Department of Mathematics and Statistics

Graduate course catalog

Some prerequisites can be overridden. If you don't meet a prerequisite, but think you are qualified to take a course, consult with one or more of: your advisor; the instructor; the department chair or associate chair.

Math 5010 [3 credit hours]

FUNCTIONS AND MODELING FOR MIDDLE GRADE MATHEMATICS

Introduction to the theory of functions through modeling. Subjects include polynomial, exponential, logarithmie and rational functions, interpolation and modeling of data sets though least squares and other methods.

Graduate math credit for education students only.

Math 5040 [3 credit hours]

CONCEPTS OF CALCULUS FOR MIDDLE GRADE MATHEMATICS

Introduction to the basic idea of calculus. Subjects include limits, continuity, the derivative and its applications, indefinite and definite integral, Fundamental Theorem of Calculus, evaluation of integrals.

Graduate math credit for education students only.

Math 5060 [3 credit hours]

NUMBER THEORY CONCEPTS FOR MIDDLE GRADE MATHEMATICS

Introduction to basic number theory. Subjects include history of number theory, prime numbers, unique factorization, Euclidean algorithm, Pythagorean relations, number systems, and transformations.

Graduate math credit for education students only.

Math 5070 [3 credit hours]

GEOMETRY CONCEPTS FOR MIDDLE SCHOOL MATHEMATICS

Descriptive geometry in 2 and 3 dimensions, use of axioms and definitions in the proof theorems, formal Euclidean geometry, transformations.

Graduate math credit for education students only.

Math 5080 [3 credit hours]

HISTORY OF MATHEMATICS FOR MIDDLE GRADE MATHEMATICS

Study of the history of mathematics from antiquity to the 20th century concentrating on the development of arithmetic, algebra, geometry and calculus.

Graduate math credit for education students only.

Math 5110 [3 credit hours]

PROBABILITY CONCEPTS FOR MIDDLE GRADE MATHEMATICS

Introduction to the theory of probability, counting principles and combinatorics, risk, coincidence, expectation and conditional probability, probability distributions.

Graduate math credit for education students only.

Math 5120 [3 credit hours]

STATISTICS CONCEPTS FOR MIDDLE GRADE MATHEMATICS

Introduction to the fundamental ideas of statistics, including sampling techniques, descriptive, variance, confidence intervals, correlation and regression.

Graduate math credit for education students only.

Math 5220 [3 credit hours]

THEORY OF INTEREST

This course covers the measurement of interest, certain annuities, yield rates, amortization and sinking funds, bonds and other securities and application of interest theory.

Math 5260 [3 credit hours]

ACTUARIAL MATHEMATICS I

Survival distributions and life tables, life insurance, life annuities, benefit premiums and reserves and multiple life functions are some topics covered in this course.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5300 [3 credit hours]

LINEAR ALGEBRA I

Theory of vector spaces and linear transformations, including such topics as matrices, determinants, inner products, eigenvalues and eigenvectors, and rational and Jordan canonical forms.

Math 5310 [3 credit hours]

LINEAR ALGEBRA II

Hermitian and normal operators, multilinear forms, spectral theorem and other topics.

Prerequiste: MATH 5300 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5330 [3 credit hours]

ABSTRACT ALGEBRA I

Arithmetic of the integers, unique factorization and modular arithmetic; group theory including normal subgroups, factor groups, cyclic groups, permutations, homomorphisms, the isomorphism theorems, abelian groups and p-groups.

Prerequiste: MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5340 [3 credit hours]

ABSTRACT ALGEBRA II

Ring theory including integral domains, field of quotients, homomorphisms, ideals, Euclidean domains, polynomial rings, vector spaces, roots of polynomials and field extensions.

Prerequiste: MATH 5330 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5350 [3 credit hours]

APPLIED LINEAR ALGEBRA

Matrices, systems of equations, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, generalized inverses, rank, numerical methods and applications to various areas of science.

Prerequiste: MATH 1890 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5380 [3 credit hours]

DISCRETE STRUCTURES AND ANALYSIS ALGORITHMS

Discrete mathematical structures for applications in computer science such as graph theory, combinatorics, groups theory, asymptotics, recurrence relations and analysis of algorithms.

Prerequiste: MATH 3320 FOR LEVEL UG WITH MIN. GRADE OF D- OR MATH 5330 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5450 [3 credit hours]

INTRODUCTION TO TOPOLOGY I

Metric spaces, topological spaces, continuous maps, bases and sub-bases, closure and interior operators, products, subspaces, sums, quotients, separation axioms, compactness and local compactness.

Prerequiste: MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5460 [3 credit hours]

INTRODUCTION TO TOPOLOGY II

Connectedness and local connectedness, convergence, metrization, function spaces. The fundamental groups and its properties, covering spaces, classical applications, e.g. Jordan Curve Theorem, Fundamental Theorem of Algebra, Brouwer's Fixed Point Theorem.

