## Fields Colloquium on Applied Mathematics

## Abstracts

### June 8, 2001

**Harvey Segur** (University of Colorado)

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

ABSTRACT:

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"

ABSTRACT:

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"

ABSTRACT:

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"

ABSTRACT:

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"

ABSTRACT:

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.

###