MATH | 410 | Advanced Calculus I | 3 OR 4 hours. | Functions of several variables, differentials, theorems of partial differentiation. Calculus of vector fields, line and surface integrals, conservative fields, Stokes's and divergence theorems. Cartesian tensors. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 210. |

MATH | 411 | Advanced Calculus II | 3 OR 4 hours. | Implicit and inverse function theorems, transformations, Jacobians. Point-set theory. Sequences, infinite series, convergence tests, uniform convergence. Improper integrals, gamma and beta functions, Laplace transform. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 410. |

MATH | 414 | Analysis II | 3 OR 4 hours. | Sequences and series of functions. Uniform convergence. Taylor's theorem. Topology of metric spaces, with emphasis on the real numbers. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 313. |

MATH | 417 | Complex Analysis with Applications | 3 OR 4 hours. | Complex numbers, analytic functions, complex integration, Taylor and Laurent series, residue calculus, branch cuts, conformal mapping, argument principle, Rouche's theorem, Poisson integral formula, analytic continuation. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade C or better in MATH 210. |

MATH | 419 | Models in Applied Mathematics | 3 OR 4 hours. | Introduction to mathematical modeling; scaling, graphical methods, optimization, computer simulation, stability, differential equation models, elementary numerical methods, applications in biology, chemistry, engineering and physics. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 220 and grade of C or better in MCS 260. |

MATH | 425 | Linear Algebra II | 3 OR 4 hours. | Canonical forms of a linear transformation, inner product spaces, spectral theorem, principal axis theorem, quadratic forms, special topics such as linear programming. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 320. |

MATH | 430 | Formal Logic I | 3 OR 4 hours. | First order logic, syntax and semantics, completeness-incompleteness. 3 undergraduate hours. 4 graduate hours. Credit is not given for MATH 430 if the student has credit for PHIL 416. Prerequisite(s): Grade of C or better in CS 202 or grade of C or better in MCS 261 or grade of C or better in MATH 215. |

MATH | 431 | Abstract Algebra II | 3 OR 4 hours. | Further topics in abstract algebra: Sylow Theorems, Galois Theory, finitely generated modules over a principal ideal domain. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 320 and grade of C or better in MATH 330. |

MATH | 435 | Foundations of Number Theory | 3 OR 4 hours. | Primes, divisibility, congruences, Chinese remainder theorem, primitive roots, quadratic residues, quadratic reciprocity, and Jacobi symbols. The Euclidean algorithm and strategies of computer programming. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 215. |

MATH | 436 | Number Theory for Applications | 3 OR 4 hours. | Primality testing methods of Lehmer, Rumely, Cohen-Lenstra, Atkin. Factorization methods of Gauss, Pollard, Shanks, Lenstra, and quadratic sieve. Computer algorithms involving libraries and nested subroutines. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 435. |

MATH | 442 | Differential Geometry of Curves and Surfaces | 3 OR 4 hours. | Frenet formulas, isoperimetric inequality, local theory of surfaces, Gaussian and mean curvature, geodesics, parallelism, and the Guass-Bonnet theorem. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 320. |

MATH | 445 | Introduction to Topology I | 3 OR 4 hours. | Elements of metric spaces and topological spaces including product and quotient spaces, compactness, connectedness, and completeness. Examples from Euclidean space and function spaces. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 313. |

MATH | 446 | Introduction to Topology II | 3 OR 4 hours. | Topics in topology chosen from the following: advanced point set topology, piecewise linear topology, fundamental group and knots, differential topology, applications to physics and biology. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 445. |

MATH | 480 | Applied Differential Equations | 3 OR 4 hours. | Linear first-order systems. Numerical methods. Nonlinear differential equations and stability. Introduction to partial differential equations. Sturm-Liouville theory. Boundary value problems and Green's functions. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 220. |

MATH | 481 | Applied Partial Differential Equations | 3 OR 4 hours. | Initial value and boundary value problems for second order linear equations. Eiqenfunction expansions and Sturm-Liouville theory. Green's functions. Fourier transform. Characteristics. Laplace transform. 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 220. |

