Every wednesday at 4pm in Eckhart 202.
We explain how we can derive rigorously MFG models from Nash equilibria letting the number of players go to infintiy. This derivation requires an analytical framework of independent interest which allows to deduce from finite dimensional equations and analyse mathematically various Partial Differential Equations on the space of probablity measures.
We present the mathematical state of the art about the new systems of Partial Differential equations known as MFG models. The issues that we discuss are: uniqueness, existence, numerical approximation and various qualitative properties of solutions.
We present a few examples of applications of MFG models like, for instance, a model for "mexican waves", for the starting time of a meeting or for growth through training and competition...
Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis." Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. Following these observations, in this talk we will present a hierarchy of mathematical models for phototaxis: a stochastic model, an interacting particle system, and a system of PDEs. We will discuss the models, their simulations, and our theorems that show how the system of PDEs can be considered as the limit dynamics of the particle system. Time-permitting, we will overview our recent results on particle, kinetic, and fluid models for phototaxis. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute), Tiago Requeijo (Math, Stanford), and Seung-Yeal Ha (Seoul, Korea).
The two processes of mutations and selection, proposed by C. Darwin, can be written in mathematical words. In a very simple, general and idealized description, the environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to multiply faster because they are better adapted to use the environment. This leads to select the 'fittest trait' in the population. On the other hand, the new-born individuals undergo small variations of the trait under the effect of mutations. In these circumstances, is it possible to observ 'speciation' and to describe the dynamical evolution of the 'fittest' traits?
We will give a class of self-contained mathematical models of such population dynamics and show that an asymptotic view allows us to formalize precisely the concepts of monomorphic or polymorphic populations. We can describe the evolution of the 'fittest traits' and various forms of branching points. We will also show that numerical solutions are consistant with individual based stochastic simulations.
Eventhough the model is very simple, its analysis relates to remarkable recent progresses of nonlinear analysis.
The Muskat problem involve the filtration of two incompressible fluids throughout a porous medium. In this talk we shall discuss in 3-D the relevance of the Rayleigh-Taylor condition, and the non-selfintersecting character of the interface, in order to have a well-possed problem.
A fundamental aspect which accounted for the success of reaction-diffusion models is concerned with the descprition of spreading phenomena at large times in unbounded domains. Estimating the spreading speeds is one of the most impoortant issues from a theoretical point of view as well as for the applications in biology and ecology. Usually, the solutions spread at a finite speed which is uniquely determined from the initial data. In this talk, I will report on some recent results obtained in colloboration with G. Nadin, L. Roques and Y. Sire which show that, even for the simplest reaction-diffusion models, spreading with a finite and uniquely determined speed may fail. Examples of propagation with an infinite speed and complex situations with non-trivial intervals of spreading speeds will be exhibited.
One aim of agent-based modeling is to see what kind of collective behavior can be created by local interactions among individual agents. I have applied such agent-based models to several complex systems, generally mimicking some kind of social dynamics among the agents. In this talk, I will describe one such application: our model for the spawning migration of the capelin, a species of fish, around Iceland. Shifting migration routes of this stock have been challenging for researchers in Iceland, who must be able to locate the stock in order to accurately assess the stock and assign reasonable fishing quotas to prevent a stock collapse. Our model accurately predicted an unusual path for 2008's yearly migration, and the same parameters were used to successfully reproduce the migration routes from two previous years. I will describe our model, including how environmental data is incorporated, and show simulation results alongside data collected by the Marine Research Institute of Iceland. The question of how the parameters in this type of model should scale with the number of particles will also be explored.
I will present results on the existence and uniqueness of fundamental solutions of positively homogeneous, fully nonlinear elliptic equations like the Bellman-Isaacs equation. The methods we use are inspired by the recently developed principal eigenvalue theory for fully nonlinear equations. Using blow-up arguments, we demonstrate that the fundamental solutions characterize the isolated singularities of solutions which are bounded on one side. As another application, we show that the "intrinsic scaling exponent" derived from the fundamental solution characterizes whether an associated stochastic process, controlled by two competing players, is recurrent or transient. This work is joint with B. Sirakov and C. K. Smart.
In the de Giorgi theory, a minimal surface is the boundary of a set whose indicator minimises, locally, the BV norm. The game here is to replace BV by $H^\alpha$, with $0<\alpha<1/2$, and the so obtained sets are called $\alpha$-minimal. We discuss in this talk the regularity of these objects. The main result is, in the spirit of de Giorgi, a conditional smoothness result: if a piece of $\alpha$-minimal surface is trapped in a sufficiently flat cylinder, then it is regular. Joint work with L. Caffarelli and O. Savin.
The lecture will discuss a class of systems of hyperbolic conservation laws in which the natural entropy fails to be convex but this deficiency is compensated by the presence of extra conservation laws. Examples will be presented from elastodynamics and electromagnetics
The G-equation is a Hamilton-Jacobi equation which is often used in the combustion community for the study of premixed flames in turbulent media. I shall present an homogenization result for this model. The surprizing fact is that homogenization holds despite the lack of standard estimates for the solutions of the equation.
In certain circumstances, elastic structures of growing tissues (such as leaves, flowers or marine invertebrates) exhibit non-zero strain at free equilibria. This phenomenon can be studied through a variational model, pertaining to the non-Euclidean version of nonlinear elasticity, where the growth changes the intrinsic metric of the tissue to a new target non-flat metric and the non-vanishing curvature is the cause of the residual stress.
We further discuss the scaling laws and $\Gamma$-convergence of the introduced 3d functional on thin plates, in the limit of vanishing thickness. Given special forms of growth tensors, we rigorously derive a version of von Karman equations with residual stress (recently proposed by Mahadevan and Liang). One important feature is the study of Sobolev spaces of isometries and infinitesimal isometries. In particular, we obtain a new condition necessary and sufficient for existence of a $W^{2,2}$ isometric immersion of a given $2d$ metric into $R^3$.
I will discuss a version of the time-dependent Ginzburg-Landau system that models a thin superconducting wire subjected to an applied voltage. Using a mixture of rigorous analysis, formal asymptotics and numerics, we analyze the behavior of solutions as the physical parameters of wire length and voltage are varied. Stable periodic solutions are shown to exist exhibiting phase slip centers (zeros of the order parameter), with period-doubling, period-tripling and chaotic behavior emerging in certain length/voltage regimes. This is joint work with Jacob Rubinstein and Junghwa Kim.
