John H. Coates

Department of Pure Mathematics and Mathematical Statistics

University of Cambridge

Cambridge, CB3 0WB, United Kingdom

jhc13@dpmms.cam.ac.uk

Stanley Osher

Department of Mathematics

University of California, Los Angeles

Box 951555 Los Angeles, CA 90095-1555, U.S.A

sjo@math.ucla.edu

(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.

Wilfried Schmid

Department of Mathematics

Harvard University

One Oxford Street

Cambridge, MA 02138, USA

schmid@math.harvard.edu

Joel Smoller

Professor of Mathematics

University of Michigan

Ann Arbor, MI 48109-1109, U.S.A

smoller@umich.edu

**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.

Leslie G. Valiant

Division of Engineering and Applied Sciences

Harvard University

33, Oxford Street, Cambridge, MA 02138, U.S.A.

valiant@deas.harvard.edu

**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 T

On 09 Dec 2004, 09:26.