Non-abelian L-functions and Arithmetic

John H. Coates
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge, CB3 0WB, United Kingdom

Abstract : The twists of the complex L-functions of elliptic curves, or more generally motives, over Q by non-abelian Artin characters of the absolute Galois group of Q have been neglected by the experts in automorphic forms, despite their great interest in arithmetic questions. The goal of my lecture will be to try and explain why this neglect seems unjustified. The first part will recall the definition of these twisted L-functions, and describe some remarkable purely arithmetic phenomena which are predicted by the study of the signs in their functional equation. The second part of the lecture will describe an approach to proving these and other related arithmetic consequences via the study of a main conjecture in non-abelian Iwasawa theory.

An Iterative Regularization Method and Inverse Scale Space for Image Restoration

Stanley Osher
Department of Mathematics
University of California, Los Angeles
Box 951555 Los Angeles, CA 90095-1555, U.S.A

(joint with Martin Burger, Donald Goldfarb, Jinjun Xu and Wotao Yin)

Abstract : We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods, by using total variation regularization. We obtain rigorous convergence results and effective criteria for the general procedure. The numerical results for denoising and deblurring appear to give significant improvement over standard models. By taking the regularization parameter very small and the number of iteration steps large we are led to a new paradigm for restoration based on inverse scale space flows, instead of variational methods.

A Conjecture on the Unitary Dual of a Reductive Lie Group

Wilfried Schmid
Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA 02138, USA

Abstract : The irreducible unitary representations of a group G are the basic building blocks of Fourier analysis on G and its quotient spaces. Collectively these representations constitute the unitary dual of G. Although a great deal of information exists about the unitary dual of a reductive Lie group, a coherent general picture has not yet emerged. I shall describe a conjecture of Kari Vilonen and myself, on whose proof we are now working.

Cosmology, Black Holes, and Shock Waves Beyond the Hubble Length

Joel Smoller
Professor of Mathematics
University of Michigan
Ann Arbor, MI 48109-1109, U.S.A

Abstract : In this talk I will describe recent work with Blake Temple where we introduce a new Cosmological Model in which the expanding Friedmann universe emerges from a time reversed Black Hole in an event more similar to a classical explosion than the standard scenario of the Big Bang. In this new model there is a shock wave at the leading edge of the expanding galaxies, and the Big-Bang is an explosion of finite total mass. We believe that General Relativity pretty much forces such a model on you as soon as you relax the assumption in the standard model that the expansion of the galaxies is of infinite mass and extent at each fixed time after the Big-Bang, assuming that the explosion is large enough to be consistent with the enormous scale on which the galaxies and the cosmic background radiation appear uniform.

Holographic Algorithms

Leslie G. Valiant
Division of Engineering and Applied Sciences
Harvard University
33, Oxford Street, Cambridge, MA 02138, U.S.A.

Abstract : Using the notion of polynomial time reduction computer scientists have discovered an astonishingly rich web of interrelationships among the myriad natural computational problems that arise in diverse applications. These relationships have been used both to give evidence of intractability, such as that of NP-completeness, as well as some surprising new algorithms.

In this talk we discuss a notion of reduction, which we call a holographic reduction, that is more general than the traditional one. Instead of locally mapping solutions one-to-one it maps them many-to-many but preserves the sum of the solutions. One application is to finding new polynomial time algorithms where none was known before. We shall give some examples of such algorithms.

A more radical potential direction is that of revisiting the currently accepted conjectures of computer science, such as that P does not equal NP, and seeing whether this new kind of reduction offers any new insights towards either positive or negative results. The talk will review complexity theory in this light.

File translated from TEX by TTH, version 2.00.
On 09 Dec 2004, 09:26.