Mat 501 Numerical Analysis (3)
Error analysis, numerical solution of nonlinear equations, interpolation
and polynomial approximation, numerical differentiation and integration,
direct methods for solving linear systems. May not be taken by students
with credit for Mat 401. Prerequisite: Mat 220 or equivalent.
Mat 502 Introduction to Maple V (3)
Introduction to basic principles of the Computer Algebra System, Maple V,
for teaching and research in mathematics. Topics include: numeric,
algebraic, and graphic capabilities; manipulating lists, arrays, and data
structure; and some programming techniques for writing procedures.
Prerequisite: Graduate status.
Mat 503A,B Life Contingencies (3,3)
Mat 503A: Treatment of the contingencies of a single life including:
mortality functions, life annuities, life insurance functions, annual
premiums, net level premium reserves, the expense factor, and more
complex benefits. Prerequisites: Mat 301 and Mat 362.
Mat 503B: Expansion of Mat 503A with emphasis on two or more lives in
combination and on multiple causes of decrement. Topics include
population theory, multi-life statuses, multi-life functions,
reversionary annuities, multiple-decrement functions, primary and
secondary decrements, and applications of multiple-decrement functions.
Prerequisites: Mat 301, Mat 362, and Mat 403A.
Mat 505 Introduction to Information and Coding Theory (3)
Coding to remove redundancy or to reduce errors due to noise and the
fundamental limitations to these processes described by Shannon's
Theorems for the binary symmetric channel. An introduction to error-
correcting codes. Prerequisite: Mat 362 or equivalent.
Mat 509 Vector Analysis (3)
Classical vector analysis presented heuristically and in physical terms.
Topics include the integral theorems of Gauss, Green, and Stokes.
Independent graduate study project required. May not be taken by students
with credit for Mat 409. Prerequisite: Mat 214.
Mat 510A,B Real Analysis (3,3)
Lebesgue measure and integration, abstract measure and integration,
product measures and Fubini's theorem, signed measures, absolute
continuity and singularity, decomposition theorems, and Radon-Nikodym
theorem. Prerequisites: Mat 511. Mat 510A is a prerequisite for 510B.
Mat 511 Foundations of Analysis (3)
The theoretical background of calculus. Real numbers, continuity, the
derivative and integral. Particularly recommended for teachers of
calculus. May not be taken by students with credit for Mat 510A.
Prerequisites: Undergraduate calculus and abstract algebra.
Mat 513A,B Complex Analysis (3,3)
Complex numbers, analytic functions, Cauchy's theorem and formula, power
series, Laurent series, normal families, conformal mapping, harmonic
functions, entire and meromorphic functions, special functions, and
analytic continuation. Prerequisites: Mat 411. Mat 513A is a prerequisite
for 513B.
Mat 515 Ordinary Differential Equations (3)
Review of first and second order differential equations. Theory of
existence and uniqueness. Dependence of solutions on initial conditions.
Linear systems with constant coefficients. Additional topics include
stability theory, series solutions, and applications. Prerequisites:
Advanced calculus and linear algebra.
Mat 516 Partial Differential Equations (3)
Properties of solution of first-order and the classical second-order
partial differential equations. Introduction to techniques in their
solution including separation of variables and Fourier series.
Prerequisites: Advanced calculus and an elementary course in ordinary
differential equations.
Mat 520A Algebra I (3)
Groups, fields, Galois theory. Prerequisite: Mat 524 or consent of
instructor.
Mat 520B Algebra II (3)
Modules over a principle ideal domain, semisimple rings and modules,
Wedderburn's theorem, tensor products, hom, projective and injective
modules, localization. Prerequisite: Mat 520A.
Mat 521 Algebra for Teachers (3)
Selected topics in discrete mathematics and algebra. Intended for
graduate students in secondary education programs. May not be taken by
students with credit in Mat 520A. Prerequisite: An undergraduate course
in abstract algebra or permission of instructor.
