MAA 5104 Advanced Calculus for Engineers and Physical Scientists I 3 Credits

Grading Scheme: Letter Grade

Advanced Calculus for Engineers and Physical Scientists I

Catalog Program Pages Referencing MAA 5104

MAA 5105 Advanced Calculus for Engineers and Physical Scientists II 3 Credits

Grading Scheme: Letter Grade

Advanced Calculus for Engineers and Physical Scientists II

Prerequisite: MAA 5104.

Catalog Program Pages Referencing MAA 5105

MAA 5228 Modern Analysis I 3 Credits

Grading Scheme: Letter Grade

Topology of metric spaces, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, the Stone-Weierstrass theorem, and the Lebesgue theory.

Prerequisite: advanced calculus.

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MAA 5229 Modern Analysis II 3 Credits

Grading Scheme: Letter Grade

Modern Analysis II

Prerequisite: MAA 5228.

Catalog Program Pages Referencing MAA 5229

MAA 5404 Introduction to Complex Variables for Engineers and Physical Scientists 3 Credits

Grading Scheme: Letter Grade

Introduction to Complex Variables for Engineers and Physical Scientists

Catalog Program Pages Referencing MAA 5404

MAA 6406 Complex Analysis I 3 Credits

Grading Scheme: Letter Grade

Rapid survey of properties of complex numbers, linear transformations, geometric forms and necessary concepts from topology. Complex integration. Cauchy's theorem and its corollaries. Taylor series and the implicit function theorem in complex form. Conformality and the Riemann-Caratheodory mapping theorem. Theorems of Bloch, Schottky, and the big and little theorems of Picard. Harmonicity and Dirichlet's problems.

Prerequisite: MAA 5229.

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MAA 6407 Complex Analysis II 3 Credits

Grading Scheme: Letter Grade

Complex Analysis II

Prerequisite: MAA 6406.

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MAA 6616 Analysis I 3 Credits

Grading Scheme: Letter Grade

Fundamentals of measure and integration theory, including Lp spaces and the Radon-Nikodym theorem. Introduction to functional analysis: Banach spaces, Hilbert spaces, and the theory of linear operators.

Prerequisite: MAA 5229.

Catalog Program Pages Referencing MAA 6616

MAA 6617 Analysis II 3 Credits

Grading Scheme: Letter Grade

Continuation of MAA 6616 Analysis I.

Prerequisite: MAA 6616.

Catalog Program Pages Referencing MAA 6617

MAA 7526 Advanced Topics in Functional Analysis I 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Algebraic and topological approach to current material and methods in analysis.

Prerequisite: MAA 6617

Catalog Program Pages Referencing MAA 7526

MAA 7527 Advanced Topics in Functional Analysis II 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Advanced Topics in Functional Analysis II

Prerequisite: MAA 7526.

Catalog Program Pages Referencing MAA 7527

MAD 6206 Combinatorial Theory I 3 Credits

Grading Scheme: Letter Grade

Matching theory, Ramsey's theorem, lattice theory, Mobius inversion, generating functions. Polya's theorem, matroids, applications, block designs, graph theory.

Catalog Program Pages Referencing MAD 6206

MAD 6207 Combinatorial Theory II 3 Credits

Grading Scheme: Letter Grade

Combinatorial Theory II

Prerequisite: MAD 6206.

Catalog Program Pages Referencing MAD 6207

MAD 6406 Numerical Linear Algebra 3 Credits

Grading Scheme: Letter Grade

Topics most useful in applications with emphasis on numerical techniques: systems of linear equations, positive definite and toeplitz systems, least squares problems, singular value decomposition, and eigenvalues. Numerical stability and efficiency of algorithms as well as effect of perturbations on the problem. Companion to MAD 6407.

Prerequisite: MAS 3114, 4105, or 4124; and programming language.

Catalog Program Pages Referencing MAD 6406

MAD 6407 Numerical Analysis 3 Credits

Grading Scheme: Letter Grade

Numerical techniques to solve systems of nonlinear equations to approximate functions, to compute derivatives, to evaluate integrals, and to integrate systems of differential equations. Introduction to numerical techniques for partial differential equations. Companion to MAD 6406.

