Fields Institute Colloquium/Seminar in Applied Mathematics 2006-2007
The Fields Institute Regional Colloquium on Applied Mathematics
is a monthly colloquium series intended to be a focal point for
mathematicians in the areas of applied mathematics and analysis.
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.
Jim Colliander (Toronto)
Walter Craig (McMaster)
Barbara Keyfitz (Fields)
Adrian Nachman (Toronto)
Mary Pugh (Toronto)
Catherine Sulem (Toronto)
May 10, 2007
Professor Adimurthi, TIFR Center, Bangalore India
Conservation Laws with Discontinous Flux
The one dimensional scalar conservation law with smooth flux
has been studied for over fifty years. There is a well developed
theory to obtain existence and uniqueness of entropy solutions,
due to Lax-Oleinik, Kruzkov and others.
If the underlying flux is discontinous the theory is not
well understood. Basically such problems arise in multi-phase
flow problems. One example is the two-phase flow problem coming
from extraction of oil by pumping water into the oil well.
Here I will discuss the existence, entropy and uniqueness
of such problems.
May 9, 2007
Computational and Applied Mathematics, Rice University
Electrical Impedance Tomography with resistor networks
We present a novel inversion algorithm for electrical impedance
tomography in two dimensions, based on a model reduction approach.
The reduced models are resistor networks that arise in five
point stencil discretizations of the elliptic partial differential
equation satisfied by the electric potential, on adaptive
grids that are computed as part of the problem. We prove the
unique solvability of the model reduction problem for a broad
class of measurements of the Dirichlet to Neumann map. The
size of the networks (reduced models) is limited by the precision
of the measurements. The resulting grids are naturally refined
near the boundary, where we make the measurements and where
we expect better resolution of the images. To determine the
unknown conductivity, we use the resistor networks to define
a nonlinear mapping of the data, that behaves as an approximate
inverse of the forward map. Then, we propose an efficient
Newton-type iteration for finding the conductivity, using
this map. We also show how to incorporate apriori information
about the conductivity in the inversion scheme.
April 11, 2007
I. Vacaru, Instituto de Matematicas y Fisica Fundamental
Curve Flows, RiemannFinsler Solitonic Hierarchies and
April 4, 2007
Georgia Institute of Technology
Variational methods for the $1$--d Euler-Poisson system
We consider the set $M$ of Borel probability measures on $R,$
of bounded second moment, endowed with the Wasserstein metric
$W_2.$ We study a specific Lagrangians $L$ defined on its tangent
bundle. If $H$ is the Hamiltonian associated to $L$, given an
initial value function $U_0$ defined on $M$ and which is $\lambda$-convex,
there is a viscosity solution to the infinite dimensional Hamilton-Jacobi
equation $\partial_t U + H(\mu, \nabla_\mu U)=0,$ for small
times $t$, with the prescribed initial value function $U_0$.
We prove that its characteristics are unique solution of the
one-dimensional Euler-Poisson system with prescribed endpoints.
These paths conserve the Hamiltonian even when the measures
has a singular part. (This is a joint work with T. Nguyen and
Mar 21, 2007
* This is a joint University of Toronto Math Department
Colloquium/Fields Institute Colloquium in Applied Mathematics
Holmes, Princeton University
From neural oscillators through stochastic dynamics to
optimal decisions, or Does math matter to gray matter?
The sequential probability ratio test (SPRT) is optimal in
that it allows one to accept or reject hypotheses, based on
noisy incoming evidence, with the minumum number of observations
for a given level of accuracy. There is increasing neural
and behavioral evidence that primate and human brains employ
a continuum analogue of SPRT: the drift-diffusion (DD) process.
I will review this and also describe how a biophysical model
of a pool of spiking neurons can be simplified to a phase
oscillator and analysed to yield spike rates in response to
stimuli. These spike rates tune DD parameters via neurotransmitter
release. This study is a small step toward the construction
of a series of models, at different time and space scales,
linking neural spikes to human decisions.
This work is joint with Eric Brown, Jeff Moehlis, Rafal
Bogacz and Jonathan Cohen at Princeton, and Garry Aston-Jones'
group at the Laboratory of Neuromodulation and Behavior, University
|Wed., Feb. 14,
Adam Oberman, Simon Fraser University
The Infinity Laplacian: from classical analysis to image
processing and random turn games.
The Infinity Laplacian equation is currently at the interface
of a number of different mathematical fields. It was first
studied in the 1950s by the Swedish mathematician Gunnar Aronsson,
motivated by classical analysis problem of building Lipschitz
extensions of a given function. While Aronsson was able to
find interesting exact solutions, progress stalled because
solutions were non-classical. It took another forty years
until analytical tools were developed to study the equation
rigorously, and computational tools were developed which made
numerical solution of the equation possible.
In the last decade, PDE theorists established existence
and uniqueness, and (quite recently) appropriate regularity
results. At the same time, the image processing community
was using the operator for edge detection, and for inpainting,
the reparation of images with damage. While the operator was
promising, they had little success, since traditional methods
for solving the equation yielded poor results.
It turns out that the right way to solve the equation is
to go back to the original Lipschitz extension problem. This
leads to a formula for the discrete operator with a simple
interpreation, and good solution properties. This formula
also leads to another surprising connection with probability
Working in the unrelated field of percolation theory, a
group of probalists (Peres-Shramm-Sheffield-Wilson) studying
a randomized version of a marble game called Hex found a connection
with the Infinity Laplacian equation. This connection gives
an interpretation of the equation as a two player random game.
I'll tell this story, and explain some of the more accessible
properties of the equation, along with pictures and numerical
|Wed., Feb. 7, 2007
||Andrei Biryuk, Instituto Superior Tecnico,
The Euler -Lagrange invariance of action minimizing measures
satisfying a holonomy constraint
|Wed., Jan 17, 2007
University of Aveiro
Billiards, optimal mass transport and problems of optimal
A body moves through a medium consisting of point particles.
The medium is very rare, so that the mutual interaction of
the particles is neglected. Interaction of the particles with
the body is absolutely elastic. We consider the following
problem: find the body, from a given class of bodies, such
that the force of resistance of the medium to its motion is
minimal or maximal.
The (minimization) problem was firstly considered by Newton
(1686) in classes of convex axially symmetric bodies. Recently,
it has been
studied by Buttazzo, Kawohl, Lachand-Robert et al (1993 .)
in classes of convex (not necessarily symmetric) bodies.
We consider this problem in wider classes of (generally
nonconvex and non-symmetric) bodies. We also study various
kinds of the body.s motion: translational motion, translation
with rotation, etc. The problem amounts to studying billiard
scattering on a compact obstacle. In several cases, the problem
can be reduced to the Monge-Kantorovich optimal mass transport
problem and then explicitly solved.
The following results will be presented: construction of
bodies of arbitrarily small resistance (case of translational
motion); .rough circles. of maximal and minimal resistance
(case of translation with slow rotation).
Possible applications may concern artificial satellites
of Earth moving on low altitudes (100 M-w 200 km) and experiencing
the drag force from the rest of the atmosphere (minimizing
the drag force); solar sails (maximizing the force of pressure
of solar photons).
Wed Nov. 29. 2006
3:10 - 4:00 pm
Soffer, Rutgers University
Soliton Dynamics and Scattering
Stewart Library, Fields Institute
|Wed. Oct. 18, 2006
|Prof. Jerry Bona, University of Illinois
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