# 6000 Level - Electives

**MTH 6310 - Commutative Rings and Modules **

Prerequisite(s): MTH 5311.

Noetherian rings, quotient rings, primary decomposition, integral dependence and valuations, Dedekind domains, and discrete valuation rings, completions, dimension theory.

**MTH 6311 - Non-Commutative Rings and Modules **

Prerequisite(s): MTH 6310.

Semi-simple rings and modules, radicals, chain conditions, decomposition of modules, Goldie's theorem, density and Morita theory.

**MTH 6312 - Abelian Group Theory**

Prerequisite(s): MTH 5311.

An introduction to the fundamental theory of torsion, torsion-free, and mixed abelian groups.

**MTH 6322 - Approximation Theory**

Prerequisite(s): MTH 4322 and 4328.

Approximation of real functions including polynomial and rational interpolation, orthogonal polynomials, Chebyshev approximation, the fast Fourier transform, splines, wavelets, and tensor product interpolation.

**MTH 6325 - Numerical Solutions of Partial Differential Equations **

Prerequisite(s): MTH 4322 and 4328.

Finite difference and finite element methods for elliptic, parabolic, and hyperbolic problems in partial differential equations.

**MTH 6340 - Compact Lie Groups **

Prerequisite(s): MTH 5310 and 5340.

Compact Lie groups, Lie algebras, representation theory, orthogonality relations, Peter Weyl theorem, structure theory, roots, Weyl character formula.

**MTH 6341 - Lie Algebras**

Prerequisite(s): MTH 5310 and 5316.

Lie algebras, semisimple Lie algebras, root systems, conjugecy theorems, classification theorem, representation theory, Chevalley algebras.

**MTH 6342 - Semisimple Lie Groups**

Prerequisite(s): MTH 6340 and 6341.

Structure theory for noncompact groups, induced representations, tempered representations, Langland's classification of irreducible admissible representations.

**MTH 6350 - Set and Model Theory**

Prerequisite(s): MTH 5311.

Propositional and predicate calculus, Loewenheim-Skolem theorems, properties of ultraproducts, model completeness, Goedel's completeness/incompleteness proofs, infinitary language, axioms of set theory, ordinal and cardinal arithmetic, models of set theory and large cardinals.

**MTH 6360 : Harmonic Analysis and Singular Integrals**

Prerequisites: MTH 5323.

Topics which may vary by semester include: basic singular integral operators (Hilbert transform, Riesz transforms, Cauchy operator, boundary layer potentials), Calderon-Zygmund theory, boundedness results on Lp spaces, and applications to boundary value problems and partial differential equations. May be repeated for credit.

**MTH 6362 - Fourier Analysis on Euclidean Spaces**

Prerequisite(s): Graduate standing.

Introduction to Fourier Analysis; singular integrals, pseudodifferential operators, Lp estimates, and applications to partial differential equations. Additional topics may vary by semester.

**MTH 6363 - Analytic Number Theory**

Prerequisite(s): Graduate standing.

Unique factorization, quadratic reciprocity, arithmetical functions, Dirichlet series, distribution of prime numbers. Additional topics may vary by semester.

**MTH 6364 - Algebraic Number Theory**

Prerequisite(s): Graduate standing.

Class field theory, cyclotomic fields, p-adic L functions, and elliptic curves. Additional topics may vary by semester.

**MTH 6365 - Topics in Combinatorics **

Prerequisite(s): Graduate standing.

Graphs, Ramsey theory, extremal set theory, generating functions, and partitions. Additional topics may vary by semester.

**MTH 6366 - Topic in Noncommutative Analysis**

Prerequisite(s): Graduate standing.

Introduction to Positive definite matrices, Matrices of the trace class and the Schatten-p classes, Lp spaces associated with von Neumann algebras, Markov semigroup of operators, Noncommutative Hardy/BMO spaces, Free Fourier Multipliers, Shannon entropy and Fisher information. Additional topics may vary by semester.

**MTH 6367 - Topics in Complex Analysis: Elliptic and Automorphic Functions**

Prerequisite(s): Graduate standing.

Topics which may vary by semester include periodic meromorphic functions, elliptic Weierstrass functions, elliptic Jacobi functions, modular functions, Picard’s theorems, modular group, automorphic functions, applications to completely integrable systems.

**MTH 6368 - Topics in Spectral Theory I **

Prerequisite(s): Graduate standing.

Maximal and minimal operators, Weyl-Titchmarsh theory, spectral functions for 2nd order ODE operators, eigenfunction expansions. Topics may vary by semester.

**MTH 6369 - Topics in Operator Theory II: Compact Operators**

Prerequisite(s): Graduate standing.

Compact operators, canonical decomposition of compact operators, singular values, l^p-based Schatten-von Neumann trace ideals, (regularized) Fredholm determinants, applications to the spectral theory of differential operators. Topics may vary by semester.

**MTH 6V00 - Graduate Research 1 to 10 sem. hrs. **

Prerequisite(s): Graduate standing.

Grants full-time status. For research credit prior to admission to candidacy for an advanced degree. May be repeated for credit through 45 hours.

**MTH 6V13 - Advanced Topics in Algebra 1 to 3 sem. hrs. **

Prerequisite(s): Consent of instructor.

May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

**MTH 6V23 - Advanced Topics in Analysis 1 to 3 sem. hrs. **

Prerequisite(s): Consent of instructor.

May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

**MTH 6V24 - Advanced Topics in Applied Mathematics 1 to 3 sem. hrs. **

Prerequisite(s): Consent of instructor.

May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

**MTH 6V28 - Advanced Topics in Numerical Analysis 1 to 3 sem. hrs. **

Prerequisite(s): Consent of instructor.

May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

**MTH 6V30 - Advanced Topics in Topology **

Prerequisite(s): Consent of instructor. Topology is the study of abstract mathematical spaces with the ultimate goal of finding invariants which are preserved under continuous transformation. Along with algebra and analysis, topology is one of the main areas of modern mathematics and as such every doctoral program in mathematics should have a course designed to cover the more advanced aspects of topology. This course would be taken primarily by doctoral candidates with a strong interest in topology.

**MTH 6V43 - Advanced Topics in Representation Theory 1 to 3 sem. hrs. **

Prerequisite(s): Consent of instructor.

May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs. Ph.D. Thesis 6V99 Dissertation 1 to 12 sem. hrs. Grants full time status. Requires passing the Preliminary Exam. Supervised research for the doctoral dissertation

## Ph.D. Thesis

6V99 Dissertation 1 to 12 sem. hrs.

Grants full time status. Requires passing the Preliminary Exam. Supervised research for the doctoral dissertation.