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Graduate Courses
MAT 511 Advanced Calculus 3 Y
Partial derivatives, implicit functions, integration in several variables, line and surface integrals. Prereq: MAT 331, and 12 credits of calculus.
MAT 512 Introduction to Real Analysis 3 Y
Real number system, set theory and elementary topological properties of the real line, continuity and differentiability, sequences and series, uniform convergence, Riemann integration, and improper integrals. Prereq: MAT 412 or permission of instructor.
MAT 513 Introduction to Complex Analysis 3 Y
Complex number system and its arithmetic, geometric representation. Linear transformations. Analytic functions and the Cauchy-Riemann equations. Integration and Cauchy's theorem, Taylor and Laurent series, singularities, poles, and residues. Applications. Prereq: MAT 412 or 511 or permission of instructor.
MAT 514 Introduction to Ordinary Differential Equations 3 S
First order differential equations. Second order linear differential equations. Power series solutions. Bessel's equations, Laplace transforms. Systems of first order differential equations. Applications. Credit will not be given for both MAT 514 and MAT 485. Prereq: 12 credits of calculus or equivalent.
MAT 517 Partial Differential Equations and Fourier Series 3 Y
Partial differential equations, boundary-value problems, Fourier series and orthogonal expansions, Bessel functions, and Legendre polynomials. Prereq: Either MAT 514 or MAT 485.
MAT 518 Topics in Applied Mathematics 3 Y
Topics may vary but may include partial differential equations, introduction to calculus of variations and Hamilton's principle, difference equations, complex variables, or integral equations. Prereq: MAT 514.
MAT 521 Introduction to Probability and Statistics 3 S
Algebra of sets. Probability in finite sample spaces. Binomial and multinomial coefficients. Random variables. Expected value and standard deviation. Density functions. Statistical applications. Prereq: 12 credits of calculus.
MAT 525 Mathematical Statistics 3 Y
Estimation and confidence intervals. Normal distribution and central limit theorem. Testing hypotheses, chi-square, t, and F distributions. Least squares, regression, and correlation. Prereq: MAT 521.
MAT 526 Probability 3 Y
Topics chosen from: Markov chains, game theory, queueing theory, information theory. Prereq: MAT 521.
MAT 531 Second Course in Linear Algebra 3 S
Abstract vector spaces and inner product spaces, linear transformations and linear operators, eigenvalues and diagonalization. Primarily for mathematics majors. Prereq: MAT 275 or CIS 275, and MAT 331.
MAT 532 Applied Linear Algebra 3 SI
Factorization of matrices, eigenvalues and eigenvectors, orthogonality. Applications of matrices to such topics as least-squares approximation, fast Fourier transform, difference and differential equations, linear programming, networks, game theory. Prereq: MAT 331 or MAT 485.
MAT 534 Introduction to Abstract Algebra 3 Y
Theory of groups, rings, and fields, including the integers and polynomial rings. Prereq: MAT 531.
MAT 541 Introduction to Number Theory 3 IR
Prime numbers, greatest common divisors, congruences. Euler's function, Fermat's theorem, primitive roots, indices, quadratic residues, Legendre and Jacobi symbols, and the quadratic reciprocity law. Prereq: MAT or CIS 275, and MAT 331 or permission of instructor.
MAT/CIS 545 Introduction to Combinatorics 3 Y
Permutations, combinations, recurrence relations, generating functions, inclusion-exclusion and applications, introductory graph theory. Prereq: MAT 275 or CIS 275.
MAT 551 Fundamental Concepts of Geometry 3 IR
Synthetic projective geometries. Coordinate systems for projective spaces. Algebraic representation of projective transformations; euclidean, non-euclidean, and affine geometries as real cases of projective geometry. Prereq: MAT 275 or CIS 275 and MAT 331.
MAT 554 Differential Geometry 3 IR
Theory of curves in three-dimensional space, including Frenet's formula, Gaussian and mean curvature, geodesics, developable surfaces, special conformal mappings. Prereq: MAT 412 or 511 or permission of instructor.