Prerequisite: MAA 4212, MAA 5105, or MAA 5229; and programming language.

Catalog Program Pages Referencing MAD 6407

MAD 7396 Topics in Combinatorial Theory I 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Topics chosen from among graph theory, coding theory, matroid theory, finite geometries, projective geometry, difference methods, and Latin squares.

Prerequisite: MAS 5312.

Catalog Program Pages Referencing MAD 7396

MAD 7397 Topics in Combinatorial Theory II 3 Credits

Grading Scheme: Letter Grade

Topics in Combinatorial Theory II

Prerequisite: MAD 7396.

Catalog Program Pages Referencing MAD 7397

MAE 6940 Supervised Teaching 1-5 Credits, Max 5 Credits

Grading Scheme: S/U

Supervised Teaching

Prerequisite: consent of graduate adviser.

Catalog Program Pages Referencing MAE 6940

MAE 6943 Internship in College Teaching 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Internship in College Teaching

Prerequisite: consent of graduate adviser.

Catalog Program Pages Referencing MAE 6943

MAP 5304 Intermediate Differential Equations for Engineers and Physical Scientists 3 Credits

Grading Scheme: Letter Grade

Intermediate Differential Equations for Engineers and Physical Scientists

Catalog Program Pages Referencing MAP 5304

MAP 5345 Introduction to Partial Differential Equations 3 Credits

Grading Scheme: Letter Grade

Introduction to Partial Differential Equations

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MAP 5489 Modeling in Mathematical Biology 3 Credits

Grading Scheme: Letter Grade

Mathematical models of biological systems. Models of growth, predator-prey populations, competition, chemostat, epidemics, excitable systems, and analytical tools such as linearization, phase-plane analysis, Poincare-Bendixson theory, Lyapunov functions, and bifurcation analysis.

Prerequisite: undergraduate course in ordinary differential equations.

Catalog Program Pages Referencing MAP 5489

MAP 6208 Numerical Optimization 3 Credits

Grading Scheme: Letter Grade

Unconstrained and constrained optimization, linear and nonlinear programming, gradient, multiplier, and quasi-Newton methods. Penalty, multiplier, and projection methods for constrained problems.

Prerequisite: MAD 6406 and MAD 6407 or consent of instructor.

Catalog Program Pages Referencing MAP 6208

MAP 6327 Applied Differential Equations I 3 Credits

Grading Scheme: Letter Grade

Theory and methods for solving linear and nonlinear systems of differential equations and partial differential equations. Applications and computer techniques included.

Prerequisite: MAA 5229.

Catalog Program Pages Referencing MAP 6327

MAP 6356 Partial Differential Equations I 3 Credits

Grading Scheme: Letter Grade

Cauchy-Kowalewski theorem, first order equations, classification of equations, hyperbolic equations, elliptic equations, parabolic equations, hyperbolic systems, nonlinear hyperbolic systems, existence theory based on functional analysis. Applications to physical sciences.

Prerequisite: MAA 5229, MAP 5345 or MAP 6506.

Catalog Program Pages Referencing MAP 6356

MAP 6357 Partial Differential Equations II 3 Credits

Grading Scheme: Letter Grade

Partial Differential Equations II

Prerequisite: MAP 6356.

Catalog Program Pages Referencing MAP 6357

MAP 6375 Numerical Partial Differential Equations 3 Credits

Grading Scheme: Letter Grade

Introduction to partial differential equations and fundamental concepts. Parabolic equations: finite differences, consistency, convergence and stability, 2- and 3-dimensional problems. Elliptic equations: finite differences, solution to linear equations, boundary integral equation methods. Hyperbolic equations: finite differences and method of characteristics. Introduction to finite elements. Methods of lines.

Prerequisite: MAD 6406 and MAD 6407 or consent of instructor.

