# Graduate Courses

# Catalog Descriptions Courses Numbered 700 or Above

701 | Topics in Mathematics for Teachers |
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Material, including both mathematical content and teaching methodology, related to classroom use at various levels, elementary through secondary. Topics may vary. May not be counted for junior-senior credit towards a major in mathematics, nor for graduate credit towards a graduate degree in mathematics. | |

Prerequisite: Permission of instructor. | |

715 | Sampling Techniques |

Statistical methodology of survey sampling. Data analysis and estimation methods for various experimental designs; fixed or random sample sizes, pre-and/or post-stratified samples, and multistage sampling. Estimates of totals, means, ratios and proportions with methods of estimating variances of such estimates. | |

Prerequisite: A post-calculus probability or statistics course. | |

717 | Nonparametric Statistics |

Methods requiring few assumptions about the populations sampled. Topics include quantile tests, tolerance limits, the sign test, contingency tables, rank-sum tests, and rank correlation. | |

Prerequisite: Math 628 or permission of the instructor. | |

722 | Mathematical Logic |

Propositional calculus. First order theories and model theory. Elementary arithmetic and Godel's incompleteness theorems. (Same as EECS 722.) | |

Prerequisite: MATH 665 or MATH 691, or equivalent evidence of mathematical maturity. | |

724 | Combinatorial Mathematics |

Counting problems, with an introduction to Polya's theory; Mobius functions; transversal theory, Ramsey's theorem; Sperner's theorem and related results. | |

725 | Graph Theory |

Graphs; trees; connectivity; Menger's theorem; eulerian and hamiltonian graphs; planarity; coloring of graphs; factorization of graphs; matching theory; alternating chain methods; introduction to matroids with applications to graph theory. | |

727 | Probability Theory |

A mathematical introduction to premeasure-theoretic probability. Topics include probability spaces, conditional probabilities and independent events, random variables and probability distributions, special discrete and continuous distributions with emphasis on parametric families used in applications, the distribution problem for functions of random variables, sequences of independent random variables, laws of large numbers, and the central limit theorem. | |

Prerequisite: Math 123 and graduate standing or permission of the instructor. | |

728 | Statistical Theory |

N Theory of point estimation and hypothesis testing with applications. Confidence region methodologies and relations to estimation and testing. | |

Prerequisite: MATH 727 or equivalent. | |

735 | Introduction to Optimal Control Theory |

An introduction to the mathematical methods of deterministic control theory is given by considering some specific examples and the general theory. The methods include dynamic programing the calculus of variations, and Pontryagin's maximum principle. Various problems of linear control systems, e.g., the linear regulator problem, are solved. | |

Prerequisite Math 320 or equivalent. | |

740 | Number Theory |

Divisibility, the theory of congruences, primitive roots and indices, the quadratic reciprocity law, arithmetical functions and miscellaneous additional topics. | |

Prerequisite: Math 123 or equivalent. | |

750 | Stochastic Adaptive Control |

The identification and control of discrete and continuous time stochastic systems is investigated. Stochastic processes (Markov chains, Brownian motion), stochastic integrals, the Ito differential rule, stochastic differential equations, martingales, and estimation techniques are introduced, as well as some elements of optimal control theory. Some specific applications and simulation results of stochastic adaptive control theory are also presented. | |

Prerequisite: Math 627. | |

765-766 | Introduction to the Theory of Functions |

Mathematics 765 and 766 are theoretical courses on the fundamental concepts of analysis and the methods of proof. These two courses include the concept of a real number; limits, continuity, and uniform convergence; derivatives and integrals of functions of one and of several real variables. | |

Prerequisite: Multivariable calculus for Math765, and Math 765 for Math 766. | |

780 | Numerical Analysis of Linear Systems |

Computational aspects of linear algebra, linear equations and matrices, direct and indirect methods, eigenvalues and eigenvectors of matrices, error analysis. | |

