2007-2008 GRADUATE COURSE DESCRIPTIONS

Measure theory and integration. Point set topology. Principles of functional analysis. L^pspaces. The Riesz representation theorem. Topics in real and functional analysis.

Graduate level-matrix theory with introduction to matrix computations. SVDs, pseudoinverses, variational characterization of eigenvalues, perturbation theory, direct and iterative methods for matrix computations.

Linear multistep methods and Runge-Kutta methods for ordinary differential equations: stability, order and convergence. Stiffness. Differential algebraic equations. Numerical solution of boundary value problems.

Finite difference methods for hyperbolic, parabolic and elliptic PDEs, with application to problems in science and engineering. Convergence, consistency, order and stability of finite difference methods. Dissipation and dispersion. Finite volume methods. Software design and adaptivity.

Weighted residual and finite element methods for the solution of hyperbolic, parabolic and elliptical partial differential equations, with application to problems in science and engineering. Error estimates. Standard and discontinuous Galerkin methods.

Group theory, ring and module theory, field theory, Galois theory, other topics.

Metric spaces, topological spaces, continuity, Hausdorff condition, compactness, connectedness, product spaces, quotient spaces. Other topics as time allows.

Homotopy groups, exact sequences, fiber spaces, covering spaces, van Kampen Theorem.

Topological manifolds, differentiable manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefschetz fixed point theorem, vector fields.

This course will be a year-long introduction to the subject of diophantine geometry, which, in its modern form, is the study of rational and integral points on algebraic varieties. This subject is over two thousand years old, and it is extremely rich, involving techniques from many different areas of pure mathematics. I have not yet decided on the exact form that the course will take; you may find it helpful to have a look at some of the following references:

J.-P. Serre, Lectures on the Mordell-Weil Theorem

M. Hindry and J. Silverman, Diophantine Geometry; An Introduction

E. Bomieri, W. Gubler, Heights in Diophantine geometry

The prerequisites for this course include a good knowledge of standard first year graduate material in pure mathematics, and a level of mathematical maturity appropriate for an advanced graduate course.

Casson's Invariant The course will develop from scratch an elegant and surprising construction, due to Andrew Casson, of an integer invariant of homology 3- spheres. This turns out to have many powerful applications, for example it can be used to show that there are topological 4-manifolds which are not homeomorphic to any simplicial complex. This invariant was subsequently the basis for an enormous amount of mathematics, including Floer homology.

Groups, Geometry, and TopologyThis course will cover various aspects of groups in relation to topology. Some possible topics:
(1) Representations of finite groups and the homology of 3-manifolds
(2) Lie groups, their representations, and geometric structures
(3) Representations of a given group into SL(2,C) and the A polynomial of a knot
(4) Relations between invariants of groups and geometry for example the volume of a closed hyperbolic 3-manifold is at most pi times the sum pf the word lengths of the relations in any presentation of the group.
I will tailor the course to the interests of those attending.

Exceptional Mathematics Many parts of mathematics have classification theorems that state that every widget belongs to one of the following infinite families of widgets or it belongs to a finite list of exceptional widgets. Following John Baez we call the infinite families \emph{classical mathematics} and the exceptions, \emph{exceptional mathematics}. The main goal of this course is to illustrate the emerging philosophy that, in the same way that the classical aspects of different mathematical disciplines are related, the exceptional aspects are also related. In particular, to first order, there is an underlying theory to exceptional mathematics, the so called exceptions are essentially all related to each other and many (if not most) of the connections hover around the octonions.

In this course I will start with the quaternions and octonions before continuing on to the exceptional Cayley plane, the E_8 lattice in R^8, the Leech lattice in R^24 and the hyperbolic reflection group closely tied to the Monster finite simple group. Other exceptional topics will be covered as time permits. Many parts of classical mathematics (such as the geometry of projective space, hyperbolic space and some other symmetric spaces) will be developed along the way.

Quantized Enveloping Algebras of Lie Algebras This course will offer an introduction to some of the core ideas in a subject called, somewhat mysteriously, "Quantum Groups", a field which arose from mathematical physics in the 1980's and has since developed many connections with areas as diverse as representation theory, noncommutative geometry, and knot theory. Its historical origin was the study of the "quantum Yang-Baxter equation" in quantum statistical mechanics, solutions to which came from the representation theory of "quantized" enveloping algebras of Lie algebras.
The prerequisite is just 220ABC; background in Lie theory, representation theory, or quantum groups is not assumed, but will be developed as needed. A sample list of topics follows.
1. Lie algebras (non-associative algebras which appear throughout mathematics) and their representations (homomorphisms into matrix algebras).
2. The enveloping algebra of a Lie algebra (an associative algebra whose modules are the representations of the Lie algebra, just as modules over a group algebra are the representations of the group).
3. Hopf structure of enveloping algebras ( a means to turn tensor products and duals of representations into representations).
4. The quantum Yang-Baxter and R-matrices
5. Construction of quantized enveloping algebras of semisimple Lie algebras.
6. Finite dimensional representations of quantized enveloping algebras.
7. The universal R-matrix of a quantized enveloping algebra (which provides a solution to the quantum Yang-Baxter equation on each finite dimensional representation).
8. Hopf duality for (quantized) enveloping algebras (by which the dual algebra becomes a (quantized) algebra of functions of a Lie Group).

Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.

During the first quarter, we will develop the basics of affine and projective geometry to study "affine algebraic groups". These are groups G which carry the structure of affine algebraic varieties, with the property that all group operations are morphisms of varieties, with the property that all group operations are morphisms of varieties. In letting G act by isomorphisms on an affine or projective variety X, one obtains an interesting interplay between properties of X and the regular functions from X to the base field which are constant on the G orbits. Applications of this theory to varieties parametrizing classes of representations of a finite dimensional algebra yield strong classification results on the representation theoretic side. Beyond classification, we will discuss degenerations of a representaion M; these are representations obtained from M by successively "breaking structural bonds" of M in a geometrically guided process. (This process consists of passing from the pints inside an orbit of a relevant group action to points on the boundary). Thus, one obtains simpler and simpler objects, which help to analyze M.

The only prerequisite is Math 220 ABC. In particular, no background in representation theory will be assumed.

Topics include geometry of surfaces, manifolds, differential forms, Lie groups, Riemannian manifolds, Levi-Civita connection and curvature, curvature and topology, Hodge theory. Additional topics such as bundles and characteristic classes, spin structures Dirac operator, comparison theorems in Riemannian geometry.

Comparison Estimate for Integral Curvature Comparison theorems usually say that if curvature is bounded from below then certain geometric quantities are bounded by that of the model space. This method is very useful in the study of interplay between geometry and topology, in particular the volume comparison and Laplace comparison of the Ricci curvature have had many topological applications. The infinite dimensional analogue of the volume comparison for the Ricci flow also gives Perelman's reduced volume monotonicity, the main tool in his work of the geometrization conjecture. Integral curvature bound is much weaker and many times more natural, we will discuss generalization of some comparison theorems to integral curvature and give many applications.

Ricci flow is an important geometric evolution equation in Riemannian Geometry. Introduced by R. Hamilton in 1982 it was used extensively by him to obtain some outstanding results on 3- manifolds and 4-manifolds. It culminated in the recent spectacular work by G. Perelman on the Poincare conjecture and the geometrization conjecture on 3-manifold.

In these two courses we'll present some material of Hamilton and Perelman's theory of the Ricci flow and its application to geometrization of 3-manifolds. Here is a tentative list of topics: short time existence of the Ricci flow, Riemannian manifolds with nonnegative curvature, the pinching results for the Ricci flow on manifolds of positive curvature, Perelman's entropy functional and the noncollapsing, the reduced distance and the reduced volume, gradient shrinking solitions, kappa-solutions, canonical neighborhoods and surgery of the Ricci flow, finite time extinction and the Poincare conjecture, emergence of hyperbolic structures.

Existence and stability of solutions, Floquet theory, Poincare-Bendixon theorem, invariant manifolds, existence and stability of periodic solutions, bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, RuelleTakens cascade.

Every non-negative polynomial on the line or on the circle is a sum of squares. This observation provides a robust algebraic criterion (called nowadays a positivity certificate) to verify on coefficients the non-negativity of a polynomial function.
In two or more variables there is a substantial gap between non-negative polynomials (on prescribed algebraic supports) and weighted sums of squares decompositions. However, the last decade has greatly improved our knowledge of this gap. The resulting certificates of positivity are very refined, and on the other hand very flexible. They have applications to every branch of mathematics where polynomial optimization with constraints is needed.

The course will offer a 10 week introduction to the subject, and will end at the edge of our collective knowledge. Even more important: this class will provide and abundant list of open problems, from very accessible to very hard, appropriate for the student interested in research in this area.

For details see the survey tibi.pdf

     A. Image Processing (Jean-Pierre Fouque)
          a. Stochastic PDEs applied to Image Processing
     B. Turbulence (Birnir)
          a. Existence of Solutions to Stochastic Nonlinear PDEs
          b. Existence and Uniqueness of Invariant Measures
          c. Applications to Turbulence
     C. Space-time White Noise (Bonnet)
          a. Stochastic PDEs with Space-time White Noise
          b. Applications to Traffic Problems

In this course we will go through the modern theory of SPDEs form three vantage points and discuss applications on three areas. The recent solutions of some turbulence problems will be explained.

GRADUATE PROGRAMS
	Application Request		PhD Programs
	General Information		Area Requirements
	The Faculty		2007-2008 Graduate Courses
	Admissions		Graduate Program Handbook [PDF]
	Microsoft Designee Program		Teaching Assistant Handbook [PDF]
	Graduate Financial Support		The Graduate Students
	Undergraduate Preparation		Graduate Program Brochure [PDF]
	Master's Degree Programs