Prerequiste: MATH 5450 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5540 [3 credit hours]

CLASSICAL DIFFERENTIAL GEOMETRY I

Smooth curves in Euclidean space including the Frenet formulae. Immersed surfaces with the Gauss map, principal curvatures and the fundamental forms. Special surfaces including ruled surfaces and minimal surfaces. Intrinsic Geometry including the Gauss Theorem Egregium.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5550 [3 credit hours]

CLASSICAL DIFFERENTIAL GEOMETRY II

Tensors, vector fields and the Cartan approach to surface theory, Bonnet's Theorem and the construction of surfaces via solutions of the Gauss Equation. Geodesics, parallel transport and Jacobi Fields. Theorems of a global nature such as Hilbert's Theorem or the Theorem of Hopf-Rinow.

Prerequiste: MATH 5540 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5600 [3 credit hours]

ADVANCED STATISTICAL METHODS I

Basics of descriptive statistics, study designs and statistical inference. Properties of, and assumptions required for, inference for means, variances, and proportions from one and two-sample paired and unpaired studies. Introduction to ANOVA with multiple comparisons and multiple regression. Model assessment and diagnostics. Statistical software will be employed. Opportunities to apply procedures to real data. Emphasis placed on the foundations to approaches in introductory statistics.

Math 5610 [3 credit hours]

ADVANCED STATISTICAL METHODS II

Statistical/biostatistical concepts and methods. Broad subject categories that may be included are study design, longitudinal data analysis, survival analysis, logistic regression, random and mixed effects models. Other topics applicable to current statistical consulting projects, or related to modern data analytics, may be introduced. Appropriate statistical software will be employed.

Prerequiste: MATH 5600 FOR LEVEL GR WITH MIN. GRADE OF C-

Math 5620 [3 credit hours]

LINEAR STATISTICAL MODELS

Multiple regression, analysis of variance and covariance, general linear models and model building for linear models. Experimental designs include one-way, randomized block, Latin square, factorial and nested designs.

Prerequiste: MATH 6650 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5630 [3 credit hours]

THEORY AND METHODS OF SAMPLE SURVEYS

The mathematical basis to estimation in various sampling contexts, including probability proportional to size sampling, stratified sampling, two-stage cluster sampling and double sampling, is developed.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5640 [3 credit hours]

STATISTICAL COMPUTING

Modern statistical computing, including programming tools, modern programming methodologies, design of data structures and algorithms, numerical computing and graphics. Additional topics selected from simulation studies, inversion of probability integral transforms, rejection sampling, importance sampling, Monte Carlo integration, bootstrapping and optimization.

Math 5680 [3 credit hours]

INTRODUCTION TO THEORY OF PROBABILITY

Probability spaces, random variables, probability distributions, moments and moment generating functions, limit theorems, transformations and sampling distributions.

Prerequiste: (MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D- AND MATH 5350 FOR LEVEL GR WITH MIN. GRADE OF D-)

Math 5690 [3 credit hours]

INTRODUCTION TO MATHEMATICAL STATISTICS

Sampling distributions, point estimation, interval estimation, hypothesis testing, regression and analysis of variance.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5710 [3 credit hours]

METHODS OF NUMERICAL ANALYSIS I

Floating point arithmetic; polynomial interpolation; numerical solution of nonlinear equations; Newton's method. Likely topics include: numerical differentiation and integration; solving systems of linear equations; Gaussian elimination; LU decomposition; Gauss-Seidel method.

Math 5720 [3 credit hours]

METHODS OF NUMERICAL ANALYSIS II

Likely topics include: Computation of eigenvalues and eigenvectors; solving systems of nonlinear equations; least squares approximations; rational approximations; cubic splines; fast Fourier transforms; numerical solutions to initial value problems; ordinary and partial differential equations.

Prerequiste: MATH 5710 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5740 [3 credit hours]

ADVANCED APPLIED MATHEMATICS I

Series and numerical solutions to ordinary differential equations, special functions, orthogonal functions, Sturm-Liouville Problems, self-adjointness, vector analysis.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5750 [3 credit hours]

ADVANCED APPLIED MATHEMATICS II

Continuation of vector analysis, introduction to complex analysis, partial differential equations, Fourier series and integrals.

Prerequiste: MATH 5740 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5780 [3 credit hours]

ADVANCED CALCULUS

Extrema for functions of one or more variables, Lagrange multipliers, indeterminate forms, inverse and implicit function theorems, uniform convergences, power series, transformations, Jacobians, multiple integrals.

Prerequiste: MATH 2850 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5800 [3 credit hours]

ORDINARY DIFFERENTIAL EQUATIONS

Modern theory of differential equations; transforms and matrix methods; existence theorems and series solutions; and other selected topics.

Prerequiste: MATH 2860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5810 [3 credit hours]

PARTIAL DIFFERENTIAL EQUATIONS

First and second order equations; numerical methods; separation of variables; solutions of heat and wave equations using eigenfunction techniques; and other selected topics.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5820 [3 credit hours]

INTRODUCTION TO REAL ANALYSIS I

A rigorous treatment of the Calculus in one and several variables. Topics to include: the real number system; sequences and series; elementary metric space theory including compactness, connectedness and completeness; the Riemann Integral.

