**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 Tianling Jin.

Parabolic equations and ergodicity (I)

From evolutionary ecology to nonlinear fractional reaction-diffusion equations.

We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law. We investigate the large-population limit with allometric demographies and derive a reaction-diffusion equation with fractional Laplacian and nonlocal nonlinearity. We then study a singular limit when the diffusion is assumed to be small. With a rescaling which differs from the classical one in the Laplacian case, we obtain a particular class of Hamilton-Jacobi equations in the limit. This singular limit has an interpretation in the biological framework of adaptive dynamics.

New bounds for some quasilinear equations

Stochastic Modeling for Darwinian Evolution

We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. We consider a multi-resources chemostat model, where the competition between bacteria results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states. An essential technical ingredient is the study of the long time behavior of a multi-resources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical equation of adaptive dynamics. From this canonical equation and still assuming small mutation steps, we give a characterization of the evolutionary branching points.

Parabolic equations and ergodicity (II)

Large time average of reachable sets and applications to homogenization of interfaces moving with oscillatory spatio-temporal velocity

We study the homogenization of a coercive level-set Hamilton-Jacobi equation in spatial-temporal oscillating environment that is periodic in space and stationary ergodic in time. This equation models fronts moving with positive oscillatory normal velocity. The main difficulties are due to the time oscillations and the linearly growth of the Hamiltonian. We first obtain the large time averages of reachable sets that are associated to certain control problems where the velocity bounds are periodic in space and stationary ergodic in time. This averaging result is then applied to homogenize the level-set equation and some other equations modeling moving interfaces. This is a joint work with Takis Souganidis and Hung Tran.

Regularity and long-time behavior of nonlocal heat flows

In this talk we consider a nonlocal parabolic system with a singular target space. Caffarelli and Lin showed that a well-known optimal eigenvalue partition problem could be reformulated as a constrained harmonic mapping problem into a singular space. We show that the heat flow corresponding to this problem is Lipschitz continuous, and study the regularity of a resulting free interface problem. We also show that the flow converges to a stationary solution of the constrained mapping problem as time approaches infinity. Time permitting, we will also discuss some related ongoing work involving more general non-smooth target spaces.

Click here for more details. Poster.

Harnack inequality for degenerate elliptic equations

Harnack and Hölder estimates for functions that solve a uniformly elliptic equation, but only where the gradient is large, were recently obtained by Imbert and Silvestre. In this talk we will discuss a new proof of these results that also gives a \(W^{1,\epsilon}\) estimate and a possible approach to the parabolic problem. We will also explain why the estimates extend to the case of unbounded drift.

Lecture 1: Robust Principal Component Analysis?

Monday, October 27, 4 - 5 PM. Ryerson 251.

Lecture 2: Modern Optimization Meets Physics: Recent Progress on the Phase Retrieval Problem

Tuesday, October 28, 4:30 - 5:30 PM. Eckhart 206.

Lecture 3: Around the Reproducibility of Scientific Research: A Knockoff Filter for Controlling the False Discovery Rate

Wednesday, October 29, 4 - 5 PM. Ryerson 251.

Existence of a solution to an equation arising from Mean Field Games

We construct a small time strong solution to a nonlocal Hamilton-Jacobi equation introduced by Lions, the so-called master equation, originating from the theory of Mean Field Games. We discover a link between metric viscosity solutions to local Hamilton-Jacobi equations studied independently by Ambrosio-Feng and G-Swiech, and the master equation. As a consequence we recover the existence of solutions to the First Order Mean Field Games equations, first proved by Lions. We make a more rigorous connection between the master equation and the Mean Field Games equations. (This talk is based on a joint work with A. Swiech).

The scaling limit of the Abelian sandpile on infinite periodic graphs

Joint work with Lionel Levine and Wesley Pegden. The Abelian sandpile is a deterministic diffusion process on graphs which, at least on periodic planar graphs, generates striking fractal configurations. The scaling limit on the two dimensional integer lattice can be described in terms of an Apollonian circle packing. General periodic graphs share some of this structure, but appear to have more complicated scaling limits in general. I will discuss both the square lattice and general cases.

Time-dependent mean-field games with logarithmic nonlinearities

In the present talk, we prove the existence of classical solutions for time dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded by below, this nonlinearity poses substantial mathematical challenges. Our result is proven by recurring to a delicate argument, which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a-priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.

Sobolev regularity for the first order Hamilton-Jacobi equation

We provide Sobolev estimates for solutions of first order Hamilton-Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse Hölder inequality.

No seminar. Happy Thanksgiving!

Rapid stabilization of time-reversible systems

In this talk we study an explicit feedback law (introduced by Komornik) that stabilizes time-reversible linear systems with arbitrarily large decay rates. We will focus on the well-posedness of the closed-loop system in the case of a boundary control and on an accurate estimation of the energy decay rate, thus explaining some previous experimental results by Bourquin et al.

For questions, contact Tianling Jin at: tj