### Algebra/Group Theory

**Math 412 Introduction to Modern Algebra (3).**- Prerequisite: Math 215 or 285; 217, 417, or 419 recommended (may be concurrent). Only 1 credit after Math 312.
- Sets, functions (mappings), relations, & the common number systems (integers to complex numbers). These are then applied to the study of groups and rings. These structures are presented as abstractions from many examples. Notions such as generator, su bgroup, direct product, isomorphism and homomorphism.

**Math 593 Algebra I (3).**- Prerequisite: Math 513.
- Rings and modules. Euclidean rings, PIDs, classification of modules over a PID. Jordan and rational canonical forms. Structure of bilinear forms. Tensor products of modules; exterior algebras.

**Math 594 Algebra II (3).**- Prerequisite: Math 593
- Group theory. Permutation representations, simplicity of alternating groups for $n > 4$. Sylow theorems. Series in groups; solvable and nilpotent groups. Jordan-Holder theorem for groups with operators. Free groups and presentations. Field extensions, norm and trace, algebraic closures. Galois theory. Transcendence degree.

**Math 613 Homological Algebra (3).**- Prerequisite: Math 590 and 594 or permission of instructor.
- Categories and functors; abelian categories and the imbedding theorem; homology and cohomology theories.

**Math 619 Topics in Algebra (3).**- Prerequisite: Math 593.
- Selected topics.

**Math 711 Advanced Algebra (3).**- Prerequisite: Math 594 and 612 or permission of instructor.
- Topics of current research interest, such as groups, rings, lattices, etc., including a thorough study of one such topic.

**Math 715 Advanced Topics in Algebra (3).**- May be taken more than once for credit.

- May be taken more than once for credit.
**Math 790 Transformation Groups (3).**- Prerequisite: Math 591.
- Selected topics from the theory of topological or differential transformation groups.

### Analysis/Functional Analysis

**Math 450 Advanced Mathematics for Engineers I (4).**- Prerequisite: Math 216, 316, or 286.
- Review of curves and surfaces in implicit, parametric, & explicit forms; differentiability and affine approx.; implicit & inverse function theorems; chain rule for 3-space; multiple integrals, scalar & vector fields; line & surface integrals; computat ions of planetary motion, work, circulation & flux over surfaces; Gauss' and Stokes' Theorems; heat equation.

**Math 451 Advanced Calculus I (3).**- Prerequisite: Math 285, or Math 215 and one subsequent course
- A rigorous single-variable calculus course, including completeness of the real numbers and various consequences such as the Bolzano-Weierstrass and Heine-Borel Theorems. Also limits, sequences, series and tests for convergence. Required for Math under graduate majors this is considered a remedial course for Masters' students.

**Math 452 Advanced Calculus II (3).**- Prerequisite: Math 217, 417, or 419 (may be taken concurrently) and Math 451.
- Partial derivatives and differentiability; gradients, directional derivatives, and the chain rule; implicit function theorem; surfaces, tangent plane; max-min theory; multiple integration, change of variable; Green's & Stokes' theorems, differential f orms, exterior surfaces; intro to differential geometry.

**Math 597 Analysis II (3).**- Prerequisite: Math 451 and 513.
- Lebesgue measure on the real line. Measurable functions and integration on $R$. Differentiation theory, fundamental theorem of calculus. Function spaces, $L^p(R)$, $C(K)$, Holder and Minkowski inequalities, duality. General measure spaces, product mea sures, Fubini's theorem. Radon-Nikodym theorem, conditional expectation, signed measures.

**Math 602 Real Analysis II (3).**- Prerequisite: Math 590 and 597.
- Introduction to functional analysis; metric spaces, completion, Banach spaces, Hilbert spaces, $L^p$ spaces; linear functionals, dual spaces, Riesz representation theorems; principle of uniform boundedness, closed graph theorem, Hahn-Banach theorem, B aire category theorem, applications to classical analysis.