MAT 562 Elementary Topology 3 IR
Metrics and metric spaces, topologies and topological spaces, separation properties, compactness, connectedness, and continuity. Prereq: MAT 275 or CIS 275 and MAT 512.
MAT 581 Numerical Methods with Programming 3 Y
Approximation methods for solution of nonlinear equations. Interpolation problems. Numerical integration. Solution of ordinary differential equations. Error analysis and writing computer programs. Primarily for mathematics and engineering students. Prereq: 12 credits of calculus and knowledge of a scientific programming language.
MAT 593 History of Mathematics 3 IR
Mathematical concepts in their historical perspective. Character and contributions of the great mathematicians and relation of mathematics to other sciences. Prereq: 12 credits of calculus and at least two 500-level mathematics courses.
MAT 599 Senior Seminar in Mathematics 3
Topic chosen by the instructor. Permission of department.
MAT 601 Fundamentals of Analysis 3 each Y
Real and complex numbers, elementary point set topology, continuity, differentiation. Sequences and series, Riemann-Stieltjes integration with respect to complex integrators.
MAT 602 Fundamentals of Analysis 3 each Y
Real and complex numbers, elementary point set topology, continuity, differentiation. Sequences and series, Riemann-Stieltjes integration with respect to complex integrators. Prereq: MAT 601, 631 or permission of instructor.
MAT 631 Introduction to Algebra 3
Linear algebra, linear transformations, eigenvectors, diagonalization, inner product spaces, groups, quotient groups, group action, Sylow theorems finitely generated Abelian groups, rings, unique factorization domains, finitely generated modules over principal ideal domains, fields, Galois theory. Prereq: MAT 531, 534.
MAT 632 Introduction to Algebra II 3
Continuation of MAT 631. Prereq: MAT 631..
MAT/CIS 645 Graph Theory 3 E
Fundamentals of graph theory and special topics including networks, matching, connectivity, planarity, and automorphism groups. Prereq: MAT 531 or equivalent.
MAT/CIS 646 Enumeration, Designs, and Matroids 3 O
Generating functions, Polya enumeration, set systems, design parameters, finite projective planes, matroids. Prereq: MAT 531 or equivalent.
MAT 651 Probability and Statistics I 3 Y
Calculus of probabilities, univariate and multivariate random variables and distribution functions,expectations and variance, condiational distributions, transformations of random variables, characteristic functions, basic limit theorems including Borel-Cantelli, Khinchin, Lindeberg-Feller. Prereq: MAT 521 or equivalent.
MAT 652 Probability and Statistics II 3 Y
Point and interval estimation, consistent, efficient, and sufficient statistics, Rao-Black-wellization, hypothesis testing, brief treatment of ranking and selection, decision theory. Prereq: MAT 651.
MAT 653 Statistical Simulation and Nonstandard Data Analysis 3 IR
Simulation and Monte Carlo techniques appropriate where statistical theory does not yet provide a solution. Design and analysis of experiments under nonstandard conditions. Prereq: MAT 651 and introductory computer programming.
MAT 654 Linear Models 3 IR
Point estimation by least squares, regression, curve fitting, testing linear hypothesis, analysis of variance, simple experimental designs.. Prereq: MAT 525, 531.
MAT 661 Introductory to Point Set Topology 3 Y
Topological spaces, continuous mappings and homeomorphisms, space-filling curves, metric spaces, Tietze extension theorem, sepaation axioms, compact spaces. Arcwise connectedness, product and quotient spaces, and the fundamental group. Prereq: MAT 601 or permission of instructor.
MAT 682 Numerical Linear Algebra 3 0
Solution of linear equations. Norms and conditioning. Calculation of eigenvalues and eigenvectors. Least squares approximation and orthogonal functions. Error analysis and writing computer programs. prereq: knowledge of linear algebra and of a scientific programming language.
