Course Descriptions (MATH)

• 511--Probability. [ = STAT 511] (3) (Prereq: MATH 241 with a grade of C or higher) 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.
• 520--Ordinary Differential Equations. (3) (Prereq: MATH 544 or 526; or consent of department) Differential equations of the first order, linear systems of ordinary differential equations, elementary qualitative properties of nonlinear systems.
• 521--Boundary Value Problems and Partial Differential Equations. (3) (Prereq: MATH 520, or 241 and 242) 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.
• 522--Wavelets. (3) (Prereq: MATH 544 or 526 or consent of instructor) Basic principles and methods of Fourier transforms, wavelets, and multiresolution analysis; applications to differential equations, data compression, signal and image processing; development of numerical algorithms. Computer implementation.
• 523--Mathematical Modeling of Population Biology. (3) (Prereq: MATH 142, BIOL 301, or MSCI 311 recommended) 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.
• 524--Nonlinear Optimization. (3) (Prereq: MATH 526 or 544 or consent of department) 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.
• 525--Mathematical Game Theory. (3) (Prereq: MATH 526 or 544) Two person zero-sum games, Minimax theorem, utility theory, n-person games, market games, stability.
• 526--Numerical Linear Algebra. (4) (Prereq: MATH 241) Matrix algebra, Gauss elimination, iterative methods; overdetermined systems and least squares; eigenvalues, eigenvectors; numerical software. Computer implementation. Three lectures and one laboratory hour per week. Credit may not be received for both MATH 526 and MATH 544.
• 527--Numerical Analysis. [ = CSCI 561] (3) (Prereq: MATH 242 or 520; or consent of department) Interpolation and approximation of functions, solution of algebraic equations, numerical differentiation and integration; solution of ordinary differential equations and boundary value problems. Computer implementation of algorithms.
• 531--Foundations of Geometry. (3) (Prereq: MATH 241) 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.
• 532--Modern Geometry. (3) (Prereq: MATH 241) Projective geometry, theorem of Desargues, conics, transformation theory, affine geometry, Euclidean geometry, non-Euclidean geometries, and topology.
• 533--Elementary Geometric Topology. (3) (Prereq: MATH 241) 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.
• 534--Elements of General Topology. (3) (Prereq: MATH 241) Elementary properties of sets, functions, spaces, maps, separation axioms, compactness, completeness, convergence, connectedness, path connectedness, embedding and extension theorems, metric spaces, and compactification.
• 540--Modern Applied Algebra. (3) (Prereq: MATH 241) Finite structures useful in applied areas. Binary relations, Boolean algebras, applications to optimization, and realization of finite state machines.
• 541--Algebraic Coding Theory. (3) (Prereq: MATH 526 or MATH 544 or consent of instructor) Error correcting codes, polynomial rings, cyclic codes; finite fields, BCH codes.
• 544--Linear Algebra. (3) (Prereq: MATH 241) Matrix algebra, solution of linear systems; notions of vector space, independence, basis, dimension; linear transformations, change of basis; eigenvalues, eigenvectors, Hermitian matrices, diagonalization; Cayley-Hamilton theorem. Credit may not be received for both MATH 526 and MATH 544.
• 546--Algebraic Structures I. (3) (Prereq: MATH 241) Permutation groups; abstract groups; introduction to algebraic structures through study of subgroups, quotient groups, homomorphisms, isomorphisms, direct product; decompositions; introduction to rings and fields.
• 547--Algebraic Structures II. (3) (Prereq: MATH 546) 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).
• 550--Vector Analysis. (3) (Prereq: MATH 241) 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.
• 551--Introduction to Differential Geometry. (3) (Prereq: MATH 241) 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.
• 552--Applied Complex Variables. (3) (Prereq: MATH 241) Complex integration, calculus of residues, conformal mapping. Taylor and Laurent Series expansions, applications.
• 554--Analysis I. (3) (Prereq: MATH 241) Least upper bound axiom, the real numbers, compactness, sequences, continuity, uniform continuity, differentiation, Riemann integral and fundamental theorem of calculus.