**Math 609 Topics in Analysis (3).**- Prerequisite: Math 451.
- Selected topics in analysis. Content and prerequisites will vary from year to year. May be taken for credit more than once.

**Math 650 Fourier Analysis (3).**- Prerequisite: Math 602 and 596.
- General properties of orthogonal systems. Convergence criteria for Fourier series. The Fourier integral, Fourier transform and Plancherel theorem. Wiener's Tauberian theorem. Elements of harmonic analysis. Applications.

**Math 701 Functional Analysis I (3).**- Prerequisite: Math 602.
- Geometry of Hilbert space; basic properties of linear operators; self-adjoint, unitary, and normal operators; spectral theorem; compact operators; unbounded operators; Banach spaces, Banach algebras, topological vector spaces.

**Math 702 Functional Analysis II (3).**- Prerequisite: Math 602 and sometimes Math 701.
- Further topics in Functional Analysis.

**Math 707 Calculus of Variations (3).**- Prerequisite: Math 597.
- Modern theory of calculus of variations. Topics will be taken from: critical point theory, Morse theory, bifurcation theory, geometric measure theory, \relax $\mathsurround =\z@ \mathinner {\ldotp \ldotp \ldotp }\mskip \thinmuskip $, etc.

**Math 710 Topics in Modern Analysis, II (3).**- Prerequisite: Math 597.
- Selected advanced topics in analysis.

### Applied Mathematics

**Math 464 Inverse Problems (3)**- Prerequisite: linear algebra and differential equations
- Solution of an inverse problem is a central component of fields such as medical tomography, geophysics, non-destructive testing, and control theory. The solution of any practical inverse problem is an interdisciplinary task. Each such problem requires a blending of mathematical constructs and physical realities. Thus, each problem has its own unique components; on the other hand, there is a common mathematical framework for these problems and their solutions. This framework is the primary content of t he course. This course will allow students interested in the above-named fields to have an opportunity to study mathematical tools related to the mathematical foundations. The course content is motivated by a particular inverse problem from a field such a s medical tomography (transmission, emission), geophysics (remote sensing, inverse scattering, tomography), or non-destructive testing. Mathematical topics include ill-posedness (existence, uniqueness, stability), regularization (e.g., Tikhonov, least squ ares, modified least squares, variation, mollification), pseudoinverses, transforms (e.g. k-plane, Radon, X-ray, Hilbert), special functions, and singular-value decomposition. Physical aspects of particular inverse problems will be introduced as needed, b ut the emphasis of the course is investigation of the mathematical concepts related to analysis and solution of inverse problems.

**Math 550 Intro to Adaptive Systems (3)**- Prerequisite: ?
- This course centers on the construction and use of agent-based adaptive models study phenomena which are prototypical in the social, biological and decision sciences. These models are "agent-based" or "bottom-up" in that the structure placed at the le vel of the individuals as basic components; they are "adaptive" in that individuals often adapt to their environment through evolution or learning. The goal of these models is to understand how the structure at the individual or micro level leads to emerg ent behavior at the macro or aggregate level. Often the individuals are grouped into subpopulations or interesting hierarchies, and the researcher may want to understand how the structure of development of these populations affects macroscopic outcomes. T he course will start with classical differential equation and game theory approaches. It will then focus on the theory and application of particular models of adaptive systems such as models of neural systems, genetic algorithms, classifier system and cel lular automata. Time permitting, we will discuss more recent developments such as sugarscape and echo.

**Math 623 Computational Finance (3)**- Prerequisite: Math 316 and Math 425 or 525
- This is a course in computational methods in finance and financial modeling. Particular emphasis will be put on interest rate models and interest rate derivatives. Specific topics include Black-Scholes theory, no-arbitrage and complete markets theory, term structure models, Hull and White models, Heath-Jarrow-Morton models, the stochastic differential equations and martingale approach, multinomial tree and Monte Carlo methods, the partial differential equations approach, finite difference methods.

