Applied Math Colloquium

We discuss several rigidity results connected to topological quantum field theories. We show that there are countably many rational TQFT's, and that the invariants of manifolds computed with them are algebraic numbers.

We describe all the theories which have a generator of quantum dimension < 2, and show that they correspond to the subgroups of quantum SU(2) at roots of 1. For instance, the quantum binary icosahedral group has 32 representations and appears at the 30th root of 1.

These subgroups arise naturally as symmetries of paths on the Coxeter ADE graphs. Their modular invariants explain the ADE classification of conformal invariants of Itzykson, Capelli and Zuber.

Recent advances in representation theory lead to combinatorial problems involving convex polytopes and their piecewise-linear transformations. We will discuss some of these problems arising in the representation theory of quantum groups, affine Hecke algebras, quivers, etc. Unexpectedly, their solution uses concepts and methods from combinatorial optimization (flows in networks) and control theory (tropical semirings).

Consider a closed finite triangulated oriented polyhedral surface S in three-space. Regard the edges of S as rigid inextendible incompressible bars at ideal universal joints, the vertices of S. There are several examples when the bar constraints on the edges allow the shape of S to change. In other words S has a non-trivial flex. We show that the volume bounded by S during such a flex is constant. This can be though of as saying that there is no exact mathematical "bellows" that can change its enclosed volume.

This result (as well the notion of bounding volume) can be generalized considerably to consider the case when S is any (possibly) singular oriented simplicial cycle of dimension n-1 in any Euclidean space of dimension n, where n is three or greater.

This is joint work with I. Sabitov and A. Walz.

The non-relativistic quantum problem of N interacting particles scales exponentially with N, making an exact solution practically out of reach even for comparatively small values of N. I will discuss some approximate but non-perturbative methods for tackling such difficult problems.

A numerical approach to Hilbert's 16th problem, which is to bound the number of limit cycles in a plane polynomial direction field, leads to the need to justify the numerical solutions.This could be done by using the approximate solutions to prove the existence of nearby exact solutions. A way to do this is shown.

Many dissipative systems, most notably turbulent fluid flows, reach a well-defined steady state when they are maintained under constant out-of-equilibrium conditions. In most cases a mechanism for selecting this state (such as a minimization principle) is not known. In some spatially extended systems, the selected state is manifested as a pattern. I will discuss experimental results in two different areas: chemical pattern formation and the instabilities of flexible objects in a laminar flow (vortex-body interactions). I first present a study of rotating spiral waves in the Belousov-Zhabotinsky chemical reaction, focusing on the spiral selection problem, and the various pattern instabilities which we have observed. I discuss the implications for planned experiments on a chemical computer. Second, I present an experiment on the transition from fluttering to tumbling motion in falling paper. Finally, I will present preliminary results from an experiment on the motion of a flexible plate in an imposed flow, with direct relevance to the question of how fish swim.

American mathematics has changed dramatically during the 20th century, maturing into one of the finest mathematics communities in the world. But many features of professional life and mathematics education have stayed the same, or cycled through familiar phases. This is a survey of American mathematics -- professional issues, education, and mathematics itself -- during the past 100 years, emphasizing both those differences and similarities.

The energetics of high-Reynolds-number turbulent flows are usually thought to be unaffected by fluid viscosity if a region very close to solid boundaries is excluded. This basic principle leads to a number of explicit results, including the so-called logarithmic law for the mean velocity distribution. This talk will provide some background on some traditional results, explain a current controversy arising from ascribing a more direct role to viscosity, and discusses a simple way of understanding the essentials of the issue.