• 555--Analysis II. (3) (Prereq: MATH 554 or consent of department) Riemann-Stieltjes integral, infinite series, sequences and series of functions, uniform convergence, Weierstrass approximation theorem, selected topics from Fourier series or Lebesgue integration.
• 561--Introduction to Mathematical Logic. (3) (Prereq: MATH 241) Syntax and semantics of formal languages; sentential logic; proofs in first order logic; Gödel's completeness theorem; compactness theorem and applications; cardinals and ordinals; the Lowenheim-Skolem-Tarski theorem; Beth's definability theorem; effectively computable functions; Gödel's incompleteness theorem; undecidable theories.
• 562--Theory of Computation. [ = CSCI 551] (3) (Prereq: CSCI 213 and CSCI 330, or MATH 526, or MATH 544, or MATH 574) Basic theoretical principles of computer science as modeled by formal languages and automata; computability and computational complexity.
• 570--Discrete Optimization. (3) (Prereq: MATH 526 or 544) Discrete mathematical models. Applications to such problems as resource allocation and transportation. Topics include linear programming, integer programming, network analysis, and dynamic programming.
• 574--Discrete Mathematics I. (3) (Prereq: MATH 142) Mathematical models; mathematical reasoning; enumeration; induction and recursion; tree structures; networks and graphs; analysis of algorithms.
• 575--Discrete Mathematics II. (3) (Prereq: MATH 574) A continuation of MATH 574. Inversion formulas; Polya counting; combinatorial designs; minimax theorems; probabilistic methods; Ramsey theory; other topics.
• 576--Combinatorial Game Theory. (3) (Prereq: MATH 526, 544 or 574) 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.
• 580--Elementary Number Theory. (3) (Prereq: MATH 241) Divisibility, primes, congruences, quadratic residues, numerical functions, Diophantine equations.
• 587--Introduction to Cryptography. {=CSCE 557} (3) (Prereq: CSCE 145, MATH 241, and either CSCE 355 or MATH 574) 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.
  
• 590--Undergraduate Seminar. (1-3) (Prereq: consent of instructor) A review of literature in specific subject areas involving student presentations. Content varies and will be announced in the Master Schedule of Classes by suffix and title. Pass-fail grading. For undergraduate credit only.
• 599--Topics in Mathematics. (1-3) Recent developments in pure and applied mathematics selected to meet current faculty and student interest.
• 650--AP Calculus for Teachers. (3) (Prereq: current secondary high school teacher certification in mathematics and at least 6 hours of calculus) 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.)
• 700--Linear Algebra. (3) Vector spaces, linear transformations, dual spaces, decompositions of spaces, and canonical forms.
• 701--Algebra I. (3) (Prereq: MATH 700) Algebraic structures, sub-structures, products, homomorphisms, and quotient structures of groups, rings, and modules.
• 703, 704--Analysis I, II. (3 each) 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.
• 705--Analysis III. (3) (Prereq: MATH 703, 704) 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.
• 706--Numerical Linear algebra (3)(Prereq: Math 700 or consent of the department) Matrix factorizations; iterative methods including pre-conditioning, iterative methods for eigenvalue problems, singular value decomposition, least squares. Includes theoretical development of concepts and practical algorithm implementation.
• 710--Probability Theory I. [ = STAT 710] (3) (Prereq: STAT 511, 512, or MATH 703) Probability spaces, random variables and distributions, characteristic functions, laws of large numbers, and the central limit theorem.
• 711--Probability Theory II. [ = STAT 711] (3) (Prereq: MATH 710) More about distributions, limit theorems, conditioning, random walks, Brownian motion, and other stochastic processes.
• 716--Selected Topics in Probability. [ = STAT 716] (3) Fields of study to be individually assigned. Primarily for doctoral candidates.
• 720--Applied Mathematics I. (3) (Prereq: MATH 555 or equivalent) Methods for solving equations from applied mathematics and the natural sciences, including a study of boundary value problems, integral equations, and eigenvalue problems using transform techniques, Green's functions, and variational principles.