Prerequisite: MATH 590 and MATH 781. | |

781 | Numerical Analysis I |

Finite and divided differences. Interpolation, numerical differentiation and integration. Gaussian quadrature. Numerical integration of ordinary differential equations. Curve fitting. (Sameas CS 781: ) | |

Prerequisite: Math 320 and knowledge of a programming language. | |

782 | Numerical Analysis II |

Direct and iterative methods for solving systems of linear equations. Numerical solution of partial differential equations. Numerical determination of eigen-vectors and eigenvalues. Solution of nonlinear equations. (Same as CS 782: ) | |

Prerequisite: Math 681 or 781. | |

783 | Applied Numerical Methods for Partial Differential Equations |

Finite Difference methods applied to particular initial-value problems (both parabolic and hyperbolic), to illustrate the concepts of convergence and stability and to provide a background for treating more complicated problems arising in engineering and physics. Finite difference methods for elliptic boundary-value problems, with a discussion of convergence and methods for solving the resulting algebraic system. Variational methods for elliptic problems. | |

Prerequisite: Math 647 or equivalent. | |

790 | Linear Algebra II |

A theoretical course on the fundamental concepts and theorems oflinear algebra. Topics covered are: vector space, basis, dimension, subspace, norm, inner product, Banach space, Hilbert space, orthonormal basis, positive definite matrix, minimal polynomial, diagonalization and other canonical forms, Cayley-Hamilton, spectral radius, dual space, quotient space. | |

Prerequisite: Math 590. | |

791-792 | Modern Algebra |

Mathematics 791 and 792 include the following topics: the number system; groups, rings and fields; matrices and linear transformations; lattices; Galois theory; linear algebras. | |

Prerequisite: Multivariable calculus for Math 791 and Math 791 for Math 792. | |

796 | Special Topics |

Arranged as needed to present appropriate material for groups of students. May be repeated for credit. | |

Prerequisite: variable. | |

799 | Directed Reading |

Directed reading on a topic chosen by the student with the advice of an instructor. May be repeated for additional credit. Consent of the department required for enrollment. | |

800 | Theory of Functions of a Complex Variable |

Cauchy's theorem and contour integration; the argument principle; maximum modulus principle; Schwarz symmetry principle; analytic continuation; monodromy theorem; applications to the gamma functions and Riemann's zeta functions; entire and meromorphic functions; conformal mapping; Riemann mapping theorem; univalent functions. | |

Prerequisite: Math 766 or concurrently with Math 766 for Math 800, and Math 800 for Math 801. | |

801 | Theory of Functions of a Complex Variable |

Continuation of MATH 800. LEC MATH 802 Set Theory. Axiomatic set theory; transfinite induction; regularity and choice; ordinal and cardinal arithmetic; miscellaneous additional topics (e.g., extra axioms such as GCH or MA; infinite combinatorics; large cardinals). | |

Prerequisite: MATH 765 or MATH 791, or concurrent enrollment in MATH 765 or MATH 791, or equivalent evidence of mathematical maturity. | |

802 | Set Theory |

Axiomatic set theory; transfinite induction; regularity and choice; ordinal and cardinal arithmetic; miscellaneous additional topics (e.g., extra axioms such as GCH or MA; infinite combinatorics; large cardinals). | |

Prerequisite: MATH 765 or MATH 791, or concurrent enrollment in Math 765 or Math 791, or equivalent evidence of mathematical maturity. | |

810-811 | Theory of Functions of a Real Variable |

Measurable spaces and functions. Measure spaces and integration. Extensions of set functions, outer measures, Lebesque measure. Signed and complex measures. Differentiation of set functions. Miscellaneous additional topics and applications. | |

Prereq: Math 766 / Math 810, Math 810 / Math 811. | |

820 | Introduction to Topology |

General topology. Set theory; topological spaces; connected sets; continuous functions; generalized convergence; product and quotient spaces; embedding in cubes; metric spaces and metrization; compact spaces; function spaces. | |

Prerequisite: Math 765. | |

821 | Algebraic Topology |

Fundamental groups; covering spaces; simplicial complexes and polyhedra; simplicial homology theory; mapping into spheres; simplicial approximation; relative homology groups; and exact sequences. | |

