
Graduate Course Descriptions
4000 Level  Electives 
MTH 4314  Abstract Algebra Prerequisite(s): A
grade of C or above in MTH 3312. Fundamentals of group, ring, and field
theory. Topics include permutation groups, group and ring homomorphisms,
direct products of groups and rings, quotient objects, integral domains,
field of quotients, polynomial rings, unique factorization domains,
extension fields, and finite fields.

MTH 4322  Numerical Analysis
(Crosslisted as CSI 4322) Prerequisite(s): A grade of C or above in MTH
2321. Numerical evaluation of derivatives and integrals, solution of
algebraic and differential equations, and approximation theory.

MTH 4326  Advanced Calculus I Prerequisite(s): A grade of C or above in
MTH 3323. Sequences and series of functions, multiple integrals,
improper multiple integrals, functions of several variables, extreme value
problems, and implicit function theorems.

MTH 4327  Advanced
Calculus II Prerequisite(s): A grade of C or above in MTH 4326.
Line and surface integrals, Green, Gauss, Stokes theorems with applications,
Fourier series and integrals, functions defined by integrals, introduction
to complex functions.

MTH 4328  Numerical Linear Algebra
(Crosslisted as CSI 4328) Prerequisite(s): A grade of C or above in
MTH 2311 and 3324. Numerical methods for solution of linear equations,
eigenvalue problems, and least squares problems, including sparse matrix
techniques with applications to partial differential equations.

MTH 4329  Theory of Functions of a Complex Variable Prerequisite(s): A
grade of C or above in MTH 2321. Number systems: the complex plane;
fractions, powers, and roots; analytic functions; elementary functions;
complex integration; power series; mapping by elementary functions; calculus
of residues.

5000 Level  Core Courses 
MTH 5310  Advanced Abstract Algebra I
Prerequisite(s): MTH 4314 and consent of the instructor. Finite groups,
Sylow theorems, nilpotent and solvable groups, principal ideal domains,
unique factorization domains, and sub rings to algebraic number fields.

MTH 5311  Advanced Abstract Algebra II Prerequisite(s): MTH 5310.
Field theory, Galois theory, modules, finitely generated modules, principal
ideal domains, homological methods, and WedderburnArtin theorems.

MTH 5323  Theory of
Functions of Real Variables I Prerequisite(s): MTH 4327. Borel
sets, measure and measurable sets, measurable functions, and the Lebesque
integral.

MTH 5324  Theory of Functions of Real Variables II
Prerequisite(s): MTH 5323. Function spaces, abstract measure, and
differentiation.

MTH 5330  Topology Prerequisite(s): Graduate standing. Topological
spaces, continuous functions, metric spaces, connectedness, compactness,
separation axioms, Tychenoff theorem, fundamental group, covering spaces,
metrization theorems.

MTH 5331  Algebraic Topology I
Prerequisite(s): MTH 5330. Homology theory, simplicial complexes,
topological invariance, relative homology, EilenbergSteenrod axioms,
singular homology, CW complexes.

MTH 5350  Complex Analysis Prerequisite(s): MTH 4327. Complex
numbers, complex functions, analytic functions, linear fractional
transformations, complex integration, Cauchy's formula, residues, harmonic
functions, series and product expansions, Gamma function, Riemann mapping
theorem, Dirichlet problem, analytic continuation.

MTH 5360  Applied Mathematics I
Prerequisite(s): Graduate standing.
Dynamical systems (ODE and PDE, discrete and continuous),
linear and nonlinear systems theory,
transform methods,
control theory and optimization,
calculus of variations,
stability theory.

MTH 5361  Applied Mathematics II
Prerequisite(s): Graduate standing.
Eigenvalue theory,
projections for linear equations
iterations and multilevel methods,
fast Fourier transforms,
approximations of differential equations,
grid adaption and numerical stability,
weak solutions and Sobelov space,
wavelets with applications.

5000 Level  Electives 
MTH 5316  Linear Algebra and Matrix Theory Prerequisite(s): MTH 3312.
Matrix calculus, eigenvalues and eigenvectors, canonical forms, orthogonal
and unitary transformations, and quadratic forms. Applications of these
concepts. A course project is required and will be specified by the
professor at the beginning of the course.

MTH 5325  Theory of Differential Equations
Prerequisite(s): MTH 3325 and 5323. Initial value problems for ordinary
differential equations: existence, uniqueness, continuous dependence,
stability analysis, oscillation theory, general linear systems, phase plane
analysis, limit cycles and periodic solutions. Topics of current interest in
dynamical systems. 
MTH 5326  Theory of Partial Differential
Equations Prerequisite(s): MTH 5324 and 5325. Linear and
quasilinear first order equations; shocks, characteristics, the Cauchy
problem, elliptic, hyperbolic, and parabolic equations, maximum principles,
Dirichlet problem, operators, Sobolev spaces, distributions. 
MTH 5332  Algebraic Topology II
Prerequisite(s): MTH 5331. Cohomology theory, homology with coefficients,
homological algebra, Kunneth theorem, duality in manifolds. 
MTH 5340  Differential Geometry Prerequisite(s): MTH 4327, 5316, and
5330. Differentiable manifolds, submanifolds, vector fields, tensor
fields, integration on manifolds, Riemannian geometry. 
MTH 5345  Functional Analysis Prerequisite(s); MTH 5324.
Banach spaces, Hilbert spaces, linear operators, and spectral theory.

MTH 5351  Applications of Complex Analysis
Prerequisite(s): MTH 5350.
Poisson summation, Mellin transformation, zeta function of Riemann, special functions, zeta
functions associated with eigenvalue problems, heat kernel, asymptotic expansion of the heat kernel,
meromorphic structure of zeta functions, theta functions, elliptic functions. 
MTH 5375  Linear Programming Prerequisite(s): MTH 2311 or consent of
instructor. Introduction to the theory and applications of linear
programming, including the simplex algorithm, duality, sensitivity analysis,
parametric linear programming, integer programming, with applications to
transportation and allocation problems and game theory. A course project is
required and will be specified by the professor at the beginning of the
course. 
MTH 5376  Nonlinear Programming Theory and
algorithms for the optimization of unconstrained problems including gradient
and QuasiNewton methods; and constrained problems to include feasible
direction methods, Lagrange multipliers, and KarushKuhnTucker conditions.
Students must have a knowledge of linear algebra, thirdsemester calculus. 
MTH 5390  Special Problems in Mathematics
Project course for the project option in the M.S. degree.< 
MTH 5V91  Special Topics in Algebra for Graduates 1 to 3 sem. hrs. May be
repeated for credit with instructor's consent. 
MTH 5V92  Special Topics in Analysis for Graduates 1 to 3 sem. hrs. May be
repeated for credit with instructor's consent. 
MTH 5V93  Special Topics in Mathematics for Education Students 1 to 3 sem. hrs.
Prerequisite(s): Consent of departmental chair and the course instructor.
May be repeated for credit for a maximum of nine semester hours if under
different topics. Maximum 9 sem. hrs. 
Master's Thesis 
MTH 5V99  Thesis 1 to 6
sem. hrs. Credit to be given for the amount of work done. In no case
will less than six semester hours be accepted. Maximum 10 sem. hrs. 
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 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. Supervised research for the
doctoral dissertation.


