# Mathematics (MATH)

### Courses

**MATH 000 Preparation for Calculus 2 Credits**

Intensive review of fundamental concepts in mathematics utilized in calculus, including functions and graphs, exponentials and logarithms, and trigonometry. This course is for students who need to take MATH 51, 81, or 21, but who require additional preparation in precalculus. The credits for this course do not count toward graduation, but do count toward GPA and current credit count. Consent of department.

**Attribute/Distribution:** MA

**MATH 005 Introduction to Mathematical Thought 3 Credits**

Meaning, content, and methods of mathematical thought illustrated by topics that may be chosen from number theory, abstract algebra, combinatorics, finite or nonEuclidean geometries, game theory, mathematical logic, set theory, topology.

**Attribute/Distribution:** MA

**MATH 009 Introduction to Finite Mathematics 4 Credits**

Systems of linear equations, matrices, introduction to linear programming. Sets, counting methods, probability, random variables, introduction to Markov chains.

**Attribute/Distribution:** MA

**MATH 012 Basic Statistics 4 Credits**

A first course in the basic concepts and methods of statistics with illustrations from the social, behavioral, and biological sciences. Descriptive statistics; frequency distributions, mean and standard deviation, two-way tables, correlation and regression; random sampling, rules of probability, probability distributions and parameters, parameter estimation, confidence intervals, hypothesis testing, statistical significance. Note: Mathematics and Statistics majors may not receive credit for both MATH 012 & ECO 045.

**Attribute/Distribution:** MA

**MATH 021 Calculus I 4 Credits**

Functions and graphs; limits and continuity; derivative, differential, and applications; indefinite and definite integrals; trigonometric, logarithmic, exponential, and hyperbolic functions.

**Attribute/Distribution:** MA

**MATH 022 Calculus II 4 Credits**

Applications of integration; techniques of integration; separable differential equations; infinite sequences and series; Taylor's Theorem and other approximations; curves and vectors in the plane.

**Prerequisites:** MATH 021 or MATH 076

**Attribute/Distribution:** MA

**MATH 023 Calculus III 4 Credits**

Vectors in space; partial derivatives; Lagrange multipliers; multiple integrals; vector analysis; line integrals; Green's Theorem, Gauss's Theorem.

**Prerequisites:** MATH 022

**Attribute/Distribution:** MA

**MATH 031 Honors Calculus I 4 Credits**

Same topics as in MATH 021, but taught from a more thorough and rigorous point of view.

**Attribute/Distribution:** MA

**MATH 032 Honors Calculus II 4 Credits**

Same topics as in MATH 022, but taught from a more thorough and rigorous point of view.

**Prerequisites:** MATH 031 or MATH 021

**Attribute/Distribution:** MA

**MATH 033 Honors Calculus III 4 Credits**

Same topics as in MATH 023, but taught from a more thorough and rigorous point of view.

**Prerequisites:** MATH 022 or MATH 032

**Attribute/Distribution:** MA

**MATH 043 Survey of Linear Algebra 3 Credits**

Matrices, vectors, vector spaces and mathematical systems, special kinds of matrices, elementary matrix transformations, systems of linear equations, convex sets, introduction to linear programming.

**Attribute/Distribution:** MA

**MATH 051 Survey of Calculus I 4 Credits**

Limits. The derivative and applications to extrema, approximation, and related rates. Exponential and logarithm functions, growth and decay. Integration. Trigonometric functions and related derivatives and integrals.

**Attribute/Distribution:** MA

**MATH 052 Survey of Calculus II 3 Credits**

Techniques of integration. Differential equations. Probability and calculus. Partial derivatives and extrema. Multiple integrals and applications.

**Prerequisites:** MATH 051 or MATH 021 or MATH 076 or MATH 081

**Attribute/Distribution:** MA

**MATH 075 Calculus I, Part A 2 Credits**

Covers the same material as the first half of MATH 021. Meets three hours per week, allowing more class time for each topic than does MATH 021.

