303 Theory of Interest and Life Insurance. (Crosslisted with Act Sci) 3 cr. Application of calculus to compound interest and insurance functions; interest compounded discretely and continuously; force of interest function; annuities payable discretely and continuously; bonds and yield rates; life tables, life annuities, single and annual premiums for insurance and annuities; reserves. P: Math 234 or con reg, or cons inst.
309 Introduction to Mathematical Statistics. (Crosslisted with Stat) 4 cr. Probability and combinatorial methods, discrete and continuous, univariate and multivariate distributions, expected values, moments, normal distribution and derived distributions, estimation. P: For majors in math and stats, Math 223 or 234.
310 Introduction to Mathematical Statistics. (Crosslisted with Stat) 4 cr. Unbiased estimation, maximum likelihood estimation, confidence intervals, tests of hypotheses, Neyman-Pearson fundamental lemma, likelihood ratio test, applications to general linear model and analysis of variance, categorical data analysis, nonparametric methods. P: For majors in Math and Stat, Math 309 or Stat 309.
319 Techniques in Ordinary Differential Equations. 3 cr. Review of linear differential equations; series solution of linear differential equations; boundary value problems; Laplace transforms; possibly numerical methods and two dimensional autonomous systems. P: Math 222.
320 Linear Algebra and Differential Equations. 3 cr. Introduction to linear algebra, including matrices, linear transformations, eigenvalues and eigenvectors. Linear systems of differential equations. Numerical aspects of linear problems. P: Math 222. Credit may not be received for both Math 320 & 340.
321 Applied Mathematical Analysis. 3 cr. Vector analysis: algebra and geometry of vectors, vector differential and integral calculus, theorems of Green, Gauss, and Stokes; complex analysis: analytic functions, complex integrals and residues, Taylor and Laurent series. P: Math 223 or 234.
322 Applied Mathematical Analysis. 3 cr. Sturm-Liouville theory; Fourier series, including mean convergence; boundary value problems for linear second order partial differential equations, including separation of variables and eigenfunction expansions. P: Math 319, 321.
331 An Introduction to Probability and Markov Chain Models. 3 cr. An overview of basic probability including discrete and continuous random variables, moment generating functions, limit theorems, conditional probability and expectations, random walks, and Markov chains. P: Math 234 or Math 222 & 240.
340 Elementary Matrix and Linear Algebra. 3 cr. Matrix algebra, linear systems of equations, vector spaces, sub-spaces, linear dependence, rank of matrices, determinants, linear transformations, eigenvalues and eigenvectors, diagonalization, inner products and orthogonal vectors, symmetric matrices. P: Math 234 or Math 222 & 240. Credit may not be received for both Math 320 & 340.
341 Linear Algebra. 3 cr. This course emphasizes the understanding of concepts in linear algebra and teaches to write and understand proofs in mathematics in general and in linear algebra in particular. P: Math 234 or cons inst. Credit may not be received for Math 341 and any of Math 340 or 375. Open to Fr.
371 Basic Concepts of Math. 3 cr. Informal treatment of propositional and first-order logic. Proof techniques. Naive set theory. Relations and function. Peano axioms. Construction of the real numbers from the natural numbers. Countable and uncountable sets. Axiom of Choice and Zorn's Lemma. P: Math 340 or con reg.
375 Topics in Multi-Variable Calculus and Linear Algebra. 5 cr. Vector spaces and linear transformations, differential calculus of scalar and vector fields, determinants, eigenvalues and eigenvectors, multiple integrals, line integrals, and surface integrals. P: Math 276 or cons inst. Stdts may not receive cr for both Math 375 & any of Math 234, 320 & 340.
376 Topics in Multi-Variable Calculus and Differential Equations. 5 cr. Topics in Multi-variable calculus and introduction to differential equiations. P: Math 375 or cons inst. Open to Fr.
415 Applied Dynamical Systems, Chaos and Modeling. 3 cr. An introduction to nonlinear dynamical systems including stability, bifurcations and chaos. The course will give underlying mathmatical ideas, but emphasize applications from many scientific fields. P: Math 319 or 320 or cons inst.
421 The Theory of Single Variable Calculus. 3 cr. This course covers material in first and second semester calculus but it is intended to teach math majors to write and understand proofs in mathematics in general and in calculus in particular. P: Math 234 or cons inst. Stdts cannot receive cr for Math 421 if they have taken Math 275-276. Open to Fr.
425 Introduction to Combinatorial Optimization. (Crosslisted with Comp Sci, ISyE) 3 cr. Exact and heuristic methods for key combinatorial optimization problems such as: shortest path, maximum flow problems, and the traveling salesman problem. Techniques include problem-specific methods and general approaches such as branch-and-bound, genetic algorithms, simulated annealing, and neural networks. P: Math 221 or Comp Sci 302 or cons inst.