Catalog Program Pages Referencing MAP 6375

MAP 6376 Finite Element Method 3 Credits

Grading Scheme: Letter Grade

Variational formulations of partial differential equations, finite element approximations; addresses theoretical framework and numerical issues. Finite element spaces in one, two, and three dimensions. Error analysis. Nonconforming finite element spaces. Isoparametric approximations to boundary conditions.

Prerequisite: MAD 6406 and MAD 6407 or consent of instructor.

Catalog Program Pages Referencing MAP 6376

MAP 6467 Stochastic Differential Equations and Filtering Theory I 3 Credits

Grading Scheme: Letter Grade

Introduction to random functions; Brownian motion process. Ito's stochastic integral; Ito's stochastic calculus; stochastic differential equations. Linear filtering; Kalman filtering; nonlinear filtering theory.

Catalog Program Pages Referencing MAP 6467

MAP 6468 Stochastic Differential Equations and Filtering Theory II 3 Credits

Grading Scheme: Letter Grade

Stochastic Differential Equations and Filtering Theory II

Prerequisite: MAP 6467.

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MAP 6472 Probability and Potential Theory I 3 Credits

Grading Scheme: Letter Grade

Random variables, independence and conditioning. Laws of large numbers and the Central Limit Theorem. Stochastic processes, martingales, Gaussian processes, Markov processes, potentials and excessive functions.

Prerequisite: MAA 5229 or STA 6326.

Catalog Program Pages Referencing MAP 6472

MAP 6473 Probability and Potential Theory II 3 Credits

Grading Scheme: Letter Grade

Probability and Potential Theory II

Prerequisite: MAP 6472

Catalog Program Pages Referencing MAP 6473

MAP 6487 Biomathematics Seminar I 3 Credits

Grading Scheme: Letter Grade

Stochastic processes, differential equations, and reaction-diffusion equations used to model various biological processes. Among the applications covered are the following: population dynamics, epidemiology, genetics, enzyme kinetics, cell differentiation and morphogenesis, nerve impulse generation, and aggregation of slime mold. The course is designed to benefit graduate students in biological sciences, as well as mathematics.

Prerequisite: MAC 2312, MAP 2302, STA 6326 or MAP 4102.

Catalog Program Pages Referencing MAP 6487

MAP 6488 Biomathematics Seminar II 3 Credits

Grading Scheme: Letter Grade

Continuation of MAP 6487.

Prerequisite: MAP 6487.

Catalog Program Pages Referencing MAP 6488

MAP 6505 Mathematical Methods of Physics and Engineering 3 Credits

Grading Scheme: Letter Grade

Orthogonal functions; theory of distributions; integral equations; eigenfunctions and Green's functions; special functions; boundary and initial value problems, with emphasis on potential theory (Laplace and Poisson equations); the wave equation; and the diffusion equation.

Prerequisite: MAA 5404, MAP 5304, MAP 5345, MAS 5157    or equivalent.

Catalog Program Pages Referencing MAP 6505

MAP 6506 Mathematical Methods of Physics and Engineering II 3 Credits

Grading Scheme: Letter Grade

Mathematical Methods of Physics and Engineering II

Prerequisite: MAP 6505.

Catalog Program Pages Referencing MAP 6506

MAP 6941 Internship in Applied Mathematics 1-5 Credits, Max 9 Credits

Grading Scheme: Letter Grade

Mathematical research on projects sponsored by a university laboratory or an off-campus industrial internship program.

Prerequisite: consent of supervisory committee chair.

Catalog Program Pages Referencing MAP 6941

MAP 7436 Seminar in Applied Mathematics I 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Various topics in applications of mathematics both classical and in areas of current research.

Catalog Program Pages Referencing MAP 7436

MAP 7437 Seminar in Applied Mathematics II 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Seminar in Applied Mathematics II

Catalog Program Pages Referencing MAP 7437

MAS 5311 Introductory Algebra I 3 Credits

Grading Scheme: Letter Grade

The basic algebraic systems: groups, rings, vector spaces, and modules. Linear transformations, matrices, and determinants.

Prerequisite: MAS 4105 and 4302.

