# Mathematics and Statistics Courses

**Mat 502 Modern Computing for Mathematicians (3) **

Introduction to (1) basic principles of computer algebra systems, (2) contemporary mathematical typesetting, and (3) mathematically literate techniques for disseminating mathematical content in both print and HTML-with-MathML forms from a single source. Several computer algebra systems will be examined with an eye toward understanding how to handle various mathematical objects and how to write procedures for tasks that are not handled natively. Written assignments will specify mathematical tasks and presentation standards. Prerequisite: Familiarity with undergraduate mathematics and some ability with computer code.

**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 522 Linear Algebra for Applications (3)**

The course’s main concentration is on theory of abstract linear spaces with applications to Numerical Analysis, probabilistic and statistical considerations including Markov chains and migration process, least squares method, etc. Such topics as Singular Value Decomposition, Numerical Rank, Power method and QR algorithm for finding eigenvalues are considered in detail using techniques of spectral theory. Prerequisites: Mat 214, or equivalent course in multivariable calculus, or permission of the instructor. Only one of Csi 501 or Mat 522 may be taken for credit.

**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: Elementary linear algebra (Mat 220 or equivalent) and classical algebra (Mat 326 or equivalent).

**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 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 532 Foundations of Geometry (3)**

Axiomatic development of absolute geometry, theory of parallels, introduction to non-Euclidean geometry, isometries of the Bolyai-Lobachevsky plane. Prerequisite: Linear Alegebra (Mat 220) or equivalent.

**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) or equivalent.

**Mat 556 (H Sta 556) Introduction to Bayesian Inference I (3) **

Topics include subjective probability, axiomatic development and applications of utility, basic concepts of decision theory, conjugate and locally uniform prior distributions. Prerequisite: Sta 555 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) Introduction to Stochastic Processes I (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 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 465. Prerequisites: An introductory course in probability or statistics, and some experience with interpretation of data in a subject matter area.

**Mat 570 Combinatorics (3) **

Enumeration techniques: recursions, generating functions, inclusion-exclusion, bijective combinatorics. Special counting sequences. Partially ordered sets. Enumeration under group action. Algebraic graph theory. Extremal combinatorics. A more advanced/specialized topic chosen from: combinatorial Hopf algebras, matroids, hyperplane arrangements, topological combinatorics, tableaux and symmetric polynomials, etc. Prerequisite: a course in abstract algebra, or Mat 328, or permission of instructor.

**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 Optimization Theory (3) **

Introduction to optimization. Constrained optimization and Lagrange multipliers. Convex sets, convex functions and conjugate functions. Fenchel duality, convex optimization, Lagrange duality, non-linear programming. Karush-Tucker conditions and calculus of variations. Prerequisites: Mat 214 and 220.

**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 581 Nonparametric Statistics (3)**

Nonparametric statistical methods are applicable to data that do not come from a specific parametric family like the normal or binomial. Such tests may assume only that the underlying distribution is continuous. The null hypothesis in a two sample nonparametric test may specify only that the two samples come from the same distribution. Specific topics include permutation tests, the sign test, the Wilcoxon signed rank test, the Mann Whitney test, the Kruskall Wallace test, the Friedman test for comparing rankings, the Kolmogorov Smirnov test, Spearman's rho and Kendall's tau. Prerequisites: At least one calculus based statistics course, e.g., AMAT 554.

**Mat 583 Topological Data Analysis I (3)**

Basic techniques and concepts of topology that are used in data analysis. This is the first of a two semester sequence in Topological Data Analysis. This subject requires knowledge of rather advanced topics in topology. This course navigates to the point where the student is ready to see the applications in data science, through a careful selection of the sequence of topics: graph theory, high-dimensional simplicial complexes, nerves of coverings, some basic general topology and homotopy theory, computational linear algebra, simplicial homology and cohomology. Prerequisites: Basic linear algebra as in AMAT 220 or equivalent.

