Kempner Colloquium Abstracts

Spring 2005

The unitary dual of a reductive group over a local field plays an important role in noncommutative harmonic analysis. Its structure is also relevant for many problems in analysis, mathematical physics and automorphic forms.

In this talk I will survey progress on the determination of the unitary dual. The spherical case refers to infinite dimensional representations which have nontrivial fixed vectors under certain maximal compact subgroups. It is somewhat more tractable, but still exhibits many of the difficulties of the general problem. I will focus in particular on relations between the real and p-adic case.

Cryptography has traditionally been the concern of military and governmental groups. However, with businesses and individuals commonly conducting sensitive transactions over a fundamentally insecure Internet, cryptography is now pervasively employed to protect the privacy and integrity of these communications. How can we be sure the algorithms employed are sound, and how do we make them fast enough that people will tolerate their use?

This talk attempts to answer these questions by sketching an overview of the area and then focussing on the author's recent work related to fast provably-secure encryption and authentication schemes.

For the past seven years, I have collaborated with some of the finest mathematical artists in the country (Ferguson, Collins, Longhurst, Grossman, Sequin, Sullivan) to produce mathematical snow sculptures at the International Snow Sculpture Championships in Breckenridge. We have created shapes that are pleasing by themselves, but also surprising to see in snow. Our work has illustrated the discovery of minimal surfaces, the sphere eversion of Morin, and the topology of knots and nonorientable surfaces, and the public has responded warmly. We have been rewarded with prizes and publicity, but the real excitement comes from the design and construction process, as one learns a lot by getting so actively involved with a three-dimensional shape. I will discuss many of the details of design, for which we now use 3-D prototyping machinery, and the construction, which is restricted to hand tools only.

A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a collection, or "system", of unitary operators defined in terms of translation and dilation operations. We will begin by describing an operator-interpolation approach to wavelet theory using the local commutant of a unitary system that was developed by the speaker and his collaborators a few years ago. This is really an abstract application of the theory of operator algebras, mainly von Neumann algebras, to wavelet theory. The concrete applications of operator-interpolation to wavelet theory include results obtained using specially constructed families of wavelet sets. In fact X. Dai and the speaker had originally developed our theory of wavelet sets specifically to take advantage of their natural and elegant relationships with these wavelet unitary systems. We will discuss some unpublished or partially published results that are due to this speaker and his former and current students.

The values of Riemann's zeta function at positive integers had been studied already by Euler. In his investigation of products of such values he introduced multiple zeta values, namely

The Mahler measure of a monic polynomial with integer (or complex) coefficents and one variable is the product of the absolute values of those roots of the polynomial that are outside the unit circle. This will be a mostly elementary survey of some well known problems and recent results about the Mahler measure. The talk is intended to be suitable for a general mathematical audience.

It is commonly believed that randomness and chaos are the antithesis of structure and form. Basic physical theories - such as quantum mechanics - suggest that in some respects the universe is intrinsically random at the atomic level. Yet there is no lack of atomic structure. Life itself, depending as it does on statistical laws of genetics and inheritance, provides another example of the structure that can occur in an apparently random process.

In the talk we illustrate how randomness and structure can coexist by exploring a number of quite simple mathematical models of chaos. The talk will include a number visualizations of the remarkable geometry and structure that can be found in chaotic systems.

The talk is intended for a general audience and will have a significant visual component. (Quantum mechanics, statistics and biology are definitely not prerequisites.)

In this talk I will describe the classical Langlands philosophy, which occupies a central place in modern number theory and representation theory, and then describe a conjectural generalization. As a first instance, I will describe in brief recent work of Bump, the speaker and Ginzburg concerning exotic theta correspondences. Exotic is to be defined in the lecture!

The notion of a differential operator is intimately connected to the infinitesimal properties of the underlying space whether it is a smooth manifold, an analytic variety, or an algebraic scheme.

The algebra of all differential operators, and its homological invariants, is thus the best reflection of the infinitesimal structure of the space.

In my talk I will shed light on several fascinating, little known, and perhaps unexpected aspects of the algebra of differential operators.

We discuss some aspect of the question (a la Mark Kac) ``Can One Hear the Shape of a Fractal Drum?''. We will focus here on the case of a drum with fractal boundary. In particular, we will pay attention to the one-dimensional case, that of fractal strings (or harps). In this situation, a suitable direct spectral problem (studied jointly with Carl Pomerance, and later on, in a brader context, with Christina He and Machiel van Frankenhuysen) is shown to be directly connected with the Riemann zeta function in the critical srip, while a corresponding inverse spectral problem is intimately connected with the Riemann hypothesis; namely, we show (jointly with Helmut Mayer) that one can hear whether a (noncritical) fractal string is Minkowski measurable if and only if the Riemann hypothesis is true.

If time allows, we will briefly explain how these intial results led naturally to a theory of complex fractal dimensions of fractal strings, which captures some of the essential features and the intrinsic osilllations of fractal and arithmetic geometries. It has since been developed in a series of papers and in the research monograph (joint with Machiel van Frankenhuysen) Fractal Geometry and Number Theory: complex dimensions of fractal strings and zeros of zeta functions (Birkhauser, Boston, 2000); second revised and enlarged edition in press, 2005, 420 pp.. The higher-dimensional theory is currently being developed with one of the speaker's Ph.D. students, Erin Pearse. Finally, in a forthcoming book/essay, entitled In Search of the Riemann Zeros: strings, fractal membranes, and noncommutative spacetimes (approx. 510 pp.), motivated in part by some aspects of the above theory and of modern theoretical physics, a theory of quantized fractal strings (or ``fractal membranes'') is provided and the parallels uncovered between fractal, self-similar geometries and arithmetic geometries are further developed to provide a general framework in which to try to understand the Riemann hypothesis and its ramifications. A rigorous and noncommutative geometric version of part of this research program is currently being provided in joint work with Ryszard Nest, with whom very recently (i.e., over the last three weeks), we are also beginnning to develop a mathematical version of ``complex homology/cohomology'', as suggested by the abovementioned two research monographs.

Most (if not all) of the talk will be dedicated to a discussion of the earlier results regarding fractal drums and strings, as described in the first paragraph of this abstract. The talk should be self-contained and understandable to graduate students.

Based on the work of C. Fefferman and R. Graham in 1985, there have been systematic studies of the conformal invariants and conformally covariant operators on manifolds of any dimensions. A special case of such an operator is a fourth order linear elliptic operator with its leading symbol the bi-Laplace operator called the Paneitz operator.

In this talk, I will discuss the study of this operator on four manifolds, its associated curvature function called the Q-curvature, the connection of Q-curvature to the study of eigenvalues of the Ricci tensor and applications to some questions about the geometry and topology of 4-manifolds. I will also discuss some open questions in the area.