Probability and independence; discrete and continuous random variables; joint, marginal, and conditional densities, moment generating functions; laws of large numbers; binomial, Poisson, gamma, univariate, and bivariate normal distributions.

Cross-listed course: STAT 511

Probability spaces. Random variables. Mean and variance. Geometric Brownian Motion and stock price dynamics. Interest rates and present value analysis. Pricing via arbitrage arguments. Options pricing and the Black-Scholes formula.

Cross-listed course: STAT 522

Convex sets. Separating Hyperplane Theorem. Fundamental Theorem of Asset Pricing. Risk and expected return. Minimum variance portfolios. Capital Asset Pricing Model. Martingales and options pricing. Optimization models and dynamic programming.

Cross-listed course: STAT 523

Differential equations of the first order, linear systems of ordinary differential equations, elementary qualitative properties of nonlinear systems.

Laplace transforms, two-point boundary value problems and Green’s functions, boundary value problems in partial differential equations, eigenfunction expansions and separation of variables, transform methods for solving PDE’s, Green’s functions for PDE’s, and the method of characteristics.

Basic principles and methods of Fourier transforms, wavelets, and multiresolution analysis; applications to differential equations, data compression, and signal and image processing; development of numerical algorithms. Computer implementation.

Applications of differential and difference equations and linear algebra modeling the dynamics of populations, with emphasis on stability and oscillation. Critical analysis of current publications with computer simulation of models.

Descent methods, conjugate direction methods, and Quasi-Newton algorithms for unconstrained optimization; globally convergent hybrid algorithm; primal, penalty, and barrier methods for constrained optimization. Computer implementation of algorithms.

Two-person zero-sum games, minimax theorem, utility theory, n-person games, market games, stability.

Matrix algebra, Gauss elimination, iterative methods; overdetermined systems and least squares; eigenvalues, eigenvectors; numerical software. Computer implementation. Credit may not be received for both MATH 526 and MATH 544. Three lectures and one laboratory hour per week.

Interpolation and approximation of functions; solution of algebraic equations; numerical differentiation and integration; numerical solutions of ordinary differential equations and boundary value problems; computer implementation of algorithms.

Cross-listed course: CSCE 561

Unconstrained and constrained optimization, gradient descent methods for numerical optimization, supervised and unsupervised learning, various reduced order methods, sampling and inference, Monte Carlo methods, deep neural networks.

The study of geometry as a logical system based upon postulates and undefined terms. The fundamental concepts and relations of Euclidean geometry developed rigorously on the basis of a set of postulates. Some topics from non-Euclidean geometry.

Projective geometry, theorem of Desargues, conics, transformation theory, affine geometry, Euclidean geometry, non-Euclidean geometries, and topology.

Topology of the line, plane, and space, Jordan curve theorem, Brouwer fixed point theorem, Euler characteristic of polyhedra, orientable and non-orientable surfaces, classification of surfaces, network topology.

Elementary properties of sets, functions, spaces, maps, separation axioms, compactness, completeness, convergence, connectedness, path connectedness, embedding and extension theorems, metric spaces, and compactification.

Finite structures useful in applied areas. Binary relations, Boolean algebras, applications to optimization, and realization of finite state machines.

Error-correcting codes, polynomial rings, cyclic codes, finite fields, BCH codes.

Vectors, vector spaces, and subspaces; geometry of finite dimensional Euclidean space; linear transformations; eigenvalues and eigenvectors; diagonalization. Throughout there will be an emphasis on theoretical concepts, logic, and methods. MATH 544L is an optional laboratory course where additional applications will be discussed.

Computer-based applications of linear algebra for mathematics students. Topics include numerical analysis of matrices, direct and indirect methods for solving linear systems, and least squares method (regression). Typical applications include theoretical and practical issues related to discrete Markov processes, image compression, and linear programming. Credit not allowed for both MATH 344L and 544L.

Permutation groups; abstract groups; introduction to algebraic structures through study of subgroups, quotient groups, homomorphisms, isomorphisms, direct product; decompositions; introduction to rings and fields.

