Mathematics Department and Graduate School Colloquia 1999-2000

Mathematics Department and Graduate School Colloquium Archive
1999-2000

May 30, 2000 Sibe Mardesic (University of Zagreb, Croatia) Shape theory

Abstract:
Shape theory is an extension of homotopy theory from the realm of CW-complexes to arbitrary topological spaces, especially metric compacta. One usually considers a 1968 paper of K. Borsuk as the beginning of the theory. The basic idea of shape theory consists in approximating spaces by polyhedra, more precisely in replacing spaces by inverse systems, and developing a homotopy theory of such systems. The use of a more sophisticated homotopy theory of systems (coherent homotopy) leads to strong shape theory. Shape theory has interesting applications in topology as well as in other areas of mathematics (C*-algebras, dynamical systems). In general topology one encounters applications of shape theory in the theory of continua, in hyperspaces, in compactifications and in the theory of fixed points. In algebraic topology shape theory led to new invariants like the homotopy and homology progroups and the shape groups. Čech homology groups are shape invariants and strong (Steenrod) homology groups are strong shape invariants.

May 23, 2000 Michael Cranston (Rochester) Dispersion of stochastic flows.

Abstract:
A problem in statistical fluid mechanics is to determine how fast a pollutant (passive tracer) will spread. We will discuss some models of motion of so-called 'passive tracers' and give some results on how far they might travel in a given amount of time. In one model each passive tracer undergoes a Brownian motion and the Brownian motions experienced by different tracers will have a given correlation. In another model the tracers move according to a random velocity field. This is the model given by Kolmogorov. The results in each case are similar and maybe a little surprising.

May 18, 2000 Kari Astala (University of Michigan) Meeting in 113 Smith Hall Quasiconformal mappings and PDE's

Abstract:
We give an overview of the connections and applications of (planar) quasiconformal mappings. Emphasis is given to the relations to differential equations. In particular we show how recent understanding of quasiconformal mappings leads to precise L^p- bounds for solutions of elliptic PDE's.

May 16, 2000 Margie Sussex (Bellevue High School) High School Mathematics from a Teacher's Perspective

Abstract:
Most people are aware by now that many changes are occurring in the teaching of pre-college mathematics. And most people are moderately confused (for good reason) about these changes. Ginger Warfield will make a bulkier-than-normal introduction giving a general outline of the situation both inside and outside of Washington. Then our speaker will give a more specific description of one particular situation, to wit, teaching from the NSF-generated Core-Plus materials. These materials were adopted throughout the Bellevue School District this year. Many teachers, including our speaker, have progressed in the course of the year from skepticism to enthusiasm.

May 9, 2000 Ruth Williams (San Diego) Dynamic control of stochastic networks in heavy traffic

Abstract:
Stochastic networks are used as models for complex manufacturing, telecommunications and computer systems. Some of these networks allow for flexible scheduling of jobs through dynamic (state-dependent) alternate routing and sequencing, hereafter collectively referred to as dynamic scheduling. Usually these models cannot be analyzed exactly, and it is a challenging problem to design dynamic scheduling policies for such networks that are simple to implement and yet are approximately optimal in an appropriate sense. As one approach to this problem, J. M. Harrison proposed the use of Brownian control problems (BCPs) as formal heavy traffic approximations to dynamic scheduling problems for stochastic networks. This talk will survey developments in the application and justification of this approach.

April 25, 2000 Michael Vogelius (Rutgers) Nonlinear boundary value problems originating from corrosion modeling

Abstract:
In this talk I will discuss some boundary value problems that naturally appear in the modeling of corrosion. I will consider the questions of existence and multiplicity of solutions, but will in particular emphasize the ``blow-up" nature of the solutions.

In the special case of a circular domain I will show how a whole family of solutions (quite surprisingly) may be expressed rather explicitly. An essential component of the derivation of these explicit formulae was a series of computational experiments---I will try to describe these as well.

April 18, 2000 Richard Stanley (MIT) Spanning trees

Abstract:
Spanning trees of graphs arise naturally in many different mathematical contexts. We will give a survey of some of these aspects of spanning trees, focusing on enumerative, algebraic, and geometric results. Some topics that we will discuss include electrical networks, volumes of convex polytopes, the Matrix-Tree Theorem, and especially a conjecture of Kontsevich on the number of zeros of a certain polynomial defined over a finite field.

