SCIENTIFIC PROGRAMS AND ACTIVITIES

Professor Adimurthi, TIFR Center, Bangalore India
Conservation Laws with Discontinous Flux
The one dimensional scalar conservation law with smooth flux has been studied for over fifty years. There is a well developed theory to obtain existence and uniqueness of entropy solutions, due to Lax-Oleinik, Kruzkov and others.

If the underlying flux is discontinous the theory is not well understood. Basically such problems arise in multi-phase flow problems. One example is the two-phase flow problem coming from extraction of oil by pumping water into the oil well. Here I will discuss the existence, entropy and uniqueness of such problems.

Liliana Borcea,
Computational and Applied Mathematics, Rice University
Electrical Impedance Tomography with resistor networks
We present a novel inversion algorithm for electrical impedance tomography in two dimensions, based on a model reduction approach. The reduced models are resistor networks that arise in five point stencil discretizations of the elliptic partial differential equation satisfied by the electric potential, on adaptive grids that are computed as part of the problem. We prove the unique solvability of the model reduction problem for a broad class of measurements of the Dirichlet to Neumann map. The size of the networks (reduced models) is limited by the precision of the measurements. The resulting grids are naturally refined near the boundary, where we make the measurements and where we expect better resolution of the images. To determine the unknown conductivity, we use the resistor networks to define a nonlinear mapping of the data, that behaves as an approximate inverse of the forward map. Then, we propose an efficient Newton-type iteration for finding the conductivity, using this map. We also show how to incorporate apriori information about the conductivity in the inversion scheme.

* This is a joint University of Toronto Math Department Colloquium/Fields Institute Colloquium in Applied Mathematics

Phil Holmes, Princeton University
From neural oscillators through stochastic dynamics to optimal decisions, or Does math matter to gray matter?
The sequential probability ratio test (SPRT) is optimal in that it allows one to accept or reject hypotheses, based on noisy incoming evidence, with the minumum number of observations for a given level of accuracy. There is increasing neural and behavioral evidence that primate and human brains employ a continuum analogue of SPRT: the drift-diffusion (DD) process. I will review this and also describe how a biophysical model of a pool of spiking neurons can be simplified to a phase oscillator and analysed to yield spike rates in response to stimuli. These spike rates tune DD parameters via neurotransmitter release. This study is a small step toward the construction of a series of models, at different time and space scales, linking neural spikes to human decisions.

This work is joint with Eric Brown, Jeff Moehlis, Rafal Bogacz and Jonathan Cohen at Princeton, and Garry Aston-Jones' group at the Laboratory of Neuromodulation and Behavior, University of Pennsylvania.

Adam Oberman, Simon Fraser University

The Infinity Laplacian: from classical analysis to image processing and random turn games.
The Infinity Laplacian equation is currently at the interface of a number of different mathematical fields. It was first studied in the 1950s by the Swedish mathematician Gunnar Aronsson, motivated by classical analysis problem of building Lipschitz extensions of a given function. While Aronsson was able to find interesting exact solutions, progress stalled because solutions were non-classical. It took another forty years until analytical tools were developed to study the equation rigorously, and computational tools were developed which made numerical solution of the equation possible.

In the last decade, PDE theorists established existence and uniqueness, and (quite recently) appropriate regularity results. At the same time, the image processing community was using the operator for edge detection, and for inpainting, the reparation of images with damage. While the operator was promising, they had little success, since traditional methods for solving the equation yielded poor results.

It turns out that the right way to solve the equation is to go back to the original Lipschitz extension problem. This leads to a formula for the discrete operator with a simple interpreation, and good solution properties. This formula also leads to another surprising connection with probability theory.

Working in the unrelated field of percolation theory, a group of probalists (Peres-Shramm-Sheffield-Wilson) studying a randomized version of a marble game called Hex found a connection with the Infinity Laplacian equation. This connection gives an interpretation of the equation as a two player random game.

I'll tell this story, and explain some of the more accessible properties of the equation, along with pictures and numerical results.

Alexander Plakhov, University of Aveiro
Billiards, optimal mass transport and problems of optimal aerodynamic resistance.
A body moves through a medium consisting of point particles. The medium is very rare, so that the mutual interaction of the particles is neglected. Interaction of the particles with the body is absolutely elastic. We consider the following problem: find the body, from a given class of bodies, such that the force of resistance of the medium to its motion is minimal or maximal.

The (minimization) problem was firstly considered by Newton (1686) in classes of convex axially symmetric bodies. Recently, it has been
studied by Buttazzo, Kawohl, Lachand-Robert et al (1993 .) in classes of convex (not necessarily symmetric) bodies.

We consider this problem in wider classes of (generally nonconvex and non-symmetric) bodies. We also study various kinds of the body.s motion: translational motion, translation with rotation, etc. The problem amounts to studying billiard scattering on a compact obstacle. In several cases, the problem can be reduced to the Monge-Kantorovich optimal mass transport problem and then explicitly solved.

The following results will be presented: construction of bodies of arbitrarily small resistance (case of translational motion); .rough circles. of maximal and minimal resistance (case of translation with slow rotation).

Possible applications may concern artificial satellites of Earth moving on low altitudes (100 M-w 200 km) and experiencing the drag force from the rest of the atmosphere (minimizing the drag force); solar sails (maximizing the force of pressure of solar photons).

Wed Nov. 29. 2006
3:10 - 4:00 pm