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.

Organizing Committee:

Walter Craig (McMaster University) - e-mail: craig@math.mcmaster.ca
Catherine Sulem (University of Toronto) - e-mail: sulem@math.toronto.edu

Schedule

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