February 22, 2000 Rob Meyerhoff Rigorous Computer-Aided Proofs in the Theory of Hyperbolic 3-Manifolds

Abstract:
Gabai, Meyerhoff, and N. Thurston obtained a rigorous computer-aided proof of the "Topological Rigidity of Hyperbolic 3-Manifolds", a result that has many implications in 3-manifold geometry/topology, and can be viewed as a piece of the Thurston Geometrization Conjecture. Because of the massive amount of computation and data which was involved, several computational difficulties had to be overcome. We will discuss some of these difficulties, and also indicate why the use of the computer in this proof and similar proofs in the theory of hyperbolic 3-manifolds is quite natural.

February 15, 2000 Note room change - 205 Smith Laci Lovasz, Microsoft Research Vector representations of graphs

Abstract:
To represent a graph in geometric way is a very natural and old problem. For example, it was proved by Steinitz early in this century that every 3-connected planar graph can be represented as the graph of vertices and edges of a (3-dimensional) polytope.

Representability of a graph in various geometric fashions turns out to be closely related to a number of basic properties of the graph. Moreover, computing these representations often helps in the design of algorithms for purely graph-theoretic problems.

With Lex Schrijver, we studied geometric representations that can be derived from a spectral invariant of a graph introduced by Colin de Verdière. For a planar graph, for example, one obtains two representations which are "dual" to each other in some sense. Linkless embeddability in 3-space can also be characterized in terms of this invariant.

February 8, 2000 Franc Forstneric, University of Wisconsin Embedding Stein Manifolds in Euclidean Spaces

Abstract:
A complex manifold is said to be a Stein manifold if it admits plenty of global holomorphic functions, in a certain precise sense. Examples of Stein manifolds include open Riemann surfaces, as well as pseudoconvex domains in complex Euclidean spaces. One characterization of Stein manifolds is that they may be represented as closed complex submanifolds of complex Euclidean spaces. In this talk I will survey results on holomorphic embeddings of Stein manifolds in Euclidean spaces, with emphasis on the results from the last decade on embeddings (and immersions) into spaces of minimal dimension. I will also describe some of the methods used in the constructions of such embeddings, in particular the connection with the Oka-Grauert principle.

February 7, 2000 (Monday) Joint Mathematics-Physics Colloquium 4:00 p.m., 120 Kane Hall Edward Witten, Institute for Advanced Study/Cal Tech Strings, Quark Confinement, and Black Holes

January 27, 2000 (Thursday) Rekha Thomas Associated Primes of Toric Initial Ideals 4:00 p.m., C-36 Padelford

Rekha Thomas is a candidate for a position in our department. Her research is in discrete mathematics.

Abstract:
Toric ideals are the defining ideals of varieties that are parameterized by monomials. In the past decade, several new connections have been established between toric ideals and convex polytopes via the computational tool of Groebner bases. In particular, toric ideals can be used to solve integer programs, a connection that has led to several new theoretical results in integer programming.

Groebner bases theory allows one to study toric ideals via their monomial initial ideals. In this talk, I will present recent results on the primary decompositions and associated primes of toric initial ideals. All of these algebraic results are proved using certain natural lattice-point-free polytopes that come from discrete optimization. They can also be reformulated as new results in the classical technique of group relaxations in integer programming.

January 25, 2000 Igor Pak Combinatorics, Probability and Computation on Finite Groups

Igor Pak is a candidate for a position in our department. His research is in discrete mathematics.

Abstract:
I will give a somewhat biased review of recent results on theoretical and practical methods for generating random group elements. Basically we will start by introducing the algorithms and discuss problems from various fields as they arise. No previous knowledge of the subject is assumed.

January 20, 2000 (Thursday) Gregery Buzzard Henon maps and structural stability 4:00 p.m., C-36 Padelford

Gregery Buzzard is a candidate for a position in our department. His research is in several complex variables and complex dynamics.

Abstract:
A Henon map is a diffeomorphism of complex 2-space having polynomial coordinate functions and interesting dynamics. The study of the dynamics of these maps has been profitably influenced by earlier work on the dynamics of diffeomorphisms of compact manifolds and on the dynamics of polynomials in one complex variable. I will explain a little about both of these influences and discuss related work on the stability of Henon maps; i.e., understanding when a small change in the given map leads to a corresondingly small change in the dynamics.

January 11, 2000 Sándor Kovács Arakelov-Parshin boundedness in higher dimensions

Sándor Kovács is a candidate for the position of assistant professor in our department. His research is in algebraic geometry.

Abstract:
At the ICM'62 in Stockholm Shafarevich conjectured the following: Let C be a fixed compact Riemann surface, Δ \subset C a fixed set of finite points and g > 1 a fixed natural number. Then the number of different smooth projective families of curves of genus g over C \ Δ is finite.