MATH | 494 | Special Topics in Mathematics | 3 OR 4 hours. | Course content is announced prior to each term in which it is given. 3 undergraduate hours. 4 graduate hours. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 496 | Independent Study | 1 TO 4 hours. | Reading course supervised by a faculty member. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the instructor and the department. |

MATH | 502 | Metamathematics I | 4 hours. | First order logic, completeness theorem and model theory. Same as PHIL 562. Prerequisite(s): MATH 430 or consent of the instructor. |

MATH | 503 | Metamathematics II | 4 hours. | Incompleteness theorems, elementary recursion theory and proof theory, first and second order arithmetic. Same as PHIL 563. Prerequisite(s): MATH 502 or PHIL 562. |

MATH | 504 | Set Theory I | 4 hours. | Naive and axiomatic set theory. Independence of the continuum hypothesis and the axiom of choice. Same as PHIL 565. Prerequisite(s): MATH 430 or MATH 502 or PHIL 562. |

MATH | 506 | Model Theory I | 4 hours. | Introduction to stability theory: categoricity, stability, forking, finite equivalence relation theorem, indiscernibles, orthogonality. Same as PHIL 567. Prerequisite(s): MATH 502 or PHIL 562. |

MATH | 507 | Model Theory II | 4 hours. | Intermediate stability theory: dependence, prime models, isolation, regular types, dimension, weight. Same as PHIL 568. Prerequisite(s): MATH 506 or PHIL 567. |

MATH | 509 | Universal Algebra I | 4 hours. | Algebraic systems, homomorphisms, congruences, subalgebras, direct and subdirect products. Equational classes, free algebras, Birkhoff's theorem. Malcev conditions, congruence distributive equational classes. Prerequisite(s): MATH 330 and MATH 425. |

MATH | 510 | Universal Algebra II | 4 hours. | Discriminator and directly representable varieties, ultraproducts and quasivarieties, finitely based equational theories, commutator and center. Prerequisite(s): MATH 509. |

MATH | 512 | Advanced Topics in Logic | 4 hours. | Advanced topics in modern logic; e.g. descriptive set theory, model theory of fields, theory of hierarchies, stable groups. Same as PHIL 569. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 513 | Advanced Topics in Universal Algebra and Lattice Theory | 4 hours. | Special topics. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 514 | Number Theory I | 4 hours. | Introduction to classical, algebraic, and analytic, number theory. Euclid's algorithm, unique factorization, quadratic reciprocity, and Gauss sums, quadratic forms, real approximations, arithmetic functions, Diophantine equations. |

MATH | 515 | Number Theory II | 4 hours. | Introduction to classical, algebraic, and analytic number theory. Algebraic number fields, units, ideals, and P-adic theory. Riemann Zeta-function, Dirichlet's theorem, prime number theorem. Prerequisite(s): MATH 514. |

MATH | 516 | Second Course in Abstract Algebra I | 4 hours. | Structure of groups, Sylow theorems, solvable groups; structure of rings, polynomial rings, projective and injective modules, finitely generated modules over a PID. Prerequisite(s): MATH 330 and MATH 425. |

MATH | 517 | Second Course in Abstract Algebra II | 4 hours. | Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems. Prerequisite(s): MATH 516. |

MATH | 518 | Representation Theory | 4 hours. | Major areas of representation theory, including structure of group algebras, Wedderburn theorems, characters and orthogonality relations, idempotents and blocks. Prerequisite(s): MATH 517. |

MATH | 519 | Algebraic Groups | 4 hours. | Classical groups as examples; necessary results from algebraic geometry; structure and classification of semisimple algebraic groups. Prerequisite(s): MATH 517. |

MATH | 520 | Commutative and Homological Algebra | 4 hours. | Commutative rings; primary decomposition; integral closure; valuations; dimension theory; regular sequences; projective and injective dimension; chain complexes and homology; Ext and Tor; Koszul complex; homological study of regular rings. Prerequisite(s): MATH 516 and MATH 517; or consent of the instructor. |

MATH | 531 | Advanced Topics in Algebra | 4 hours. | Researchlevel topics such as groups and geometries, equivalencies of module categories, representations of Lie-type groups. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 533 | Real Analysis I | 4 hours. | Introduction to real analysis. Lebesgue measure and integration, differ entiation, L-p classes, abstract integration. Prerequisite(s): MATH 411 or MATH 414 or the equivalent. |