Rings, ideals, polynomial rings, unique factorization domains; structure of finite groups; topics from: fields, field extensions, Euclidean constructions, modules over principal ideal domains (canonical forms).

Polynomials and affine space, Grobner bases, elimination theory, varieties, and computer algebra systems.

Vector fields, line and path integrals, orientation and parametrization of lines and surfaces, change of variables and Jacobians, oriented surface integrals, theorems of Green, Gauss, and Stokes; introduction to tensor analysis.

Parametrized curves, regular curves and surfaces, change of parameters, tangent planes, the differential of a map, the Gauss map, first and second fundamental forms, vector fields, geodesics, and the exponential map.

Complex integration, calculus of residues, conformal mapping, Taylor and Laurent Series expansions, applications.

Least upper bound axiom, the real numbers, compactness, sequences, continuity, uniform continuity, differentiation, Riemann integral and fundamental theorem of calculus.

Riemann-Stieltjes integral, infinite series, sequences and series of functions, uniform convergence, Weierstrass approximation theorem, selected topics from Fourier series or Lebesgue integration.

Syntax and semantics of formal languages; sentential logic, proofs in first order logic; Godel’s completeness theorem; compactness theorem and applications; cardinals and ordinals; the Lowenheim-Skolem-Tarski theorem; Beth’s definability theorem; effectively computable functions; Godel’s incompleteness theorem; undecidable theories.

Basic theoretical principles of computing as modeled by formal languages and automata; computability and computational complexity.

Cross-listed course: CSCE 551

Discrete mathematical models. Applications to such problems as resource allocation and transportation. Topics include linear programming, integer programming, network analysis, and dynamic programming.

Mathematical models; mathematical reasoning; enumeration; induction and recursion; tree structures; networks and graphs; analysis of algorithms.

A continuation of MATH 574. Inversion formulas; Polya counting; combinatorial designs; minimax theorems; probabilistic methods; Ramsey theory; other topics.

Winning in certain combinatorial games such as Nim, Hackenbush, and Domineering. Equalities and inequalities among games, Sprague-Grundy theory of impartial games, games which are numbers.

Divisibility, primes, congruences, quadratic residues, numerical functions. Diophantine equations.

Design of secret codes for secure communication, including encryption and integrity verification: ciphers, cryptographic hashing, and public key cryptosystems such as RSA. Mathematical principles underlying encryption. Code-breaking techniques. Cryptographic protocols.

Cross-listed course: CSCE 557

Recent developments in pure and applied mathematics selected to meet current faculty and student interest.

This course is designed for middle-level pre-service mathematics teachers. This course covers geometric reasoning, Euclidean geometry, congruence, area, volume, similarity, symmetry, vectors, and transformations. Dynamic software will be utilized to explore geometry concepts. This course cannot be used for credit toward a major in mathematics.

This course introduces basic concepts in number theory and modern algebra that provide the foundation for middle level arithmetic and algebra. Topics include: algebraic reasoning, patterns, inductive reasoning, deductive reasoning, arithmetic and algebra of integers, algebraic systems, algebraic modeling, and axiomatic mathematics. This course cannot be used for credit towards a major in mathematics.

A thorough study of the topics to be presented in AP calculus, including limits of functions, differentiation, integration, infinite series, and applications. Not intended for degree programs in mathematics.

Vector spaces, linear transformations, dual spaces, decompositions of spaces, and canonical forms. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Algebraic structures, sub-structures, products, homomorphisms, and quotient structures of groups, rings, and modules. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

