# MATH – Mathematics

Graduate offerings in mathematics are limited to those courses for which there is sufficient student demand as determined by the Department of Mathematics.

MATH 490G MATHEMATICS IN SECONDARY SCHOOLS (3-0-3)(F). Objectives, content, and methods of secondary school mathematics programs. PREREQ: MATH 370 and six hours of mathematics completed at or above the 300-level or PERM/INST.

MATH 501 FOUNDATIONS OF MATHEMATICS (3-0-3)(SU). The language and methods of reasoning used throughout mathematics, and selected topics in discrete mathematics. PREREQ: MATH 143.

MATH 502 LOGIC AND SET THEORY (3-0-3)(S). Structured as three five-week components: formal logic, set theory, and topics to be determined by the instructor. The logic component includes formalization of language and proofs, the completeness theorem, and the Lowenheim-Skolem theorem. The set theory component includes orderings, ordinals, the transfinite recursion theorem, and the Axiom of Choice and some of its equivalents. PREREQ: MATH 314.

MATH 503 LINEAR ALGEBRA (3-0-3)(F). Concepts of linear algebra from a theoretical perspective. Topics include vector spaces and linear maps, dual vector spaces and quotient spaces, eigenvalues and eigenvectors, diagonalization, inner product spaces, adjoint transformations, orthogonal and unitary transformations, Jordan normal form. PREREQ: MATH 187 or MATH 189, and MATH 301.

MATH 505 ABSTRACT ALGEBRA (3-0-3)(F)(Odd years). Topics in group theory, ring theory and field theory with emphasis on finite and solvable groups, polynomials and factorization, extensions of fields. PREREQ: MATH 301 and MATH 305.

MATH 506 ADVANCED ALGEBRA (3-0-3)(S)(Even years). The study of algebraic topics taken from mappings, semi-groups, groups, Sylow Theorems, group actions, rings, ascending and descending chain conditions, polynomial rings, fields, field extensions, Galois theory, Modules, Tensor products. PREREQ: MATH 405 or MATH 505.

MATH 507 ADVANCED NUMBER THEORY (3-0-3)(F)(Even years). Arithmetic functions, Mobius Inversion, Fundamental algorithm, Prime numbers, Factoring, quantification of number theoretic results. PREREQ: MATH 406 or PERM/INST.

MATH 508 ADVANCED ASYMMETRIC CRYPTOGRAPHY AND CRYPTANALYSIS (3-0-3)(F). An in-depth study of asymmetric cryptography, post-quantum cryptography, digital signatures, and analysis of cryptographic security. Elliptic curves and elliptic curve isogenies. Lattices and the shortest vector problem. Post-quantum cryptosystems such as NTRU and cryptographis schemas based on Short Integer Solution and Learning With Errors. PREREQ: CS 567 or MATH 305 or MATH 307 or MATH 308.

MATH 509 SYMMETRIC KEY CRYPTOGRAPHY AND CRYPTANALYSIS (3-0-3)(S). An in-depth study of modern block and stream ciphers, lightweight cryptography, hash functions, analysis cryptographic security, and current advances in cryptanalysis. Finite fields, vector spaces, enumerative combinatorics. Algebraic structures of symmetric key cryptosystems such as DES, Rijndael, PRESENT, GIFT and Gröstl. Security proofs including algebraic, linear, and differential cryptanalysis. Analysis of side-channel attacks. PREREQ: CS 567 or MATH 305 or MATH 307 or MATH 308.

MATH 511 INTRODUCTION TO TOPOLOGY (3-0-3)(F)(Even years). Sets, metric and topological spaces, product and quotient topology, continuous mappings, connectedness and compactness, homeomorphisms, fundamental group, covering spaces. PREREQ: MATH 314.

MATH 512 ADVANCED TOPOLOGY (3-0-3)(S)(Odd years). Introduction into concepts of algebraic and geometric topology: homotopy and homology groups, cohomology, manifolds, duality theorems, special topics. PREREQ: MATH 411 or MATH 511 or PERM/INST.

MATH 514 REAL ANALYSIS (3-0-3)(S). An advanced course in real analysis: Riemann integration, the fundamental theorem of calculus, sequences and series of functions, multivariable calculus. Additional topics may include Fourier series, analysis of metric spaces, the Baire property, and advanced topology of Euclidean space. PREREQ: MATH 275 and MATH 314.