MATH | 534 | Real Analysis II | 4 hours. | Continuation of MATH 533. Prerequisite(s): MATH 417. |

MATH | 535 | Complex Analysis I | 4 hours. | Analytic functions as mappings. Cauchy theory. Power Series. Partial fractions. Infinite products. Prerequisite(s): MATH 411. |

MATH | 536 | Complex Analysis II | 4 hours. | Normal families, Riemann mapping theorem. Analytic continuation, Harmonic and subharmonic functions, Picard theorem, selected topics. Prerequisite(s): MATH 535. |

MATH | 537 | Introduction to Harmonic Analysis I | 4 hours. | Fourier transform on L(p) spaces, Wiener's Tauberian theorem, Hilbert transform, Paley Wiener theory. Prerequisite(s): MATH 533; and MATH 417 or MATH 535. |

MATH | 539 | Functional Analysis I | 4 hours. | Topological vector spaces, Hilbert spaces, Hahn-Banach theorem, open mapping, uniform boundedness principle, linear operators in a Banach space, compact operators. Prerequisite(s): MATH 533. |

MATH | 541 | Partial Differential Equations I | 4 hours. | Theory of distributions; fundamental solutions of the heat equation, wave equation, and Laplace equation. Harmonic functions. Cauchy problem for the wave equation. Prerequisite(s): MATH 417. |

MATH | 542 | Partial Differential Equations II | 4 hours. | Cauchy problem for hyperbolic equations. Propagation of singularities. Boundary value problems for elliptic equations. Prerequisite(s): MATH 541. |

MATH | 546 | Advanced Topics in Analysis | 4 hours. | Subject may vary from semester to semester. Topics include partial differential equations, several complex variables, harmonic analysis and ergodic theory. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 547 | Algebraic Topology I | 4 hours. | The fundamental group and its applications, covering spaces, classification of compact surfaces, introduction to homology, development of singular homology theory, applications of homology. Prerequisite(s): MATH 330 and MATH 445. |

MATH | 548 | Algebraic Topology II | 4 hours. | Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics. Prerequisite(s): MATH 547. |

MATH | 549 | Differentiable Manifolds I | 4 hours. | Smooth manifolds and maps, tangent and normal bundles, Sard's theorem and transversality, embedding, differential forms, Stokes's theorem, degree theory, vector fields. Prerequisite(s): MATH 445; and MATH 310 or MATH 320 or the equivalent. |

MATH | 550 | Differentiable Manifolds II | 4 hours. | Vector bundles and classifying spaces, lie groups and lie algbras, tensors, Hodge theory, Poincare duality. Topics from elliptic operators, Morse theory, cobordism theory, deRahm theory, characteristic classes. Prerequisite(s): MATH 549. |

MATH | 551 | Riemannian Geometry | 4 hours. | Riemannian metrics and Levi-Civita connections, geodesics and completeness, curvature, first and second variation of arc length, comparison theorems. Prerequisite(s): MATH 442 and MATH 549. |

MATH | 552 | Algebraic Geometry I | 4 hours. | Basic commutative algebra, affine and projective varieties, regular and rational maps, function fields, dimension and smoothness, projective curves, schemes, sheaves, and cohomology, posiive characteristic. |

MATH | 553 | Algebraic Geometry II | 4 hours. | Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces. Prerequisite(s): MATH 552. |

MATH | 554 | Complex Manifolds I | 4 hours. | Holomorphic functions in several variables, Riemann surfaces, Sheaf theory, vector bundles, Stein manifolds, Cartan theorem A and B, Grauert direct image theorem. Prerequisite(s): MATH 517 and MATH 535. |

MATH | 555 | Complex Manifolds II | 4 hours. | Dolbeault Cohomology, Serre duality, Hodge theory, Kadaira vanishing and embedding theorem, Lefschitz theorem, Complex Tori, Kahler manifolds. Prerequisite(s): MATH 517 and MATH 535. |

MATH | 568 | Topics in Algebraic Topology | 4 hours. | Homotopy groups and fibrations. The Serre spectral sequence and its applications. Classifying spaces of classical groups. Characteristic classes of vector bundles. May be repeated. Students may register in more than one section per term. Prerequisite(s): MATH 548 or consent of the instructor. |