Mat 524 Advanced Linear Algebra (3)
Brief review of elementary linear algebra. Duality, quadratic forms,
inner product spaces, and similarity theory of linear transformations. A
term paper or other additional work is required. Prerequisite: One course
in algebra beyond elementary linear algebra.
Mat 525 Number Theory (3)
Divisibility, congruences, quadratic reciprocity, diophantine equations,
sums of squares, cubes, continued fractions, and algebraic integers. May
not be taken by students with credit for Mat 324 or 425. An independent
graduate study project is required. Prerequisite: Any 300-level or higher
course in algebra.
Mat 526 Geometric Constructions (3)
An inquiry into the correspondence between various geometric construction
tools and fields of algebraic numbers. Recommended for secondary teachers
and graduate students in secondary education program.
Mat 531 Transformation Geometry (3)
Transformation geometry, tessellations, and the ornamental groups of the
plane. Recommended for secondary teachers and graduate students in
teacher education programs. Not open to students with credit in Mat 231
or 331.
Mat 538 Differential Geometry (3)
The classical theory of curves and surfaces. Fundamental forms,
geodesics, the Gauss-Bonnett Theorem and its applications and exponential
mapping. Prerequisite: Vector calculus.
Mat 540A Topology I (3)
General theory of topological and Hausdorf spaces, metric spaces and
Euclidean spaces. Topics include metrization theorems, continuous curves,
arc-wise connectivity, and topological characterizations of certain
spaces. Prerequisite: Consent of instructor.
Mat 540B Topology II (3)
The study of complexes, simplicial homology, and cohomology ring
structure. Duality theorems. Lefschetz fixed-point theorem. Applications.
Prerequisite: Mat 540A.
Mat 551 Set Theory and Foundations of Mathematics (2)
Essentials of cardinal and ordinal arithmetic, transfinite induction,
Zorn's Lemma, and the Axiom of Choice. May not be taken by students with
credit for Mat 511.
Mat 552 History of Mathematics (3)
History of the development of mathematics, emphasizing the contributions
of outstanding persons and civilizations. Not open to students with
credit for Mat 452. Recommended for secondary teachers and graduate
students in the secondary education program. Prerequisites: Undergraduate
courses in abstract algebra and geometry, or by permission.
Mat 554 (H Sta 554) Introduction to Theory of Statistics (3)
A mathematical treatment of principles of statistical inference. Topics
include probability, random variables and random vectors, univariate and
multivariate distributions and an introduction to estimation. Appropriate
for graduate students in other disciplines and for preparation for the
second actuarial examination. Prerequisite: Calculus or linear algebra.
Mat 555 (H Sta 555) Introduction to Theory of Statistics II (3)
Continuation of Mat 554 (H Sta 554). Topics include methods of
estimation, theory of hypothesis testing, sufficient statistics,
efficiency and linear models. Appropriate for graduate students in other
disciplines and for preparation for the second actuarial examination.
Prerequisite: Mat 554 (H Sta 554), Mat 557A or equivalent.
Mat 557 (H Sta 557) Introduction to Bayesian Inference II (3)
Continuation of Mat 556 (H Sta 557). Topics include limiting posterior
distributions, estimation and hypothesis testing, preposterior
distribution and their application to the design of statistical
investigations. Prerequisite: Mat 556 (H Sta 556) or equivalent.
Mat 558 (H Sta 558) Methods of Data Analysis I (3)
Statistical methodology emphasizing exploratory approaches to data.
Elementary notions of modeling and robustness. Overview of inferential
techniques in current use. Criteria for selection and application of
methods. Use of computing facilities to illustrate and implement methods.
Regression and analysis of variance are primary topics. Prerequisite: Mat
554 (H Sta 554) or equivalent.
Mat 559 (H Sta 559) Methods of Data Analysis II (3)
Continuation of Mat 558 (H Sta 558). Topics will include clustering,
multivariate analyses, sequential and nonparametric methods.