431 Introduction to the Theory of Probability. (Crosslisted with Stat) 3 cr. Probability in discrete sample spaces; combinatorial analysis; conditional probabilities, stochastic independence, Laplace limit theorem, Poisson distribution, laws of large numbers, random variables, central limit theorem, applications. P: Math 223 or 234.
435 Introduction to Cryptography. (Crosslisted with Comp Sci, ECE) 3 cr. Cryptography is the art and science of transmitting digital information in a secure manner. This course will provide an introduction to its technical aspects. P: Math 320 or 340 or cons inst. Open to Fr.
441 Introduction to Modern Algebra. 3 cr. The integers, emphasizing general group and ring properties. Permutation groups, symmetry groups, polynomial rings, leading to notions of abstract groups and rings. Congruences, computations, including finite fields and applications. Emphasis on concepts and concrete examples and computations, not complicated proofs. P: Math 340. Stdts who have passed Math 541 are not permitted to take Math 441 for credit.
443 Applied Linear Algebra. 3 cr. Review of matrix algebra. Simultaneous linear equations, linear dependence and rank, vector space, eigenvalues and eigenvectors, diagonalization, quadratic forms, inner product spaces, norms, canonical forms. For students whose main field of interest is not pure mathematics. Discussion of numerical aspects and applications in the sciences. P: Math 320 or 340 or cons inst.
461 College Geometry I. 3 cr. An introduction to Euclidean or non-Euclidean geometry at the college level. P: Math 223 or 234.
473 History of Mathematics. (Crosslisted with Hist Sci) 3 cr. An historical survey of the main lines of mathematical development. P: Cons inst.
475 Introduction to Combinatorics. (Crosslisted with Stat, Comp Sci) 3 cr. Problems of enumeration, distribution, and arrangement. Inclusion-exclusion principle. Generating functions and linear recurrence relations. Combinatorial identities. Graph coloring problems. Finite designs. Systems of distinct representatives and matching problems in graphs. Potential applications in the social, biological, and physical sciences. Puzzles. Problem solving. P: Math 320 or 340 or cons inst.
490 Undergraduate Seminar. 1-3 cr. Problem solving techniques. P: So st and cons inst.
491 Topics in Undergraduate Mathematics. 3 cr. Topics will vary. P: Math 223 or 234 & cons inst.
513 Numerical Linear Algebra. (Crosslisted with Comp Sci) 3 cr. Direct and iterative solution of linear and nonlinear systems and of eigenproblems. LU and symmetric LU factorization. Complexity, stability, and conditioning. Nonlinear systems. Iterative methods for linear systems. Qr-factorization and least squares. Eigenproblems: local and global methods. P: Math 340 or equiv, Comp Sci 302 or equiv.
514 Numerical Analysis. (Crosslisted with Comp Sci) 3 cr. Polynomial forms, divided differences. Polynomial interpolation. Polynomial approximation: uniform approximation and Chebyshev polynomials, least-squares approximation and orthogonal polynomials. Numerical differentiation and integration. Splines, B-splines and spline approximation. Numerical methods for solving initial and boundary value problems for ordinary differential equations. P: Math 340 or equiv, Comp Sci 302 or equiv.
515 Introduction to Splines and Wavelets. (Crosslisted with Comp Sci) 3 cr. Introduction to Fourier series and Fourier transform; time-frequency localization; wavelets and frames. Applications: denoising and compression of signals and images. Interpolation and approximation by splines: interpolation, least-squares approximation, smoothing, knot insertion and subdivision; splines in Cagd. P: Comp Sci 302 or equiv, Math 340 or equiv.
519 Ordinary Differential Equations. 3 cr. Provides a rigorous introduction to ordinary differential equations and dynamical systems. Intended for math majors and advanced (or graduate) students in other disciplines. P: Math 319 or 320 or 340, & Math 421 or 521; or cons inst.
521 Advanced Calculus. 3 cr. Fundamental notions of limits, continuity, differentiation, and integration, for functions of one or more variables, convergence and uniform convergence of infinite series, and improper integrals. P: Math 340 or con reg.
522 Advanced Calculus. 3 cr. Differentials and Jacobians, transformation of coordinates and of multiple integrals, line and surface integrals. P: Math 521.
525 Linear Programming Methods. (Crosslisted with Comp Sci, ISyE, Stat) 3 cr. Real linear algebra over polyhedral cones; theorems of the alternative for matrices. Formulation of linear programs. Duality theory and solvability. The simplex method and related methods for efficient computer solution. Perturbation and sensitivity analysis. Applications and extensions, such as game theory, linear economic models, and quadratic programming. P: Math 443 or 320 or 340 or cons inst.