An introduction to algebraic structures; group theory including subgroups, quotient groups, homomorphisms, isomorphisms, decomposition; introduction to rings and fields. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Fields and field extensions. Galois theory, topics from, transcendental field extensions, algebraically closed fields, finite fields. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Theory of rings including ideals, polynomial rings, and unique factorization domains; structure of finite groups; fields; modules. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Compactness, completeness, continuous functions. Outer measures, measurable sets, extension theorem and Lebesgue-Stieltjes measure. Integration and convergence theorems. Product measures and Fubini’s theorem. Differentiation theory. Theorems of Egorov and Lusin. Lp-spaces. Analytic functions: Cauchy-Riemann equations, elementary special functions. Conformal mappings. Cauchy’s integral theorem and formula. Classification of singularities, Laurent series, the Argument Principle. Residue theorem, evaluation of integrals and series. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The real numbers and least upper bound axiom; sequences and limits of sequences; infinite series; continuity; differentiation; the Riemann integral. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Compactness, completeness, continuous functions. Outer measures, measurable sets, extension theorem and Lebesgue-Stieltjes measure. Integration and convergence theorems. Product measures and Fubini’s theorem. Differentiation theory. Theorems of Egorov and Lusin. Lp-spaces. Analytic functions: Cauchy-Riemann equations, elementary special functions. Conformal mappings. Cauchy’s integral theorem and formula. Classification of singularities, Laurent series, the Argument Principle. Residue theorem, evaluation of integrals and series. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Sequences and series of functions; power series, uniform convergence; interchange of limits; limits and continuity in several variables. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Signed and complex measures, Radon-Nikodym theorem, decomposition theorems. Metric spaces and topology, Baire category, Stone-Weierstrass theorem, Arzela-Ascoli theorem. Introduction to Banach and Hilbert spaces, Riesz representation theorems. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Approximation of functions by algebraic polynomials, splines, and trigonometric polynomials; numerical differentiation; numerical integration; orthogonal polynomials and Gaussian quadrature; numerical solution of nonlinear systems, unconstrained optimization. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Vectors and matrices; QR factorization; conditioning and stability; solving systems of equations; eigenvalue/eigenvector problems; Krylov subspace iterative methods; singular value decomposition. Includes theoretical development of concepts and practical algorithm implementation. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Probability spaces, random variables and distributions, expectations, characteristic functions, laws of large numbers, and the central limit theorem. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Cross-listed course: STAT 810

More about distributions, limit theorems, Poisson approximations, conditioning, martingales, and random walks. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Cross-listed course: STAT 811

This course will include a study of permutations and combinations; probability and its application to statistical inferences; elementary descriptive statistics of a sample of measurements; the binomial, Poisson, and normal distributions; correlation and regression. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Modeling and solution techniques for differential and integral equations from sciences and engineering, including a study of boundary and initial value problems, integral equations, and eigenvalue problems using transform techniques, Green’s functions, and variational principles. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Foundations of approximation of functions by Fourier series in Hilbert space; fundamental PDEs in mathematical physics; fundamental equations for continua; integral and differential operators in Hilbert spaces. Basic modeling theory and solution techniques for stochastic differential equations. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Topics in optimization; includes linear programming, integer programming, gradient methods, least squares techniques, and discussion of existing mathematical software. Graduate standing or consent of the department. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Elliptic equations: fundamental solutions, maximum principles, Green’s function, energy method and Dirichlet principle; Sobolev spaces: weak derivatives, extension and trace theorems; weak solutions and Fredholm alternative, regularity, eigenvalues and eigenfunctions. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Detailed study of the following topics: method of characteristics; Hamilton-Jacobi equations; conservation laws; heat equation; wave equation; linear parabolic equations; linear hyperbolic equations. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Approximation of functions; existence, uniqueness and characterization of best approximants; Chebyshev’s theorem; Chebyshev polynomials; degree of approximation; Jackson and Bernstein theorems; B-splines; approximation by splines; quasi-interpolants; spline interpolation. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Elliptic equations: fundamental solutions, maximum principles, Green’s function, energy method and Dirichlet principle; Sobolev spaces: weak derivatives, extension and trace theorems; weak solutions and Fredholm alternative, regularity, eigenvalues and eigenfunctions. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Ritz and Galerkin weak formulation. Finite element, mixed finite element, collocation methods for elliptic, parabolic, and hyperbolic PDEs, including development, implementation, stability, consistency, convergence analysis, and error estimates.

All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Nonlinear approximation from piecewise polynomial (spline) functions in the univariate and multivariate case, characterization of the approximation spaces via Besov spaces and interpolation, Newman’s and Popov’s theorems for rational approximation, characterization of the approximation spaces of rational approximation, nonlinear n-term approximation from bases in Hilbert spaces and from unconditional bases in Lp (p>1), greedy algorithms, application of nonlinear approximation to image compression.

