The Fields Institute Colloquium/Seminar in 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.

Schedule

**PAST SEMINARS July 2005-June 2006**

**May 15, 2006**

**Martin Feinberg**, Chemical Engineering & Mathematics,
The Ohio State University

*Understanding Bistability in Complex Enzyme-Driven Reaction
Networks*

Abstract. In nature there are millions of distinct networks of
chemical reactions that might present themselves for study at
one time or another. Each network gives rise to its own system
of differential equations. These are usually large and almost
always nonlinear. Nevertheless, the reaction network induces the
corresponding differential equations (up to parameter values)
in a precise way. This raises the possibility that qualitative
properties of the induced differential equations might be tied
directly to reaction network structure.

Chemical reaction network theory has as its goal the development
of powerful but readily implementable tools for connecting reaction
network structure to the qualitative capacity for certain phenomena.
The theory goes back at least to the 1970s*. It has not been specific
to biology, but, for obvious reasons, there is now growing interest
in biological applications. Very recent work (with Gheorghe Craciun)
has been dedicated specifically to biochemical networks driven
by enzyme-catalyzed reactions. In particular, it is now known
that there are remarkable and quite subtle connections between
properties of reaction diagrams of the kind that biochemists normally
draw and the capacity for biochemical switching. My aim in this
talk will be to explain, for an audience unfamiliar with chemical
reaction network theory, those tools that have recently become
available

**March 22, 2006**

**Gian Michele Graf**, ETH Zurich

*Transport in adiabatic quantum pumps *

A quantum pump is an externally driven device coupled to reservoirs
at equilibrium with one another. We consider transport phenomena
when the electrons are independent and the driving is slow compared
to the dwell time of particles in the pump. The charge transport
associated with a given change of pump parameters is characterized
in terms of S-matrices pertaining to time-independent junctions.
In fact, several transport properties (charge, dissipation and,
at positive temperature, noise and entropy production) may be
expressed in terms of the matrix of energy shift which, like Wigner's
time delay to which it is dual, is determined by the S-matrix.
We discuss transport at a semiclassical level, including geometric
aspects of transport such as the question of charge quantization.
On the analytical side we present an adiabatic theorem on transport
for open gapless systems.

**March 15, 2006**

**H. Mete Soner**, Koc University, Istanbul

*Backward stochastic differential equations and fully nonlinear
PDE's.*

In the early 90's Peng and Pardoux discovered a striking connection
between semilinear parabolic PDE's and backward stochastic differential
equations (BSDE in short). This connection and the BSDE's have
been extensively studied in the last decade and a deep theory
of BSDEs have been developed. However, the PDE's that are linked
to BSDE's are necessarily semilinear. In joint work with Patrick
Cheredito (Princeton) Nizar Touzi (CREST, Paris), Nic Victoir
(Oxford) we extended the theory of BSDE's by adding an equation
for the second order term, which we call 2BSDE in short. Through
this extension we are able to show that all fully nonlinear, parabolic
equations can be represented via 2BSDE's. In this talk, I will
describe this theory and possible numerical implications for the
fully nonlinear PDE's.

**March 8, 2006**

**Michael Shearer**, North Carolina State University

*Thin film equations for fluid motion driven by surfactants.*

In the lubrication approximation, the motion of a thin liquid
film is described by a single fourth-order partial differential
equation that models the evolution of the height of the film.
When the fluid is driven by a Marangoni force generated by a distribution
of insoluble surfactant, the thin film equation is coupled to
an equation for the concentration of surfactant. In this talk,
I show the basic structure of this system, and begin an analysis
of wave-like solutions in the specific context of a thin film
flowing down an inclined plane. Numerical simulations reveal an
array of traveling waves, which persists when capillarity and
surface diffusion are neglected. The limiting system is of mixed
hyperbolic and degenerate parabolic type, and supports a variety
of special solutions that can be combined to solve prototype initial
value problems, and to study long-time behavior of general solutions.

**November 23, 2005**

**Henri Berestycki**, Ecole des hautes études en sciences
sociales Paris and University of Chicago

*Fronts and propagation speed for reaction-diffusion equations
in non homogeneous media*

I will review some recent works about reaction-diffusion equations
which are not homogeneous. These are nonlinear parabolic equations
with a dependence on the space variable or problems set in unbounded
domains with boundaries. In this context, one wishes to understand
how to extend the notion of travelling front solution and also
to determine the asymptotic speed of spreading in the case of
a Fisher type nonlinearity. I will discuss the case of a periodic
environment for which one defines "pulsating travelling fronts"
and then mention some results about general non homogeneous problems.
I report here on several joint works with François Hamel,
Nikolai Nadirashvili and Hiroshi Matano.

**November 16, 2005**

**Irene Fonseca,** Carnegie Mellon University

*Variational Methods in the Study of Imaging, Foams, Quantum
Dots ... and More.*

Several questions in applied analysis motivated by issues in computer
vision, physics, materials sciences and other areas of engineering
may be treated variationally leading to higher order variational
problems and to models involving lower order density measures.
Their study often requires state-of-the-art techniques, new ideas,
and the introduction of innovative tools in partial differential
equations, geometric measure theory, and calculus of variations.
In this talk it will be shown how some of these questions may
be reduced to well understood first order problems, while in others
the higher order plays a fundamental role.

Applications to phase transitions, to the equilibrium of foams
under the action of surfactants, imaging, micromagnetics and thin
films will be addressed.

back to top