June 25, 2018

Fields Colloquium on Applied Mathematics


June 8, 2001

Harvey Segur (University of Colorado)

TITLE: "A search for stable patterns of water waves"

Waves on the surface of water have been observed and studied since ancient times. An important question in this study is whether stable wave forms exist for water waves, because such stable wave forms might serve as the basic building blocks of a practical, approximate model of water waves. The governing equations are nonlinear and water's surface is two-dimensional, so we seek a family of exact solutions of the nonlinear equations, with surface patterns that are genuinely two-dimensional and are stable to small perturbations. This talk describes a search for such stable wave forms, involving several people using many tools: rigorous analysis, approximate perturbation theory, laboratory and numerical experiments. No prior knowledge of water wave theory will be required for this talk.

April 11, 2001

Claude Bardos (Universite de Paris and Ecole Normale Sperieure - Cachan)

TITLE: "Mathematical Analysis of the Time Reversal Method"

A mathematical analysis is given of the `time reversal mirror' used in particular in ultrasonic acoustics with applications in nondestructive testing, medical techniques (lithotripsy and hyperthermia) or underwater acoustics. The present analysis is done in the frame of linear theory. No new theorems are proven but it turns out that many of the tools of modern analysis, including relation between classical and quantum ergodicity, find applications in the field.

Robert McCann (University of Toronto)

TITLE: "Kinetic Equilibration Rates for Granular Media"

This joint work with Jose Carillo and Cedric Villani provides an algebraic decay rate bounding the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. The rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional Riemannian manifold. An abstract theory is developed for gradient flows which shows how degenerate convexity (or even non-convexity) --- if uniformly controlled --- will quantify contractivity of the flow.

February 7, 2001

Joceline LEGA (University of Arizona, Tucson)

TITLE: "Hydrodynamics of bacterial colonies: a model"

Remarkably rich behaviors have been observed in bacterial colonies which are forced to develop on top of a porous medium (agar) saturated with nutrients. Indeed, depending on the wetness of the growth medium and on the nutrient concentration, the colony boundary may take fascinating shapes, which are reminiscent of fractal structures. Recent experiments performed in the group of N. Mendelson at Arizona have shown that, in wet conditions, strains of Bacillus subtilis growing on an agar plate may form eddies and jets of bacteria. Such structures appear in the wetter regions of the colony and have a size which is intermediate between that of a single bacterium and that of the entire colony. In this talk, I will first summarize experimental observations of bacterial colonies on agar plates. I will then introduce a hydrodynamic model which describes the mixture of bacteria and the water they swim in as a two-phase fluid. Finally, I will discuss how classical models of chemotaxis can be recovered from this hydrodynamic description.

Nick ERCOLANI (University of Arizona at Tucson)

TITLE: "Landau Theory for Irrotational Vector Fields"

The singular perturbation of the potential energy $\int(1-u^2)^2$ by $\epsilon^2 |\nabla u|^2$ is a classical model for phase transitions. The extension of this problem from scalar fields $u$ to gradient vector fields has until recently resisted analysis. In this talk we will review some of the physical motivations for this latter problem coming, primarily, from the modelling of defects in pattern formation. We will also describe the derivation of the associated variational models and aspects of their singular limits.