541 Modern Algebra. 3 cr. Groups, normal subgroups, Cayley's theorem, rings, ideals, homomorphisms, polynomial rings, abstract vector spaces. P: Math 320 or 340 or cons inst.
542 Modern Algebra. 3 cr. Field extensions, roots of polynomials, splitting fields, simple extensions, linear transformations, matrices, characteristic roots, canonical forms, determinants. P: Math 541.
551 Elementary Topology. 3 cr. Topological spaces, connectedness, compactness, separation axioms, metric spaces. P: Math 223 or 234.
552 Elementary Geometric and Algebraic Topology. 3 cr. Introduction to algebraic topology. Emphasis on geometric aspects, including two-dimensional manifolds, the fundamental group, covering spaces, basic simplicial homology theory, the Euler-Poincare formula, and homotopy classes of mappings. P: Math 551 and 542.
561 Differential Geometry. 3 cr. Theory of curves and surfaces by differential methods. P: Math 320 or 340; and Math 521.
567 Elementary Number Theory. 3 cr. Fundamental theorem of arithmetic, quadratic residues and quadratic reciprocity, number-theoretic functions, certain diophantine equations, Farey fractions, continued fractions. P: Math 340 or con reg.
571 Mathematical Logic. (Crosslisted with Philos) 3 cr. Basics of logic and mathematical proofs; propositional logic; first order logic; undecidability. P: Math 223 or 234 or equiv.
615 Introduction to Applied Mathematics and Numerical Analysis. 3 cr. P: Cons inst.
623 Complex Analysis. 3 cr. Elementary functions of a complex variable; conformal mapping; complex integrals; the calculus of residues. P: Math 321 or 521.
627 Introduction to Fourier Analysis. 3 cr. Fourier series and integrals, and their applications. P: Math 521.
629 Introduction to Measure and Integration. 3 cr. Lebesgue integral and measure, abstract measure and integration, differentiation, spaces of integrable functions. P: Math 522.
632 Introduction to Stochastic Processes. (Crosslisted with Stat, ISyE, OTM) 3 cr. Markov chains: classification, recurrence, transcience, limit theory. Renewal theory, Markov processes, birth-death processes. Applications to queueing, branching, and other models in science, engineering and business. Topics drawn from semi-Markov processes, martingales, Brownian motion. P: Math 431, or Stat 309 & 310, or Stat 311 & 312, or Stat 313 or 314.
633 Queuing Theory and Stochastic Modeling. (Crosslisted with ISyE, OTM) 3 cr. Reliability theory; coherent systems and reliability bounds. Markovian queues and Jackson networks. Steady-state behavior of general service time queues. Priority queues. Approximation methods and algorithms for complex queues. Simulation. Dynamic programming; applications to inventory and queueing. P: Math, Ind Engr 632 or cons inst.
635 An Introduction to Brownian Motion and Stochastic Calculus. 3 cr. This course presents an introduction to Brownian motion and its application to stochastic calculus. Sample path properties of Brownian motion, Ito stochastic integrals, Ito's formula, stochastic differential equations and properties of their solutions, and various applications will be included. P: Math 521 & 632.
641 Introduction to Error-Correcting Codes. (Crosslisted with ECE) 3 cr. A first course in coding theory. Codes (linear, Hamming, Golay, dual); decoding-encoding; Shannon's theorem; sphere-packing; singleton and Gilbert-Varshamov bounds; weight enumerators; MacWilliams identities; finite fields; other codes (Reed-Muller, cyclic, BCH, Reed-Solomon) and error-correction algorithms. P: Math 320 or 340, and Math 541 or cons inst.
642 Linear Algebra. 3 cr. A rigorous course in linear algebra. Topics: finite and infinite dimensional vector spaces over division rings; determinants; diagonalization; canonical forms; inner products (Euclidean, Hermitian); bilinear and quadratic forms over various fields; orthogonal and symplectic geometries over finite fields; linear groups. P: Math 340 & 541 or cons inst.
699 Directed Study. 1-6 cr. P: Jr or Sr st. Graded on a lettered basis; requires cons inst.
701 Mathematical Methods in Physics and Engineering. 3 cr. Green's functions; distributions with applications to Fourier series; Fourier and Laplace transforms; differential equations; boundary-value problems. Metric and Hilbert spaces; linear operator theory including compact operators; integral equations. P: Math 321-322, or equivalent linear algebra, complex variables, and partial differential equations. Not suitable for students in Math Ph.D. program.