Prerequisite: Math 791 and 820. | |

822 | Algebraic Topology |

Review of simplicial homology; Lefschetz fixed point theorem and degree theory; singular, cellular, and axiomatic homology; Jordan Brouwer separation theorems; universal coefficient theorems, products in cohomology, homotopy groups, and the Hurewicz Theorem. | |

Prerequisite: MATH 821. | |

830-831 | Abstract Algebra |

A study of some structures, theorems, and techniques in algebra whose use has become common in many branches of mathematics. | |

Prerequisite:Math 792 for Math 830 and Math 830 for Math 831. | |

840 | Differentiable Manifolds |

Multilinear algebra of finite dimensional vector spaces over fields; differentiable structures and tangent and tensor bundles; differentiable mappings and differentials; exterior differential forms; curves and surfaces as differentiable manifolds; affine connections and covariant differentiation; Riemannian manifolds. | |

Prerequisite: MATH 765 and MATH 792. | |

850 | Differential Equations and Dynamical Systems |

Discrete and differentiable dynamical systems with an emphasis on the qualitative theory. Topics to be covered include review of linear systems, existence and uniqueness theorems, flows and discrete dynamical systems, linearization (Hartman-Grobman theorem), stable and unstable manifolds, Poincare sections, normal forms, Hamiltonian systems, and an introduction to bifurcation theory and chaos. | |

851 | Topics in Dynamical Systems |

Topics to be covered include complex dynamical systems, perturbation theory, nonlinear analysis of time series, chaotic dynamical systems, and numerical methods as dynamical systems. This course may be repeated for credit. | |

865 | Introduction to Stochastic Processes |

Markov chains; Markov processes; diffusion processes; stationary processes. Emphasis is placed on applications: random walks; branching theory; Brownian motion; Poisson process; birth and death processes. | |

Prerequisites: Math 627 and Math 765. | |

866 | Stochastic Processes II |

This is a second course in stochastic processes, focused on stochastic calculus with respect to a large class of semi-martingales and its applications to topics selected from classical analysis (linear PDE), finance, engineering, and statistics. The course will start with basic properties of martingales and random walks and then develop into the core program on Itoøs stochastic calculus and stochastic differential equations. These techniques provide useful and important tools and models in many pure and applied areas. | |

Prerequisite: MATH 727 and MATH 865. | |

870 | The Analysis of Variance |

The general linear hypothesis with fixed effects; the Gauss-Markovtheorem, confidence ellipsoids, and tests under normal theory; multiple comparisons and the effect of departures from the underlying assumptions; analysis of variance for various experimental designs and analysis of covariance. | |

Prerequisites: Math 628 and either linear algebra or Math 792. | |

872 | Multivariate Statistical Analysis |

The multivariate normal distribution; tests of hypotheses on means and covariance matrices; estimation; correlation; multivariate analysis of variance; principal components; canonical correlation. | |

Prerequisite: Math 628 and either Math 590 or Math 792. | |

874 | Statistical Decision Theory |

Game theory, admissible decision functions and complete class theorems; Bayes and minimax solutions; sufficiency; invariance; multiple decision problems; sequential decision problems. | |

Prerequisite: MATH 628 and MATH 766. | |

881 | Advanced Numerical Linear Algebra |

Advanced topics in numerical linear algebra including pseudo-spectra, rounding error analysis and perturbation theory, numerical methods for problems with special structure, and numerical methods for large scale problems. | |

Prerequisite: Math 781, 782, 790, or permission of the instructor. | |

882 | Advanced Numerical Differential Equations |

Advanced course in the numerical solution of ordinary and partial differential equations including modern numerical methods and the associated analysis. | |

Prerequisite: MATH 781, 782, 783, or permission of the instructor. | |

896 | Master's Research Component |

899 | Master's Thesis |

905 | Several Complex Variables |

Holomorphic functions in several complex variables, Cauchyâ€™s integral for poly-discs, multivariable Taylor series, maximum modulus theorem. Further topics may include: removable singularities, extension theorems, Cauchy-Riemann operator, domains of holomorphy, special domains and algebraic properties of rings of analytic functions. | |