MATH | 569 | Advanced Topics in Geometric and Differential Topology | 4 hours. | Topics from areas such as index theory, Lefschetz theory, cyclic theory, KK theory, non-commutative geometry, 3-manifold topology, hyperbolic manifolds, geometric group theory, and knot theory. Prerequisite(s): Approval of the department. |

MATH | 570 | Advanced Topics in Differential Geometry | 4 hours. | Subject may vary from semester to semester. Topics may include eigenvalues in Riemannian geometry, curvature and homology, partial differential relations, harmonic mappings between Riemannian manifolds hyperbolic geometry, arrangement of hyperplanes. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 571 | Advanced Topics in Algebraic Geometry | 4 hours. | Various topics such as algebraic curves, surfaces, higher dimensional geometry, singularities theory, moduli problems, vector bundles, intersection theory, arithematical algebraic geometry, and topologies of algebraic varieties. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 574 | Applied Optimal Control | 4 hours. | Introduction to optimal control theory; calculus of variations, maximum principle, dynamic programming, feedback control, linear systems with quadratic criteria, singular control, optimal filtering, stochastic control. Prerequisite(s): MATH 411 or consent of the instructor. |

MATH | 575 | Integral Equations and Applications | 4 hours. | Fredholm and Volterra equations, Fredholm determinants, separable and symmetric kernels, Neumann series, transform methods, Wiener-Hopf method, Cauchy kernels, nonlinear equations, perturbation methods. Prerequisite(s): MATH 411 and MATH 417 and MATH 481; or consent of instructor. |

MATH | 576 | Classical Methods of Partial Differential Equations | 4 hours. | First and second order equations, method of characteristics, weak solutions, distributions, wave, Laplace, Poisson, heat equations, energy methods, regularity problems, Green functions, maximum principles, Sobolev spaces, imbedding theorems. Prerequisite(s): MATH 410 and MATH 481 and MATH 533; or consent of instructor. |

MATH | 577 | Advanced Partial Differential Equations | 4 hours. | Linear elliptic theory, maximum principles, fixed point methods, semigroups and nonlinear dynamics, systems of conservation laws, shocks and waves, parabolic equations, bifurcation, nonlinear elliptic theory. Prerequisite(s): MATH 533 and MATH 576 or consent of the instructor. |

MATH | 578 | Asymptotic Methods | 4 hours. | Asymptotic series, Laplace's method, stationary phase, steepest descent method, Stokes phenomena, uniform expansions, multi-dimensional Laplace integrals, Euler-MacLaurin formula, irregular singular points, WKBJ method. Prerequisite(s): MATH 417 and MATH 481; or consent of instructor. |

MATH | 579 | Singular Perturbations | 4 hours. | Algebraic and transcendental equations, regular perturbation expansions of differential equations, matched asymptotic expansions, boundary layer theory, Poincare-Lindstedt, multiple scales, bifurcation theory, homogenization. Prerequisite(s): MATH 481 or consent of the instructor. |

MATH | 580 | Mathematics of Fluid Mechanics | 4 hours. | Development of concepts and techniques used in mathematical models of fluid motions. Euler and Navier Stokes equations. Vorticity and vortex motion. Waves and instabilities. Viscous fluids and boundary layers. Asymptotic methods. Prerequisite(s): Grade of C or better in MATH 410 and grade of C or better in MATH 417 and grade of C or better in MATH 481. |

MATH | 581 | Special Topics in Fluid Mechanics | 4 hours. | Geophysical fluids with applications to oceanography and meteorology, astrophysical fluids, magnetohydrodynamics and plasmas. Prerequisite(s): Grade of C or better in MATH 580. |

MATH | 582 | Linear and Nonlinear Waves | 4 hours. | Derivation and analysis of models for linear and nonlinear wave propagation, including acoustic, hydrodynamic, and eletromagnetic waves. Analytical techniques include exact formulas and asymptotic methods. Prerequisite(s): MATH 480 and MATH 481; or consent of the instructor. |

