The Fields Institute Regional Colloquium on Applied Mathematics is
a monthly colloquium series intended to be a focal point for mathematicians
in the area of applied mathematics and the analysis of partial differential
equations. The series consists of talks by internationally recognized
experts in the field, some of whom reside in the region and others who
are invited to visit especially for the colloquium.

In recent years, there have been numerous dramatic successes in mathematics
and its applications to diverse areas of science and technology; examples
include super-conductivity, nonlinear wave propagation, optical fiber
communications, and financial modeling. The intent of the Colloquium
series is to bring together the applied mathematics community on a regular
basis, to present current results in the field, and to strengthen the
potential for communication and collaboration between researchers with
common interests. We meet for one session per month during the academic
year, for an afternoon program of two colloquium talks.

**Walter Craig** *(McMaster University)* - e-mail: craig@math.mcmaster.ca

**Catherine Sulem** *(University of Toronto)* - e-mail: sulem@math.toronto.edu

**SPEAKERS**

**March 26, 2002**

3:00 p.m. **Paul Rabinowitz**, University of Wisconsin - Madison

*Mixed states for an Allen-Cahn type equation *

**March 6, 2002**

1:30 p.m. **Irene Gamba**, University of Texas - Austin

*On the time evolution and diffusive steady states for inelastic Boltzmann
equations.*

Kinetic models with inelastic collisions provide an approach to understanding
regimes of rapid granular flows. One of the interesting features that
can be addressed by means of kinetic theory is the deviations from equilibrium
Maxwell distributions in the steady regimes of granular systems. We
study a model for a granular gas based on the inelastic homogeneous
Boltzmann equation for hard spheres. We address the issues of existence
solutions in C^\infty(R^3)\cap L^1_k, uniqueness and large velocity
behavior of the solutions. In particular we show that steady solutions
in the diffusive regime are bounded below by A exp(- B|v|^{3/2} ) with
computable constants A and B.

This is a joint work with V. Panferov and C. Villani.

3:00 p.m. **Luis Caffarelli**, University of Texas - Austin

*Fully non linear equations in random media*

We discuss the problem of constructing homogenization limits for fully
non linear equations in random media: What are fully non linear equations,
how the random media is described, and why limits exists.

**December 12, 2001**

2:00 p.m. **A. Ruzmaikina**, University of Virginia

*Quasi-stationary states of the Indy-500 model*

We consider the space of configurations of infinitely many particles
on the negative real line. The particles in each configuration perform
independent identically distributed jumps at each time step. After each
time step the configuration is shifted so that the leading particle
is at 0. We prove that the stationary measure of this stochastic process
is supported on Poisson processes with densities "a exp(-ax)",
where a>0 is a parameter.

**November 22, 2001**

2:00 p.m. **J. Tom Beale**, Duke University

*Computational Methods for Singular and Nearly Singular Integrals*

Mathematical models of many problems in science can be formulated in
terms of singular integrals. The representation of a harmonic function
as a single or double layer potential is a familiar example. Thus there
is a need for accurate and efficient numerical methods for calculating
such integrals. We will describe one approach, in which we replace a
singularity, or near singularity, with a regularized version, compute
a sum in a standard way, and then add correction terms, which are found
by asymptotic analysis near the singularity. We have used this approach
to design a convergent boundary integral method for three-dimensional
water waves. Boundary integral methods of this type have been used for
some time; they are based on singular integrals arising from potential
theory. The choice of discretization influences the numerical stability
of the time-dependent method. In related work we have developed a method
for computing a double layer potential on a curve, evaluated at a point
near the curve. Thus values at grid points inside the curve can be found
in a routine way, even for points near the boundary. This method can
be used to solve the Dirichlet problem in an irregular region with smooth
boundary. It may offer a way to compute the influence of a moving boundary
in viscous, incompressible fluid flow.

3:30 p.m. **MinOo**, McMaster University

*Dimensional asymptotics for spin chains*

**October 25, 2001**

1:00 p.m.** Rob Almgren, **University of Toronto

*Optimal Glider Flying*

2:00 p.m. **Constantine Dafermos**, Brown University

Progress on hyperbolic conservation laws