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  NonCommutative Rings and Modules
Prerequisite(s): MTH 6310.
Semisimple 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, torsionfree, 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, LoewenheimSkolem 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 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, padic 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 Schattenp 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, WeylTitchmarsh 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^pbased Schattenvon Neumann trace ideals, (regularized) Fredholm determinants, applications to the spectral theory of differential operators. Topics may vary by semester. 
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. 