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.

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