Prerequisite: Mat 558 (H Sta 558) or equivalent.
Mat 560 (H Sta 560) Statistics Lab (3)
An introduction to applied stochastic processes. Topics include Markov
chains, queuing theory, renewal theory, Poisson processes and extensions,
epidemic and disease models. Prerequisite: Mat 555 (H Sta 555) or an
introductory probability course.
Mat 562 (H Sta 562) Design of Experiments I (3)
Principles in the design and analysis of controlled experiments. Topics
include general linear hypotheses, multiple classifications, Latin
Squares and factorial designs. Prerequisite(s): Mat 555 (H Sta 555) or
equivalent.
Mat 564 (H Sta 564) Sample Survey Methodology I (3)
Principles of survey sampling and analysis. Topics include simple random
sampling, stratified sampling, cluster sampling and multistage sampling.
Prerequisite: Mat 555 (H Sta 564) or equivalent.
Mat 565 Applied Statistics (3)
A course in statistical methods for students with some knowledge of
statistics. Topics include multiple regression, analysis of variance and
nonparametric statistical techniques. Emphasis on data analysis and
statistical methodology. May not be taken for credit by students with
credit for Mat 308 or 465. Prerequisites: An introductory course in
probability or statistics, and some experience with interpretation of
data in a subject matter area.
Mat 566 (H Sta 566) Analysis of Categorical Data I (3)
Introduction to the analysis of categorical data. Topics include rates,
ratios and proportions, relative risk, Cochran-Mantel-Haenszel
procedures, linear and log-linear models for categorical data, maximum
likelihood estimation and tests of hypotheses. Prerequisite: Mat 555 (H
Sta 555) or equivalent.
Mat 568 (H Sta 568) Statistical Ecology (3)
Density estimates for closed and open populations using simple and
multiple marking methods. Mortality and survival estimation, population
dynamics. Spatial patterns in one and two-species populations.
Characterization of many-species populations. Prerequisite: Mat 555 (H
Sta 555) or equivalent.
Mat 569 (H Sta 569) Survey of Statistics (3)
Survey of hypothesis testing and estimation theory. Recommended for
secondary teachers and graduate students in mathematical education.
Prerequisites: Statistics course and consent of the instructor.
Mat 570 Combinatorics (3)
Principle of inclusion and exclusion, Ramsey's theorem, orthogonal Latin
squares, difference sets, combinatorial designs.
Mat 572 Linear Programming (3)
Mathematical foundations of linear programming: existence, duality,
computational methods. Connections with game theory, transportation
problems, and network flows.
Mat 575 Optimal Control Theory (3)
Single integral optimization problems, relation to mathematical
programming, and gradient algorithms. Prerequisites: Linear algebra and
differential equations.
Mat 576 Game Theory (3)
A survey of various game models and solution concepts, including two-
person games in various forms, nonzero sum games, bimatrix games, Nash
theory of bargaining, multistage games, and models of n-person games,
both cooperative and noncooperative, with and without side payments.
Prerequisite: Mat 372, 572, or consent of the instructor.
Mat 577 Chaos and Complexity (3)
Exploring chaos and complexity as motivated principally by computer
experiments with the logistic equation and Conway's cellular automaton
Life. Intended for secondary teachers and graduate students in secondary
education programs. Prerequisite: Graduate student status.
Mat 580 Proseminar (2)
Students prepare and present papers from the mathematics and statistics
literature. Prerequisite: Admission to a graduate degree program in
mathematics and statistics.
Mat 587 Institute in Mathematics (1-3)
A variety of special courses are offered under the institute. Topics may
vary from semester to semester. Prerequisite: Graduate status.
Mat 611 Functional Analysis: Basic Principles (3)
Topological vector spaces, Hahn-Banach theorem, principle of uniform
boundedness and closed graph theorem. Banach and Hilbert spaces; duality.