**Attribute/Distribution:** MA

**MATH 076 Calculus I, Part B 2 Credits**

Continuation of MATH 075, covering the second half of MATH 021. Meets three hours per week.

**Prerequisites:** MATH 075

**Attribute/Distribution:** MA

**MATH 081 Calculus with Business Applications I 4 Credits**

Limits and continuity; exponential, logarithmic and trigonometric functions; derivatives; extrema; approximations; indefinite and definite integrals. Applications with emphasis on business and economics.

**Attribute/Distribution:** MA

**MATH 082 Calculus with Business and Economics Applications II 4 Credits**

Integration by parts, Riemann sums; differential equations; series; Taylor series. Vectors, inner products and projections; functions of several variables, partial derivatives. Multiple integrals; vector-valued functions. Applications with emphasis on finance and economics.

**Prerequisites:** MATH 081 or MATH 021 or MATH 076 or MATH 051

**Attribute/Distribution:** MA

**MATH 114 (PHIL 114) Symbolic Logic 4 Credits**

A first course in logical theory, introducing the notions of logical consequence and proof, as well as related concepts such as consistency and contigency. Formal systems taught may include: term, sentence logic, and predicate logic.

**Attribute/Distribution:** MA

**MATH 130 (BIOS 130) Biostatistics 4 Credits**

Elements of statistics and probability with emphasis on biological applications. Statistical analysis of experimental and observational data.

**Prerequisites:** MATH 052 or MATH 022

**MATH 163 Introduction to Mathematical Reasoning 3 Credits**

An introduction to the discipline of mathematics for students considering a major in mathematics. Provides an introduction to rigorous mathematical reasoning, including basic proof techniques (e.g., basic propositional calculus, induction, contradiction) and key concepts which recur throughout mathematics (e.g., universal and existential quantifiers, equivalence classes, basic set theory). Students majoring in mathematics should complete this course before MATH 242, MATH 243 or MATH 301 and are encouraged to complete this course in the first or second year of study.

**Prerequisites:** MATH 021

**MATH 171 Readings 1-3 Credits**

Study of a topic in mathematics under individual supervision. Intended for students with specific interests in areas not covered in the listed courses. Consent of department chair required.

**Attribute/Distribution:** MA

**MATH 201 Problem Solving 1 Credit**

Practice in solving challenging mathematics problems using a variety of techniques. Permission of instructor required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** MA

**MATH 202 Actuarial Exam I 1 Credit**

Preparation for the first actuarial exam – probability. Problems in calculus and probability with insurance applications.

**Prerequisites:** (MATH 023 or MATH 052 or MATH 082) and (MATH 231 or MATH 263)

**MATH 203 Actuarial Exam II - Financial Mathematics 2 Credits**

Preparation for the second actuarial exam - financial mathematics. Mathematics of interest and investments, interest rate measurement, present value, annuities, loan repayment schemes, bond valuation, introduction to derivative securities. Practice in solving problems from past exams.

**Prerequisites:** MATH 022 or MATH 082

**Attribute/Distribution:** MA

**MATH 205 Linear Methods 3 Credits**

Linear differential equations and applications; matrices and systems of linear equations; vector spaces; eigenvalues and application to linear systems of differential equations.

**Prerequisites:** MATH 022

**MATH 208 Complex Variables 3 Credits**

Functions of a complex variable; calculus of residues; contour integration; applications to conformal mapping and Laplace transforms.

**Prerequisites:** MATH 023

**MATH 214 (PHIL 214) Topics in Philosophical Logic 4 Credits**

Topics may include the many systems of non-classical logic, truth theory, the impact of incompleteness and undecidability results on philosophy, the foundational projects of various philosopher/mathematicians, or the work of an important figure in the history of philosophical logic. Consent of instructor required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** MA

**MATH 229 Geometry 3-4 Credits**

Discussion of geometry as an axiomatic system. Euclid's postulates. History of and equivalent versions of Euclid's fifth postulate. Finite projective geometries. NonEuclidean geometries based upon negation of the fifth postulate: Geometry on the sphere; Hyperbolic and elliptic geometries. Examination of the concepts of “straight”, angle, parallel, symmetry and duality in each of these geometries. Applications of the different geometries will be considered.