**Math 651 Topics in Applied Mathematics I (3).**- Prerequisite: Math 451, 555 and one other 500-level course in analysis or differential equations.
- Topics such as celestial mechanics, continuum mechanics, control theory, general relativity, nonlinear waves, optimization, statistical mechanics.

**Math 652 Topics in Applied Mathematics II (3).**- Prerequisite: Math 451, 555 and one other 500-level course in analysis or differential equations.
- Topics such as celestial mechanics, continuum mechanics, control theory, general relativity, nonlinear waves, optimization, statistical mechanics.

### Combinatorics

**Math 565 Combinatorics and Graph Theory (3).**- Prerequisite: Math 412 or 451 or equivalent experience with abstract mathematics.
- Graph Theory: trees, k-connectivity; Eulerian & Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, 5-color theorem, Kuratowski's theorem & matrix-tree theorem; Enumeration: fundamental principles, bijections, generating functions, binomial theorem, partitions and q series, linear recurrences, generating functions and Polya theory.

**Math 566 Combinatorial Theory (3).**- Prerequisite: Math 216, 286, 316 o permission of instructor.
- Permutations, combinations, generating functions, and recurrence relations. The existence and enumeration of finite, discrete configurations. Systems of representatives, Ramsey's theorem, and extremal problems. Construction of combinatorial designs.

**Math 664 Combinatorial Theory I (3).**- An introduction to the techniques of enumeration. Basic material for first half of this course is found in Stanley's ``Enumerative Combinatorics, Vol. I''. The second half consists of topics such as ordinary and exponential generating functions, Sieve methods, partitions and $q$-series, Polya Theory and other optional topics as time permits.

**Math 665 Combinatorial Theory II (3).**- This is a new course which is a continution of Math 664 and will be taught Winter 1991. A description will be available in Nov. 1990.

**Math 669 Topics in Combinatorial Theory (3).**- Prerequisite: Math 565 or 566 or 664 or permission of instructor.
- Selected topics from the foundations of combinatorics, including the analysis of general partially ordered sets, combinatorial designs in loops and structures in abstract systems, enumeration under group action, combinatorial aspects of finite simple groups.

### Commutative Algebra/Algebraic Geometry

**Math 614 Commutative Algebra I (3).**- Prerequisite: Math 593.
- Review of commutative rings and modules. Local rings and localization. Noetherian and Artinian rings. Integral independence. Valuation rings, Dedekind domains, completions, graded rings. Dimension theory.

**Math 615 Commutative Algebra II (3).**- Prerequisite: Math 614 or permissions of instructor.
- This is a continuation of Math 614: structure of complete local rings, regular, Cohen-Macaulay, and Gorenstein rings, excellent rings, Henselian rings, etale maps, equations over local rings.

**Math 631 Algebraic Geometry I. (3).**- Prerequisite: Math 594 or 614 or permission of instructor).
- Theory of algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, singular points, divisors, differentials, intersections. Schemes, cohomology, curves and surfaces, varieties over the complex numbers.

**Math 632 Algebraic Geometry II. (3).**- Prerequisite: Math 631).
- Continuation of Math 631.

**Math 731 Topics in Algebraic Geometry I (3).**- Selected topics in algebraic geometry.

**Math 732 Topics in Algebraic Geometry II (3).**- Prerequisite: Math 631 or 731.
- Selected topics in algebraic geometry.

### Complex Analysis

**Math 555 Introduction to Complex Variables (3; 1 for students with credit for Math 455 or 554).**- Prerequisite: Math 450 or 451. Intended primarily for students of engineering and of other cognate subjects. Doctoral students of mathematics elect Math 596.
- Complex numbers, continuity; derivative; conformal representation, integration; Cauchy theorems, power series; singularities; application to engineering and mathematical physics.

**Math 596 Analysis I (3).**- Prerequisite: Math 451. 2 hours credit for those with credit for 555.
- Review of analysis in $R^2$ including metric spaces, differentiable maps, Cauchy-Riemann equations, automophisms. Analytic functions, Cauchy integral formula. Power series and Laurent expansions, fundamental theorem of algebra, harmonic functions. Fun ctions analytic in a disk. Global properties of analytic functions. Riemann mapping theorem. Normal families.

