Stelson Lectures
Feb 3-4, 2000
Robert D. MacPherson
(Institute for Advanced Studies)
"Spaces of Lattices"
"Combinatorics of Polyhedra"
May 26-27, 1999
Richard Stanley
(Massachusetts Institute of Technology)
"characteristic polynomials of hyperplane arrangements"
ABSTRACT
By a hyperplane arrangement A, we mean a finite collection of affine hyperplanes in Rn. The characteristic polynomial c(A,q) is a polynomial in q of degree n that is a fundamental combinatorial invariant of A. We will survey some connections between c(A,q) and the combinatorial, topological, and algebraic properties of A. A number examples will be presented, including in particular the braid and Linial arrangements.
"volumes, mixed volumes, and lattice enumerators of convex polytopes"
ABSTRACT
This will be a survey talk devoted to exact formulas for volumes and lattice point enumerators of interesting classes of convex polytopes. Examples will include the polytope of degree sequences, the Catalanotope, flow polytopes, and the parking function polytope.
1998
Stephan Luckhaus
(Max Planck Institute, Bonn)
"the Stefan problem as a model for changes of phase"
ABSTRACT
In a continuum physics setting, phase changes that involve either a latent heat or concentration jumps across the phase interface have been modelled by the Stefan problem. In more than one dimension the resulting equation or system can only be solved excluding metastable phenomena like undercooling. In order to explain these phenomena, surface tension, i.e., interface curvature and velocity, has been introduced into the equations. This goes under the name Gibbs Thompson law. Methods from minimal surface theory are used to construct solutions for these systems. But again there are phenomena, nucleation for example, which are not explained by the theory. The hope is that using stochastic lattice models in approprite scaling limits one may find a solution to this problem, but this has not yet been done.
1997
Halil Mete Soner
(Carnegie Mellon University)
"reaction diffusion equations, mean curvature flow, and supercooled solidification"
ABSTRACT
Mathematical models for phase transitions postulate that there are several regions corresponding to different phases, and they are separated by interfaces. In sharp interface models, interfaces are assumed to be hypersurfaces, while the diffused interface theories use an additional field variable called the order parameter to model the interface. Typically, in the second approach, the order parameter is modelled as a solution to a reaction-diffusion equation with a small parameter.
In this talk, I will discuss both the sharp and the diffused interface models for the supercooled solidification. These models are natural extensions of the classical Stefan problem, and the sharp interface model is the asymptotic limit of the one with diffused interfaces. I will outline the methods used in this asymptotic result and, then, I will show the connection between these models and some geometric evolution problems, such as the mean curvature flow.
"Ginzburg-Landau model for superconductivity"
ABSTRACT
The Ginzburg and Landau theory for superconductivity is a phenomological model that was proposed in the late '50s. Mathematically, it is a variational problem for the magnetic vector potential and the order parameter which takes values in the unit ball of the complex plane. The length of the order parameter is proportional to the density of superconducting electrons. The Euler-Lagrange equations are the Maxwell's system coupled to a reaction diffusion equation, and it is very similar to the phase field model for supercooled solidification, which will be discussed in the first talk.
After a historical introduction, I will discuss recent results of Bethuel, Brezis, Helein, Jerrard, Lin and Struwe on a related variational problem. I will then outline some asymptotic results for the evolutionary Ginzburg-Landau system that I have obtained with R. Jerrard of the University of Illinois. In one particular limit, normal regions reduce to several points called vortices, and the dynamics of these vortices is governed by a system of ordinary differential equations.
1996
Gilles Pisier
(Universite Paris VI and Texas A&M University)
"similarity problems for representations of groups or operator algebras"
ABSTRACT
We will survey some recent progress on certain similarity problems which can be posed in various contexts. In the group case the problem has a long history going back to Sz. Nagy and Dixmier and may be formulated as: on which groups G is every uniformly bounded representation p : G Æ B(H) "unitarizable" (i.e. similar to a unitary one)? For the disc algebra, the analogous problem was posed by Halmos in 1970 and was recently solved: there is a polynomially bounded operator which is not similar to a contraction. For C*-algebras, the corresponding problem (posed in 1955 by Kadison) is still open: is every bounded homomorphism on a C*-algebra similar to a contractive one? The more recent concept of a "completely bounded map" allows us to treat these problems in a common framework. In each case (either groups, uniform algebras, or C*-algebras), the appropriate notion of "amenability" plays an important role in the recent developments that will be presented.
1995
Stanley Osher
(UCLA)
"subscale capturing in numerical analysis"
ABSTRACT
In these two talks we shall describe numerical methods which were devised for the purpose of computing small scale behavior without either fully resolving the whole solution or explicitly tracking certain singular parts of it. Techniques developed for this purpose include shock capturing, front capturing, and multiscale analysis. Areas in which these methods have recently proven useful include image processing, computer vision, and differential geometry, as well as more traditional fields of physics and engineering.
1993
Carsten Thomassen
(Technical University of Denmark)
"maps and graphs on surfaces"
ABSTRACT
In 1890, P. J. Heawood gave an upper bound for the number of colors needed to color a map on the orientable surface of genus g, i.e., the sphere with g handles added. About 80 years later, G. Ringel and J. W. T. Youngs showed that Heawood's bound (which tends to infinity as g tends to infinity) is best possible. For example, on the torus, seven colors are sufficient and, in some cases, also necessary for coloring a map.
In this lecture we describe a 5-color theorem for each surface: if every noncontractible curve intersects many countries of the map, then 5 colors are sufficient. There is no 4-color theorem of this type.
"sign-nonsingular matrices, permanents, and directed graphs"
ABSTRACT
A square matrix is sign-nonsingular if each term in the standard expression of the determinant is nonnegative. The problem of recognizing such a matrix is important in studying sign-solvability of linear systems of equations.
The permanent of a square matrix is defined as the determinant except that the alternating factor is omitted. The permanent is difficult to compute. In 1913 Polya suggested that the permanents of certain matrices can be computed by transforming them into determinants of sign-nonsingular matrices by multiplying some entries by -1. We show how these (and other) problems are related to a fundamental problem in graph theory, namely that of finding a directed cycle of even length in a directred graph.
1993
Persi Diaconis
(Harvard University)
"the mathematics of mixing things up"
ABSTRACT
I will argue that it takes seven ordinary shuffles to mix up 52 cards. The argument gives Hodge decompositions for Hochshild decompositions. (Joint work with David Bayer).
1989
Ciprian Foias
(Indiana University)
"the existence and the application of inertial sets in the study of dissipative partial differential equations"