**Attribute/Distribution:** MA

**MATH 230 Numerical Methods 3 Credits**

Representation of numbers and rounding error; polynomial and spline interpolation; numerical differentiation and integration; numerical solution of nonlinear systems; numerical solution of initial and boundary value problems; Monte Carlo methods. Knowledge of MATLAB or PYTHON or C required.

**Prerequisites:** MATH 205 or MATH 241 or MATH 242

**MATH 231 Probability and Statistics 3 Credits**

Probability and distribution of random variables; populations and random sampling; chi-square and t distributions; estimation and tests of hypotheses; correlation and regression theory of two variables. Not available for credit to students who have completed both MATH 263 and MATH 264.

**Prerequisites:** MATH 022 or MATH 052 or MATH 082

**MATH 234 Fractal Geometry 3 Credits**

Metric spaces and iterated function systems; various types of fractal dimension; Julia and Mandelbrot sets. Other topics such as chaos may be included. Small amount of computer use.

**Prerequisites:** MATH 023

**MATH 241 Applied Linear Algebra 3,4 Credits**

The theoretical basis for applying linear algebra in other fields, including statistics. Topics will include systems of equations, vector spaces, matrices, and linear transformations. Additional topics will include matrix factorizations (including LU, QR, eigen-decomposition, and SVD) and how they can be used in computer analysis of data sets. Some students may optionally choose to take MATH 205 as preparation for this course. Not available for credit to students who have completed MATH 242 or STAT 342.

**Prerequisites:** MATH 022 or MATH 082

**MATH 242 Linear Algebra 3-4 Credits**

An introduction to the study of vector spaces and linear transformations, with emphasis on mathematical rigor. Not available for credit to students who have completed MATH 241 / STAT 342.

**Prerequisites:** MATH 022 and MATH 163

**MATH 243 Algebra 3,4 Credits**

Introduction to basic concepts of modern algebra: groups, rings, and fields.

**Prerequisites:** MATH 242

**MATH 252 Introduction to Combinatorics and Graph Theory 3 Credits**

Topics in combinatorics and graph theory chosen to introduce the subjects and some of their common proof techniques. Sequences and recursive formulas; counting formulas; bijections; inclusion/exclusion; the Pigeonhole Principle; generating functions; equivalence relations. Graph theory topics include trees, connectivity, traversability, matching and coloring. Not available for credit to students who have completed MATH 305.

**Prerequisites:** MATH 022

**MATH 261 (CSE 261) Discrete Structures 3 Credits**

Topics in discrete mathematical structures chosen for their applicability to computer science and engineering. Sets, propositions, induction, recursion; combinatorics; binary relations and functions; ordering, lattices and Boolean algebra; graphs and trees; groups and homomorphisms.

**Prerequisites:** MATH 021 or MATH 076

**MATH 263 Introduction to the Theory of Probability 3 Credits**

An introduction to the basics of Calculus-based theory of Probability. Includes combinatorial techniques, events, independence, and conditional probability; most important discrete and continuous probability distributions, expectation and variance; joint distributions and covariance; moment generating functions; basic form of the Laws of Large Numbers and the Central Limit Theorem. Focuses on use of concepts to solve problems. Prior knowledge of Probability not required. Not available for credit to students who have completed (MATH 231 and MATH 264) or MATH 309.

**Prerequisites:** MATH 023 or MATH 052 or MATH 082

**Can be taken Concurrently:** MATH 023

**MATH 264 Introduction to Statistical Reasoning and Methods 4 Credits**

Introduction to the basic concepts, logic and issues involved in statistical reasoning and statistical methods used to analyze data and evaluate studies. Topics include descriptive statistics and exploratory data analysis; elementary probability and statistical inference. Examples drawn from various areas of application. Use of computer software (e.g., Minitab, R) to facilitate understanding and to complete data analysis. Three lectures and one computer laboratory. Not available for credit to students who have completed both MATH 231 and MATH 263.

