Kempner Colloquium Abstracts

Fall 2004

Suppose that a finite group G can be embedded in the group of nonsingular n × n complex matrices. What can we say about G? Jordan showed that, modulo an abelian normal subgroup, the order is bounded; Frobenius gave a specific bound, but of too great an order of magnitude.

Boris Weisfeiler, who disappeared about twenty years ago, left a near-complete manuscript in which he almost gave the correct generic bound, with extensions to arbitrary algebraically closed fields in place of the complex numbers. This uses the classification of finite simple groups.

I will discuss a reworking of Weisfeiler's work in which more precise information is given. My talk will not assume any technical knowledge of group theory.

The Ricci flow was introduced in 1982 by Richard Hamilton to study Riemannian metrics. It acts as a heat equation, averaging out curvature to make a manifold more and more uniform. Such a flow can start with a complicated manifold and flow it to a very simple one such as a sphere or flat torus. For this reason Ricci flow has found applications toward classifying manifolds and smoothing metrics in geometry and physics. Recent work by Perelman has overcome many of the obstacles in Hamilton's program to classify closed three-dimensional manifolds using Ricci flow. This also solves the Poincaré conjecture, giving a purely topological result. We shall look at the Ricci flow, some of the problems for which it is used (primarily geometrization of three-manifolds), and look at some important techniques in studying the flow, most notably ways of attacking solutions which become singular.

Many epidemics, particularly those related to sexually transmitted diseases, have a core group in which the epidemic is particularly prevalent. When the resources for controlling the epidemic are limited, the extent to which these means are allotted to the various groups is a problem, which is studied here by means of mathematical models. It is shown that the core group should get the lion's share, much more than is currently done, e.g. with HIV.

In this talk we will revise several results about complete spacelike hypersurfaces in the de Sitter space. In particular, we will characterize the totally umbilical round spheres under hypothesis relative to their higher order mean curvatures, hyperbolic image and volume. We will also study the case of space like surfaces in the 3-dimensional de Sitter space, revising some rigidity results for totally umbilical round spheres under assumptions relative to the curvatures of their first and second fundamental forms.

Classical Schur-Weyl duality is a beautiful segment of invariant theory in which one considers the dual actions of the general linear group and symmetric group on tensors. I will try to explain how this works, and then discuss various generalizations, ending up with a very recent result which has close connections with the combinatorics of derangements.

In recent years Voiculescu's free probability has made a great deal of progress in operator algebras. It has also made connections to many different areas of mathematics including combinatorics, random matrices, probability, operator algebras, fractal geometry, L² Betti numbers, and homology. I will discuss some of these connections, paying special attention to its operator algebra applications.

In recent years finitely generated groups, such as fundamental groups of compact manifolds, have been studied via their representation theory, (i.e. harmonic analysis), but also as metric spaces, (i.e. geometric group theory). Studying the relation between these points of view has proved to be useful for both. In this talk we will try to describe the analytic and geometric settings carefully and give a survey results relating them. We will also discuss applications to topological rigidity theorems.

The talk will be devoted to the best possible estimate of the deviation of a partial sum of an asymptotic series from a function for the moderate (not necessarily large) values of the independent variable. The result will be illustrated by the classical Stirling formula for n!.

Ask most any cognitive scientist working today if a digital computational system could develop aesthetic sensibility and you will likely receive the optimistic reply that this remains an open empirical question. However, I attempt to show, while drawing upon the later Wittgenstein, that the correct answer is in fact available. And it is a negative a priori. It would seem, for example, that recent computational successes in textual attribution, most notably those of Donald Foster (famed finder of Ted Kaczynski a.k.a. "the Unabomber") speak favorably of the digital model's capacity to overcome the "aspect blindness" handicap in this domain. I argue however that such results are only achievable when rigid input-to-output parameters are given, and that this element is precisely what is absent in standard examples of aesthetic judgment. I will thus conclude that while the connectionist model anticipated by Turing may provide the best approach for the AI project, its capacity for meeting its own sufficiency requirements is necessarily crippled by its inability to share in what can be generally referred to as the collective engagements of human solidarity.