Kempner Colloquium Abstracts

Spring 2004

Over the past twenty years, wavelets have gained popularity as bases for transforms used in image and signal processing. This talk will begin with a brief introduction to how wavelets arise naturally in this context. We will then show how the tools of abstract harmonic analysis and spectral multiplicity theory can be used to build and classify wavelets. In particular, we present results from joint work with L. Baggett, P. Jorgensen, H. Medina and J. Packer that extend the classical techniques of Mallat and Meyer to construct wavelets using generalized multi-resolution analyses and generalized filters.

The dynamics of opinion transformation is modeled by a neural network with a nonnegative matrix of connections. Noise is introduced at each site, and the limit of the stationary distributions of the resulting Markov chains as the noise goes to zero is taken as an indication of what configurations will be seen. An algorithm for computing this limit is given, and a number of examples are worked out. Some of the mathematical ideas developed, such as visible states, time scales, and a calculus of indexed probabilities, are of independent interest.

The box principle of Dirichlet is the fact that if several objects are distributed among several boxes with the number of objects exceeding the number of boxes, then there is at least one pair of these objects that lie in the same box. Although simple, this principle has strong limitations in the field of Diophantine approximation. For example, it is often at the very basis of a transcendence proof. Moreover, one expects that the conclusions derived from this principle are essentially optimal, except in obvious cases.

The goal of this talk is to present: 1) several results that can be derived from the box principle, 2) important instances where such results appear to be essentially optimal and 3) other instances where they fail to be so (an example of Cassels in several variables and more recent examples in one variable).

We consider a geometric existence and uniqueness problem for the geodesic PDE on the path space of the Euclidean line (a nonlinear type changing hyperbolic-parabolic evolution equation). Our objective is to construct the smoothest possible global single-valued solutions for which the length functional grows linearly with the curve's deformation parameter. The solutions will be constructed as Lagrangian surfaces in the cotangent bundle of the plane, and will be smooth-immersed on the complement of the singular curves. These well behaved singularities will carry topological invariants for the Lagrangian surface, and will occur exactly where the surface intersects the parabolic locus of the PDE. The construction is based on the observations that this PDE admits an infinite dimensional group of automorphisms, and intermediate (Hamilton-Jacobi) PDE.

In the talk, we will recall some connections between the theory of C^*-algebras and some applications, with special emphasis on wavelets and quantum information theory. In the wavelet part, we will touch on joint work with Larry Baggett, Kathy Merrill, and Judy Packer.

On one side, the focus will be on examples, or rather classes of C^*-algebras; especially the Cuntz algebras, but also the deformation C^*-algebras, such as the rotation algebras, and the q-deformation algebras derived from the Fermion/Boson algebras. We will stress the following key issues on C^*-algebras: their isomorphism classes, their representations, and some theorems on stability of C^*-isomorphism classes. This is relevant for particle physics, for example for quons and the Gibbs' paradox.

Representations of the Cuntz algebras, or the Cuntz relations, play a key role in analysis/synthesis filters in signal processing, both for transmission of speech and of images. They are used in compression in wavelet algorithms, and, at the same time, in a different guise, in quantum programs from quantum computation. While the famous factoring algorithm of P. Shor and the search algorithm of L. Grover are the two known quantum algorithms which are closest to being "practical", and at the same time in showing dramatic speedup compared to the corresponding classical algorithms, there are others, and the wavelet algorithm is one.

In the talk we will compare the wavelet algorithms in the two cases, classical and quantum. The role played by quantum error-correction codes will be touched on.

In 1884 Schwarz proved that a round soap bubble provides the least-area way to enclose a given volume of air. In 2002 Hutchings, Morgan, Ritoré, and Ros proved the Double Bubble Conjecture, which says that the familiar double soap bubble provides the least-area way to enclose and separate two given volumes of air. We'll discuss the history, the proof, open questions, and more recent results, some by students. No prerequisites.

In 1882 a brilliant young Dutch mathematician began work on a doctoral thesis problem suggested by his mentor Ch. Hermite. The Problem: Summation of divergent series; that is, finding an analytic function g(z) for which the series is its asymptotic expansion.

Stieltjes' work on this problem culminated in an 1886 thesis and an 1894 landmark paper in which he posed and solved the Stieltjes moment problem, introduced a new (Stieltjes) integral and gave two representations of g(z): the first a Stieltjes integral transform and the second a (Stieltjes) continued fraction. He also laid foundations for the general theory of orthogonal polynomials, linear functionals and normal families of analytic functions.

This expository talk will include biographical material on Stieltjes' life and work and an extension of Stieltjes' methods based on recent research on orthogonal polynomials. A specific application to the gamma function proves a conjecture made in 1982 by J. Cizek and E. R. Vrscay on the asymptotic behavior of the S-fraction coefficients associated with the function gamma(z).

The term periods is used in Algebraic Geometry and Number Theory to denote integrals of differential forms over cycles. K.T. Chen has shown how to extend this theory to Iterated Integrals, defined exactly as in elementary Calculus, and made applications to Topology. We will apply iterated integrals to complex manifolds that have Kähler metrics.

A variety of results/methods/problems will be presented in which representation theoretic methods are used to construct and characterize incidence structures possessing a high degree of regularity. A good example is Smith's disproof of a long standing conjecture of Storer that non-abelian difference sets exist only with parameters that are feasible for abelian difference sets.

To be successful such constructions require methods that keep track of more information than the underlying linear algebra. Successful approaches include exploitation of "cone conditions" having the flavor of linear programing and exploitation of integrality, leading to some very explicit number theoretic questions.

I will conclude with a list of some of these number theoretic questions.

This talk is concerned with frames, which are particular types of sequences in a Hilbert space that have useful basis-like properties even though they may be redundant, or overcomplete. Such redundant systems offer many advantages in applications, such as extra design flexibility or increased stability against noise or data loss. However, while redundancy has a clear qualitative meaning, quantifying redundancy turns out to be a difficult problem. We will review the basic properties of frames, and then introduce the concept of "localized" frames and relate redundancy to localization. As particular corollaries we recover the Nyquist density phenomena for windowed exponentials and Gabor frames, but furthermore extend these to new situations and derive some new properties of those systems.