**Math 604 Complex Analysis II (3).**- Prerequisite: Math 596.
- Selected topics such as potential theory, geometric function theory, analytic continuation, Riemann surfaces, uniformization and analytic varieties.

**Math 605 Several Complex Variables (3).**- Prerequisite: Math 604 or consent of instructor.
- Power series in several complex variables, domains of holomorphy, pseudo convexity, plurisubharmonic functions, the Levi problem. Domains with smooth boundary, tangential Cauchy-Riemann equations, the Lewy and Bochner extension theorems. The $\overlin e {\partial }$-operator and Hartog's Theorem, Dol beault-Grothendieck lemma, theorems of Runge, Mittag-Leffler and Weierstrass. Analytic continuation, monodromy theorem, uniformization and Koebe's theorem, discontinuous groups.

**Math 606 Riemann Surfaces (3).**- Prerequisite: Math 590, 604, and some knowledge of group theory.
- Introduction to the theory of Riemann surfaces. The Riemann surface of an analytic function. Covering surfaces, monodromy theorem, groups of cover transformations, uniformization theorem. Differentials and integrals, Riemann-Roch theorem.

**Math 703 Topics in Complex Function Theory I (3).**- Prerequisite: Math 604.
- Selected advanced topics in function theory. May be taken for credit more than once, as the content will vary from year to year.

**Math 704 Topics in Complex Function Theory II (3).**- Prerequisite: Math 604.
- Selected advanced topics in function theory. May be taken for credit more than once, as the content will vary from year to year.

### Differential Equations

**Math 454 Boundary Value Problems for Partial Differential Equations (3 or 1).**- Prerequisite: Math 216, 316, or 286.
- Classical representation & convergence theorems for Fourier series; separation of variables for the 1-dim heat & wave eqn; heat & wave eqns in higher dim; spherical & cylindrical Bessel functions; Legendre polynomials; asymptotic integrals; discrete F ourier transform; applications to linear input-output systems, etc.

**Math 558 Ordinary Differential Equations (3).**- Prerequisite: Math 450 or 451.
- Existence and uniqueness theorems for flows, linear systems, Floquet theory, Poincar{\accent 19 e}-Bendixson theory, Poincar{\accent 19 e} maps, periodic solutions, stability theory, Hopf bifurcations, chaotic dynamics.

**Math 656 Introduction to Partial Differential Equations (3).**- Prerequisite: Math 558, 596 and 597 or permission of instructor.
- Characteristics, heat, wave and Laplace's equation, energy methods, maximum principles, distribution theory.

**Math 657 Nonlinear Partial Differential Equations (3).**- Prerequisite: Math 656 or permission of instructor.
- A survey of ideas and methods arising in the study of nonlinear partial differential equations, nonlinear variational problems, bifurcation theory, nonlinear semigroups, shock waves, dispersive equations.

**Math 756 Advanced Topics in Partial Differential Equations (3).**- May be taken more than once for credit.

### Differential Geometry

**Math 433 Introduction to Differential Geometry (3).**- Prerequisite: Math 215.
- Curves and surfaces in 3-space, using calculus. Curvature and torsion of curves. Curvature, covariant differentiation, parallelism, isometry, geodesics, and area on surfaces. Gauss Bonnet Theorem. Minimal surfaces.