**Prerequisites:** MATH 021 or MATH 051 or MATH 081

**MATH 271 Readings 1-3 Credits**

Study of a topic in mathematics under individual supervision. Intended for students with specific interests in areas not covered in the listed courses. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** MA

**MATH 291 Undergraduate Research 1-4 Credits**

Research in mathematics or statistics under the direction of a faculty member. Department permission required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** ND

**MATH 300 Apprentice Teaching 1-4 Credits**

**Repeat Status:** Course may be repeated.

**MATH 301 Principles of Analysis I 3-4 Credits**

Existence of limits, continuity and uniform continuity; HeineBorel Theorem; existence of extreme values; mean value theorem and applications; conditions for the existence of the Riemann integral; absolute and uniform convergence; emphasis on theoretical material from the calculus of one variable.

**Prerequisites:** MATH 023

**MATH 302 Principles of Analysis II 3-4 Credits**

Continuation of MATH 301. Functions of several variables; the implicit function theorem, and further topics with applications to analysis and geometry.

**Prerequisites:** MATH 301

**Attribute/Distribution:** MA

**MATH 303 (PHIL 303) Mathematical Logic 3-4 Credits**

Detailed proofs are given for the basic mathematical results relating the syntax and semantics of firstorder logic (predicate logic): the Soundness and Completeness (and Compactness) Theorems, followed by a brief exposition of the celebrated limitative results of Gödel, Turing, and Church on incompleteness and undecidability. The material is conceptually rigorous and mathematically mature; the necessary background is a certain degree of mathematical sophistication or a basic knowledge of symbolic logic. Consent of instructor required.

**Attribute/Distribution:** MA

**MATH 304 Axiomatic Set Theory 3-4 Credits**

A development of set theory from axioms; relations and functions; ordinal and cardinal arithmetic; recursion theorem; axiom of choice; independence questions. Consent of instructor required.

**Attribute/Distribution:** MA

**MATH 305 Enumerative Combinatorics 3 Credits**

An introduction to basic theoretical results and techniques of enumerative combinatorics such as combinatorial identities, generating functions, inclusion/exclusion, recurrence relations, bijective proofs and permutations. Additional topics will be covered as time permits.

**Prerequisites:** MATH 242

**MATH 307 General Topology I 3-4 Credits**

An introductory study of topological spaces, including metric spaces, separation and countability axioms, connectedness, compactness, product spaces, quotient spaces, function spaces.

**Prerequisites:** MATH 301

**Attribute/Distribution:** MA

**MATH 309 Probability with Applications and Simulations 3 Credits**

Foundations of Probability; Random Variables; Probability Models; Expectations and Moment Generating Functions; Joint and Conditional Distributions; Functions of Random Variables. Introduction to fundamental ideas and techniques of stochastic modeling, with an emphasis on the applications. The last part of the course is devoted to techniques and methods of Monte Carlo simulation. R or other software will be used in this course.

**Prerequisites:** MATH 023 or MATH 052 or MATH 082

**MATH 310 Random Processes and Applications 3-4 Credits**

Theory and applications of stochastic processes. Limit theorems, introduction to random walks, Markov chains, Poisson processes, birth and death processes, and Brownian motion. Applications to financial mathematics, biology, business and engineering.

**Prerequisites:** MATH 263 or MATH 309 or (MATH 231 and (MATH 205 or MATH 241), )

**MATH 311 Graph Theory 3 Credits**

An introduction to basic theoretical results and techniques of graph theory such as trees, connectivity, matchings, coloring, planar graphs and Hamiltonicity. Additional topics will be covered as time permits.

**Prerequisites:** MATH 163 or MATH 252 or CSE 140

**MATH 312 Statistical Computing and Applications 3,4 Credits**

Use of statistical computing packages; exploratory data analysis; Monte Carlo methods; randomization and resampling, application and interpretation of a variety of statistical methods in real world problems.

**Prerequisites:** MATH 012 or MATH 231 or MATH 264 or ECO 045

**MATH 316 Complex Analysis 3-4 Credits**

Concept of analytic function from the points of view of the CauchyRiemann equations, power series, complex integration, and conformal mapping.