702 Mathematical Methods in Physics and Engineering. 3 cr. Continuation of 701. P: Math 701 or equiv.
703 Methods of Applied Mathematics 1. 3 cr. Study of the linear algebraic structure underlying discrete equilibrium problems. Boundary value problems for continuous equilibria: Sturm-Liouville equations, Laplace's equation, Poisson's equation, and the equations for Stokes flow. Contour integration and conformal mapping. Applications of dynamics leading to initial value problems for ODEs and PDEs. Green's functions for ODEs and introduction to asymptotic methods for ODEs, e.g. WKB analysis. Separation of variables and eigenfunction expansions for linear PDEs. Examples from physics and engineering throughout. P: Math 340, 521-22, and 623 (or 623 concur), or equiv. Not open to stdts with cr for Math 701/702.
704 Methods of Applied Mathematics 2. 3 cr. Derivation, nature and solution of canonical partial differential equations of applied mathematics. Conservation laws, advection, diffusion. First order PDEs, characteristics, shocks. Traffic flow, eikonal and Hamilton-Jacobi equations. Higher order PDEs: classification, Fourier analysis, well-posedness. Series solutions and integral transforms. Green's functions and distributions. Similarity solutions. Asymptotics of Fourier integrals. Laplace's method, stationary phase. Ship waves. Perturbation methods. P: Math 340, 521-22, and 623 (or 623 concur), or equiv. Not open to stdts with cr for Math 701-702.
705 Mathematical Fluid Dynamics. 3 cr. Advanced introduction to fluid dynamics. Basic concepts; elementary viscous flow; Navier-Stokes equations. Elementary airfoil theory; boundary layers. Vortex motion. Waves. Very viscous flow. Compressible flows. Instabilities, bifurcations, turbulence. Requires working knowledge of multivariate calculus, differential equations and mechanics. P: Cons inst.
707 Theory of Elasticity. 3 cr. P: Cons inst.
709 Mathematical Statistics. (Crosslisted with Stat) 4 cr. Introduction to measure theoretic probability; derivation and transformation of probability distributions; generating functions and characteristic functions; conditional expectation, sufficiency, and unbiased estimation; methods of large sample theory including laws of large numbers and central limit theorems; order statistics. P: Cons inst or one yr adv calculus and Math, Stat 431, Math, Stat 310.
710 Mathematical Statistics. (Crosslisted with Stat) 4 cr. Estimation, efficiency, Neyman-Pearson theory of hypothesis testing, confidence regions, decision theory, analysis of variance, and distribution of quadratic forms. P: Stat, Math 709.
712 Finite Difference Methods. (Crosslisted with Comp Sci) 3 cr. Development of finite difference methods for initial and boundary value problems for hyperbolic, parabolic, and elliptic partial differential equations. Analysis of accuracy and stability of difference schemes. Direct and iterative methods for solving elliptic difference schemes. Applications from science and engineering. P: Comp Sci 302, 412, Math 419 or equiv or cons inst.
716 Ordinary Differential Equations. 3 cr. Existence, uniqueness, and continuous dependence theorems, linear systems, stability, singular points, and boundary value problems. Qualitative behavior of nonlinear equations, stability, Lyapunov functions, invariant manifolds, bifurcation theory, periodic orbits, and connecting orbits. P: Senior (600) level analysis crse or cons inst.
717 Numerical Functional Analysis. (Crosslisted with Comp Sci) 3 cr. Fundamentals of normed spaces and linear operators; analysis of nonlinear operators; existence of and iterative methods for solutions of linear and nonlinear operator equations, error estimation; variational theory and minimization problems; monotonocity theory. Development of abstract tools and application of them to the general analysis of numerical methods for such problems as differential and integral equations. P: Comp Sci 513-514 & Math 234 or cons inst.
721 A First Course in Real Analysis. 3 cr. A first course in real analysis concentrating on measures, integration, and differentiation and including an introduction to Hilbert spaces. P: Math 522.
722 Complex Analysis. 3 cr. The basic theory of functions of one complex variable including Cauchy formula, singularities and residues, meromorphic functions, conformal mappings, harmonic functions, approximation and the nonhomogeneous d-bar equation. P: Math 721; con reg in 701 or 721.
723 Advanced Topics in the Theory of One Complex Variable. 3 cr. Topics may include: entire functions, value distribution theory, Riemann surfaces and complex dynamics in one variable. P: Math 722.
725 A Second Course in Real Analysis. 3 cr. An introduction to further topics in real analysis: Banach spaces, Fourier transforms, elements of distribution theory, and applications. P: Math 721.
726 Nonlinear Programming Theory and Applications. (Crosslisted with Comp Sci, ISyE, Stat) 3 cr. Separation theorems and other properties of convex sets in finite-dimensional spaces. Formulation of nonlinear programming problems. Saddle point (Lagrangian) optimality criteria for convex nonlinear programs. Duality theorems for convex programs. First- and second-order Kuhn-Tucker stationary-point theory for differentiable non-convex programs. Perturbation and sensitivity analysis. Applications and extensions. P: Familiarity with basic mathematical analysis and either Math 443 or 320; or cons inst.