Topological spaces, filters, compact spaces, connected spaces, uniform spaces, complete spaces, topological groups, function spaces. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Topological spaces, filters, compact spaces, connected spaces, uniform spaces, complete spaces, topological groups, function spaces. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The fundamental group, homological algebra, simplicial complexes, homology and cohomology groups, cup-product, triangulable spaces. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The fundamental group, homological algebra, simplicial complexes, homology and cohomology groups, cup-product, triangulable spaces. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Differentiable manifolds; classical theory of surfaces and hypersurfaces in Euclidean space; tensors, forms and integration of forms; connections and covariant differentiation; Riemannian manifolds; geodesics and the exponential map; curvature; Jacobi fields and comparison theorems, generalized Gauss-Bonnet theorem. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Manifolds; topological groups, coverings and covering groups; Lie groups and their Lie algebras; closed subgroups of Lie groups; automorphism groups and representations; elementary theory of Lie algebras; simply connected Lie groups; semisimple Lie groups and their Lie algebras. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Synthetic and analytic projective geometry, homothetic transformations, Euclidean geometry, non-Euclidean geometries, and topology. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Algebraic geometry over the complex numbers, using ideas from topology, complex variable theory, and differential geometry.

Course content varies and will be announced in the schedule of classes by title. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Algebraic geometry over the complex numbers, using ideas from topology, complex variable theory, and differential geometry.

Theory of groups, rings, modules, fields and division rings, bilinear forms, advanced topics in matrix theory, and homological techniques.

Representation and character theory of finite groups (especially the symmetric group) and/or the general linear group, Young tableaux, the Littlewood Richardson rule, and Schur functors.

Sublattices, homomorphisms and direct products of lattices; freely generated lattices; modular lattices and projective geometries; the Priestley and Stone dualities for distributive and Boolean lattices; congruence relations on lattices. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Extremal properties of positive definite and hermitian matrices, doubly stochastic matrices, totally non-negative matrices, eigenvalue monotonicity, Hadamard-Fisher determinantal inequalities. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Prime spectrum and Zariski topology; finite, integral, and flat extensions; dimension; depth; homological techniques, normal and regular rings. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Properties of affine and projective varieties defined over algebraically closed fields, rational mappings, birational geometry and divisors especially on curves and surfaces, Bezout’s theorem, Riemann-Roch theorem for curves. All Non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Course content varies and will be announced in the schedule of classes by title. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The Fourier transform on the circle and line, convergence of Fejer means; Parseval’s relation and the square summable theory, convergence and divergence at a point; conjugate Fourier series, the conjugate function and the Hilbert transform, the Hardy-Littlewood maximal operator and Hardy spaces. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The L1 and L2 theory of the Fourier transform on the line, bandlimited functions and the Paley-Weiner theorem, Shannon-Whittacker Sampling Theorem, Riesz systems, Mallat-Meyer multiresolution analysis in Lebesgue spaces, scaling functions, wavelet constructions, wavelet representation and unconditional bases, nonlinear approximation, Riesz’s factorization lemma, and Daubechies’ compactly supported wavelets.

Normal families, meromorphic functions, Weierstrass product theorem, conformal maps and the Riemann mapping theorem, analytic continuation and Riemann surfaces, harmonic and subharmonic functions. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Properties of analytic functions, complex integration, calculus of residues, Taylor and Laurent series expansions, conformal mappings. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Properties of holomorphic functions of several variables, holomorphic mappings, plurisubharmonic functions, domains of convergence of power series, domains of holomorphy and pseudoconvex domains, harmonic analysis in several variables. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Banach spaces, Hilbert spaces, spectral theory of bounded linear operators, Fredholm alternatives, integral equations, fixed point theorems with applications, least square approximation. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Linear topological spaces; Hahn-Banach theorem; closed graph theorem; uniform boundedness principle; operator theory; spectral theory; topics from linear differential operators or Banach algebras. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Linear topological spaces; Hahn-Banach theorem; closed graph theorem; uniform boundedness principle; operator theory; spectral theory; topics from linear differential operators or Banach algebras. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Course content varies and will be announced in the schedule of classes by title. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