Catalog Program Pages Referencing MAS 5311

MAS 5312 Introductory Algebra II 3 Credits

Grading Scheme: Letter Grade

Introductory Algebra II

Prerequisite: MAS 5311.

Catalog Program Pages Referencing MAS 5312

MAS 6331 Algebra I 3 Credits

Grading Scheme: Letter Grade

Solvable and nilpotent groups, Jordan-Holder theorem, abelian groups, Galois theory, Noetherian rings, Dedekind domains, Jacobson radical, Jacobson density theorem, Wedderburn-Artin theorem.

Prerequisite: MAS 5312.

Catalog Program Pages Referencing MAS 6331

MAS 6332 Algebra II 3 Credits

Grading Scheme: Letter Grade

Algebra II

Prerequisite: MAS 6331.

Catalog Program Pages Referencing MAS 6332

MAS 7215 Theory of Numbers I 3 Credits

Grading Scheme: Letter Grade

Introduction to theory of numbers; theorems on divisibility; congruence, number-theoretic functions; primitive roots and indices; quadratic reciprocity law; Diophantine equations and continued functions.

Prerequisite: 2 of MAA 6407, MAA 6617, MAS 6332.

Catalog Program Pages Referencing MAS 7215

MAS 7216 Theory of Numbers II 3 Credits

Grading Scheme: Letter Grade

Theory of Numbers II

Prerequisite: MAS 7215.

Catalog Program Pages Referencing MAS 7216

MAS 7396 Advanced Topics in Algebra I 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Current topics in algebra.

Prerequisite: MAA 6407, MAA 6617, MAS 6332 or MTG 6347.

Catalog Program Pages Referencing MAS 7396

MAS 7397 Topics in Algebra II 3 Credits

Grading Scheme: Letter Grade

Topics in Algebra II

Catalog Program Pages Referencing MAS 7397

MAT 6905 Individual Work 3 Credits, Max 9 Credits

Grading Scheme: Letter Grade

Individual Work

Catalog Program Pages Referencing MAT 6905

MAT 6910 Supervised Research 1-5 Credits, Max 5 Credits

Grading Scheme: S/U

Supervised Research

Catalog Program Pages Referencing MAT 6910

MAT 6932 Special Topics in Mathematics 3 Credits, Max 9 Credits

Grading Scheme: Letter Grade

Special Topics in Mathematics

Prerequisite: consent of graduate adviser, who should be consulted well in advance of registration.

Catalog Program Pages Referencing MAT 6932

MAT 6971 Research for Master's Thesis 1-15 Credits

Grading Scheme: S/U

Research for Master's Thesis

Catalog Program Pages Referencing MAT 6971

MAT 7979 Advanced Research 1-12 Credits

Grading Scheme: S/U

Research for doctoral students before admission to candidacy. Designed for students with a master's degree in the field of study or for students who have been accepted for a doctoral program. Not appropriate for students who have been admitted to candidacy.

Catalog Program Pages Referencing MAT 7979

MAT 7980 Research for Doctoral Dissertation 1-15 Credits

Grading Scheme: S/U

Research for Doctoral Dissertation

Catalog Program Pages Referencing MAT 7980

MHF 5107 Introduction to Set Theory 3 Credits

Grading Scheme: Letter Grade

Basic axioms and concepts of set theory, axiom of choice, Zorn's lemma, Schroder-Bernstein theorem, cardinal numbers, ordinal numbers, and the continuum hypothesis.

Catalog Program Pages Referencing MHF 5107

MHF 5207 Foundations of Mathematics 3 Credits

Grading Scheme: Letter Grade

Models and proofs. Foundations of the real and natural number systems. Algorithms. Turing Machines, undecidability and independence. Examples and applications in algebra, analysis, geometry, and topology.

Catalog Program Pages Referencing MHF 5207

MHF 6306 Mathematical Logic I 3 Credits

Grading Scheme: Letter Grade

Languages, models, and theories; Godel's completeness and incompleteness theorems; formal number theory and axiomatic set theory; applications to other areas of mathematics.