**Math 537 Introduction to Differentiable Manifolds (3).**- Prerequisite: Math 590 and 513.
- Manifolds, differential forms, Stokes' theorem, Lie group basics, Riemannian metrics, Levi-Civita connection, geodesics, Riemann curvature tensor, Jacobi fields

**Math 635 Differential Geometry (3).**- Prerequisite: Math 537 or permission of instructor.
- Second fundamental form, Hadamard manifolds, spaces of constant curvature, first and second variational formulas, Rauch comparision theorem, and other topics chosen by the instructor

**Math 636 Topics in Differential Geometry (3).**- Prerequisite: Math 635.

### Lie Theory/Representation Theory

**Math 612 Lie Algebras and Their Representations.**

- Prerequisite: Math 593 and 594 or consent of instructor.
- Representation Theory of semisimple Lie algebras over the complex numbers. Weyl's Theorem, root systems, Harish Chandra's Theorem, Weyl's formulae and Kostant's Multiplicity Theorem. Lie groups, their Lie algebras and further examples of representatio ns.

### Logic and Foundations

**Math 481 Introduction to Mathematical Logic (3).**- Prerequisite: Math 412 or 451 or equivalent experience with abstract mathematics.
- In the first third of the course the notion of a formal language is introduced and propositional connectives, tautologies and tautological consequences are studied. The heart of the course is the study of 1st order predicate languages and their models . New elements here are quantifiers. Notions of truth, logical consequences, & provability lead to completeness & compactness. Applications.

**Math 582 Introduction to Set Theory (3).**- Prerequisite: Math 412 or 451 or equivalent experience with abstract mathematics.
- The main topics are set algebra (union, intersection), relations and functions, orderings (partial-, linear-, well-), the natural numbers, finite and denumerable sets, the Axiom of Choice, and ordinal and cardinal numbers.

**Math 681 Mathematical Logic (3).**- Prerequisite: Mathematical maturity appropriate to a 600-level course. (No previous knowledge of mathematical logic is needed.)
- Syntax and semantics of first-order logic. Formal deductive systems. Soundness and completeness theorems. Compactness principle and applications. Decision problems for formal theories. Additional topics may include non-standard models and logical syst ems other than classical first-order logic.

**Math 682 Set Theory (3).**- Prerequisite: Math 681 or Equivalent.
- Axiomatic development of set theory including cardinal and ordinal numbers. Constructible sets and the consistency of the axiom of choice and the generalized continuum hypothesis. Forcing and the independence of choice and the continuum hypothesis. Ad ditional topics may include combinatorial set theory, descriptive set theory, or further independence results.

**Math 683 Model Theory (3).**- Prerequisite: Math 681 or equivalent.
- Connections between classes of mathematical structures and the sentences (in first-order logic) describing them. Definable sets within structures; definable classes of structures. Methods for producing structures with prescribed properties. Categorica l and complete theories. Methods for analyzing the first-order properties of structures. Introduction to some concepts of classification theory.

**Math 684 Recursion Theory (3).**- Prerequisite: Math 681 or equivalent.
- Elementary theory of recursive functions, sets, and relations and recursively enumerable sets and relations. Definability and incompleteness in arithmetic. Godel's incompleteness theorems. Properties of r.e. sets. Relative recursiveness, degrees of un solvability and the jump operator. Oracle constructions. The Friedberg-Muchnik Theorem and the priority method.

**Math 781 Topics in Mathematical Logic (3).**- Prerequisite: Varies according to content.
- Advanced topics in mathematical logic. Content will vary from year to year. May be repeated for credit.

### Mathematical Physics

**Math 556 Methods of Applied Mathematics I (3).**- Prerequisites: Math 451, 513, and 555.
- Fourier analysis, Hilbert space, Sturm-Liouville problems, partial differential equations, Green's functions, distributions, weak solutions, eigenfunction expansions, orthogonal polynomials, special functions, scattering.

**Math 557 Methods of Applied Mathematics II (3).**- Prerequisites: Math 451, 513, and 555.
- Asymptotic expansions, regular and singular perturbation theory, nonlinear stability theory, bifurcations, applications to differential equations and spectral theory.

### Number Theory

**Math 475 Elementary Number Theory (3).**- Theory of congruences, Euler's function, Diophantine equations, quadratic domains. Intended primarily for students interested in secondary and collegiate teaching.