MATH 515 REAL AND LINEAR ANALYSIS (3-0-3)(F). Lebesgue measure on the reals, construction of the Lebesgue integral and its basic properties. Advanced linear algebra and matrix analysis. Fourier analysis, introduction to functional analysis. PREREQ: MATH 414 or MATH 514.

MATH 522 ADVANCED SET THEORY (3-0-3)(F). Topics in modern set theory may be drawn from forcing, choiceless set theory, infinitary combinatorics, set-theoretic topology, descriptive set theory, inner model theory, and alternative set theories. PREREQ: MATH 402 or MATH 502 or PERM/INST.

MATH 526 COMPLEX VARIABLES (3-0-3)(S)(Odd years). Complex numbers, functions of a complex variable, analytic functions, infinite series, infinite products, integration, proofs and applications of basic results of complex analysis. Topics include the Cauchy integral formulas, the residue theorem, the Riemann mapping theorem and conformal mapping. PREREQ: MATH 275.

MATH 527 INTRODUCTION TO APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS (3-0-3)(F). Introduction to applied mathematics in science and engineering: Vector calculus, Fourier series and transforms, series solutions to differential equations, Sturm-Liouville problems, wave equation, heat equation, Poisson equation, analytic functions, and contour integration. PREREQ: MATH 275 and MATH 333.

MATH 533 ORDINARY DIFFERENTIAL EQUATIONS (3-0-3)(S)(Odd years). Theory of linear and nonlinear ordinary differential equations and their systems, including Dynamical systems theory. Properties of solutions including existence, uniqueness, asymptotic behavior, stability, singularities and boundedness. PREREQ: MATH 333.

MATH 536 PARTIAL DIFFERENTIAL EQUATIONS (3-0-3)(S)(Even years). Theory of partial differential equations and boundary value problems with applications to the physical sciences and engineering. Detailed analysis of the wave equation, the heat equation, and Laplace’s equation using Fourier series and other tools. PREREQ: MATH 275 and MATH 333, or PERM/INST.

MATH 537 PRINCIPLES OF APPLIED MATHEMATICS (3-0-3)(S). Finite and infinite dimensional vector spaces, spectral theory of differential operators, distributions and Green’s functions applied to initial and boundary value problems. Potential theory, and conformal mappings. Asymptotic methods and perturbation theory. Exact content determined by the instructor. PREREQ: MATH 427 or MATH 527 or PERM/INST.

MATH 547 HISTORY OF MATHEMATICS (3-0-3)(F/S/SU). The course is designed for mathematics teachers in the secondary school. The course consists of two parts: the first part traces the development of algebra, geometry, analytic geometry and calculus to the 19th century; the second part gives a brief introduction to, and history of, some of the developments in mathematics during the last century. May not be used for the Master’s degree in Mathematics. PREREQ: PERM/INST.

MATH 556 LINEAR PROGRAMMING (3-0-3)(SU)(On Demand). Linear optimization problems and systems of linear inequalities. Algorithms include simplex method, two-phase method, duality theory, and interior point methods. Programming assignments. PREREQ: MATH 301.

MATH 562 PROBABILITY AND STATISTICS (3-0-3)(F). Provides a solid foundation in the mathematical theory of statistics. Topics include probability theory, distributions and expectations of random variables, transformations of random variables, moment-generating functions, basic limit concepts and brief introduction to theory of estimation and hypothesis testing: point estimation, interval estimation and decision theory. PREREQ: MATH 275, MATH 301, and MATH 361.

MATH 564 MATHEMATICAL MODELING (3-0-3)(F/SU). Introduction to mathematical modeling through case studies. Deterministic and probabilistic models; optimization. Examples will be drawn from the physical, biological, and social sciences. A modeling project will be required. May not be used for the master’s degree in Mathematics. PREREQ: MATH 361 or PERM/INST.

MATH 565 (CS 565) NUMERICAL METHODS I (3-0-3)(F). Approximation of functions, solutions of equations in one variable and of linear systems. Polynomial, cubic spline, and trigonometric interpolation. Optimization. Programming assignments. May be taken for CS or MATH credit, but not both. PREREQ: MATH 365 or PERM/INST.