Prerequiste: MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5830 [3 credit hours]

INTRODUCTION TO REAL ANALYSIS II

Differentiable functions on Rn; the Implicit and Inverse Function Theorems; sequences and series of continuous functions; Stone-Weierstrass Theorem; Arsela-Ascoli Theorem; introduction to measure theory; Lebesgue integration; the Lebesgue Dominated Convergence Theorem.

Prerequiste: MATH 5820 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5860 [3 credit hours]

CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY I

Conditions for an extreme (Euler's equations, Erdman corner conditions, conditions of Legendre, Jacobi and Weierstrass, fields of extremals, Hilbert's invariant integral); ); Raleigh-Ritz method; isoperimetric problems; Lagrange, Mayer-Bolza problems. Recommended: MATH 5820.

Prerequiste: MATH 1890 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5870 [3 credit hours]

CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY II

Pontryagin's maximum principle; necessary and sufficient conditions for optimal control, controllability, time optimal control, existence of optimal controls, relationship to the calculus of variations.

Prerequiste: MATH 5860 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 5880 [3 credit hours]

COMPLEX VARIABLES

Analytic functions; Cauchy's theorem; Taylor and Laurent series; residues; contour integrals; conformal mappings, analytic continuation and applications.

Prerequiste: MATH 2860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 5970 [1 credit hours]

INDUSTRIAL MATH PRACTICUM

Students must submit for approval by their adviser a report on the solution of a practical problem involving mathematics. The problem must be drawn from a company, university department of government unit.

Math 5980 [3 credit hours]

TOPICS IN MATHEMATICS

Special topics in mathematics.

Math 6180 [3 credit hours]

LINEAR AND NONLINEAR PROGRAMMING

Simplex algorithm, ellipsoidal algorithm, Karmarkar's method, interior point methods, elementary convex analysis, optimality conditions and duality for smooth problems, convex programming, algorithms and their convergence.

Prerequiste: MATH 5820 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6190 [3 credit hours]

INFINITE DIMENSIONAL OPTIMIZATION

Introduction to nonlinear analysis, abstract optimization problems on abstract spaces, applications to calculus of variations, optimal control theory and game theory.

Math 6300 [3 credit hours]

ALGEBRA I

Group actions, Sylow's theorems, permutation groups, nilpotent and solvable groups, abelian groups, rings, unique factorization domains, fields.

Prerequiste: MATH 5340 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6310 [3 credit hours]

ALGEBRA II

Field extensions, Galois theory, modules, Noetherian and Artinian rings, tensor products, primitive rings, semisimple rings and modules, the Wedderburn-Artin theorem.

Prerequiste: MATH 6300 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6400 [3 credit hours]

TOPOLOGY I

Topological spaces, continuous functions, compactness, product spaces, Tychonov's theorem, quotient spaces, local compactness, homotopy theory, the fundamental group, covering spaces.

Prerequiste: MATH 4450 FOR LEVEL UG WITH MIN. GRADE OF D- OR MATH 5450 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 7450 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6410 [3 credit hours]

TOPOLOGY II

Homology theory, excision, homological algebra, the Brouwer fixed point theorem, cohomology, differential manifolds, orientation, tangent bundles, Sard's theorem, degree theory.

Prerequiste: MATH 6400 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6440 [3 credit hours]

DIFFERENTIAL GEOMETRY I

Introduction to differential geometry. Topics include differentiable manifolds, vector fields, tensor bundles, the Frobenius theorem, Stokes' theorem, Lie groups.

Prerequiste: MATH 6410 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6450 [3 credit hours]

DIFFERENTIAL GEOMETRY II

Topics include connections on manifolds, Riemannian geometry, the Gauss-Bonnet theorem. Further topics may include: homogeneous and symmetric spaces, minimal surfaces, Morse theory, comparison theory, vector and principal bundles.

Prerequiste: MATH 6440 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6500 [3 credit hours]

ORDINARY DIFFERENTIAL EQUATIONS

Existence, uniqueness and dependence on initial conditions and parameter, nonlinear planar systems, linear systems, Floquet theory, second order equations, Sturm-Liouville theory.

Math 6510 [3 credit hours]

PARTIAL DIFFERENTIAL EQUATIONS

First order quasi-linear systems of partial differential equations, boundary value problems for the heat and wave equation, Dirichlet problem for Laplace equation, fundamental solutions for Laplace, heat and wave equations.

Math 6520 [3 credit hours]

DYNAMICAL SYSTEMS I

Topic include the flow-box theorem, Poincare maps, attractors, w limit sets, Lyapunov stability, invariant submanifolds, Hamiltonian systems and symplectic manifolds.

Prerequiste: MATH 6500 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6600 [1-5 credit hours]

STATISTICAL CONSULTING

Real data applications of various statistical methods, project design and analysis including statistical consulting experience. May be repeated for credit.