Prerequisite: 510A.
Mat 612 Classical Harmonic Analysis (3)
Fourier series on the unit circle, classical Banach spaces of functions,
convergence and summability, the conjugate function, connections with
Taylor series and Complex Analysis, lacunary trigonometric series and the
Weierstrass function, Fourier transforms on the line, harmonic functions,
boundary behavior, theorem of the brothers Riesz, introduction to Hardy
spaces. Prerequisite(s): Mat 513B and Mat 611 or consent of instructor.
Mat 613 Geometric Function Theory (3)
Schwarz's Lemma and its generalizations, hyperbolic metric and
applications to complex analysis, harmonic and subharmonic functions,
harmonic measure, subordination, extremal length, Ahlfors' distortion
theorem, symmetrization, logarithmic capacity. Prerequisite: Mat 513B.
Mat 615 Introduction to Multi-dimensional Complex Analysis (3)
Holomorphic functions, power series and holomorphic maps in several
variables, extension phenomena, domains of holomorphy, pseudoconvexity,
holomorphic convexity, plurisubharmonic functions, inhomogeneous Cauchy
Riemann equations, Levi's problem, analytic sheaves, global meromorphic
functions with prescribed local parts. Prerequisite: Mat 513A,B.
Mat 616 Introduction to Ergodic Theory (3)
Mixing conditions, ergodic theorems, entropy, isomorphisms, Kolmogorov
automorphisms, orbit equivalence and applications to number theory.
Prerequisite: Mat 510A.
Mat 617 Introduction to Dynamical Systems (3)
Smooth, continuous, and discrete dynamical systems, orbit structure in
the phase space, limit behavior, stability properties, invariant and
minimal sets, connections with differential equations. Prerequisites: Mat
510A and Mat 540A.
Mat 620 Representation Theory of Finite Groups (3)
Characters and representations of finite groups, induced modules, Artin's
and Brauer's theorems. Prerequisite: Mat 520B.
Mat 625 Algebraic Curves (3)
Subjects covered are taken from the theory of divisors (Riemann-Roch
theorem, the Jacobian manifold, Abel's theorem...), the theory of moduli,
the theory of elliptic curves, the theory of linear systems of curves on
an algebraic surface, and related topics.
Mat 640 Introduction to Combinatorial Group Theory (3)
Groups given by generators and relations with emphasis on free groups,
free products with amalgamations and HNN extensions. Geometric methods
stressed. Prerequisites: Mat 520A, Mat 540A.
Mat 645 Introduction to Algebraic Topology (3)
Singular homology, CW complexes and cohomology theory. Cup and cap
products, universal coefficient theorems. Lens spaces, projective spaces
and manifolds. Prerequisites: Mat 520A, Mat 540B.
Mat 646 Introduction to Differentiable Manifolds (3)
Basic properties of differentiable manifolds. Tangent and normal bundles,
imbeddings and immersions: approximation theorems, forms, vectors fields
and Stiefel-Whitney classes. Prerequisite: Mat 540B.
Mat 654 (H Sta 654) Probability and Theory of Statistical Inference I (3)
Univariate and multivariate distribution theory, properties of
estimators, large sample theory, confidence intervals and theory of
tests. Prerequisite: Mat 555 (H Sta 555) or equivalent.
Mat 655 (H Sta 655) Probability and Theory of Statistical Inference II
(3)
Continuation of Mat 654 (H Sta 654). Advance theory of tests, decision
theory and other topics. Prerequisite: Mat 654 (H Sta 654) or equivalent.
Mat 660 (H Sta 660) Linear Models I (3)
Topics include the theory of least squares, distribution of quadratic
forms, G-inverse, general Gauss-Markov model, estimation, hypothesis
tests, confidence intervals for unrestricted models, regression models
and analysis of variance. Prerequisite: Mat 555 (H Sta 555) or
equivalent.
