and Fourier transforms. Sobolev spaces, embedding theorems.

0410 - 616 : q-Analysis

CR

:

3

q-binomial theorem. Basic hypergeometric series and summation formula. q-analogues of the classical orthogonal polynomials. Applications in analysis, number theory, combinatorics, physics and computer algebra.

0410 - 617 : Methods of Mathematical  Physics

CR

:

3

Linear transformations and quadratic forms. Orthogonal systems. Fourier series and the Fourier integral. Linear integral equations. Calculus of variations. Eigenvalue problems. Special functions defined by eigenvalue problems and fractional calculus. Heat and wave equations.

0410 - 618 : Advanced topics in Analysis

CR

:

3

Advanced topics in Analysis will include recent advances.

0410 - 623 : Advanced topics in Applied Mathematics

CR

:

3

Advanced topics in Applied Mathematics will include recent advances.

0410 - 625 : Topology

CR

:

3

Advanced topics in function spaces, uniform spaces, compactifications and applications. Surfaces, fundamental groups and covering spaces.

0410 - 626 : Algebraic Topology

CR

:

3

Simplicial homology, homotopy theory, homotopy groups. Cohomology, singular homology.

0410 - 635 : Advanced topics in Geometry and Topology

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3

Advanced topics in Geometry and Topology will include recent advances.

0410 - 636 : Graph Theory

CR

:

3

Matchings in bipartite graphs, perfect matchings, Konig’s theorem. Matchings in a general graph, Tutte’s theorem, f-factor theorem. Colorings of graphs, vertex coloring, Brook’s theorem, edge colorings, Vising’s theorem. Ramsey theory and other extremal problems.

0410 - 637 : Theory of Linear Programming

CR

:

3

Theory of linear optimization and modeling techniques. Convex analysis, duality theory and KKT optimality conditions. Sensitivity analysis, primal, dual and primal-dual simplex algorithm. Convergence and implementation issues.

0410 - 638 : Advanced topics in Discrete Mathematics

CR

:

3

Advanced topics in Discrete Mathematics, in particular in combinatorics, graph theory and/or linear programming, will include recent advances.

0410 - 650 : Approximation and Optimization

CR

:

3

Formulation of approximation and optimization problem in function spaces. Normed spaces and dual spaces. Convex sets, separation theorems and differentiability properties of convex functions. Optimality criteria (Karush-Kuhn-Tucker variational inequalities). Least square problems in Hilbert spaces, finite dimensional approximation, and Chebychev approximation. Duality and saddle points.

0410 - 661 : Numerical Solution of Partial Differential Equations

CR

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3

Characteristics and boundary conditions for first and second order PDEs. Finite difference methods. Semi-discretization (MOL). Finite elements, Petrov-Galerkin methods. Discretization of eigenvalue problems.

0410 - 668 : Advanced topics in Numerical Mathematics

CR

:

3

Advanced topics in Numerical Mathematics will include recent advances.

0410 - 690 : Seminar in Mathematics

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:

(1-3)

The aim of the seminar in mathematics is to have the student learn about the state of the art of current research in a specific area of Mathematics and gain experience in presenting and discussing material assigned by the seminar instructor.

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