727 Calculus of Variations and Related Topics. 3 cr. Introduction to some classical methods in the calculus of variations and continues on to modern direct methods. Techniques for establishing and studying minima and other critical points will be developed. Applications will be chosen from a wide variety of fields: geometry, differential equations, classical mechanics, geodesics, and optimal control theory. Background material in the areas of application will be provided. P: Cons inst.
733 Computational Methods for Large Sparse Systems. (Crosslisted with ECE, Comp Sci) 3 cr. Sparse matrices in engineering and science. Sparsity preservation. Numerical error control. Transversal algorithms, Tarjan's algorithm, Tinney's algorithms, minimum degree, banding methods, nested dissection, frontal methods. Linear and nonlinear equation solving. Compensation. Sparse vector methods. Iterative methods. ODE and PDE applications. P: Comp Sci 367 & ECE 334; or Comp Sci 367, 412 & Math 340; or cons inst.
735 Stochastic Analysis. 3 cr. Foundations of continuous time stochastic processes, semimartingales and the semimartingale integral, Ito's formula, stochastic differential equations, stochastic equations for Markov processes, application in finance, filtering, and control. P: Math 431 & cons inst.
737 Introduction to Stochastic Control. 3 cr. Control of systems with dynamics given by stochastic differential equations; optimal control for continuous time Markov processes; classical and weak solutions for Hamilton-Jacobi-Bellman equations; completely and partially observed systems; applications selected from the social sciences and engineering. P: Math 735, Math 831, or Math 632 & cons inst.
740 Enumerative Combinatorics/Symmetric Functions. 3 cr. Introductory graduate course on algebraic combinatorics. Topics: inclusion-exclusion principle, permutation statistics, sieve methods, unimodal sequences, posets, lattice theory, Mobius functions, generating functions, bases and transition matrices for symmetric functions, Young tableaux, plane partitions, polytopes, poset homology, Stanley-Reisner rings. P: Math 541 & 542 or cons inst.
741 Abstract Algebra. 3 cr. The basic prerequisite for all advanced graduate courses in algebra. Usually a study of finite groups and noncommutative rings. Group theoretic topics may include: permutation groups, Lagrange's theorem, Cauchy's theorem and the Sylow theorems, solvable and nilpotent groups. Ring theoretic topics may include: Artinian rings and modules, the Wedderburn theorems, the Hopkins-Levitzki theorem, the Jacobson radical and density theorem. P: Math 542 or equiv.
742 Abstract Algebra. 3 cr. Continuation of 741. Usually the study of commutative rings and fields. Ring theoretic topics may include: modules over PIDs, Noetherian rings and the Hilbert basis theorem, the Lasker-Noether theorem, the Krull intersection theorem, integrality and the Hilbert Nullstellensatz. Field theoretic topics may include: algebraic extensions, Galois theory, solvability of polynomials and classical constructability problems. P: Math 741.
743 Linear Algebra and Matrix Theory. 3 cr. Classical and modern matrix theory: canonical forms, Perron-Frobenius-Wielandt theory of nonnegative matrices, matirx commutators, elementary divisor theory, matrix equations, tensor product, linear preservers, hermitian matrices, eigenvalues bounds, inertia theorems, field of values, doubly stochastic matrices, combinatorial matrix theory. P: Math 340 & Math 541, or cons inst.
744 Algebraic Graph Theory. 3 cr. Algebraic techniques in graph theory. Adjacency matrices; spectral properties; chromatic polynomials; edge coloring; Tutte polynomials; isomorphism; symmetry (line, Cayley, regular, strongly-regular); two graphs and switching. Possible additional topics: spanning trees, matchings, extremal graph theory, Ramsey theory, association schemes. P: Math 541 and 542, or cons inst.
745 Theory of Groups. 3 cr. Usually covers a selection of topics in finite group theory including: subnormality, coprime actions, the Schur-Zassenhaus theorem, solvable and nilpotent groups, transfer theory and its applications, and normal p-complement theorems. Occasionally, the focus may be on some aspects of infinite group theory. P: Cons inst.
746 Topics in Ring Theory. 3 cr. Will alternate between commutative and noncommutative ring theory. Commutative topics include localization; local rings; dimension theory; Cohen-Macaulay rings. Noncommutative topics include projective modules; injective modules; flat modules; homological and global dimension; Wedderburn and Goldie rings. P: Math 741, 742 or equiv.
747 Lie Algebras. 3 cr. Lie algebras and matrix groups. Topics: tangent spaces; exponentials; Baker-Campbell-Hausdorff formula; (nilpotent, solvable, semisimple) Lie algebras; Engel's and Lie's theorems; Levi decomposition; Killing form; sl(2)-representations; root systems; Dynkin diagrams; Weyl groups; Cartan and Borel subalgebras; Serre's theorem. P: Math 541 & 542 or cons inst.
