# Mathematics (MTH)

MTH 4314  Abstract Algebra  (3)
Pre-requisite(s): A grade of C or above in MTH 2311 and MTH 3312; or consent of instructor

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  (3)
Cross-listed as CSI 4322
Pre-requisite(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  (3)
Pre-requisite(s): A grade of C or above in MTH 2321 and MTH 3323 or consent of instructor

The real and complex number systems, basic topology, numerical sequences and series, continuity, differentiation, integration, sequences and series of functions.

MTH 4327  Advanced Calculus II  (3)
Pre-requisite(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  (3)
Cross-listed as CSI 4328
Pre-requisite(s): A grade of C or above in MTH 2311

Numerical methods for solution of linear equations, eigenvalue problems, and least squares problems, including sparse matrix techniques with applications to partial equations.

MTH 4329  Theory of Functions of a Complex Variable  (3)
Pre-requisite(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.

MTH 5100  Teaching Mathematics  (1)

This course equips graduate students in mathematics with pedagogical strategies and techniques for teaching mathematics effectively in various settings. Topics covered include instructional design, assessment methods, communication skills, and addressing the needs of diverse learners.

MTH 5310  Advanced Abstract Algebra I  (3)
Pre-requisite(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  (3)
Pre-requisite(s): MTH 5310

Field theory, Galois theory, modules, finitely generated modules, principal ideal domains, homological methods, and Wedderburn-Artin theorems.

MTH 5316  Linear Algebra and Matrix Theory  (3)
Pre-requisite(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 5323  Theory of Functions of Real Variables I  (3)
Pre-requisite(s): MTH 4327

Borel sets, measure and measurable sets, measurable functions, and the Lebesque integral.

MTH 5324  Theory of Functions of Real Variables II  (3)
Pre-requisite(s): MTH 5323

Function spaces, abstract measure, and differentiation.

MTH 5325  Theory of Differential Equations  (3)
Pre-requisite(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  (3)
Pre-requisite(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 5330  Topology  (3)

Topological spaces, continuous functions, metric spaces, connectedness, compactness, separation axioms, Tychenoff theorem, fundamental group, covering spaces, metrization theorems.

MTH 5331  Algebraic Topology I  (3)
Pre-requisite(s): MTH 5330

Homology theory, simplicial complexes, topological invariance, relative homology, Eilenberg-Steenrod axioms, singular homology, CW complexes.

MTH 5332  Algebraic Topology II  (3)
Pre-requisite(s): MTH 5331

Cohomology theory, homology with coefficients, homological algebra, kunneth theorem, duality in manifolds.

MTH 5340  Differential Geometry  (3)
Pre-requisite(s): MTH 4327, 5316, and 5330

Differentiable manifolds, submanifolds, vector fields, tensor fields, integration on manifolds, Riemannian geometry.

MTH 5344  Probability Theory  (3)
Pre-requisite(s): MTH 5323

Topics include the law of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic Theorems, and Brownian motion.

MTH 5345  Functional Analysis  (3)
Pre-requisite(s): MTH 5324

Banach spaces, Hilbert spaces, linear operators, and spectral theory.

MTH 5350  Complex Analysis  (3)

Comoplex 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 5351  Applications of Complex Analysis  (3)
Pre-requisite(s): MTH 5350

Poisson summation, Mellin transformation, zeta function of Riemann, special functions, zeta functions associated with ezjen value problems, heat kernel, asymptotic expansion of the heat kernel, metamorphic structure of zeta functions, theta functions, elliptic functions.

MTH 5360  Applied Mathematics I  (3)

Dynamical systems (discrete and continuous), linear and nonlinear systems theory, transform methods, control theory and optimization, calculus of variations, stability theory.

MTH 5361  Applied Mathematics II  (3)

Eigenvalue theory, projections for linear equations, iteration and multilevel methods, fast Fourier Transforms, approximations of differential equations, grid adaptation and numerical stability, weak solutions and Sobolev space, wavelets.