**Prerequisites:** MATH 301

**Attribute/Distribution:** MA

**MATH 319 Introduction to Differential Equations 3 Credits**

An introductory, yet rigorous treatment of topics in differential equations chosen to prepare students for advanced work in mathematics or applied mathematics. Homogeneous and non-homogeneous linear differential equations, existence and uniqueness theorems, Gronwall's inequality; systems of first order linear differential equations; autonomous first-order systems: critical points, stability, bifurcation; series and periodic solutions, Fourier series and their convergence; introduction to numerical simulation methods.

**Prerequisites:** MATH 242 or MATH 205 or MATH 241

**MATH 320 Ordinary Differential Equations 3-4 Credits**

The analytical and geometric theory of ordinary differential equations, including such topics as linear systems, systems in the complex plane, oscillation theory, stability theory, geometric theory of nonlinear systems, finite difference methods, general dynamical systems.

**Prerequisites:** MATH 023 and (MATH 205 or MATH 319)

**MATH 321 Topics in Discrete Mathematics 3 Credits**

Selected topics in areas of discrete mathematics. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** MA

**MATH 322 Methods of Applied Analysis I 3 Credits**

Fourier series, eigenfunction expansions, SturmLiouville problems, Fourier integrals and their application to partial differential equations; special functions. Emphasis is on a wide variety of formal applications rather than logical development.

**Prerequisites:** MATH 205 or MATH 319

**MATH 323 Methods of Applied Analysis II 3 Credits**

Green's functions; integral equations; variational methods; asymptotic expansions, method of saddle points; calculus of vector fields, exterior differential calculus.

**Prerequisites:** MATH 322

**Attribute/Distribution:** MA

**MATH 327 Groups and Rings 3-4 Credits**

An intensive study of the concepts of group theory including the Sylow theorems, and of ring theory including unique factorization domains and polynomial rings.

**Prerequisites:** MATH 242 and MATH 243

**Attribute/Distribution:** MA

**MATH 329 Computability Theory 3-4 Credits**

Core development of classical computability theory: enumeration, index and recursion theorems, various models of computation and Church's Thesis, uncomputability results, introduction to reducibilities and their degrees (in particular, Turing degrees, or degrees of uncomputability), computable operators and their fixed points.

**Attribute/Distribution:** MA

**MATH 331 Differential Geometry of Curves and Surfaces 3 Credits**

Local and global differential geometry of curves and surfaces in Euclidean 3space. Frenet formulas for curves, isoperimetric inequality, 4vertex theorem; regular surfaces, first fundamental form, Gauss map, second fundamental form; curvatures for curves and surfaces and their relations; The GaussBonnet theorem.

**Prerequisites:** MATH 023 and (MATH 205 or MATH 242)

**MATH 334 Mathematical Statistics 3-4 Credits**

Populations and random sampling; sampling distributions; theory of statistical estimation; criteria and methods of point and interval estimation; theory of testing statistical hypotheses.

**Prerequisites:** MATH 263 or MATH 309

**MATH 338 Linear Models in Statistics with Applications 3,4 Credits**

Least square principles in multiple regression and their interpretations; estimation, hypotheses testing, confidence and prediction intervals, modeling, regression diagnostic, multicollinearity, model selection, analysis of variance and covariance; logistic regression. Introduction to topics in time series analysis such as ARMA, ARCH, and GARCH models. Applications to natural sciences, finance and economics. Use of computer packages.

**Prerequisites:** (MATH 012 or MATH 231 or MATH 264) and (MATH 043 or MATH 205 or MATH 241 or MATH 242 or STAT 342)

**MATH 339 3,4 Credits**

This course introduces the student to the statistical analysis of time series data and useful models: autocorrelation, stationarity, trend removal, and seasonal adjustment, basic time series models like AR, MA, ARMA; estimation, forecasting, and GARCH models; multivariate models, and factor models. The course emphasizes the main ideas and the most popular and widely used methods, and the use of a computer to practice the methods. Knowledge of scientific programming in a language such as R required.