An axiomatic development of set theory: sets and classes; recursive definitions and inductive proofs; the axiom of choice and its consequences; ordinals; infinite cardinal arithmetic; combinatorial set theory. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Models of computation; recursive functions, random access machines, Turing machines, and Markov algorithms; Church’s Thesis; universal machines and recursively unsolvable problems; recursively enumerable sets; the recursion theorem; the undecidability of elementary arithmetic. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

First order predicate calculus; elementary theories; models, satisfaction, and truth; the completeness, compactness, and omitting types theorems; countable models of complete theories; elementary extensions; interpolation and definability; preservation theorems; ultraproducts. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Course content varies and will be announced in the schedule of classes by title. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The application and analysis of algorithms for linear programming problems, including the simplex algorithm, algorithms and complexity, network flows, and shortest path algorithms. No computer programming experience required. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

An introduction to the theory and applications of discrete mathematics. Topics include enumeration techniques, combinatorial identities, matching theory, basic graph theory, and combinatorial designs. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

A continuation of MATH 774. Additional topics will be selected from: the structure and extremal properties of partially ordered sets, matroids, combinatorical algorithms, matrices of zeros and ones, and coding theory. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The study of the structure and extremal properties of graphs, including Eulerian and Hamiltonian paths, connectivity, trees, Ramsey theory, graph coloring, and graph algorithms. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Continuation of MATH 776. Additional topics will be selected from: reconstruction problems, independence, genus, hypergraphs, perfect graphs, interval representations, and graph-theoretical models. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Course content varies and will be announced in the schedule of classes by title. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Diophantine equations, distribution of primes, factoring algorithms, higher power reciprocity, Schnirelmann density, and sieve methods. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Elementary properties of integers, Diophantine equations, prime numbers, arithmetic functions, congruences, and the quadratic reciprocity law. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The prime number theorem, Dirichlet’s theorem, the Riemann zeta function, Dirichlet’s L-functions, exponential sums, Dirichlet series, Hardy-Littlewood method partitions, and Waring’s problem. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

The prime number theorem, Dirichlet’s theorem, the Riemann zeta function, Dirichlet’s L-functions, exponential sums, Dirichlet series, Hardy-Littlewood method partitions, and Waring’s problem. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Algebraic integers, unique factorization of ideals, the ideal class group, Dirichlet’s unit theorem, application to Diophantine equations. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Thue-Siegel-Roth theorem, Hilbert’s seventh problem, diophantine approximation. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Course content varies and will be announced in the schedule of classes by title. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

Although this course is required of all candidates for the master’s degree it is not included in the total credit hours in the master’s program. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

First of two required math pedagogy courses for graduate assistants in the department. Pedagogical topics include assessment theory, discourse, theory, lesson planning, and classroom management. Applications assist graduate students with syllabusnesson/assessment creation, teacher questioning, midcourse evaluations, and student learning and engagement. This course will replace the University's requirement for GRAD 701. Restricted to Mathematics graduate students teaching at some capacity.

Second of two required math pedagogy courses for graduate assistants in the department. Pedagogical topics include student-learning and reflection theories, sociomathematical norms, and constructivism. Applications assist graduates with lesson/revision/reflection, student-centered investigations, curriculum problem solving and metacognition. This course will replace the University's requirement for GRAD 701. Restricted to Mathematics graduate students teaching at some capacity.

The exposition of advanced mathematics emphasizing the organization of proofs and the formulation of concepts; computer typesetting systems for producing mathematical theses, books, and articles.

Full admission to graduate study in mathematics. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

For master’s candidates. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

A review of current literature in specified subject areas involving student presentations. Content varies and will be announced in the schedule of classes by title. Minimum of 3 credit hours required of all doctoral students. All non-degree students should request permission to register from the Graduate Director in the Mathematics Department.

For doctoral candidates.