MTH 5375  Linear Programming  (3)
Pre-requisite(s): MTH 2311 and FORTRAN, or consent of instructor

Introduction to the theory and applications of linear programming, including the simplex algorithm, duality, sensitivity 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  (3)

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 Kuhn-Tucker conditions. Students must have a knowledge of linear algebra, third-semester calculus, and FORTRAN.

MTH 5380  Statistical Methods for Research  (3)
Pre-requisite(s): For graduate students from various disciplines

Introduction to the more common statistical concepts and methods. Emphasis is placed on proper applications of statistical tools. Topics include: interval estimation, tests of hypotheses, linear regression and correlation, categorical data analysis, design of experiments and analysis of variance, and the use of computer packages.

MTH 5390  Special Problems in Mathematics  (3)

Project course for the project option in the M.S. degree.

MTH 5V91  Special Topics in Algebra for Graduates  (1-3)

May be repeated for credit up to 18 hours.

MTH 5V92  Special Topics in Analysis for Graduates  (1-3)

May be repeated for credit up to 18 hours.

MTH 5V93  Special Topics in Mathematics for Education Students  (1-3)
Pre-requisite(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.

MTH 5V95  Special Topics in Topology-Geometry  (1-3)

May be repeated for credit for a maximum of 9 semester hours.

MTH 5V96  Special Topics in Graph Theory  (1-3)

Introduction to graph theory; Euler tours, matching, connectivity, planar graphs, coloring, and random graphs. Additional topics may vary by semester.

MTH 6310  Commutative Rings and Modules  (3)
Pre-requisite(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  (3)
Pre-requisite(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  (3)
Pre-requisite(s): MTH 5311

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

MTH 6315  Homological Algebra  (3)
Pre-requisite(s): MTH 5311 or consent of instructor

Categories, chain complexes, homology and cohomology, and derived functors. Detailed examination of Ext, Tor, adjoint functors, and direct and inverse limits for categories of modules. Kunneth formula and universal coefficient theorems. Cohomology of groups.

MTH 6322  Approximation Theory  (3)
Pre-requisite(s): MTH 4322 and 4328

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

MTH 6325  Numerical Solutions of Partial Differential Equations  (3)
Pre-requisite(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  (3)
Pre-requisite(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  (3)
Pre-requisite(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  (3)
Pre-requisite(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  (3)
Pre-requisite(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 cardinals arithmetic, models of set theory and large cardinals.

MTH 6362  Fourier Analysis on Euclidean Spaces  (3)

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  (3)

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

MTH 6364  Algebraic Number Theory  (3)

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

MTH 6365  Topics in Combinatorics  (3)

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

MTH 6366  Topic in Noncommutative Analysis  (3)

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  (3)

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, and applications to completely integrable systems

MTH 6368  Topics in Spectral Theory I  (3)

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

MTH 6369  Topics in Operator Theory II: Compact Operators  (3)

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.

For research credit prior to admission to candidacy for an advanced degree. Credit will be given for the amount of work done. May be repeated for credit through 45 hours.

MTH 6V13  Advanced Topics in Algebra  (1-3)
Pre-requisite(s): Consent of instructor

May be repeated for credit up to 18 hours.

MTH 6V23  Advanced Topics in Analysis  (1-3)
Pre-requisite(s): Consent of instructor

May be repeated for credit up to 18 hours.

MTH 6V24  Advanced Topics in Applied Mathematics  (1-3)
Pre-requisite(s): Consent of instructor

May be repeated for credit up to 18 hours.

MTH 6V28  Advanced Topics in Numerical Analysis  (1-3)
Pre-requisite(s): Consent of instructor

May be repeated for credit up to 18 hours.

MTH 6V30  Advanced Topics in Topology  (1-3)
Pre-requisite(s): Consent of instructor

Topology is the study of abstract mathematical spaces with the ultimate goal of finding invariants that are preserved under continuous transformation. This course is intended for doctoral candidates with a strong interest in topology. May be repeated for credit.

MTH 6V43  Advanced Topics in Representation Theory  (1-3)
Pre-requisite(s): Consent of instructor

May be repeated for credit up to 18 hours.

MTH 6V99  Dissertation  (1-12)

Supervised research for the doctoral dissertation.