**Prerequisites:** (MATH 264 or MATH 312) and (MATH 263 or MATH 309)

**MATH 340 (CSE 340) Design and Analysis of Algorithms 3 Credits**

Algorithms for searching, sorting, manipulating graphs and trees, finding shortest paths and minimum spanning trees, scheduling tasks, etc.: proofs of their correctness and analysis of their asymptotic runtime and memory demands. Designing algorithms: recursion, divide-and-conquer, greediness, dynamic programming. Limits on algorithm efficiency using elementary NP-completeness theory.

**Prerequisites:** (MATH 021 or MATH 031 or MATH 076) and CSE 140 and CSE 017

**MATH 341 Mathematical Models and Their Formulation 3 Credits**

Mathematical modeling of engineering and physical systems with examples drawn from diverse disciplines. Emphasis is on building models of real world problems and the analysis as well as numerical simulations of the models.

**Prerequisites:** MATH 205 or MATH 241 or MATH 242

**MATH 342 Number Theory 3-4 Credits**

Basic concepts and results in number theory, including such topics as primes, the Euclidean algorithm, Diophantine equations, congruences, quadratic residues, quadratic reciprocity, primitive roots, number-theoretic functions, distribution of primes, Pell’s equation, Fermat’s theorem, partitions. Consent of instructor required.

**Attribute/Distribution:** MA

**MATH 343 Introduction To Cryptography 3,4 Credits**

Classical elementary cryptography: Caesar cipher, other substitution ciphers, block ciphers, general linear ciphers. Fast random encryption (DES and AES: Advanced Encryption Standard). Public key systems (RSA and discrete logs). Congruences, modular arithmetic, fast exponentiation, polynomials, matrices. Distinction between polynomial time (primality), Subexponential time (factoring) and fully Exponential computation (elliptic curves). Introduction to sieving and distributed computation. Consent of instructor required.

**Attribute/Distribution:** MA

**MATH 350 Special Topics 3 Credits**

A course covering special topics not sufficiently covered in listed courses. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** MA

**MATH 365 Statistical Machine Learning 3,4 Credits**

This course provides a broad introduction to concepts, methods, and practices of statistical machine learning: parametric and nonparametric regression, logistic regression, classification, and basic neural networks; kernel and nearest neighbor estimation, clustering, Bayesian and mixture models. In addition, we will explore selected topics like model selection, cross-validation; PCA, dimension reduction, regularized regression; trees, and ensemble learning. Knowledge of scientific programming in a language such as R required.

**Prerequisites:** (MATH 205 or MATH 241 or MATH 242) and (MATH 264 or MATH 312) and (MATH 263 or MATH 309)

**MATH 371 Readings 1-3 Credits**

The study of a topic in mathematics under appropriate supervision, designed for the individual student who has studied extensively and whose interests lie in areas not covered in the listed courses. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** MA

**MATH 374 Statistical Project 3 Credits**

Supervised field project or independent reading in statistics or probability. Consent of department chair required.

**Attribute/Distribution:** MA

**MATH 381 Undergraduate Research 1-4 Credits**

Research in mathematics or statistics under the direction of a faculty member. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 391 Senior Honors Thesis 3 Credits**

Independent research under faculty supervision, culminating in a thesis presented for departmental honor. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**Attribute/Distribution:** MA

**MATH 401 Real Analysis I 3 Credits**

Set theory, real numbers; introduction to measures, Lebesgue measure; integration, general convergence theorems; differentiation, functions of bounded variation, absolute continuity; Lp spaces.

**Prerequisites:** MATH 301

**MATH 402 Real Analysis II 3 Credits**

Metric spaces; introduction to Banach and Hilbert space theory; Fourier series and Fejer operators; general measure and integration theory, RadonNikodym and Riesz representation and theorems; LebesgueStieljtes integral.

**Prerequisites:** MATH 307 or MATH 401

**MATH 403 Topics in Real Analysis 3 Credits**

Intensive study of topics in analysis with emphasis on recent developments. Requires permission of the department chair.