748 Algebraic Number Theory. 3 cr. An introductory graduate level course on algebraic number theory. Topics: a rigorous introduction to the arithmetic of number fields; algebraic integers, geometry of numbers, Dirichlet's Unit Theorem, ideal class groups, first case of Fermat's Last Theorem; prime decompositions, Galois automorphisms. P: Math 741-742 or equiv or cons inst.
749 Analytic Number Theory. 3 cr. An introduction to (abelian) Hecke L-functions and their arithmetic applications to topics such as the distribution of primes and the study of ideal class groups. P: Math 741-742 or equiv or cons inst.
750 Homological Algebra. 3 cr. A first course in homological algebra. Topics include: complexes, cohomology, double complexes, spectral sequences; abelian categories, derived categories, derived functors; Tor and Ext, Koszul complexes; group cohomology; sheaf cohomology, hypercohomology. P: Math 741-742 or equiv or cons inst. May be taken concurrently with Math 742.
751 Introductory Topology I. 3 cr. An introduction to algebraic and differential topology. Elements of homotopy theory, fundamental group, covering spaces. Differentiable manifolds, tangent vectors, regular values, transversality, examples of compact Lie groups. Homological algebra, chain complexes, cell complexes, singular and cellular homology, calculations for surfaces, spheres, projective spaces, etc. P: Math 541 and 551 or equiv.
752 Introductory Topology II. 3 cr. Continuation of 751. Cohomology, Universal Coefficient Theorem, Kunneth Formula, cup and cap products, applications to manifolds, orientability, Poincare Duality. Differential forms, integration and Stokes Theorem, De Rham Theorem. Calculations, further duality theorems, Euler class, Lefschetz Fixed-Point Theorem. P: Math 751 or equiv.
753 Algebraic Topology I. 3 cr. Higher homotopy groups, elements of obstruction theory, fibrations, bundle theory, classifying spaces, applications to smooth mainfolds, differential forms, vector bundles, characteristic classes, cobordism, applications and calculations. P: Math 752 or equiv.
754 Algebraic Topology II. 3 cr. Continuation of 753. Topics include: spectral sequences and their applications, topology of Lie Groups, H-spaces, Hopf Algebras, homotopy classification of bundles, the Steenrod Algebra and its applications, introduction to generalized cohomology theories, spectra, elements of K-theory. P: Math 753 or equiv.
755 Topology and its Applications. 3 cr. Deals with applications of topology to manifolds and other spaces of interest in mathematics. Possible topics include surgery, algebraic and topological K-theory, group actions, cyclic homology, topological methods in mathematical physics, knot invariants, gauge theory and its applications, loop spaces, configuration spaces, moduli spaces. P: Math 754 or equiv.
757 Geometric Topology. 3 cr. Piecewise linear topology, knot theory, topology of 3-manifolds, taming theorems, approximation theorems, handle body theory, theory of retracts. P: Cons inst.
761 Differentiable Manifolds. 3 cr. Differentiable manifolds, vector bundles, implicit function theorem, submersions and immersions, vector fields and flows, foliations and Frobenius theorem, differential forms and exterior calculus, integration and Stoke's theorem, De Rham theory, Riemannian metrics. P: Math 522 or equiv.
762 Differential Topology. 3 cr. This course introduces the fundamental techniques and theorems of differential topology. The following topics will be covered: Submersions and immersions, Jet bundles, approximation, Sard's theorem, Whitney embedding theorem, transversality, intersection theory, Poincare-Hopf theorem, isotopy, Hopf-degree theorem, Corbodism, Morse theory, classification of two-manifolds. P: Math 761.
763 Introduction to Algebraic Geometry. 3 cr. Algebraic preliminaries, including local rings; valuation theory, and power series rings; geometry of algebraic varieties with emphasis on curves and surfaces. P: Math 741-742.
764 Introduction to Algebraic Geometry. 3 cr. Continuation of Math 763. P: Math 763.
765 Differential Geometry. 3 cr. This course covers the metric properties of Riemannian manifolds. The following topics will be covered: Vector bundles and connections, Riemannian metrics, submanifolds and second fundamental form, first variation of arc length, geodesics, Hopf-Rinow theorem, second variation of arc length, Jacobi fields and index lemmas, Bonnet-Meyer theorem, Rauch comparison theorem, spaces of constant curvature, Hodge-de Rham theory. P: Math 761.