Math 6610 [3 credit hours]

STATISTICAL CONSULTING II

Real data applications of various statistical methods, project design and analysis including statistical consulting experience.

Math 6620 [3 credit hours]

CATEGORICAL DATA ANALYSIS

Important methods and modeling techniques using generalized linear models and emphasizing loglinear and logit modeling.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6630 [3 credit hours]

NONPARAMETRIC STATISTICS

Statistical methods based on counts and ranks; methods designed to be effective in the presence of contaminated data or error distribution misspecification.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF C-

Math 6640 [3 credit hours]

TOPICS IN STATISTICS

Topics selected from an array of modern statistical methods such as survival analysis, nonlinear regression, Monte Carlo methods, etc.

Math 6650 [3 credit hours]

STATISTICAL INFERENCE

Estimation, hypothesis testing, prediction, sufficient statistics, theory of estimation and hypothesis testing, simultaneous inference, decision theoretic models.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6670 [3 credit hours]

MEASURE THEORETIC PROBABILITY

Real analysis, probability spaces and measures, random variables and distribution functions, independence, expectation, law of large numbers, central limit theorem, zero-one laws, characteristic functions, conditional expectations given a s-algebra, martingales.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6680 [3 credit hours]

THEORY OF STATISTICS

Exponential families, sufficiency, completeness, optimality, equivariance, efficiency. Bayesian and minimax estimation. Unbiased and invariant tests, uniformly most powerful tests. Asymptotic properties for estimation and testing. Most accurate confidence intervals.

Prerequiste: MATH 5960 FOR LEVEL GR WITH MIN. GRADE OF D- OR (MATH 6650 FOR LEVEL GR WITH MIN. GRADE OF D- AND MATH 6670 FOR LEVEL GR WITH MIN. GRADE OF D-)

Math 6690 [3 credit hours]

MULTIVARIATE STATISTICS

Multivariate normal sampling distributions, T tests and MANOVA, tests on covariance matrices, simultaneous inference, discriminant analysis, principal components, cluster analysis and factor analysis.

Prerequiste: MATH 5690 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 6650 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6720 [3 credit hours]

METHODS OF MATHEMATICAL PHYSICS I

Analytic functions, residues, method of steepest descent, complex differential equations, regular singularities, integral representation, real and complex vector spaces, matrix groups, Hilbert spaces, coordinate transformations.

Math 6730 [3 credit hours]

METHODS OF MATHEMATICAL PHYSICS II

Self-adjoint operators, special functions, orthogonal polynomials, partial differential equations and separation of variables, boundary value problems, Green¿s functions, integral equations, tensor analysis, metrics and curvature, calculus of variations, finite groups and group representations.

Prerequiste: MATH 6720 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6800 [3 credit hours]

REAL ANALYSIS I

Completeness, connectedness and compactness in metric spaces, continuity and convergence, the Stone-Weierstrass Theorem, Lebesgue measure and integration on the real line, convergence theorems, Egorov's and Lusin's theorems, derivatives, functions of bounded variation.

Prerequiste: MATH 4830 FOR LEVEL UG WITH MIN. GRADE OF D- OR MATH 5830 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6810 [3 credit hours]

REAL ANALYSIS II

The Vitali covering theorem, absolutely continuous functions, Lebesgue-Stieltjes integration, the Riesz representation theorem , Banach spaces, Lp-spaces, abstract measures, the Radon-Nikodym theorem, measures on locally compact Hausdorff spaces.

Prerequiste: MATH 6800 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6820 [3 credit hours]

FUNCTIONAL ANALYSIS I

Topics include Topological vector spaces, Banach spaces, convexity, the Hahn-Banch theorem, weak and strong topologies, Lp spaces and duality.

Prerequiste: MATH 6810 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6830 [3 credit hours]

FUNCTIONAL ANALYSIS II

Topics include the Mackey-Ahrens Theorem, Banach algebras, spectra in Banach algebras, commutative Banach algebras, unbounded operators, the spectral theorem, topics in functional analysis.

Prerequiste: MATH 6820 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6840 [3 credit hours]

COMPLEX ANALYSIS I

Elementary analytic functions, complex integration, the residue theorem, infinite sequences of analytic functions, Laurent expansions, entire functions.

Prerequiste: MATH 6800 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6850 [3 credit hours]

COMPLEX ANALYSIS II

Meromorphic functions, conformal mapping, harmonic functions and the dirichlet problem, the Riemann mapping theorem, monodromy, algebraic functions, Riemann surfaces, elliptic functions and the modular function.

Prerequiste: MATH 6840 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 6860 [3 credit hours]

MEASURE THEORETIC PROBABILITY I

Focus on measure theory and probability. Measures and their extensions, integration, convergence theorems, product measures. Probability spaces, random variables and distribution functions, independence, expectation, law of large numbers, central limit theorem, zero-one laws, characteristic functions.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Corequisite: MATH 6800

Math 6930 [1 credit hours]

COLLOQUIUM

Lectures by visiting mathematicians and staff members on areas of current interest in mathematics.

