Titles and Abstracts

Stochastic Homogenization of viscous HJ equations in 1d

Andrea Davini (Università di Roma, La Sapienza, Italy)

In this talk I will present some new results I have recently obtained about homogenization of viscous Hamilton-Jacobi equations in dimension one in stationary ergodic environments with nonconvex Hamiltonians. In the non-degenerate case, i.e., when the diffusion coefficient is strictly positive, homogenization is established for superlinear Hamiltonians of fairly general type. This closes a long standing question. When, on the other hand, the diffusion coefficient degenerates, meaning that it is zero at some points or on some regions of the real line, homogenization is proved for Hamiltonians that are additionally assumed quasiconvex in the momentum variable. Furthermore, the effective Hamiltonian is shown to be quasiconvex. This latter result is new even in the periodic setting, despite homogenization being known for quite some time.

Monotonicity Methods for Mean Field Games: A Functional Analytic Perspective

Diogo Gomes (King Abdullah University of Science and Technology, Saudi Arabia)

Abstract: This presentation explores mean field games (MFGs) through the lens of functional analysis, focusing on the role of monotonicity methods in understanding their properties and deriving solutions.

Monotonicity operators emerge as a central tool in our analysis. We establish the connection between monotone operators and variational inequalities, showcasing how the latter offers a flexible framework for addressing situations where traditional solutions may not exist.

Building on this foundation, we extend our discussion to the Banach space setting, examining monotone operators between a Banach space and its dual. We present existence theorems and regularization methods tailored to this context. We conclude by exploring the concept of weak-strong uniqueness, which establishes conditions under which weak and strong solutions of MFGs coincide.

Metric-measure spacetimes: a nonsmooth approach to Einstein's theory of gravity

Robert McCann (University of Toronto, Canada)

Abstract: While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. In a setting which relaxes local compactness, we describe a differential calculus for monotone curves and functions, and applications including a notion of infinitesmal Minkowskianity (that distinguishes Lorentz from Lorentz-Finsler norms), and a $p$$p$-d'Alembert comparison theorem which allows us to prove a Lorentzian splitting theorem for Lorentzian manifolds with limited regularity $g_{ij}\in C^1$$g_{ij} \in C^1$. Based on joint works with Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Argam Ohanyan, Felix Rott and Clemens Saemann.

Nash equilibria, Mather measures and ergodic Mean-field games

Cristian Mendico (Institut de Mathématiques de Bourgogne, France)

Abstract: In this presentation, we will analyze the various domains in which the ergodic mean field game (MFG) system arises. Specifically, we will explore how weak KAM theory can be used to study this system and derive results regarding long-time behavior or the approximation of Nash equilibria. Finally, we will introduce a quasi-stationary system—a model in which, at each moment, agents optimise their expected future cost under the assumption that their environment remains static.

Periodic homogenization of first and second order Hamilton--Jacobi equations: recent progresses

Hung V. Tran (University of Wisconsin Madison, US)

I will give a quick introduction to the periodic homogenization theory of Hamilton--Jacobi equations and related problems. I will then describe some recent results on optimal convergence rates for both first and second order cases. Afterwards, I will explain the key ideas in the proof of the first order convex PDE.

Long-time behavior of the contact Hamilton-Jacobi semi-flow

Kaizhi Wang (Shanghai Jiao Tong University, China)

As we all know, the long-time behavior of the Hamilton-Jacobi semi-flow on the space of continuous functions on a closed manifold, where the Hamiltonian satisfies Tonelli conditions, is simple. In this talk I will introduce several recent results on the long-time behavior of the contact Hamilton-Jacobi semi-flow, where the situation is quite different from the classical case. This talk is based on joint works with Jun Yan, Yuqi Ruan and Kai Zhao.

Existence and nonexistence of effective burning velocity under the curvature G-equation model

Yifeng Yu (University of California Irvine, US)

G-equation is a well known level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when the curvature effect is considered:

In this talk, I will show the existence of effective burning velocity under the above curvature G-equation model when $V$$V$ is a two dimensional cellular flow, which can be extended to more general two dimensional incompressible periodic flows. Our proof combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation based on the two dimensional structures. In three dimensions, the effective burning velocity will cease to exist even for simple periodic shear flows when the flow intensity surpasses a bifurcation value. The existence result is based on joint work with Hongwei Gao, Ziang Long and Jack Xin, while the non-existence result is from collaboration with Hiroyoshi Mitake, Connor Mooney, Hung Tran, and Jack Xin.

Integrability of conformally symplectic Hamiltonian flows

Maxime Zavidovique (Sorbonne Université, France)

The goal of this talk is to introduce some notions of integrability for conformally symplectic flows associated to Tonelli Hamiltonians. We are guided by the famous result of Burago and Ivanov, that Riemannien metrics without conjugate points are flat on the torus. We prove that our conformally symplectic flows, when they don’t have conjugate points, are what we call Hopf integrable. This is joint work with Marie-Claude Arnaud and Xifeng Su.

Convergence Rate of Homogenization For quasi-periodic Hamilton-Jacobi Equations

Jianlu Zhang (Chinese Academy, China)

In 1987, Lions firstly proposed the homogenization for Hamilton-Jacobi equations, which revealed the significance of effective Hamiltonian in controlling the large time behavior of solutions. The quantitative estimate of such a homogenization was studied in recent years, which mainly answers the convergence rate for compact case. In this talk, we will introduce a novel quasi-periodic approach, which reveals the relation between the smoothness of effective Hamiltonian and the convergence rate.