**Mat 584 Topological Data Analysis II (3)**

An introduction to the two main areas of Topological Data Analysis, the persistent homology and the Mapper algorithm. This is the second of a two semester sequence in Topological Data Analysis. In this course, the students will learn to apply homology computations to filtered metric spaces producing the main topological signature of a data set called persistent homology. In the second half of the course, the Mapper will be used as an illustration of topology based dimension reduction techniques which produce a one-dimensional summary, a graph, of the multi-dimensional data set. Case studies with real world applications are included to illustrate the theory. Prerequisites: AMAT 583 or permission of instructor.

**Mat 585 Practical methods in topological data analysis (3)**

This class is the practicum, the final installment in the Topological Data Analysis sequence (AMAT 583-584-585). Each participant will select a data science problem whose solution will employ the use of topological data analysis, design the project and successfully complete it. The instructor will provide close supervision at all stages of this project. Prerequisites: AMAT 583-584 or permission of the instructor.

**Mat 587 Topics in Modern Mathematics (3)**

Selected topics in mathematics. The topic of the course will be indicated in the course schedule and in departmental announcements. The course may be repeated for credit when the topic differs. Prerequisite: Graduate status.

**Mat 590 Function Theory and Functional Analysis for Applications (3)**

This course covers function analytic aspects necessary for applications in various areas of science and engineering, notably in Data Science. Among main topics of the course are: elementary theory of Lebesgue measure and integration, spaces of Lebesgue integrable functions, Banach spaces and Hanh-Banach theorem, duality in Banach spaces, Hilbert spaces, reproducing kernel Hilbert spaces, non-linear analysis in Banach spaces. Prerequisites: Basic linear algebra, e.g., AMAT 220; calculus of several variables, e.g., AMAT 214.

**Mat 591 Optimization Methods and Nonlinear Programming (3)**

Modern methods in convex optimization and nonlinear programming. Newton's method, gradient descent, linear programming, quadratic optimization, semidefinite programming and related topics. Prerequisites: AMAT590.

**Mat 592 Machine Learning (3)**

The primary goal of this course is to provide students with statistical tools and mathematical principles needed to solve both the traditional and modern data science problems encountered in practice. In particular, the course covers a wide variety of topics in machine learning. It introduces the key terms, concepts and methods in machine learning, with an emphasis on developing critical analytical skills through hands-on exercises of actual data analysis tasks. At the same time, it will cover modern machine learning topics such as boosting and online learning for large-scale data analysis. In addition, the students will practice basic programming skills to use software tools in machine learning. Prerequisites: AMAT591 and AMAT 554.

**Mat 593 Practical methods in machine learning (3)**

This course is the final installment and capstone for the sequence of Machine Learning (MAT 590-593). The student will select a real-world data analysis project which can be effectively addressed by employing machine learning methods. It involves a principled data-analytical process including data preprocessing, the detailed design of the project, comparison of different machine learning methods, and critical analysis of the obtained results. The instructor will provide close guidance throughout the project. Prerequisites: AMAT 590-592 or permission of the instructor.

**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 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 621 Lie Algebras and Their Representations (3)**

Basic notions in Lie theory. The universal enveloping algebra. Nilpotent and solvable Lie algebras. Semisimple Lie algebras and the Cartan decomposition. The root system and the Weyl group. Classification of finite-dimensional complex simple Lie algebras. The classical simple Lie algebras. Verma modules. Irreducible modules for the semisimple Lie algebras. Character formulas (Weyl, Freudenthal). Prerequisite: Mat 520A.

**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 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 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 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 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 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 (Sta 760) Basic Probability Theory (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 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 894 Directed Readings in Mathematics (1-5) **

May be repeated for credit. Prerequisite: Consent of instructor.

**Mat 897 Independent Study and Research (1-5) **

Independent study at the doctoral level under the direction of a member of the mathematics faculty. May be repeated for credit. Prerequisite: Consent of instructor.

**Mat 899 Doctoral Dissertation (1) **

May be repeated for credit. Course grading is Load Only and does not earn credit. Prerequisite: Consent of dissertation director. Registration for this course is limited to doctoral students who have been admitted to candidacy.