**Math 476 Computational Laboratory in Number Theory (3).**- Prerequisite: Math. 475 or 575; may be concurrently.
- Taken by students enrolled in a first course in number theory. Students conduct numerical explorations on IBM-PC-clones, using software t ailored for the purpose.

**Math 575 Introduction to Theory of Numbers (3; 1 hour credit for students with credit for Math 475.)**- Prerequisite: Math 451 and 513, or permission of instructor.
- Elementary theory of congruences. The quadratic reciprocity law. Properties of number theoretic functions.

**Math 675 Analytic Theory of Numbers (3).**- Prerequisite: Math 575, 596.
- Theory of the Riemann zeta-function and the L-functions, distribution of primes, Dirichlet's theorem on primes in a progression, quadratic forms, transcendental numbers.

**Math 676 Theory of Algebraic Numbers (3).**- Prerequisite: Math 575, 594.
- Arithmetic of algebraic extensions, the basis theorems for units, valuation and ideal theory.

**Math 677 Diophantine Problems (3).**- Prerequisite: Math 575.
- Topics in diophantine approximation, diophantine equations and transcendence.

**Math 678 Modular Forms (3).**- Prerequisite: Math 596 and 575.
- A basic introduction to modular functions, modular forms, modular groups. Hecke operators, Selberg trace formula. Applications to theory of partitions, quadratic forms, class field theory and elliptic curves.

**Math 679 Arithmetic of Elliptic Curves (3).**- Topics in the theory of elliptic curves.

**Math 775 Topics in Analytic Number Theory (3).**- Prerequisite: Math 675.
- Selected topics in analytic number theory.

**Math 776 Topics in Algebraic Number Theory (3).**- Prerequisite: Math 676.
- Selected topics in algebraic number theory.

**Math 777 Topics in Diophantine Problems (3).**- Prerequisite: Math 677.

### Numerical Analysis

**Math 403 Mathematical Modeling using Computer Algebra Systems (3).**- Prerequisite: one year of calculus.
- This course is designed to give students comprehensive training in the use of Mathematica or another Computer Algebra System. The course carries the student from from the rudiments of Mathematica, through its use in solving equations of all sorts, and finally to the point where the student is able to simulate realistic problems from their field of interest. Overview of Mathematica; manipulations; functions; evaluation of expressions; conditional function definitions; recursion; iteration; modeling in the physical and social sciences.

**Math 471 Introduction to Numerical Methods (3).**- Prerequisite: Math. 216, 316, or 286; 217, 417, or 419; and a working knowledge of one high-level computer language.
- Computer arithmetic; Newton's method for nonlinear equations; polynomial interpolation, numerical integration, systems of linear eqns; initial value problems for ordinary diff. eqns; quadrature; partial pivoting, spline approximations, partial differe ntial eqns; Monte Carlo methods.

**Math 571 Numerical Methods for Scientific Computing I (3).**- Prerequisite: Math 217, 419, or 513 and 454 or permission.
- Systems of linear equations, eigenvalue problems, direct and iterative methods, multigrid, conjugate gradient method, two-point boundary value problems, elliptic boundary value problems, finite-difference and finite-element methods.

**Math 572 Numerical Methods for Scientific Computing II (3).**- Prerequisite: Math 217, 419, or 513 and 454 or permission.
- Initial-value problems, ordinary differential equations, Runge-Kutta methods, multistep methods, stiff systems, heat and wave equations, finite-difference schemes, consistency, stability, convergence, CFL condition, Lax equivalence theorem, von Neuman n stability, nonlinear hyperbolic equations.

**Math 671 Analysis of Numerical Methods I (3).**- Prerequisite: Math 571, 572, or permission of instructor
- This is a course on special topics in numerical analysis and scientific computing. Subjects of current research interest will be included. Recent topics have been: Finite difference methods for hyperbolic problems, Multigrid methods for elliptic bound ary value problems. Students can take this class for credit repeatedly.