Math 6940 [1-5 credit hours]

PROSEMINAR

Problems and techniques of teaching elementary college mathematics, supervised teaching, seminar in preparation methods.

Math 6960 [3-6 credit hours]

MASTER THESIS

.

Math 6980 [3 credit hours]

TOPICS IN MATHEMATICAL SCIENCES

Special topics in Mathematics or Statistics.

Math 6990 [1-5 credit hours]

READINGS IN MATHEMATICS

Readings in areas of Mathematics of mutual interest to the student and the professor.

Math 7300 [3 credit hours]

LINEAR ALGEBRA I

Theory of vector spaces and linear transformations, including such topics as matrices, determinants, inner products, eigenvalues and eigenvectors, and rational and Jordan canonical forms.

Math 7310 [3 credit hours]

LINEAR ALGEBRA II

Hermitian and normal operators, multilinear forms, spectral theorem and other topics.

Prerequiste: MATH 5300 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7330 [3 credit hours]

ABSTRACT ALGEBRA I

Arithmetic of the integers, unique factorization and modular arithmetic; group theory including normal subgroups, factor groups, cyclic groups, permutations, homomorphisms, the isomorphism theorems, abelian groups and p-groups.

Prerequiste: MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7340 [3 credit hours]

ABSTRACT ALGEBRA II

Ring theory including integral domains, field of quotients, homomorphisms, ideals, Euclidean domains, polynomial rings, vector spaces, roots of polynomials and field extensions.

Prerequiste: MATH 5330 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7350 [3 credit hours]

APPLIED LINEAR ALGEBRA

Matrices, systems of equations, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, generalized inverses, rank, numerical methods and applications to various areas of science.

Prerequiste: MATH 1890 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7380 [3 credit hours]

DISCRETE STRUCTURES AND ANALYSIS ALGORITHMS

Discrete mathematical structures for applications in computer science such as graph theory, combinatorics, groups theory, asymptotics, recurrence relations and analysis of algorithms.

Prerequiste: MATH 3320 FOR LEVEL UG WITH MIN. GRADE OF D- OR MATH 5330 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7450 [3 credit hours]

INTRODUCTION TO TOPOLOGY I

Metric spaces, topological spaces, continuous maps, bases and sub-bases, closure and interior operators, products, subspaces, sums, quotients, separation axioms, compactness and local compactness.

Prerequiste: MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7460 [3 credit hours]

INTRODUCTION TO TOPOLOGY II

Connectedness and local connectedness, convergence, metrization, function spaces. The fundamental groups and its properties, covering spaces, classical applications, e.g. Jordan Curve Theorem, Fundamental Theorem of Algebra, Brouwer's Fixed Point Theorem.

Prerequiste: MATH 5450 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7540 [3 credit hours]

CLASSICAL DIFFERENTIAL GEOMETRY I

Smooth curves in Euclidean space including the Frenet formulae. Immersed surfaces with the Gauss map, principal curvatures and the fundamental forms. Special surfaces including ruled surfaces and minimal surfaces. Intrinsic Geometry including the Gauss Theorem Egregium.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7550 [3 credit hours]

CLASSICAL DIFFERENTIAL GEOMETRY II

Tensors, vector fields and the Cartan approach to surface theory, Bonnet's Theorem and the construction of surfaces via solutions of the Gauss Equation. Geodesics, parallel transport and Jacobi Fields. Theorems of a global nature such as Hilbert's Theorem or the Theorem of Hopf-Rinow.

Prerequiste: MATH 5540 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7600 [3 credit hours]

ADVANCED STATISTICAL METHODS I

Basics of descriptive statistics, study designs and statistical inference. Properties of, and assumptions required for, inference for means, variances, and proportions from one and two-sample paired and unpaired studies. Introduction to ANOVA with multiple comparisons and logistic and multiple regression. Model assessment and diagnostics. Statistical software will be employed. Opportunities to apply procedures to real data. Emphasis placed on the foundations to approaches in introductory statistics.

Math 7610 [3 credit hours]

ADVANCED STATISTICAL METHODS II

Statistical/biostatistical concepts and methods. Broad subject categories that may be included are study design, longitudinal data analysis, survival analysis, logistic regression, random and mixed effects models and Bayesian Statistics. Other topics applicable to current statistical consulting projects, or related to modern data analytics, may be introduced. Appropriate statistical software will be employed.

Prerequiste: MATH 5600 FOR LEVEL GR WITH MIN. GRADE OF C-

Math 7620 [3 credit hours]

LINEAR STATISTICAL MODELS

Multiple regression, analysis of variance and covariance, general linear models and model building for linear models. Experimental designs include one-way, randomized block, Latin square, factorial and nested designs.

Prerequiste: MATH 6650 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7630 [3 credit hours]

THEORY AND METHODS OF SAMPLE SURVEYS

The mathematical basis to estimation in various sampling contexts, including probability proportional to size sampling, stratified sampling, two-stage cluster sampling and double sampling, is developed.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7640 [3 credit hours]

STATISTICAL COMPUTING

Modern statistical computing, including programming tools, modern programming methodologies, design of data structures and algorithms, numerical computing and graphics. Additional topics selected from simulation studies, inversion of probability integral transforms, rejection sampling, importance sampling, Monte Carlo integration, bootstrapping and optimization.