This was confirmed by Arakelov in 1971 following some ideas of Parshin. Their method was to divide the problem into two: "boundedness" and "rigidity". Boundedness means that the parameter space of such families is of finite type, in particular it has finitely many components, while rigidity means that there exist no non-trivial deformations of these families, i.e., the components of the parameter space are 0-dimensional. Together they imply that the parameter space is finite.

Higher dimensional generalizations of the boundedness part will be discussed in the lecture as well as applications of the results to other questions such as the Catanese-Schneider conjecture on the minimal number of singular fibers of a family of minimal varieties of general type (higher dimensional analogues of curves of genus greater than 1) and Shokurov's conjecture on the algebraic hyperbolicity of certain fine moduli spaces.

January 6, 2000 Peter Ebenfelt, Sweden Title to be announced Peter Ebenfelt is a candidate for a position in the Department of Mathematics.

November 30, 1999 Paul Gunnells, Columbia University Modular forms, toric varieties, and nonvanishing of $L$ -functions

Abstract:
A modular form is a holomorphic function that satisfies a certain symmetry with respect to a discrete group action. Modular forms are important in number theory because they package arithmetic information into an analytic object. For example, the Shimura-Taniyama-Weil conjecture asserts that if a modular form

f

satisfies certain conditions, then its Fourier coefficents equal the number of points of an elliptic curve

E

over finite fields. Another example concerns the $L$ -function

L (f,s)

associated with a modular form. According to the Birch and Swinnerton-Dyer conjecture, the order of vanishing of

L (f,s)

s=1

is related to the complexity of the set of rational points of

E

. A toric variety is an algebraic variety built out of the combinatorial data of a collection of cones in a lattice. Examples include affine space

C

ⁿ and projective space

P

ⁿ (C). In contrast to an elliptic curve, a toric variety is a much less complicated variety from a number-theoretic point of view. For example, one can easily count the number of points of a toric variety over a finite field. In this talk we present work with Lev Borisov connecting toric varieties to modular forms. We construct a subring

T (l) of the modular forms, the toric modular forms. Our main result is that

T (l) is a natural subring, in the sense that it is stable under various operations from the classical theory of modular forms. We also characterize the space of toric modular forms associated to toric surfaces: it coincides (modulo Eisenstein series) with the space of cusp forms with nonvanishing

L

-functions. The talk will be directed to a general mathematical audience. In particular, we will not assume that anyone has ever seen a modular form, an

L

-function, or a toric variety before.

November 16, 1999 Mario Bonk, Tech. Univ. Braunschweig Negative curvature in analysis

Abstract:
Gromov introduced a notion of hyperbolicity for general metric spaces. This concept captures some of the essential features of negatively curved spaces, and gives a general framework for phenomena like Mostow rigidity.

In this talk I will give a brief survey of this theory and show how the concept of Gromov hyperbolicity is useful in various areas of analysis. Examples include the theory of Martin boundaries, invariant metrics on pseudoconvex domains, quasiconformal mappings, and geometric function theory.

The talk will be understandable for non-experts.

October 26, 1999 Michael Freedman, Microsoft Theory Group Mueller 153 Computer Science in the Early Universe

Abstract:
The Shor factoring algorithm suggests that standard finite dimensional quantum mechanical systems can be manipulated to perform computations which would be too (exponentially) slow on a classical computer. It is natural to wonder if the more exotic mathematical progeny of theoretical physics: modular functors, TQFTs, mirror symmetry, etc..., support computational models of even greater power. We will answer the first two questions in the negative by showing that the action a Unitary topological modular functor can be efficiently simulated in the state space of an (ordinary) quantum computer.

October 19, 1999 Rafe Mazzeo, Stanford Kähler-Einstein cone metrics and critical exponent elliptic problems

Abstract:
The geometric problem for which I'll describe the solution is the existence of a Kähler-Einstein metric on a compact Kähler manifold which has a specified edge singularity along a smooth divisor. After some discussion of the geometric background and analysis required in the proof, I'll indicate how this problem leads to a new viewpoint on critical growth phenomena in nonlinear elliptic equations, and will also touch on a few other equations where the techniques developed here are useful.

September 28, 1999 Richard Melrose, MIT Toeplitz Operators, Bundles of Algebras and Index Theorems

Abstract:
I will discuss the notion of a Toeplitz operator on a contact manifold, due to Boutet de Monvel and Guillemin, and relate it to the Heisenberg algebra of Dynin, Beals and Greiner, and Taylor. In this way I will indicate how the index theorems of Atiyah and Singer and of Boutet de Monvel are related and how this has led, in recent work with C. L. Epstein, to a conformation of the index formula, for Fourier integral operators, proposed by Atiyah and Weinstein.