Catalog Program Pages Referencing MHF 6306

MTG 5316 Introduction to Topology I 3 Credits

Grading Scheme: Letter Grade

Basic axioms and concepts of point-set topology, compactness, connectedness, separation axioms, metric spaces, metrization. Tietze extension theorem. Urysohn lemma, Tychonoff theorem, fundamental group.

Catalog Program Pages Referencing MTG 5316

MTG 5317 Introduction to Topology II 3 Credits

Grading Scheme: Letter Grade

Introduction to Topology II

Prerequisite: MTG 5316.

Catalog Program Pages Referencing MTG 5317

MTG 5411 Introduction to Fractal Geometry 3 Credits

Grading Scheme: Letter Grade

Introduction to techniques for generating and analyzing fractal sets. Hausdorff dimension, self-similarity, and iterated function systems. If time permits, Brownian paths, Julia sets, and Mandelbrot set.

Prerequisite: advanced calculus or consent of instructor.

Catalog Program Pages Referencing MTG 5411

MTG 5412 Introduction to Dynamical Systems and Chaos 3 Credits

Grading Scheme: Letter Grade

Introduction to nonlinear dynamical systems and chaos. One-dimensional systems, bifurcation theory, symbolic dynamics, Sarkovskii's theorem, Schwarzian derivative, Bernoulli shifts and subshifts of finite type, and kneading theory. If time permits, toral automorphisms, Henon map and complex dynamics.

Prerequisite: advanced calculus or consent of instructor.

Catalog Program Pages Referencing MTG 5412

MTG 6256 Differential Geometry I 3 Credits

Grading Scheme: Letter Grade

Foundations of the theory of smooth manifolds, vector fields, and differential forms. Topics chosen from a list including differential topology, Lie groups, symplectic geometry, Riemannian geometry, and applications to physics.

Prerequisite: consent of instructor.

Catalog Program Pages Referencing MTG 6256

MTG 6257 Differential Geometry II 3 Credits

Grading Scheme: Letter Grade

Differential Geometry II

Prerequisite: MTG 6256.

Catalog Program Pages Referencing MTG 6257

MTG 6346 Topology I 3 Credits

Grading Scheme: Letter Grade

A basic introduction to advanced topology. Topics covered include general topology, algebraic topology, homotopy theory and topology of manifolds.

Prerequisite: MTG 5317.

Catalog Program Pages Referencing MTG 6346

MTG 6347 Topology II 3 Credits

Grading Scheme: Letter Grade

Topology II

Prerequisite: MTG 6346.

Catalog Program Pages Referencing MTG 6347

MTG 6401 Ergodic Theory and Dynamical Systems I 3 Credits

Grading Scheme: Letter Grade

Periodic points, recurrence, nonwandering and chain recurrent sets, topological conjugacy, minimal sets. Topological entropy, metric entropy. Measure preserving transformations, ergodicity, mixing. Birkhoff's ergodic theorem. Bernouilli shifts. Anosov diffeomorphisms, structural stability, hyperbolic sets. Basic sets, symbolic dynamics, Markov partitions. Lyapunov exponents, KAM (Kolmogorov, Arnold, Moser) theory.

Prerequisite: MTG 5317, MAA 6617, or consent of instructor.

Catalog Program Pages Referencing MTG 6401

MTG 6402 Ergodic Theory and Dynamical Systems II 3 Credits

Grading Scheme: Letter Grade

Continuation of MTG 6401.

Prerequisite: MTG 6401.

Catalog Program Pages Referencing MTG 6402

MTG 7396 Advanced Topics in Topology I 3 Credits, Max 6 Credits

Grading Scheme: Letter Grade

Topics change yearly.

Prerequisite: MTG 6347.

Catalog Program Pages Referencing MTG 7396

MTG 7397 Advanced Topics in Topology II 3 Credits

Grading Scheme: Letter Grade

Discussion of advanced topics in topology and its applications.

Prerequisite: MTG 7396.

Catalog Program Pages Referencing MTG 7397