770 Foundations of Mathematics. 3 cr. First-order logic syntax and semantics, Completeness and Compactness Theorems, Lowenheim-Skolem Theorem, computable and computably enumerable sets, Incompleteness Theorem, axioms of Zermelo-Fraenkel set theory with choice, ordinal and cardinal arithmetic.
771 Set Theory. 3 cr. Martin's Axiom, Suslin and Aronszajn trees, diamond principle, absoluteness and reflection, constructible universe, and one-step forcing constructions. P: Math 770.
773 Computability Theory. 3 cr. Turing degree and jump, strong reducibilities, arithmetic hierarchy, index sets, simple and (hyper)hypersimple sets, easy forcing arguments in computability theory, finite and infinite injury, Friedberg-Muchnik and Sacks Splitting Theorem, Sacks Jump and Sacks Density Theorems, computable ordinals. P: Cons inst.
776 Model Theory. 3 cr. Review of compactness and some consequences. Quantifier elimination with examples. The omitting types theorem. Categoricity. Baldwin-Lachlan theory. Strongly minimal and o-minimal theories. Saturated models. Morley's theorem. P: Math 770.
777 Nonlinear Dynamics, Bifurcations and Chaos. (Crosslisted with ECE, CBE) 3 cr. Advanced interdisciplinary introduction to qualitative and geometric methods for dissipative nonlinear dynamical systems. Local bifurcations of ordinary differential equations and maps. Chaotic attractors, horseshoes and detection of chaos. P: Cons inst.
801 Topics in Applied Mathematics. 3 cr. Selected topics in applied mathematics, applied analysis or numerical analysis and scientific computing. P: Cons inst.
803 Experimental Design I. (Crosslisted with Stat) 3 cr. Summary of matrix algebra required, theory of estimable functions, incomplete blocks, balanced incomplete block designs, partially balanced incomplete block designs. P: Stats 310 or cons inst.
805 Special Functions. 3 cr. Special functions arising from mathematics, physics, and engineering, their series and integral representations, differential and other functional equations, generating functions, and orthogonality . P: Cons inst.
806 Integral Transforms and Their Applications. 3 cr. P: Cons inst.
807 Dynamical Systems. 3 cr. Treats the qualitative behavior of continuous and discrete dynamical systems, including Hamiltonian systems of differential equations. Typical topics include periodic and almost periodic solutions, the fixed point theorem of Poincare and Birkhoff, invariant curves and Kam theory, celestial mechanics, and chaotic behavior. P: Cons inst.
812 Advanced Methods of Applied Mathematics. 3 cr. Differential equations; asymptotic methods in complex analysis; problems of matching; integral transforms; integral equations; introduction to spectral theory; calculus of variations; tensor analysis. P: Cons inst.
819 Partial Differential Equations. 3 cr. Classical theory of partial differential equations, together with an introduction to the modern theory based on functional analysis. P: Cons inst.
820 Partial Differential Equations. 3 cr. Continuation of Math 819. P: Math 819.
821 Advanced Topics in Real Analysis. 3 cr. P: Cons inst.
823 Advanced Topics in Complex Analysis. 3 cr. Several complex variables. Basic several complex variables or more special topics. P: Cons inst.
825 Selected Topics in Functional Analysis. 3 cr. Topics will vary and may include spectral theory, nonlinear functional analysis or abstract harmonic analysis. P: Cons inst or Math 722.
826 Advanced Topics in Functional Analysis and Differential Equations. 3 cr. Continuation of Math 825. P: Cons inst.
827 Fourier Analysis. 3 cr. Introduction to Fourier analysis in Euclidean spaces and related topics that may include singular and oscillatory integrals and trigonometric series. P: Cons inst.
828 Advanced Topics in Harmonic Analysis. 3 cr. Continuation of Math 827. Advanced topics related to research in harmonic analysis. P: Cons inst.
831 Theory of Probability. (Crosslisted with Stat) 3 cr. An introduction to measure theoretic probability and stochastic processes. Topics include foundations, independence, zero-one laws, laws of large numbers, convergence in distribution, characteristic functions, central limit theorems, random walks, conditional expectations. P: Math 629, 721, or con reg in 721, or con inst.
832 Theory of Probability. (Crosslisted with Stat) 3 cr. Continuation of 831. Possible topics include martingales, weak convergence of measures, introduction to Brownian motion. P: Cons inst.
833 Topics in the Theory of Probability. (Crosslisted with Stat) 3 cr. Topics in probability and stochastic processes. P: Cons inst.
835 Topics in Mathematical Systems Theory. 3 cr. A broadly based course in the mathematical theory of control and information systems. Subjects from: control theory of finite or infinite dimensional systems; optimal control and filtering; state estimation and system identification; two-dimensional filtering and image processing; information theory. Consult Timetable footnotes or instructor for specific content in a given semester. P: Cons inst.