MATH | 583 | Topics in Wave Propagation | 4 hours. | Rigorous, asymptotic, and numerical analysis of mathematical models for linear and nonlinear waves. Techniques include inverse scattering, asymptotic analysis, and finite-difference and spectral methods. Prerequisite(s): MATH 480 and MATH 481; consent of the instructor. |

MATH | 584 | Applied Stochastic Models | 4 hours. | Applications of stochastic models in chemistry, physics, biology, queueing, filtering, and stochastic control, diffusion approximations, Brownian motion, stochastic calculus, stochastically perturbed dynamical systems, first passage times. Prerequisite(s): MATH 417 and MATH 481 and STAT 401, or consent of the instructor. |

MATH | 586 | Computational Finance | 4 hours. | Introduction to the mathematics of financial derivatives; options, asset price random walks, Black-Scholes model; partial differential techniques for option valuation, binomial models, numerical methods; exotic options, interest-rate derivatives. Prerequisite(s): Grade of C or better in MATH 220 and grade of C or better in STAT 381; or consent of the instructor. |

MATH | 587 | Nonlinear Dynamics, Chaos and Applications | 4 hours. | Introduction to nonlinear dynamics, bifurcations, chaotic dynamics, and strange attractors. Linear response to small external fluctuations. Related numerical methods. Prerequisite(s): Grade of C or better in MATH 480 and Grade of C or better in MCS 471; or consent of the instructor. |

MATH | 589 | Teaching and Presentation of Mathematics | 2 hours. | Strategies and techniques for effective teaching in college and for mathematical consulting. Observation and evaluation, classroom management, presenting mathematics in multidisciplinary research teams. Required for teaching assistants in MSCS. No graduation credit awarded for students enrolled in the Master of Science in the Teaching of Mathematics degree program. |

MATH | 590 | Advanced Topics in Applied Mathematics | 4 hours. | Topics from areas such as: elastic scattering, nonlinear problems in chemistry and physics, mathematical biology, stochastic optimal control, geophysical fluid dynamics, stability theory, queueing theory. Prerequisite(s): Approval of the department. |

MATH | 591 | Seminar on Mathematics Curricula | 4 hours. | Examination of research and reports on mathematics curricula. Analysis of research in teaching and learning mathematics. Developments in using technology in mathematics teaching. Prerequisite(s): Enrollment in the Doctor of Arts program in mathematics or consent of the instructor. |

MATH | 592 | Seminar on Mathematics: Philosophy and Methodology | 4 hours. | Problems related to teaching and learning mathematics. Analysis of work of Piaget, Gagne, Bruner, Ausabel, Freudenthal, and others and their relation to mathematics teaching. Prerequisite(s): Enrollment in the Doctor of Arts program in mathematics or consent of instructor. |

MATH | 593 | Graduate Student Seminar | 1 hours. | For graduate students who wish to receive credit for participating in a learning seminar whose weekly time commitment is not sufficient for a reading course. This seminar must be sponsored by a faculty member. Satisfactory/Unsatisfactory grading only. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 594 | Internship in Mathematics | 0 TO 8 hours. | Under the direction of a faculty advisor, students work in government or industry on problems related to their major field of interest. At the end of internship, the student must present a seminar on the internship experiences. Satisfactory/Unsatisfactory grading only. May be repeated to a maximum of 8 hours. Only 4 credit hours count toward the 32 credit hours required for the MS in MISI degree. Does not count toward the 12 credit hours of 500-level courses requirement. Prerequisite(s): Completion of the core courses in the degree program in which the student is enrolled and approval of the internship program by the graduate advisor and the graduate studies committee. |

MATH | 595 | Research Seminar | 1 hours. | Current developments in research with presentations by faculty, students, and visitors. Satisfactory/Unsatisfactory grading only. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |

MATH | 596 | Independent Study | 1 TO 4 hours. | Reading course supervised by a faculty member. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the instructor and the department. |

MATH | 598 | Master's Thesis | 0 TO 16 hours. | Research work under the supervision of a faculty member leading to the completion of a master's thesis. Satisfactory/Unsatisfactory grading only. Prerequisite(s): Approval of the department. |

MATH | 599 | Thesis Research | 0 TO 16 hours. | Research work under the supervision of a faculty member. Satisfactory/Unsatisfactory grading only. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. |