 |
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
(Cross-listed 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
(Cross-listed 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 Wedderburn-Artin 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, Eilenberg-Steenrod 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 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 Quasi-Newton methods; and constrained problems to include feasible
direction methods, Lagrange multipliers, and Karush-Kuhn-Tucker conditions.
Students must have a knowledge of linear algebra, third-semester 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 - 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 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.
|
|
 |