MATH 566 (CS 566) NUMERICAL METHODS II (3-0-3)(S). Matrix theory and computations including eigenvalue problems, least squares, QR, SVD, and iterative methods. The discrete Fourier transform and nonlinear systems of equations. Programming assignments. May be taken for CS or MATH credit, but not both. PREREQ: CS 565 or MATH 465 or MATH 565 or PERM/INST.

MATH 567 NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS (3-0-3)(F). Numerical techniques for initial and boundary value problems.  Elliptic, parabolic, hyperbolic, and functional differential equations. Finite difference, finite volume, finite element, and spectral methods.  Efficiency, accuracy, stability and convergence of algorithms. Programming assignments. PREREQ: MATH 333, and MATH 465 or MATH 565, or PERM/INST.

MATH 571 DATA ANALYSIS (3-0-3)(F). Applications of statistical data analysis in various disciplines, introduction to statistical software, demonstration of interplay between probability models and statistical inference. Topics include introduction to concepts of random sampling and statistical inference, goodness of fit tests for model adequacy, outlier detection, estimation and testing hypotheses of means and variances, analysis of variance, regression analysis and contingency tables. PREREQ: MATH 361.

MATH 572 COMPUTATIONAL STATISTICS (3-0-3)(S). Introduction to the trend in modern statistics of basic methodology supported by state-of-art computational and graphical facilities, with attention to statistical theories and complex real world problems. Includes: data visualization, data partitioning and resampling, data fitting, random number generation, stochastic simulation, Markov chain Monte Carlo, the EM algorithm, simulated annealing, model building and evaluation. A statistical computing environment will be used for students to gain hands-on experience of practical programming techniques. PREREQ: MATH 361 or PERM/INST.

MATH 573 TIME SERIES ANALYSIS (3-0-3)(S)(Even years). Introduction to time series analysis with an emphasis on application to interdisciplinary projects using SAS/ETS; autoregressive-moving average models, seasonal models, model identification, parameter estimation, model checking, forecasting, estimation of trends and seasonal effects, transfer function models, and spectral analysis. PREREQ: MATH 361 or PERM/INST.

MATH 574 LINEAR MODELS (3-0-3)(S)(Odd years). Introduction to the Gauss-Markov model with use of relevant statistical software. Includes linear regression, analysis of variance, parameter estimation, hypothesis testing, model building and variable selection, multicollinearity, regression diagnostics, prediction, general linear models, split plot designs, repeated measures analyses, random effects models. PREREQ: MATH 361.

MATH 579 TEACHING COLLEGE MATHEMATICS (1-0-1)(F,S,SU). Development of skills in the teaching of college mathematics. Effective use of class time, syllabus and test construction, learning styles, and disability issues. Lecturing, use of group work, and other teaching techniques. (Pass/Fail.) PREREQ: PERM/INST.

SELECTED TOPICS (1-3 Variable). To be offered as staff availability permits:

MATH 580 SET THEORY

MATH 581 LOGIC

MATH 582 TOPOLOGY

MATH 583 COMPUTATIONAL MATHEMATICS

MATH 584 COMPUTATIONAL ALGEBRA

MATH 585 CRYPTOLOGY

MATH 586 STATISTICS

MATH 587 DIFFERENTIAL EQUATIONS

MATH 588 INVERSE THEORY

MATH 598 SEMINAR IN MATHEMATICS (1-0-1)(F/S). Seminars by mathematicians on a wide range of subjects, including advanced mathematical topics selected from texts, mathematical journals, and current research. Format may include student presentation and discussion. Students will attend seminars, write summaries, and search for relevant literature. May be repeated once for credit. (Pass/Fail.) PREREQ: PERM/INST.

MATH 667 (CS 667) ADVANCES IN APPLIED CRYPTOGRAPHY (3-0-3)(S)(Even Years). Secure two-party and multiparty computation, proof by simulation, cryptographic commitments, sigma protocols, zero-knowledge proofs, advanced authenticated key exchange protocols, identification protocols and their security. PREREQ: CS 567 or MATH 508 or MATH 509, and regular admission to Doctor of Philosophy in Computing or Master of Science in Computer Science or Master of Science in Mathematics.

Refer to the University-wide Graduate Courses section in this catalog for additional course offerings.