MAT 683,684 Methods of Numerical Analysis I and II 3 each E
Numerical methods of interpolation, approximation, integration, and differentiation, solutions of nonlinear equations, ordinary and partial differential equations. Error analysis and writing computer programs. Prereq: MAT 511,531 or equivalent. Prereq` for MAT 684: MAT 683 or permission of instructor.
MAT 687 Introduction to Applied Mathematics 3 IR
Mathematical model building, dimensional analysis, scaling, and perturbation theory. Models selected from the natural and social sciences according to the interests of instructor and students. Examples are: planetary orbits, fluid flow, isomers in organic chemistry, biological competition, biochemical kinetics, and physiological flow. Prereq: knowledge of advanced calculus and differential equations.
MAT 690 Independent Study 3 SI
MAT 701 Real Variables I 3 Y
Measure and integration, including basic theorems on integration and differentiation of sequences of functions; modes of convergence, product measures. Prereq: MAT 602
MAT 702 Functional Analysis I 3 Y
Norms, seminorms, and inner products on linear spaces. Standard theorems on linear functionals and operations. Dual spaces and weak topologies, classical spaces and their duals. Applications. Prereq: MAT 631, 661, 701.
MAT 704 Differential Equations 3 IR
Existence theorems for ordinary differential equations, linear differential equations and systems, Euler variational equations, typical Cauchy and boundary-value problems for partial differential equations. Prereq: MAT 632,701.
MAT 705 Calculus on Manifolds 3 IR
Differentiable manifolds, differential forms, exterior calculus, integration over manifolds, Stokes's theorem, other selected topics. Prereq: MAT 602, 661,632 or permission of instructor.
MAT 712 Functions of a Complex Variable I 3 Y
Cauchy theory, power series, analytic continuation, entire functions, the residue theorem, contour integration, maximum modulus theorem and applications, conformal representation. Dirichlet series, special functions. Prereq: MAT 602.
MAT 721 Probability I 3 Y
Measure and integration. Random variables, their distributions and transforms. Modes of convergence. Classical limit laws. Markov chains. Prereq: MAT 602, 701.
MAT 722 Probability II 3 Y
Conditional expectation. Martingales. Bownian motion. Ergodic theorem. Random walks. Prereq: MAT 721.
MAT 731 Rings and Modules 3
Submodules, factor modules, chain conditions, Hilbert basis theorem, division rings, Schur's lemma. Jacobson density theorem, semi-simple modules, socies, Jacobson radical, semi primitive rings, Artin-Wedderburn theorem, integral extensions, completions, localization. Prereq: MAT 631, 632.
MAT 732 Homological Algebra 3
Free projective and injective resolutions, Tor and Ext functors, flat modules, regular sequences, regular rings, complete intersections, Koszul complexes, dimension, depth, Cohen-Macaulay rings, Gorenstein rings. Prereq: MAT 631,632,731.
MAT 738 Introduction to Algebraic Geometry 3 IR
The study of the zeros of polynomials. Classical algebraic varieties in affine and projective space, followed by introduction to modern theory of sheaves, schemes, and cohomology. Prereq: MAT 632.
MAT/CIS 740 Topics in Combinatorics 3 IR
Prereq: MAT/CIS 645 or MAT/CIS 646.
MAT/CIS 750 Statistical Consulting 3 IR
Experience in working with real data taken from current projects in the statistical laboratory and from published papers.
MAT 752 Statistical Ranking, Selection, and Multiple Comparisons 3 IR
Statistical selection of the best category or population. Preference-zone and subset formulations. Multivariate preferences and populations. Applications. Recent developments, including Multiple Comparisons with the Best (MCB) and the Heteroscedastic Method (HM). Prereq: MAT 652 or permission of instructor.
MAT 753 Decision Theory 3 IR
Minimax theorems, completeness of the class of Bayes procedures. Invariance. Criteria for admissibility. Prereq: MAT 525 or permission of instructor.