Mat 662 (H Sta 662) Multivariate Analysis I (3)
Topics include basic properties of multivariate normal distributions and
other related distributions, inference in multivariate cases and
principle component analysis. Prerequisite: Mat 555 (H Sta 555) or the
consent of the instructor.
Mat 664 (H Sta 664) Time Series Analysis I (3)
Topics include the study of inference, estimation, prediction,
parsimonious description for univariate time-ordered data, various models
including Box-Jenkins and classical stationary processes with rational
spectral densities. Prerequisite(s): Mat 555 (H Sta 555) and Mat 559 (H
Sta 559) or consent of instructor.
Mat 665 (H Sta 665) Time Series Analysis II (3)
Continuation of Mat 664 (H Sta 664). Advance topics include study of
univariate and multivariate time-ordered data, various models including
Box-Jenkins and classical stationary processes with rational spectral
densities.
Mat 680 Master's Seminar: General (3)
Selected topics in mathematics. This or Mat 683 is required of all
candidates for a master's degree in the general program, except those who
write a master's thesis.
Mat 681 Master's Seminar: Teaching (3)
Selected topics in mathematics. Required of all candidates for a master's
degree in the teaching program, except those who write a master's thesis.
Mat 682 Master's Seminar: Statistics (3)
Selected topics in statistics. Required of all candidates for a master's
degree in the statistics program, except those who write a master's
thesis.
Mat 683 Master's Seminar: Actuarial Mathematics (3)
Selected topics in actuarial mathematics. This or Mat 680 is required of
all candidates for the master's degree in the general program, except for
those who write a master's thesis.
Mat 697 Independent Study and Research (1-5)
Independent study at the master's level under faculty direction. May be
repeated once for credit. Prerequisite: Consent of instructor.
Mat 699 Master's Thesis (1-5)
May be repeated for credit. Prerequisite: Consent of thesis director.
Mat 711 Functional Analysis: Topics (3)
Compact operators, operators on Hilbert space, spectral theory, Banach
algebras, distributions. Prerequisites: Mat 513A, Mat 611.
Mat 712 Spaces of Analytic Functions (3)
Analytic and harmonic functions in the unit disk, the Cauchy and Poisson
kernels, functions of bounded Nevanlinna characteristic, HP spaces, the
disk algebra, factorization, inner and outer functions, invariant
subspaces for the shift operator. Ha as a Banach algebra, the Corona
theorem. Bergman spaces: zero sets, factorization, inner and outer
functions. Prerequisite: Mat 612 or consent of instructor.
Mat 713 Univalent Functions (3)
A study of the theory of univalent functions, including elementary
distortion theorems and coefficient estimates, special classes of
univalent functions, Lowner's parametric representation, variational
techniques, Baernstein's symmetrization theorem, extreme points of
families, de Branges' theorem. Prerequisite: Mat 513B.
Mat 714 Abstract Harmonic Analysis (3)
Harmonic analysis for non-commutative groups. Unitary representations,
irreducibility, Schur's lemma, decomposition of unitary representations,
Peter-Weyl theorem and Plancheral theorem for compact groups;
distributional character for infinite dimensional unitary
representations, generalized lemma, direct integral decomposition of
unitary representations, and Plancherel theorem for non-compact groups.
Prerequisite(s): May 611 and Mat 612 or consent of instructor.
Mat 715 Function Theory in Several Complex Variables (3)
Bochner-Martinelli formula, Cauchy integral on convex domains, integral
representations and estimates for solutions of d-bar on strictly
pseudoconvex domains, approximation theorems in spaces of holomorphic
functions, Bergman projection, boundary regularity of biholomorphic maps.
Prerequisite: Mat 615.
Mat 716 Partial Differential Equations and Several Complex Variables (3)
Distributions, Partial Differential Equations, L2-estimates for d-bar, d-
bar Neumann problem, Condition R, CR-functions, CR-extension problem,
extension of biholomorphisms. Prerequisite: Mat 615.