**Repeat Status:** Course may be repeated.

**MATH 404 Topics in Mathematical Logic 3 Credits**

Intensive study of topics in mathematical logic. Consent of instructor required.

**Repeat Status:** Course may be repeated.

**MATH 405 Partial Differential Equations I 3 Credits**

Classification of partial differential equations; methods of characteristics for first order equations; methods for representing solutions of the potential, heat, and wave equations, and properties of the solutions of these equations; maximum principles.

**Prerequisites:** MATH 319 or MATH 320

**MATH 406 Partial Differential Equations II 3 Credits**

Continuation of MATH 405. Emphasis on second order equations with variable coefficients and systems of first order partial differential equations.

**Prerequisites:** MATH 405

**MATH 408 Algebraic Topology I 3 Credits**

Polyhedra; fundamental groups; simplicial and singular homology.

**MATH 409 Mathematics Seminar 1-6 Credits**

An intensive study of some field of mathematics not offered in another course. Consent of department chair required.

**MATH 410 Independent study 1-4 Credits**

The study of a topic in mathematics under appropriate supervision, designed for the individual student who has studied extensively and whose interests lie in areas not covered in the listed courses. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 421 Introduction To Wavelets 3 Credits**

Continuous and discrete signals; review of Fourier analysis; discrete wavelets; time frequency spaces; Haar and Walsh systems; multiresolution analysis; Hilbert spaces; quadratic mirror filters; fast wavelet transforms; computer code; applications to filtering, compression, and imaging.

**Prerequisites:** ECE 108 or MATH 205

**MATH 423 Differential Geometry I 3 Credits**

Differential manifolds, tangent vectors and differentials, submanifolds and the implicit function theorem. Lie groups and Lie algebras, homogeneous spaces. Tensor and exterior algebras, tensor fields and differential forms, de Rham cohomology, Stokes' theorem, the Hodge theorem. Must have completed MATH 301, or MATH 243 or MATH 205 with permission of instructor.

**MATH 424 Differential Geometry II 3 Credits**

Curves and surfaces in Euclidean space; mean and Gaussian curvatures, covariant differentiation, parallelism, geodesics, GaussBonnet formula. Riemannian metrics, connections, sectional curvature, generalized GaussBonnet theorem. Further topics.

**Prerequisites:** MATH 423

**MATH 428 Fields And Modules 3 Credits**

Field theory, including an introduction to Galois theory; the theory of modules, including tensor products and classical algebras.

**Prerequisites:** MATH 327

**MATH 430 Numerical Analysis 3 Credits**

Multistep methods for ordinary differential equations; finite difference methods for partial differential equations; numerical approximation of functions. Use of computer required.

**Prerequisites:** MATH 230

**MATH 435 Functional Analysis I 3 Credits**

Banach spaces and linear operators; separation and extension theorems; open mapping and uniform boundedness principles; weak topologies; local convexity and duality; Banach algebras; spectral theory of operators; and compact operators.

**Prerequisites:** MATH 307 and MATH 401

**MATH 441 (CSE 441) Advanced Algorithms 3 Credits**

Algorithms for searching, sorting, manipulating graphs and trees, scheduling tasks, finding shortest path, matching patterns in strings, cryptography, matroid theory, linear programming, max-flow, etc., and their correctness proofs and analysis of their time and space complexity. Strategies for designing algorithms, e.g. recursion, divide-and-conquer, greediness, dynamic programming. Limits on algorithm efficiency are explored through NP completeness theory. Quantum computing is briefly introduced. Credit will not be given for both CSE 340 (MATH 340) and CSE 441 (MATH 441).

**MATH 444 Algebraic Topology II 3 Credits**

Continuation of MATH 408. Cohomology theory, products, duality.

**Prerequisites:** MATH 408

**MATH 445 Topcs in Algebraic Topology 3 Credits**

Selected topics reflecting the interests of the professor and the students.