Math 7680 [3 credit hours]

INTRODUCTION TO THEORY OF PROBABILITY

Probability spaces, random variables, probability distributions, moments and moment generating functions, limit theorems, transformations and sampling distributions.

Prerequiste: MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7690 [3 credit hours]

INTRODUCTION TO MATHEMATICAL STATISTICS

Sampling distributions, point estimation, interval estimation, hypothesis testing, regression and analysis of variance.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7710 [3 credit hours]

METHODS OF NUMERICAL ANALYSIS I

Floating point arithmetic; polynomial interpolation; numerical solution of nonlinear equations; Newton's method. Likely topics include: numerical differentiation and integration; solving systems of linear equations; Gaussian elimination; LU decomposition; Gauss-Seidel method.

Math 7720 [3 credit hours]

METHODS OF NUMERICAL ANALYSIS II

Likely topics include: Computation of eigenvalues and eigenvectors; solving systems of nonlinear equations; least squares approximations; rational approximations; cubic splines; fast Fourier transforms; numerical solutions to initial value problems; ordinary and partial differential equations.

Prerequiste: MATH 5710 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7740 [3 credit hours]

ADVANCED APPLIED MATHEMATICS I

Series and numerical solutions to ordinary differential equations, special functions, orthogonal functions, Sturm-Liouville Problems, self-adjointness, vector analysis.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7750 [3 credit hours]

ADVANCED APPLIED MATHEMATICS II

Continuation of vector analysis, introduction to complex analysis, partial differential equations, Fourier series and integrals.

Prerequiste: MATH 5740 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7800 [3 credit hours]

ORDINARY DIFFERENTIAL EQUATIONS

Modern theory of differential equations; transforms and matrix methods; existence theorems and series solutions; and other selected topics.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7810 [3 credit hours]

PARTIAL DIFFERENTIAL EQUATIONS

First and second order equations; numerical methods; separation of variables; solutions of heat and wave equations using eigenfunction techniques; and other selected topics.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7820 [3 credit hours]

INTRODUCTION TO REAL ANALYSIS I

A rigorous treatment of the Calculus in one and several variables. Topics to include: the real number system; sequences and series; elementary metric space theory including compactness, connectedness and completeness; the Riemann Integral.

Prerequiste: MATH 3190 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7830 [3 credit hours]

INTRODUCTION TO REAL ANALYSIS II

Differentiable functions on Rn; the Implicit and Inverse Function Theorems; sequences and series of continuous functions; Stone-Weierstrass Theorem; Arsela-Ascoli Theorem; introduction to measure theory; Lebesgue integration; the Lebesgue Dominated Convergence Theorem.

Prerequiste: MATH 5820 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7860 [3 credit hours]

CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY I

Conditions for an extreme (Euler's equations, Erdman corner conditions, conditions of Legendre, Jacobi and Weierstrass, fields of extremals, Hilbert's invariant integral); Raleigh-Ritz method; isoperimetric problems; Lagrange, Mayer-Bolza problems.

Prerequiste: MATH 5820 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7870 [3 credit hours]

CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY II

Pontryagin's maximum principle; necessary and sufficient conditions for optimal control, controllability, time optimal control, existence of optimal controls, relationship to the calculus of variations.

Prerequiste: MATH 5860 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 7880 [3 credit hours]

COMPLEX VARIABLES

Analytic functions; Cauchy's theorem; Taylor and Laurent series; residues; contour integrals; conformal mappings, analytic continuation and applications.

Prerequiste: MATH 3860 FOR LEVEL UG WITH MIN. GRADE OF D-

Math 7980 [3 credit hours]

TOPICS IN MATHEMATICS

Special topics in mathematics.

Math 8180 [3 credit hours]

LINEAR AND NONLINEAR PROGRAMMING

Simplex algorithm, ellipsoidal algorithm, Karmarkar's method, interior point methods, elementary convex analysis, optimality conditions and duality for smooth problems, convex programming, algorithms and their convergence.

Prerequiste: MATH 5820 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 7820 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8190 [3 credit hours]

INFINITE DIMENSIONAL OPTIMIZATION

Introduction to nonlinear analysis, abstract optimization problems on abstract spaces, applications to calculus of variations, optimal control theory and game theory.

Prerequiste: MATH 6150 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 6810 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8150 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8810 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8300 [3 credit hours]

ALGEBRA I

Group actions, Sylow's theorems, permutation groups, nelpotent and solvable groups, abelian groups, rings, unique factorization domains, fields.

Prerequiste: MATH 5340 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 7340 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8310 [3 credit hours]

ALGEBRA II

Field extensions, Galois theory, modules, Noetherian and Artinian rings, tensor products, primitive rings, semisimple rings, and modules, the Wedderburn-Artin theorem.