Prerequisite: MATH 800. | |

910 | Algebraic Curves |

Algebraic sets, varieties, plane curves, morphisms and rational maps, resolution of singularities, Riemann-Roch theorem. | |

Prerequisite: MATH 791 and MATH 792. | |

915 | Introduction to Homological Algebra |

Injective and projective resolutions, homological dimension, chain complexes and derived functors (including Tor and Ext). | |

Prerequisite: MATH 830 and MATH 831, or consent of instructor. | |

920 | Lie Groups and Lie Algebras |

General properties of Lie groups, closed subgroups, one-parameter subgroups, homogeneous spaces, Lie bracket, Lie algebras, exponential map, structure of semisimple Lie algebras, invariant forms, Maurer-Cartan equation, covering groups, spinor groups. | |

Prerequisite: MATH 791 and MATH 820. | |

930 | Topics in General Topology |

Various topics will be discussed. These topics will include some of the following: Cardinal functions of topological spaces, trees and linear orders, normality of products, subspaces of Stone-Cech remainders, Infinitary Ramsey theory, generalizations of metrizability and paracompactness, and pathological subsets of the real line. | |

Prerequisite: Math 820. | |

940 | Advanced Probability |

Probability measures, random variables, distribution functions, characteristic functions, types of convergence, central limit theorem. Laws of large numbers and other limit theorems. Conditional probability, Markov processes, and other topics in the theory of stochastic processes. | |

Prerequisite: Math 811. | |

950 | Partial Differential Equations |

This course is a classical introduction to the subject of Partial Differential Equations including first-order equations by the method of characteristics and the linear second-order equations which arise in mathematical physics: the wave equation, the Laplace equation, and the heat equation. The class will also provide some exposure to a variety of more modern methods, especially the use of functional analysis, semigroups and fixed point techniques for studying nonlinear partial differential equations. | |

Prerequisite: Math 766 | |

951 | Partial Differential Equations II |

The course uses functional analytic techniques to further develop various aspects of the modern framework of linear and nonlinear partial differential equations. Sobolev spaces, distributions and operator theory are used in the treatment of linear second- order elliptic, parabolic, and hyperbolic equations. In particular we discuss the kind of potential, diffusion and wave equations that arise in inhomogeneous media, with an emphasis on the solvability of equations with different initial/boundary conditions. Then, we will survey the theory of semigroup of operators, which is one of the main tools in the study of the long-time behavior of solutions to nonlinear PDE. The theories and applications encountered in this course will create a strong foundation for studying nonlinear equations and nonlinear science in general. | |

Prerequisites: Math 950 | |

960-961 | Functional Analysis |

Topological vector spaces, Banach spaces, basic principles of functional analysis. Weak and weak* topologies, operators and adjoints. Hilbert spaces, elements of spectral theory. Locally convex spaces. Duality and related topics. Applications. | |

Prerequisite: Math 810 and Math 820 or concurrent with Math 820 for Math 960, and Math 960 for Math 961. | |

963 | C*-Algebras |

The basics of C*-algebras, approximately finite dimensional C*-algebras, irrational rotation algebras, C*-algebras of isometries, group C*-algebras, crossed products C*-algebras, extensions of C*-algebras and the BDF theory. | |

Prerequisite: MATH 811 or MATH 960, or consent of instructor. | |

970 | Analytic K-Theory |

K0 for rings, spectral theory in Banach algebras, K1 for Banach algebras, Bott periodicity and six-term cyclic exact sequence. | |

Prerequisite: MATH 792 and MATH 960. | |

990 | Seminar |

993 | Readings in Mathematics |

996 | Special Topics |

Courses offered in the recent past include: Several Complex Variables, Commutative Algebra, C*-Algebras, Stochastic Control, Fourier Analysis, Topological Groups, Harmonic Analysis. | |

999 | Doctoral Dissertation |