SCIENTIFIC PROGRAMS AND ACTIVITIES

2009-10 Past Seminars

Alex Tamasan (University of Central Florida)
Recovering the electrical conductivity from interior data

In this talk I will present a new method to reconstruct the conductivity of a body from interior knowledge of the magnitude of the current density field generated while maintaining a specific boundary voltage. The problem reduces to the Dirichlet problem for the 1-Laplacian (a degenerate elliptic equation), and it is equivalent to finding specific minimal surfaces in a conformal metric. An implicit compatibility relation between the interior data and the boundary data is not captured by the degenerate elliptic PDE. Instead, we formulate and study a minimization problem. Under certain conditions, level sets of minimizers of the functional are area minimizers (in the particular metric), fact which allows us to treat the case when partial interior data is available. In the end I will show some numerical results based on implementation of the theory. This is joint work with Adrian Nachman of U Toronto and Alexander Timonov of U South Carolina Upstate.

PAST TALKS

Diffraction has been the mainstay of experimental crystallography for nearly a hundred years. Recent interest in quasicrystals and aperiodic tilings has brought fresh insights into the nature of diffraction and its relation to symmetry, especially in the case of pure point diffraction.

In this talk I will try to make a case for diffraction as an encoding of symmetry and then delve into the famous inverse problem of unravelling the information about a structure from information about its diffraction.

The diffraction is a measure. Which pure point measures can occur as diffraction patterns and given such a measure how does one find and classify all the structures that could have produced it? This is the homometry problem. In answering it we arrive naturally in the setting of certain stochastic processes. The complexity of the classification revolves around the set of extinctions in the diffraction.

The talk will be aimed at a general mathematical audience.

The increase of entropy was regarded as perhaps the most perfect and unassailable law in physics and it was even supposed to have philosophical import. Einstein, like most physicists of his time, regarded the second
law of thermodynamics as one of the major achievements of the field, and it entered his work in several ways. The essence of the second law is the statement that all processes can be quantified by an entropy
function whose increase is a necessary and sufficient condition for a process to occur. As a fundamental physical law no deviation, however tiny, is permitted and its consequences are far-reaching. Current wisdom regards the second law as a consequence of statistical mechanics but the entropy principle, which was discovered before statistical mechanics was invented, ought to be derivable from a few logical principles without recourse to Carnot cycles, ideal gases and other assumptions about such things as 'heat', 'hot' and 'cold', 'temperature', 'reversible processes', etc. Like conservation of energy (the ``first'' law), the existence of a law so precise and so model-independent must have a logical foundation that is independent of the details of the constitution of matter. In this lecture the foundations of the subject and the construction (with J. Yngvason) of entropy from a few simple principles will be presented. (No previous familiarity with the subject is required.)

A summary can be found in:
"A Guide to Entropy and the Second Law of Thermodynamics",
Notices of the Amer. Math. Soc. vol 45 571-581 (1998).
http://www.ams.org/notices/199805/lieb.pdf.
This paper received the American Mathematical Society 2002 Levi Conant prize for ``the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding five years''.

Jose Francisco Rodrigues (University of Lisbon / CMAF)

Constrained Reaction-Diffusion and Transport Systems: the N-membrane and Multiphase Problems

We analyse vector valued diffusion and transport equations with a class of constraints of unilateral and bilateral type. Using the variational inequality approach we characterize explicitly the associated Lagrange multipliers by reducing the problems to semi-linear systems coupled through the characteristic functions of the coincident sets of the N-membranes problem, analogously to the obstacle problem. In collaboration with Lisa Santos, we obtain new results to the system associated with the Gibbs simplex for multiphase problems. We also discuss the stability of the solutions and their coincident sets, in particular, the asymptotic behaviour in time for the respective evolution problems.

Ivana Alexandrova (East Carolina University)
Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation

We consider the problem of quantum resonances in magnetic scattering by two solenoidal fields at large separation in two dimensions. We study the distribution of resonances near the real axis when the distance between two centers of fields goes to infinity. We give a sharp lower bound on resonance widths in terms of backward amplitude calculated explicitly for scattering by each solenoidal field. The study is based on a new type of complex scaling method. As an application, we also discuss the relation to semiclassical resonances in scattering by two solenoidal fields. This is joint work with Hideo Tamura.

Andrew Belmonte
http://www.math.psu.edu/belmonte/
W. G. Pritchard Laboratories
Department of Mathematics
Penn State University, USA

Sinking Amid Bubbles
The transient and steady state motion of a solid sphere falling through a fluid depends to a large degree on the material properties of the fluid medium, be it Newtonian, Stokes, viscoelastic, or something more complicated. A field of rising bubbles provides a convenient way to slow down or even reverse the sedimentation of a heavy sphere, as utilized in some industrial situations. I will present an experimental and mathematical study of a single sphere descending through such a bubbly fluid (Reynolds numbers around 1000) in a quasi-2D geometry, focusing on two transitions: from falling to floating, and the onset of a diffusive lateral motion. This is joint work with Michael Higley (now at NJIT).

Eric Carlen (Rutgers)
http://www.math.rutgers.edu/~carlen/
Rate of relaxation to stable profiles for some fourth order evolution equations equations

We will explain recent work on obtaining strong stability results, with rate of relaxation bounds, on stationary profiles for a class of forth order equations of thin film and Cahn-Hilliard type. The talk is based on joint work with Carvalho, Orlandi, and Suleyman.

Jeff Schenker, Michigan State University
TBA

Albert Fathi (Ecole Normale Superieure de Lyon)
Denjoy-Schwartz and Hamilton-Jacobi

Given a C2 Hamiltonian H(xp), C2-strictly convex in the moment variable, it has been shown by Patrick Bernard that one can always find C1 strict subsolutions with locally Lipschitz derivative of the Hamilton-Jacobi equation. After explaining the general background, the talk will concentrate on the constraints imposed on smoother critical subsolutions by the implications of the classical Denjoy-Schwartz theory of Dynamical Systems on surfaces.

Dr Garry Newsam
Defence Science and Technology Organisation (DSTO) AUSTRALIA

New Theories of Imagery and Implications for Image Segmentation

Analysis has constructed a rich hierarchy of function spaces but it is not often obvious where real objects fit within it. The talk will review the evolution of views on where one particular class of objects, images, sits within the hierarchy and consider the implications for one particular image processing problem, image segmentation. The nub of the talk is that the last decade has seen the emergence of an surprising consensus: images are not in any of the standard function spaces but are really distributions. This unexpected characterisation appears to be the only way to accommodate the consistent empirical observations that key properties of images are essentially scale-invariant. The result has strong implications for how image processing problems such as deblurring or computation of depth from motion should be formulated; the talk will conclude with a brief discussion of what it might mean for image segmentation based on minimising the Mumford-Shah functional.