**Prerequisites:** MATH 444

**MATH 449 Topics In Algebra 3 Credits**

Intensive study of topics in algebra with emphasis on recent developments. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 450 Special Topics 3 Credits**

Intensive study of some field of the mathematical sciences not covered in listed courses. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 455 Topics In Number Theory 3 Credits**

Selected topics in algebraic and/or analytic number theory. Consent of instructor required.

**Repeat Status:** Course may be repeated.

**MATH 461 Topics In Mathematical Statistcs 3 Credits**

An intensive study of one or more topics such as theory of statistical tests, statistical estimation, regression, analysis of variance, nonparametric methods, stochastic approximation, and decision theory.

**Repeat Status:** Course may be repeated.

**Prerequisites:** MATH 334 and MATH 401

**MATH 462 Modern Nonparametric Methods in Statistics 3 Credits**

Classical and modern methods of nonparametric statistics; order and rank statistics; tests based on runs, signs, ranks, and order statistics; distribution free statistical procedures for means, variances, correlations, and trends; relative efficiency; KolmogorovSmirnov statistics; statistical applications of Brownian process; modern techniques such as robust methods, nonparametric smoothing, and bootstrapping; additional topics such as nonparametric regression and dimension reduction.

**Prerequisites:** (MATH 334 or STAT 334) and (MATH 338 or STAT 338)

**MATH 463 (STAT 463) Advanced Probability 3 Credits**

Measure theoretic foundations; random variables, integration in a measure space, expectations; convergence of random variables and probability measures; conditional expectations; characteristic functions; sums of random variables, limit theorems.

**Prerequisites:** MATH 309 and MATH 401

**MATH 464 Advanced Stochastic Process 3 Credits**

Theory of stochastic processes; stopping times; martingales; Markov processes; Brownian motion; stochastic calculus; Brownian bridge, laws of suprema; Gaussian processes.

**Prerequisites:** MATH 309 and MATH 401

**MATH 465 Topics in Probability 3 Credits**

Selected topics in probability. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 467 Stochastic Calculus 3 Credits**

Brownian Motion, Martingales. Introduction to the theory of Stochastic Calculus, Itô Formula, and Stochastic Differential Equations, Black-Scholes model. Development of the Martingale Representation Theorem and Girsanov's theorem for change of measure. Multidimensional Stochastic Calculus. Applications to different problems from finance, physics, biology, etc.

**Prerequisites:** MATH 231 or MATH 309

**MATH 468 Financial Stochastic Analysis 3 Credits**

Basic mathematical concepts behind pricing of derivative securities. Hedging and pricing by arbitrage in the setting of binary trees and Black-Scholes model. Application of Stochastic Calculus to the pricing of a variety of financial instruments: multiple stock models, American and exotic options, and foreign currency interest rate. Heath-Jarrow-Morton model for the term structure of interest rates and short rate models. Applications of the theory to a variety of interest rates contracts including swaps, caps, floors, swap options.

**Prerequisites:** MATH 467

**MATH 470 Proseminar 1-3 Credits**

Preparation for entering the mathematics profession. Topic of emphasis typically include methods of teaching mathematics, commonly available research tools and the publication process, the responsibilities of an academic position, and searching for a job. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 471 Homological Algebra 3 Credits**

Modules, tensor products, categories and functors, homology functors, projective and injective modules.

**Prerequisites:** MATH 428

**MATH 472 Group Representations 3 Credits**

Linear representations and character theory with emphasis on the finite and compact cases.

**Prerequisites:** MATH 428

**MATH 475 Topics in Geometry 3 Credits**

Selected topics in geometry, such as geometric analysis, algebraic geometry, complex geometry, characteristic classes, geometric flows or geometric measure theory, with emphasis on recent developments. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 485 Topics in Financial Mathematics 3 Credits**

Selected topics in financial mathematics. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 490 Thesis 1-6 Credits**

**MATH 491 Research 1-4 Credits**

Research in mathematics or statistics under the direction of a faculty member. Consent of department chair required.

**Repeat Status:** Course may be repeated.

**MATH 499 Dissertation 1-15 Credits**

**Repeat Status:** Course may be repeated.