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

MATH 503 ADVANCED LINEAR ALGEBRA (3-0-3)(S). 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 314.

MATH 505 ABSTRACT ALGEBRA (3-0-3)(F)(Odd-numbered 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-numbered 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 Algorithms, Prime numbers, Factoring, quantification of number theoretic results. PREREQ: MATH 306.

MATH 508 ADVANCED PUBLIC KEY CRYPTOLOGY (3-0-3)(F). Galois Fields, Vector Spaces and Lattices. Group based and lattice asymmetric cryptographic primitives based. Security models for public key cryptosystems. The study of security foundations of current public key cryptosystems. PREREQ: MATH 305 or MATH 307 or MATH 308.

MATH 509 SYMMETRIC KEY CRYPTOLOGY (3-0-3)(S).  Combinatorics, Galois Fields and Extensions, and Vector Spaces. One-way functions, Hash functions, and pseudo-random number generators. Data Encryption Standard, Rijndael and other symmetric key cryptosystems and their cryptanalysis. PREREQ: 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 511 or PERM/INST.

MATH 514 ADVANCED CALCULUS (4-0-4)(F).Introduction to fundamental elements of Analysis on Euclidean spaces including the basic differential and integral calculus. Topics include: Infinite series, sequences and series of function, uniform convergences, theory of integration, implicit function theorem and applications. PREREQ: MATH 275, MATH 301, 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 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 instructor permission.

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 333.

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 will be determined by the instructor. PREREQ: MATH 427 or MATH 527 or permission of the instructor.

MATH 562 PROBABILITY AND STATISTICS II(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 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. PREREQ: MATH 365 or PERM/INST.

MATH 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. PREREQ: 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)(S)(Even years). Provides an application of the various disciplines in statistics to data analysis, 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)(F). Introduction to the trend in modern statistics of basic methodology supported by state-of-art computational and^Mgraphical 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.

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.

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). 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. Graded Pass/Fail. PREREQ: PERM/INST.

SELECTED TOPICS (Variable credit) In depth study of advanced topics in targeted areas of mathematics.

MATH 580 TOPICS IN SET THEORY

MATH 581 TOPICS IN LOGIC

MATH 582 TOPICS IN TOPOLOGY

MATH 583 TOPICS IN COMPUTATIONAL MATHEMATICS

MATH 584 TOPICS IN COMPUTATIONAL ALGEBRA

MATH 585 TOPICS IN CRYPTOLOGY

MATH 586 TOPICS IN STATISTICS

MATH 587 TOPICS IN DIFFERENTIAL EQUATIONS

MATH 588 TOPICS IN INVERSE THEORY

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