MAT 754 Sequential Analysis 3 IR
General sequential decision problems, sequential probability ratio test, sequential test among three hypotheses, sequential estimation, optimal stopping, Wald's identity. Generalized SPRT's, Cox's theorem,sequential regression, functional equations, dynamic programming, sequential choice of experiments. Prereq: MAT 525.
MAT 755 Multivariate Statistical Analysis 3 IR
Multivariate normal distribution, conditional densities, partial correlation, multiple correlation, regression coefficiencts, maximum-likelihood estimates, Hotelling's statistic, Wishart distribution, tests of hypotheses, and linear discriminant functions. Prereq: MAT 525 or 532.
MAT 761 Introduction to Algebraic Topology 3 Y
Fundamental group covering spaces, chain complexes, simplicial or singular homology and cohomology theory, exact sequences, and the eilenberg-Steenrod axioms. Prereq: MAT 632, 661.
MAT 762 Algebraic Topology 3 IR
Homology, cohomology ring, universal coefficient theorem, duality, homotopy, theory, selected topics. Prereq: MAT 632,761 or permission of instructor.
MAT 771 Differential Geometry 3 IR
Differential manifolds, tensor fields and mappings, differential forms and Stokes's theorem, affine connections, exponential mapping, covariant differentiation, torsion and curvature tensors, Riemannian connections, complete Riemannian manifolds, other modern topics. Prereq: MAT 602,661,632 or permission of instructor.
MAT 781 Advanced Numerical Methods 3
each IR Numerical methods of linear algebra. Numerical treatment of large linear systems. Matrix inversions and eigenvalue analysis. Boundary and eigenvalue problems for ordinary and partial differential equations. Laboratory. Prereq: MAT 632,684,704 or permission of instructor.
MAT 782 Advanced Numerical Methods 3
each IR Numerical methods of linear algebra. Numerical treatment of large linear systems. Matrix inversions and eigenvalue analysis. Boundary and eigenvalue problems for ordinary and partial differential equations. Laboratory. Prereq: MAT 781.
MAT 800 Topics in Analysis 3 IR
MAT 802 Real Variables II 3 IR
Topics in real variables and measure theory, such as differentiation theory in euclidean and abstract spaces, generalized derivatives and integrals, ergodic theory, martingales, surface area. Prereq: MAT 701.
MAT 804 Functional Analysis II 3 IR
Abstract integration, Radon-Nikodym theorem. Representation of set functions by integrals. Ergodic theorems. Duality. Weak topologies, convex sets, and extreme points. Elements of spectral theory. Prereq: MAT 702.
MAT 805 Partial Differential Equations 3 IR
First-order linear equations. Hamilton, Jacobi, and Lie transformations. Classifications of second-order linear equations. Boundary- and initial-value problems. Sturm-Liouville problems and connections with integral equations. Nonlinear equations. Prereq: permission of instructor.
MAT 807 Integral Equations 3 IR
Classification and examples of regular integral equations. Fredholm's theorems, Hilbert-Schmidt theory. Applications to differential equations. Nonlinear integral equations. Connections with general functional analysis. Prereq: MAT 804, or permission of instructor.
MAT 812 Functions of a Complex Variable II 3 IR
Continuation of MAT 712. Prereq: MAT 602.
MAT 820 Topics in Probability 3
MAT 830 Topics in Modern Algebra 3 IR
MAT/CIS 840 Topics in Combinatorics 3
Prereq: MAT/CIS 645 or MAT/CIS 646.
MAT 850 Topics in Statistics 3 IR
Contents vary from semester to semester. May be repeated for credit with permission.
MAT 860 Topics in Topology 3 IR
MAT 880 Topics in Numberical Analysis and Applied mathematics 3
MAT 890 Advanced Seminar 3 IR
For advanced graduate students and staff members; credit determined by extent of participation in the seminar. May be repeated for credit with permission of department. Prereq: permission of instructor.
MAT 999 Research 0-15 S Research work on a doctoral dissertation, under supervision of some member of the graduate staff. Credit depends on amount of time devoted to the work; course may be repeated up to a maximum of 30 credits. Prereq: permission of department.