837 Topics in Numerical Analysis. (Crosslisted with Comp Sci) 3 cr. From advanced areas. Contents may vary. P: Cons inst.
840 Topics in Reflection Groups and Symmetric Groups. 3 cr. Reflection groups: root systems, hyperplanes, invariants, complex reflection groups, Iwahori-Hecke algebras, Kazhdan-Lusztig polynomials, Bruhat order.
Symmetric groups: tableaux, Specht modules, branching rules, Kostka numbers, formulas (hook, determinantal, Littlewood-Richardson, Murnaghan-Nakayama), Robinson-Schensted algorithm, Viennot's construction, symmetric functions.
841 Character Theory. 3 cr. Introduction to classical character and representation theory of finite groups. Group algebras; orthogonality relations, integrality; character degrees; Burnside's theorem; products of characters; induced characters; Frobenius reciprocity; Clifford theory; Frobenius' theorem. Possible additional topics: modular representations, block theory, symmetric group representations. P: Cons inst.
842 Topics in Applied Algebra. (Crosslisted with ECE) 3 cr. Applied topics with emphasis on algebraic constructions and structures. Examples include: algebraic coding theory; codes (algebraic-geometric, convolutional, low-density-parity-check, space-time); curve and lattice based cryptography; watermarking; computer vision (face recognition, multiview geometry). P: Cons inst.
843 Representation Theory. 3 cr. Introduction to the representation theory of Lie groups and their combinatorics. Universal enveloping algebras, highest weight modules, induction, restriction, weights, characters, multiplicity formulas, tensor products, Shapovalov forms, filtrations, Kazhdan-Zetlin patterns, Littelmann paths. P: Cons inst.
844 Elliptic Curves and Modular Forms. 3 cr. An introduction to the theory of elliptic curves and modular forms. Topics include Fourier expansions of modular forms, Hecke theory, L-functions, the Mordell-Weil theorem, and Selmer and Shafarevich-Tate groups. P: Cons inst.
845 Class Field Theory. 3 cr. Introduction to local and global class field theory. Theory of local fields; local and global class field theory; complex multiplication, adeles, ideles, idele class characters, Tchebotarev's Density Theorem, CM elliptic curves, construction of class fields of imaginary quadratic fields . P: Math 748 or 749 or cons inst.
846 Topics in Combinatorics. 3 cr. Topics in algebraic combinatorics such as (but not limited to) association schemes, hypergeometric series, classical orthogonal polynomials, codes, lattices, invariant theory, alternating sign matrices and domino tilings, statistical mechanical models, 6j-symbols, buildings and diagram geometries, matroids. P: Cons inst.
847 Topics in Algebra. 3 cr. Topics may include: Lie groups, algebraic groups, Chevalley groups, simple groups and associated geometries, group cohomol-ogy, group rings, Hopf algebras, enveloping algebras, quantum groups, infinite-dimensional Lie algebras, Hecke algebras, automorphic forms, Galois representations, zeta and L-functions, abelian varieties. P: Cons inst.
851 Topics in Geometric Topology. 3 cr. P: Cons inst.
853 Topics in Algebraic Topology. 3 cr. P: Cons inst.
856 Topics in Differential Topology. 3 cr. The theory of differential manifolds such as differential forms and de Rham theorem, cobordism groups, Lie groups, homogeneous spaces, fiber bundles, characteristic classes. P: Cons inst.
863 Advanced Topics in Algebraic Geometry. 3 cr. Geometry of several complex variables; algebraic groups, abelian varieties; topological aspects of algebraic geometry, including sheaf theory and homology theory; advanced theory of local rings; intersection theory of algebraic varieties.
865 Advanced Topics in Geometry. 3 cr. Selected from advanced projective geometry, non-Euclidean geometry, Riemannian geometry, distance geometry and the geometry of convex surfaces, geometry of numbers. P: Cons inst.
873 Advanced Topics in Foundations. 3 cr. From all areas of mathematical logic. P: Cons inst.
887 Approximation Theory. (Crosslisted with Comp Sci) 3 cr. Interpolation and approximation by means of interpolation; uniform approximation; best approximation; approximation in normed linear spaces; spline functions; orthogonal polynomials; degree of approximation; computational procedures. P: Cons inst.
903 Seminar in Mathematics Education. 1-3 cr.
911 Seminar in College Mathematics Teaching. 1 cr. P: Cons inst.
921 Seminar in Analysis. 1-3 cr. P: Cons inst.
941 Seminar—Algebra. 1-3 cr. P: Cons inst.
951 Seminar in Topology. 1-3 cr. P: Cons inst.
975 Seminar—The Foundations of Mathematics. 1-3 cr. From all areas of mathematical logic. P: Cons inst.
990 Reading and Research. 1-3 cr. P: Cons inst.