### Probability Theory

**Math 425 (Stat. 425) Introduction to Probability (3).**- Prerequisite: Math 215.
- This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theo ry developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of Math 116 and 215. Math c oncentrators should be sure to elect sections of the course that are taught by mathematics (not Statistics) faculty. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent even ts, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis.

**Math 525 (Stat. 525). Probability Theory (3).**- Prerequisite: Math 450 or 451; or permission of instructor.
- Axiomatic probability; combinatorics; random variables and their distributions; expectation; the mean, variance, and moment generating function; induced distributions; sums of independent random variables; the law of large numbers; the central limit t heorem. Optional topics drawn from: random walks, Markov chains, and/or martingales.

**Math 526 (Stat. 526). Discrete State Stochastic Processes (3).**- Prerequisite: Math 525 or EECS 501.
- Review of discrete distributions generating functions; compound distributions, renewal theorem, modelling of systems as Markov chains; Markov chains: first properties; Chapman-Kolmogorov equations; return and first passage times; classification of sta tes and periodicity; absorption probabilities and the forward equation; stationary distributions and the backward equation; ergodicity; limit properties; application to branching and queueing processes; examples from engineering, biological, and social sc iences; Markov chains in continuous time; embedded chains; the M/G/1 queue; Markovian decision processes, application to inventory problems; other topics at the instructor's option.

**Math 625 (Math. Stat. 625) Probability and Random Processes I (3).**- Prerequisite: Math 597.
- Axiomatics; measures and integration in abstract spaces. Fourier analysis, characteristic functions. Conditional expectation, Kolmogoroff extension theorem. Stochastic processes; Wiener-Levy, infinitely divisible, stable. Limit theorems, law of the it erated logarithm.

**Math 626 (Math. Stat 626) Probability and Random Processes II (3).**- Prerequisite: Math 625.
- Selected topics from among: diffusion theory and partial differential equations; spectral analysis; stationary processes, and ergodic theory; information theory; martingales and gambling systems; theory of partial sums.

### Topology

**Math 590 An Introduction to Topology (3).**- Prerequisite: Math 451.
- Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra and other topics.

**Math 591 General and Differential Topology (3).**- Prerequisite: Math 451.
- Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differential manifolds, tangent spaces, vector fields, submanifolds, inverse f unction theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces.

**Math 592 An Introduction to Algebraic Topology (3).**- Prerequisite: Math 591.
- Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Maclane axioms, Brouwer's and Lefschetz' fixed point theorems and other topics.

**Math 690 Topological Groups (3).**- Prerequisite: Math 590.
- Group theory, general topology, integration. Elementary properties, Haar measure, representation (Peter-Weyl), positive definite functions, Fourier transforms.

**Math 691 Combinatorial and Geometric Topology I (3).**- Prerequisite: Math 591.
- Selected topics in the theory of piecewise linear and topological manifolds.

**Math 692 Combinatorial and Geometric Topology II (3).**- Prerequisite: Math 691.
- Selected topics in the theory of piecewise linear and topological manifolds.

**Math 694 Differential Topology (3).**- Prerequisite: Math 537 and 591 or permission of instructor.
- Transversality, embedding theorems, vector bundles and selected topics from the theories of cobordism, surgery, and characteristic classes.

**Math 695 Algebraic Topology I (3).**- Prerequisite: Math 591 or permission of instructor.
- Cohomology Theory, the Universal Coefficient Theorems, Kunneth Theorems (product spaces and their homology and cohomology), fiber bundles, higher homotopy groups, Hurewicz' Theorem, Poincar{\accent 19 e} and Alexander duality.

**Math 696 Algebraic Topology II (3).**- Prerequisite: Math 695 or permission of instructor.
- Further topics in algebraic topology typically taken from: obstruction theory, cohomology operations, homotopy theory, spectral sequences and computations, cohomology of groups, characteristic classes.

**Math 697 Topics in Topology (3).**- An intermediate level topics course.

**Math 791 Advanced Topics in the Topology of Manifolds I (3).**- Prerequisite: Permission of instructor.