• 721--Applied Mathematics II. (3) (Prereq: MATH 720) Topics in partial differential equations with emphasis on the equations of the natural sciences; includes classifications of higher order equations, existence and uniqueness of solutions, theory of characteristics, basic properties of elliptic and parabolic equations, Dirichlet and Neumann problems, and the Cauchy problem for hyperbolic equations.
• 722--Numerical Optimization. (3) (Prereq: graduate standing or consent of the department) Topics in optimization; includes linear programming, integer programming, gradient methods, least squares techniques, and discussion of existing mathematical software.
• 723--Advanced Differential Equations. (3) (Prereq: MATH 721 or consent of instructor) Advanced topics in ordinary and partial differential equations.
• 724--Numerical Differential Equations. (3) Techniques for numerically solving differential equations; includes finite difference methods, Galerkin methods, finite element method, and collocation.
• 725--Approximation Theory. (3) (Prereq or coreq: MATH 703) 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.
• 726--Numerical Analysis I. (3) (Prereq: MATH 554 (or equivalent) and Math 706) Error analysis; approximation of functions by algebraic polynomials, splines, and trigonometric polynomials; divided differences; numerical differentiation; quadrature including Gaussian and Romberg integration; a thorough study of numerical ODEs.
• 727--Numerical Analysis II. [ = CSCI 761] (3) (Prereq: MATH 726) Continuation of MATH 726.
• 728--Selected Topics in Applied Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
• 730, 731--General Topology I, II. (3 each) Topological spaces, filters, compact spaces, connected spaces, uniform spaces, complete spaces, topological groups, function spaces.
• 732, 733--Algebraic Topology I, II. (3 each) (Prereq: MATH 730 or 705, and 701) The fundamental group, homological algebra, simplicial complexes, homology and cohomology groups, cup-product, triangulable spaces.
• 734--Differential Geometry. (3) (Prereq: MATH 550) 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.
• 735--Lie Groups. (3) (Prereq: MATH 705 or 730) 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.
• 738--Selected Topics in Geometry and Topology. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
• 740--Algebra II. (3) (Prereq: MATH 701) Fields and field extensions Galois theory, topics from: transcendental field extensions, algebraically closed fields, finite fields.
• 741, 742 Algebra III, IV. (3 each) (Prereq: MATH 740) Theory of rings, modules, fields, bilinear forms, and advanced topics in matrix theory.
• 743--Lattice Theory. (3) (Prereq: MATH 740) 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.
• 744--Matrix Theory. (3) (Prereq: MATH 700) Extremal properties of positive definite and hermitian matrices, doubly stochastic matrices, totally non-negative matrices, eigenvalue monotonicity, Hadamard-Fisher determinantal inequalities.
• 746--Commutative Algebra. (3) (Prereq: MATH 701) Prime spectrum and Zariski topology; finite, integral, and flat extensions; dimension; depth; homological techniques, normal and regular rings.
• 747--Algebraic Geometry. (3) (Prereq: MATH 701) 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.
• 748--Selected Topics in Algebra. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
• 750--Fourier Analysis. (3) (Prereq: MATH 703 and 704) 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.
• 751--The Mathematical Theory of Wavelets. (3) (Prereq: MATH 750) The L¹ and L² theory of the Fourier transform on the line, bandlimited functions and the Paley-Wiener 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.
• 752--Complex Analysis. (3) (Prereq: MATH 703, 704) Normal families, meromorphic functions, Weierstrass product theorem, conformal maps and the Riemann mapping theorem, analytic continuation and Riemann surfaces, harmonic and subharmonic functions.
• 754--Several Complex Variables. (3) (Prereq: MATH 703 and 704) 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.
• 755--Applied Functional Analysis. (3) (Prereq: MATH 703) Banach spaces, Hilbert spaces, spectral theory of bounded linear operators, Fredholm alternatives, integral equations, fixed point theorems with applications, least square approximation.
• 756, 757--Functional Analysis I, II. (3 each) (Prereq: MATH 704) Linear topological spaces; Hahn-Banach theorem; closed graph theorem; uniform boundedness principle; operator theory; spectral theory; topics from linear differential operators or Banach algebras.