Prerequiste: MATH 6300 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8300 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8320 [3 credit hours]

RING THEORY I

Radical theory, rings of quotients, Goldie's Theorem, chain conditions, dimensions of rings, module theory, topics in commutative rings.

Prerequiste: MATH 6310 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8310 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8330 [3 credit hours]

RING THEORY II

Advanced topics in ring theory. Possible topics include group rings, enveloping algebras, almost split sequences, PI-rings, division rings, self-injective rings, and ordered rings.

Prerequiste: MATH 6310 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8310 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8340 [3 credit hours]

GROUP THEORY I

Fundamental topics in group theory. Possible topics include free groups, presentations, free products and amalgams, permutation groups, abelian groups, nilpotent and solvable groups, subnormality, extensions, the Schur-Zassenhaus theorem, the transfer homomorphism, linear methods, local analysis.

Prerequiste: MATH 6310 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8310 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8350 [3 credit hours]

GROUP THEORY II

Advanced topics in group theory. Possible topics include cohomolgy of groups, locally finite groups, character theory, modular representation theory, representation theory of symmetric and classical groups, finite simple groups, geometric group theory.

Prerequiste: MATH 6310 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8310 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8400 [3 credit hours]

TOPOLOGY I

Topological spaces, continuous functions, compactness, product spaces, Tychonov's theorem, quotient spaces, local compactness, homotopy theory, the fundamental group, covering spaces.

Prerequiste: MATH 7450 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 4450 FOR LEVEL UG WITH MIN. GRADE OF D- OR MATH 5450 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8410 [3 credit hours]

TOPOLOGY II

Homology theory, excision, homological algebra, the Brouwer fixed point theorem, cohomology, differential manifolds, orientation, tangent bundles, Sard' theorem, degree theory.

Prerequiste: MATH 6400 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8400 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8440 [3 credit hours]

DIFFERENTIAL GEOMETRY I

Introduction to differential geometry. Topics include differentiable manifolds, vector fields, tensor bundles, the Frobenius theorem, Stokes' theorem, Lie groups.

Prerequiste: MATH 6410 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8410 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8450 [3 credit hours]

DIFFERENTIAL GEOMETRY II

Topics include connections on manifolds, Riemannian geometry, the Gauss-Bonnet theorem. Further topics may include: homogeneous and symmetric spaces, minimal surfaces. Morse theory, comparison theory, vector and principal bundles.

Prerequiste: MATH 6440 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8440 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8500 [3 credit hours]

ORDINARY DIFFERENTIAL EQUATIONS

Existence, uniqueness and dependence on initial conditions and parameter, nonlinear planar systems, linear systems, Floquet theory, second order equations, Sturm-Liouville theory.

Math 8510 [3 credit hours]

PARTIAL DIFFERENTIAL EQUATIONS

First order quasi-linear systems of partial differential equations, boundary value problems for the heat and wave equation, Dirichlet problem for Laplace equation, fundamental solutions for Laplace, heat and wave equations.

Math 8520 [3 credit hours]

DYNAMICAL SYSTEMS I

Topic include the flow-box theorem, Poincare maps, attractors, w-limit sets, Lyapunov stability, invariant submanifolds, Hamiltonian systems and symplectic manifolds.

Prerequiste: MATH 6500 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8500 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8540 [3 credit hours]

PARTIAL DIFFERENTIAL EQUATIONS I

Possible topics may include: the Cauchy-Kovalevskaya Theorem, nonlinear partial differential equations of the first order, theory of Sobolev spaces, linear second order PDE's of elliptic, hyperbolic and parabolic type.

Prerequiste: MATH 6510 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8510 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8550 [3 credit hours]

PARTIAL DIFFERENTIAL EQUATIONS II

Selected topics in Partial Differential Equations of current interest emphasizing nonlinear theory. Possible topics may include: Minimal surfaces, applications of the Hopf maximum principle, free boundary value problems, harmonic maps, geometric evolution equations and the Navier-Stokes equation.

Prerequiste: MATH 6540 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8540 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8600 [1-5 credit hours]

STATISTICAL CONSULTING

Real data applications of various statistical methods, project design and analysis including statistical consulting experience. May be repeated for credit.

Math 8610 [2 credit hours]

STATISTICAL CONSULTING I AND II

Real data applications of various statistical methods, project design and analysis including statistical consulting experience.

Math 8620 [3 credit hours]

CATEGORICAL DATA ANALYSIS

Important methods and modeling techniques using generalized linear models and emphasizing loglinear and logit modeling.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 7680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8630 [3 credit hours]

NONPARAMETRIC STATISTICS

Statistical methods based on counts and ranks; methods designed to be effective in the presence of contaminated data or error distribution misspecification.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF C- OR MATH 7680 FOR LEVEL GR WITH MIN. GRADE OF C-

Math 8640 [3 credit hours]

TOPICS IN STATISTICS

Topics selected from an array of modern statistical methods such as survival analysis, nonlinear regression, Monte Carlo methods, etc.