Mat 721 Homological Algebra (3)
Introduction to categories and functors, homology theories and sheaves.
Prerequisite: Consent of instructor.
Mat 722 Theory of Algebraic Numbers (3)
Basic theory of global and local algebraic number fields, including the
Dirichlet units theorem and finiteness of the class number. Prerequisite:
Mat 520B.
Mat 724 Commutative Algebra (3)
Rings, primary decomposition, localization, chain conditions, integral
dependence, discrete valuation rings, completions, dimension theory.
Prerequisite: Mat 520B.
Mat 725 Algebraic Geometry (3)
Subjects covered are taken from the following: the theory of schemes, the
use of transcendental methods in algebraic geometry, the theory of
abelian varieties, the theory of algebraic surfaces, intersection theory,
desingularization theory, deformations and degenerations of algebraic
varieties, and arithmetic algebraic geometry. Prerequisite: Mat 625 or
724.
Mat 731A,B Theory of Complex Analytic Varieties (3,3)
Basic properties of n-dimensional complex algebraic varieties, including
the Nullstellensatz, a study of multiplicities and singularities.
Extension of the results of the first semester to complex-analytic
varieties, including the Weierstrass theorems, the proper mapping
theorem, facts on tangent spaces and cones, and analytic sheaves.
Prerequisites: For Mat 731A - Mat 520B and Mat 615; for Mat 731B - Mat
731A.
Mat 740 Advanced Combinatorial Group Theory (3)
Decision problems, groups acting on trees and growth of groups. Metric
properties of presentations including small cancellation hypotheses.
Prerequisite: Mat 640.
Mat 745 Advanced Algebraic Topology (3)
Fibrations, spectral sequences and groups acting on spaces; homotopy
theory and characteristic classes. Manifolds and duality theorems.
Prerequisite: Mat 645.
Mat 760A,B (Sta 760, 761) Basic Probability Theory (3,3)
Measure theoretic foundations of probability, distribution functions,
sums of independent random variables, laws of large numbers,
characteristics functions, central limit theorem, conditional
expectation, martingales, stationary processes, random walks, Markov
chains, Brownian motion, law of the iterated logarithm. Prerequisites:
Mat 510. Mat 760A is a prerequisite for 760B.
Mat 780 Seminar in Mathematics (3)
Selected topics chosen from the various fields of mathematics. May be
repeated for credit. Prerequisite: Consent of instructor.
Mat 800 Colloquium in Mathematics and Statistics (1)
The department's regular colloquium, supplemented by a seminar in which
the subject of each colloquium lecture is introduced or discussed.
Prerequisite: Admission to the Ph.D. program in mathematics or consent of
instructor.
Mat 810 Topics in Analysis (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 812 Seminar in Analysis (1-4)
May be repeated for credit.
Mat 815 Topics in Complex Analysis (1-4)
May be repeated for credit.
Mat 817 Seminar in Complex Analysis (1-4)
May be repeated for credit.
Mat 820 Topics in Algebra (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 822 Seminar in Algebra (1-4)
May be repeated for credit.
Mat 824 Topics in Algebraic Number Theory (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 825 Topics in Algebraic Geometry and Commutative Algebra (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 840 Topics in Topology (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 842 Seminar in Topology (1-4)
May be repeated for credit.
Mat 860 (Sta 860) Topics in Probability (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 862 (Sta 862) Seminar in Probability (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 865 (Sta 865) Topics in Statistics (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 867 (Sta 867) Seminar in Statistics (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 870 Topics in Applied Mathematics (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 872 Seminar in Applied Mathematics (1-4)
May be repeated for credit. Prerequisite: Consent of instructor.
Mat 894 Directed Readings in Mathematics (1-5)
May be repeated for credit. Prerequisite: Consent of instructor.