**The seminar meets regularly on Wednesdays** at **4pm** in **Eckhart 202**. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Will Feldman.

Recent developments in Mean Field Game theory and related topics Part I

Recent developments in Mean Field Game theory and related topics Part II

October 14

October 21

Sharp interface limit in a phase field model of cell motility

We consider a system of two parabolic PDEs arising in modeling of motility of eukaryotic cells on substrates. The two key properties of this system are (i) presence of gradients in the coupling terms (gradient coupling) and (ii) mass (volume) preservation constraints. We derive the equation of the motion of the cell boundary, which is the mean curvature motion perturbed by a novel nonlinear term and prove that the sharp interface property of initial conditions is preserved in time. We next show that this novel term leads to surprising features of the motion of the interface such as discontinuities of the interface velocity and hysteresis.

Because of the properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why recently developed Γ-convergence techniques also can not be used. A special form of asymptotic expansion is introduced to reduce analysis to a single nonlinear PDE: a one- dimensional model problem. Stability analysis reveals a qualitative change in the behavior of the system depending on the main physical parameter. This is joint work with V. Rybalko and M. Potomkin.

The master equation and the convergence problem in mean field games

in this joint work with F. Delarue, J.-M. Lasry and P.L. Lions, we study the convergence, as $N$ tends to infinity, of a system of $N$ coupled Hamilton-Jacobi equations, the Nash system. This system arises in differential game theory. We describe the limit problem in terms of the so-called ``master equation", a kind of second order partial differential equation stated on the space of probability measures. Our first main result is the well-posedness of the master equation. To do so, we first show the existence and uniqueness of a solution to the "mean field game system with common noise", which consists in a coupled system made of a backward stochastic Hamilton-Jacobi equation and a forward stochastic Kolmogorov equation and which plays the role of characteristics for the master equation. Our second main result is the convergence, in average, of the solution of the Nash system and a propagation of chaos property for the associated "optimal trajectories".

Neumann homogenization via integro-differential methods

We study the homogenization of fully nonlinear elliptic equations with oscillatory Neumann data in half-space type domains. We employ new methods to prove the averaging, by recasting the original boundary equation as a global interior homogenization problem involving an integro-differential operator on the boundary itself. This is done by using the Dirichlet-to-Neumann mapping for the fully nonlinear equation in the the interior of the domain. This is joint work with Nestor Guillen.

Stochastic scalar conservation laws

In this talk we will review recent results for stochastic scalar conservation laws with random flux. In the first part we will focus on a well-posedness theory for the case of spatially inhomogeneous, random fluxes as they appear in mean field games. In the second part we will investigate the long time behavior and regularity of solutions to stochastic scalar conservation laws on the torus. In particular, we will observe a certain regularizing effect due to the noise.

November 11

How long does it take to compute the eigenvalues of a random symmetric matrix?

In the early days of scientific computing, Goldstine and Von Neumman suggested that it would be fruitful to study the "typical" performance of Gaussian elimination on random input. This approach lay dormant for decades until Alan Edelman’s 1989 thesis on the condition numbers of random matrices. Since then numerical linear algebraists have made basic contributions to random matrix theory and the study of condition numbers of random matrices has proven to be a rich subject.

We approach the symmetric eigenvalue problem from a similar viewpoint. The underlying mathematical issue is to analyze the number of iterations required for an eigenvalue algorithm to converge. We study the QR algorithm, the Toda algorithm and a matrix-sign algorithm. These algorithms have a beautiful structure, intimately tied to completely integrable Hamiltonian systems.

Our main results are an empirical discovery of "universality in numerical computation", and some explanations for it.

This is joint work with Percy Deift and Tom Trogdon (Courant Institute), Christian Pfrang (JP Morgan) and Enrique Pujals (IMPA).

Parabolic equations with Sobolev regular data: the final cut

(Based on works with and by PL Lions) We present a general theory of existence and uniqueness of linear parabolic equations with Lebesgue/Sobolev regular coefficients and initial conditions. Applications to the theory of stochastic differential equations are also discussed.

On stable melting and freezing rates in the Stefan problem

We consider the two dimensional Stefan problem describing the evolution of a spherically symmetric ice ball. Motivated by the pioneering analysis of Herrero and Velazquez we prove the existence of STABLE non self-similar finite time melting regimes, and compute the deviation from self-similarity. We introduce a new and canonical functional framework for the study of type II (i.e. non self similar) blow up for a class of problems including a related construction for the harmonic heat flow studied by Raphael and Schweyer. This is a joint work with P. Raphael.

Partial differential equation continuum limits for discrete sorting problems

Many problems in science and engineering involve the sorting, or ordering, of large amounts of multi-variate data. A common sorting technique is to arrange the data into layers by repeatedly removing the set of extremal points. Different notions of extremality lead to different sorting algorithms. Two common examples are non-dominated sorting, and convex hull peeling, which are widely used in multi-objective optimization, machine learning, and robust statistics. In this talk, I will present a Hamilton-Jacobi equation continuum limit for nondominated sorting, and a conjectured partial differential equation (PDE) continuum limit for convex hull peeling. I will also present some new numerical schemes for the Hamilton-Jacobi equation, and show how to design very fast approximate sorting algorithms based on numerical solving the continuum PDE.

For questions, contact Will Feldman at: feldman