Math 8650 [3 credit hours]

STATISTICAL INFERENCE

Estimation, hypothesis testing, prediction, sufficient statistics, theory of estimation and hypothesis testing, simultaneous inference, decision theoretic models.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 7680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8670 [3 credit hours]

MEASURE THEORETIC PROBABILITY

Real analysis, probability spaces and measures, random variables and distribution functions, independence, expectation, law of large numbers, central limit theorem, zero-one laws, characteristic functions, conditional expectations given a s-algebra, martingales.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 7680 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8680 [3 credit hours]

THEORY OF STATISTICS

Exponential families, sufficiency, completeness, optimality, equivariance, efficiency. Bayesian and minimax estimation. Unbiased and invariant tests, uniformly most powerful tests. Asymptotic properties for estimation and testing. Most accurate confidence intervals.

Math 8690 [3 credit hours]

MULTIVARIATE STATISTICS

Multivariate normal sampling distributions, T tests and MANOVA, tests on covariance matrices, simultaneous inference, discriminant analysis, principal components, cluster analysis and factor analysis.

Prerequiste: MATH 5690 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 6650 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8650 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8720 [3 credit hours]

METHODS OF MATHEMATICAL PHYSICS I

Analytic functions, residues, method of steepest descent, complex differential equations, regular singularities, integral representation, real and complex vector spaces, matrix groups, Hilbert spaces, coordinate transformations.

Math 8730 [3 credit hours]

METHODS OF MATHEMATICAL PHYSICS II

Self-adjoint operators, special functions, orthogonal polynomials, partial differential equations and separation of variables, boundary value problems, Green¿s functions, integral equations, tensor analysis, metrics and curvature, calculus of variations, finite groups and group representations.

Prerequiste: MATH 6720 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8720 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8800 [3 credit hours]

REAL ANALYSIS I

Completeness, connectedness and compactness in metric spaces, continuity and convergence, Stone-Weierstrass Theorem, Lebesgue measure and integration on the real line, convergence theorems, Egorov's and Lusin's theorems, derivatives, functions of bounded variation.

Prerequiste: MATH 7830 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 4830 FOR LEVEL UG WITH MIN. GRADE OF D- OR MATH 5830 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8810 [3 credit hours]

REAL ANALYSIS II

The Vitali covering theorem, absolutely continuous functions, Lebesgue-Stieltjes integration, the Reisz representation theorem, Banach spaces, Lp-spaces, abstract measures, the Radon-Nikodym theorem, measures on locally compact Hausdorff spaces.

Prerequiste: MATH 6800 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8800 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8820 [3 credit hours]

FUNCTIONAL ANALYSIS I

Topics include Topological vector spaces, Banach spaces, convexity, the Hahn-Banach theorem, weak and strong topologies, Lp spaces and duality.

Prerequiste: MATH 6810 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8810 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8830 [3 credit hours]

FUNCTIONAL ANALYSIS II

Topics include the Mackey-Ahrens Theorem, Banach algebras, spectra in Banach algebras, commutative Banach algebras, unbounded operators, the spectral theorem, topics in functional analysis.

Prerequiste: MATH 6820 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8820 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8840 [3 credit hours]

COMPLEX ANALYSIS I

Elementary analytic functions, complex integration, the residue theorem, infinite sequences of analytic functions, Laurent expansions, entire functions.

Prerequiste: MATH 6800 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8800 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8850 [3 credit hours]

COMPLEX ANALYSIS II

Meromorphic functions, conformal mapping, harmonic functions and the Dirichlet problem, the Riemann mapping theorem, monodromy, algebraic functions, Riemann surfaces, elliptic functions and the modular function.

Prerequiste: MATH 6840 FOR LEVEL GR WITH MIN. GRADE OF D- OR MATH 8840 FOR LEVEL GR WITH MIN. GRADE OF D-

Math 8870 [3 credit hours]

MEASURE THEORETIC PROBABILITY II

Focus on stochastic processes.Conditional expectations, martingales, random walks, markov chains, ergodic theorem, brownian motion.

Prerequiste: MATH 5680 FOR LEVEL GR WITH MIN. GRADE OF D- AND MATH 6860 FOR LEVEL GR WITH MIN. GRADE OF D- AND MATH 8860 FOR LEVEL GR WITH MIN. GRADE OF D-

Corequisite: MATH 6800

Math 8930 [1 credit hours]

COLLOQUIUM

Lectures by visiting mathematicians and staff members on areas of current interest in mathematics.

Math 8940 [1-5 credit hours]

PROSEMINAR

Problems and techniques of teaching elementary college mathematics, supervised teaching, seminar in preparation methods.

Math 8960 [3-6 credit hours]

DISSERTATION

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Math 8980 [3 credit hours]

TOPICS IN MATHEMATICAL SCIENCES

Special topics in Mathematics or Statistics.

Math 8990 [1-5 credit hours]

READINGS IN MATHEMATICS

Readings in areas of Mathematics of mutual interest to the student and the professor.

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