• 758--Selected Topics in Analysis. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
• 760--Set Theory. (3) 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.
• 761--The Theory of Computable Functions. (3) 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.
• 762--Model Theory. (3) 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.
• 768--Selected Topics in Foundations of Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
• 770--Discrete Optimization. (3) 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.
• 774--Discrete Mathematics I. (3) An introduction to the theory and applications of discrete mathematics. Topics include enumeration techniques, combinatorial identities, matching theory, basic graph theory, and combinatorial designs.
• 775--Discrete Mathematics II. (3) (Prereq: MATH 774 or consent of the instructor) 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.
• 776--Graph Theory I. (3) The study of the structure and extremal properties of graphs, including Eulerian and Hamiltonian paths, connectivity, trees, Ramsey theory, graph coloring, and graph algorithms.
• 777--Graph Theory II. (3) (Prereq: MATH 776 or consent of instructor) Continuation of MATH 776. Additional topics will be selected from: reconstruction problems, independence, genus, hypergraphs, perfect graphs, interval representations, and graph-theoretical models.
• 778--Selected Topics in Discrete Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
• 780--Elementary Number Theory. (3) Diophantine equations, distribution of primes, factoring algorithms, higher power reciprocity, Schnirelmann density, and sieve methods.
• 782, 783--Analytic Number Theory I, II. (3) (Prereq: MATH 580 and 552) 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.
• 784--Algebraic Number Theory. (3) (Prereq: MATH 546 and 580) Algebraic integers, unique factorization of ideals, the ideal class group, Dirichlet's unit theorem, application to Diophantine equations.
• 785--Transcendental Number Theory. (3) (Prereq: MATH 580) Thue-Siegel-Roth theorem, Hilbert's seventh problem, diophantine approximation.
• 788--Selected Topics in Number Theory. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
• 790--Graduate Seminar. (1) (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.)
• 797--Mathematics into Print. (3) 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.
• 798--Directed Readings and Research. (1-6) (Prereq: full admission to graduate study in mathematics).
• 799--Thesis Preparation. (1-9) For master's candidates.
• 890--Graduate Seminar. (1-3) (Prereq: consent of instructor) A review of current literature in specified subject areas involving student presentations. Content varies and will be announced in the schedule of classes by suffix and title. Minimum of 3 credit hours required of all doctoral students. (Pass/Fail Grading)
• 899--Dissertation Preparation. (1-12) For doctoral candidates.

Note: The courses listed below are for candidates for the degree of Master of Mathematics or Master of Arts in Teaching, or for special graduate students interested in secondary school teaching.

• 701-I--Foundations of Algebra I. (3) (Prereq: MATH 241 or equivalent) An introduction to algebraic structures; group theory including subgroups, quotient groups, homomorphisms, isomorphisms, decomposition; introduction to rings and fields.
• 702-I--Foundations of Algebra II. (3) (Prereq: MATH 701-I or equivalent) Theory of rings including ideals, polynomial rings, and unique factorization domains; structure of finite groups; fields; modules.
• 703-I--Foundations of Analysis I. (3) (Prereq: MATH 241 or equivalent) The real numbers and least upper bound axiom; sequences and limits of sequences; infinite series; continuity; differentiation; the Riemann integral.
• 704-I--Foundations of Analysis II. (3) (Prereq: MATH 703-I or equivalent) Sequences and series of functions; power series, uniform convergence; interchange of limits; limits and continuity in several variables.
• 712-I--Probability and Statistics. (3) 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.
• 736-I--Modern Geometry. (3) (Prereq: MATH 241 or equivalent) Synthetic and analytic projective geometry, homothetic transformations, Euclidean geometry, non-Euclidean geometries, and topology.
• 752-I--Complex Variables. (3) (Prereq: MATH 241 or equivalent) Properties of analytic functions, complex integration, calculus of residues, Taylor and Laurent series expansions, conformal mappings.
• 780-I--Theory of Numbers. (3) (Prereq: MATH 241 or equivalent) Elementary properties of integers, Diophantine equations, prime numbers, arithmetic functions, congruences, and the quadratic reciprocity law.