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.

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.

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.