Amick Lectures in Applied Mathematics

2011-2012 Speaker: Jeffrey Rauch (University of Michigan)



The three lectures are independent. All are concerned with phenomena of resonance. They report on joint works with P. Donnat, J.-L. Joly, G. Metivier, F. Colombini, and V. Petkov.

Lecture 1: Modeling the dispersion of light

Tuesday May 29, 4:30 PM - 5:30 PM in Eckhart 202

The problem addressed here is to find adequate models that describe the fact that light is split into colors by a prism. It is split because in glass high frequencies go slower than low frequencies so are bent more on passing through an interface.

We are all taught this and also that light has only one speed.

The short version of the answer is that the slowing is because of a resonant interaction between the light and atoms. The physical description dates to Lorenz and the mathematical version to Bloembergen who revisited the problem when faced with modelling nonlinear dispersion.



Lecture 2: Parametric resonance for the wave equation

Wednesday May 30, 4 PM - 5 PM in Eckhart 202

Parametric resonance is the phenomenon whereby differential equations with time periodic coefficients each of whose frozen time equations is conservative can lead to exponential growth. This phenomenon occurs for the wave equation in unbounded domain with positive, smooth, compactly supported potential periodic in time (conjecture of Cooper-Strauss).



Lecture 3: Dense oscillations for the Euler equations

Thursday May 31, 4:30 PM - 5:30 PM in Eckhart 202

Nonlinear resonance occurs when oscillations interact when they would be independent from the linear perspective. This can lead to cascades whereby few oscillations can create more, that create more, ... etc. Such a situation is sometimes called a cascade.

For the equations of two dimensional inviscid compressible fluid dynamics we study such a cascade showing that three incoming oscillatory wave trains can and do produce an infinite family of nonlinearly interacting waves sufficiently rich so that there are waves moving in all but a finite set of the rational directions on the unit circle.

Previous years

2010-2011 Speaker: L. Mahadevan (Harvard)



Soft interfaces: statics and dynamics

May 9 at 4pm in E 202

The boundaries between fields is often a fertile source of problems and ideas. Similarly, interfaces are rich sources of physical phenomena and questions. I will discuss some simple quantitative approaches to understand the behavior of unstructured and structured soft interfaces. These include the creasing of soft interfaces, the peeling and healing of thin adherent films, and the mechanics and self-organization of a hairy interface. I hope to show that even minimal models for the phenomena lead to interesting mathematical problems and challenges.


Morphogenesis I: geometry, physics and biology at the cellular scale

May 10 at 4:30pm in E 206

See below for abstract.


Morphogenesis II: geometry, physics and biology at the tissue scale

May 11 at 4pm in E 202

The growth and form of a soft solid pose a range of problems that combine aspects of mathematics, physics and biology. I will discuss some examples of growth and form in the plant and animal world motivated by qualitative and quantitative biological observations. In the first lecture, I will discuss problems at the macromolecular and cellular scale. These include the shape of bundles of filaments, the dynamic instability of a polymerizing microtubule and the shape of a freely growing pollen tube. In the second lecture, I will discuss problems at the tissue scale. These include the undulating fringes on a leaf, the blooming of a flower, and the looping morphogenesis of the vertebrate gut. In each case, we will see how a combination of physical experiments, mathematical models and simple computations allow us to unravel the basis for the diversity and complexity of biological form, while suggesting a rich new lode of problems in geometry and analysis.

2009-2010 Speaker: Claude le Bris

Randomness in Multiscale Computational Mechanics and how to deal with it

October 4, 2010 at 4pm

The common denominator of all the contributions presented is random modelling for materials. Models for some complex fluids (typically solutions of flexible polymers) and models for some composite solid materials will be presented. In each category, emphasis will be laid upon the random component of the modelling, how this component is derived from the microscale and formalized, how it affects the overall mathematical nature of the problem giving birth to new questions of interest, how its coupling with the deterministic part of the system is dealt with computationally. The talk will overview works by, and joint works with various coworkers: PL. Lions (College de France), X. Blanc (CEA, Paris), A. Anantharaman, S. Boyaval , R. Costaouec, B. Jourdain, T. Lelievre, F. Legoll, G. Stoltz, F. Thomines (ENPC, Paris).


Molecular dynamics and computational statistical mechanics: some mathematical approaches

October 6, 2010 at 4pm

Molecular dynamics is a commonly used technique to compute ensemble averages, and beyond, thermodynamic quantities. Mathematically, molecular dynamics imply ordinary and stochastic differential equations, possibly multiscale in nature. A selection of various relevant mathematical questions will be addressed: long-time integration of, possibly highly oscillatory, Hamiltonian systems; simulation of constrained stochastic differential equations, long-time integration of Langevin type equations for metastable systems, large deviation theory for coarse-grained models, etc. The talk is based upon a series of works by, and joint works with X. Blanc (Paris 6 and CEA), M. Dobson, F. Legoll, T. Lelievre, G. Stoltz, (ENPC and INRIA).


Numerical approaches for non periodic and random composite materials: some recent progress

October 8, 2010 at 4:30pm

The talk will overview some recent contributions on several theoretical aspects and numerical approaches in stochastic homogenization. In particular, some variants of the theory of classical stochastic homogenization will be introduced. The relation between such homogenization problems and other multiscale problems in materials science will be emphasized. On the numerical front, some approaches will be presented, for acceleration of convergence in stochastic homogenization (variance reduction issues, etc) as well as for approximation of the stochastic problem when the random character is only a perturbation of a deterministic model. The talk is based upon a series of joint works with X. Blanc (CEA, Paris), PL. Lions (College de France, Paris), and A. Anantharaman, R. Costaouec, F. Legoll, F. Thomines (ENPC, Paris).

2008-2009 Speaker: Pierre-Louis Lions

On Mean Field Games

Thursday, October 2, 4:30 PM, Eckhart 133, 1118 E. 58th St.

This talk will be a general presentation of Mean Field Games (MFG in short), a new class of mathematical models and problems introduced and studied in collaboration with Jean-Michel Lasry. Roughly speaking, MFG are mathematical models that aim to describe the behavior of a very large number of "agents" who optimize their decisions while taking into account and interacting with the other agents. The derivation of MFG, which can be justified rigorously from Nash equilibria for N player games, letting N go to infinity, leads to new nonlinear systems involving ordinary differential equations or partial differential equations. Many classical systems are particular cases of MFG like, for example, compressible Euler equations, Hartree equations, porous media equations, semilinear elliptic equations, Hamilton-Jacobi-Bellman equations, Vlasov-Boltzmann models... In this talk we shall explain in a very simple example how MFG models are derived and present some overview of the theory, its connections with many other fields and its applications.


Symmetric functions of a large number of variables

Friday, October 3, 4:00 PM, Eckhart 202, 1118 E. 58th St.

In this talk, we present the general mathematical tools needed to justify the derivation of Mean Field Games models. It turns out that these tools have many other applications : Large deviations for Stochastic Partial Differential Equations and applications to Physics, interacting systems of stochastic particles, Nonlinear Partial Differential Equations (NLPDE in short) in large dimensions, mass transportation theory... We shall present the natural setup for all these asymptotic problems is the space of probability measures (the so-called Wasserstein space), the differential calculus and NLPDE on this space deduced from such limits.


Mean Field Games : Analysis and Applications

Monday, October 6, 4:00 PM, Eckhart 202, 1118 E 57th St.

In this talk, we present some mathematical results on MFG systems both for Ordinary Differential Equations and Partial Differential Equations (existence, uniqueness, stability, approximation...). And we discuss briefly some applications to Economics and Finance.