Martino Bardi (Università di Padova, Italy)
Several interesting asymptotic properties of Hamilton-Jacobi equations are based on the so-called critical value of the Hamiltonian
I will present some recent improvements holding for non-coercive Hamiltonians arising from the optimal control of affine systems, possibly with an uncontrolled drift term. Different from the previous theory, I use in a crucial way viscosity solutions that can be discontinuous. I will also mention some partial results for convex-concave Hamiltonians arising from 2-person, 0-sum differential games.
Finally, I will briefly describe some similar problems and results for the first order systems of backward-forward PDEs arising in the theory of deterministic Mean Field Games.
Shibing Chen (University of Science and Technology of China, China)
It is well known that the optimal transport problem can develop singularities when the target domain is non-convex. In this talk, I will discuss some results concerning the regularity of the singular set in optimal transportation.
Marco Cirant (Università di Padova, Italy)
We discuss the sharp rate of convergence in sup-norm of the vanishing viscosity limit of Hamilton-Jacobi equations with convex Hamiltonians, with periodic data. We focus first on uniformly convex Hamiltonians, then on strictly convex Hamiltonians. The results are based on the nonlinear adjoint method. Joint work with A. Goffi (University of Firenze)
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.
Jinqiao Duan (Great Bay University, China)
Dynamical systems are often under the influence of random fluctuations. In particular, stochastic Hamiltonian dynamical systems have attracted a lot of attention recently. The interactions between Hamiltonian structures and uncertainty are leading to fascinating dynamical phenomena.
The speaker will overview recent advances or research issues in stochastic Hamiltonian dynamics, including impact of noise on Hamiltonian structure, symplectic vs contact structures, and stochastic Hamilton-Jacobi equation.
Nicola Gigli (SISSA, Italy)
The concepts of Sobolev functions, elliptic operators and Banach spaces are central in modern geometric analysis. In the setting of Lorentzian geometry, however, unless one restricts the attention to Cauchy hypersurfaces these do not have a clear analogue, due to the signature of the metric tensor. Aim of the talk is to discuss some recent observations in this direction centered around the fact that for
The talk is mostly based on joint project with Beran, Braun, Calisti, McCann, Ohanyan, Rott, Saemann.
Diogo Gomes (King Abdullah University of Science and Technology, Saudi Arabia)
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.
Bangxian Han (Shandong University, China)
Wasserstein barycenter problem, which arises from the optimal transport theory, has got a lot of attention in various fields in the last decade. I will introduce some recent progress about Wasserstein barycenter problem in the setting of extended metric measure spaces. I will focus on the key roles of curvature and differential structure in the study of this problem. Based on joint works with Dengyu Liu and Zhuonan Zhu.
Jiahui Hong (Nanjing University of Aeronautics and Astronautics, China)
In this talk, we begin with the problem of propagation of singularities for Hamilton-Jacobi equation. We introduce a novel theory of maximal slope curves for any pair
Liang Jin (Nanjing University of Science and Technology, China)
Let
Wenjia Jing (Tsinghua University, China)
Homogenization of Hamilton-Jacobi (inviscid) equations and Hamilton-Jacobi-Bellman (viscous) equations in periodic and in stationary ergodic media has received a lot of attention since the pioneer work of Lions, Papanicolaou and Varadhan. These problems arise naturally in applications such as optimal control and combustion in heterogeneous environments. In this talk I will review some key results for the stochastic homogenization of HJB equation, and discuss some recent progress we have on the stochastic homogenization of HJ equation with a vanishing non-local term modeling jump-diffusion. Our approach is by an adaption of the method developed by Kosygina, Rezakhanlou and Varadhan (CPAM 2006). This work is joint with Dr. Qi Zhang from BIMSA.
Konstantin Khanin (University of Toronto, Canada)
Shuhei Kitano (Waseda University, Japan)
The Calderón-Zygmund estimate states that the
Kang Liu (Institut de Mathématiques de Bourgogne, France)
In this presentation, we address the numerical resolution of potential mean-field games (MFGs). (1) For non-degenerate MFGs, we employ a Theta-scheme for discretization, which preserves the variational structure of the continuous system. Combined with the Frank-Wolfe (FW) algorithm, this approach yields a mesh-independent convergence rate for solving the original problem. (2) We also consider first-order MFGs from a Lagrangian perspective, leading to a mean field optimization (MFO) problem. We analyze the stability of the MFO with respect to the initial distribution, which allows us to quantify the error introduced when the system is discretized using a many-particle approximation. Once again, the FW algorithm is used to solve the resulting many-particle system and provides robust approximation of the solution of first-order MFGs.
Xinan Ma (University of Science and Technology of China, China)
we consider the Dirichlet problem for the homogeneous
To solve this problem, we consider the Dirichlet problem of the approximating
Ezequiel Maderna (Universidad de la República, Uraguy)
The lack of global classical solutions to the Hamilton-Jacobi equation in the
Robert McCann (University of Toronto, Canada)
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
Cristian Mendico (Institut de Mathématiques de Bourgogne, France)
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.
Panrui Ni (Shanghai Mathematical Center, China)
We study the asymptotic behavior of solutions of an equation of the form
on a closed Riemannian manifold
This is the first time that converging and diverging families of solutions are shown to coexist in such a generality. This is a joint work with Andrea Davini, Jun Yan and Maxime Zavidovique.
Antonio Siconolfi (Università di Roma, La Sapienza, Italy)
We consider a Hamiltonian
In this setting, we find through a Lagrangian approach mild/variational solutions of the corresponding time-dependent first-order MFG system, by suitably using Kakutani–Glicksberg–Fan Theorem.
Further assuming the Hamiltonian to be differentiable in the moment variable, we prove that the mild solutions of above are actually solutions of the system in the usual sense, with continuity equation driven by a vector field depending on the solution
Alfonso Sorrentino (Università di Roma, Tor Vergata, Italy)
In recent years, there has been growing interest in the study of Hamilton-Jacobi equations on networks and related problems. In this talk, I will present a homogenization result for a family of time-dependent Hamilton-Jacobi equations posed on a periodic network and rescaled by a vanishing parameter. A suitable notion of periodicity, tailored to the network setting, will be introduced. As the rescaling parameter tends to zero, we obtain a limiting Hamilton-Jacobi equation defined on a Euclidean space. Notably, the dimension of this effective space is determined by the topological complexity of the network and is independent of the dimension of the ambient space in which the network is embedded. This talk is based on joint work with Marco Pozza and Antonio Siconolfi.
Xifeng Su (Beijing Normal University)
For any exact twist map
Philippe Thieullen (Université de Bordeaux, France)
A discrete weak KAM solution is a potential function that highlights the ground state configurations at zero temperature of an infinite chain of atoms interacting with a periodic or quasi-periodic substrate. It is well known that weak KAM solutions exist for periodic substrates as in the Frenkel–Kontorova model. Weak solutions may not exist in the almost periodic setting as in the theory of stationary ergodic Hamilton– Jacobi equations (where they are called correctors). For linearly repetitive quasi-periodic substrates, we show that equivariant interactions that fulfill a twist condition and a nondegenerate property always admit sublinear weak KAM solutions.We moreover classify all possible types of weak KAM solutions and calibrated configurations according to an intrinsic prefered order. The notion of prefered order is new even in the classical periodic case.
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.
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.
Lin Wang (Beijing Institute of Technology, China)
For two-parameter families of dissipative twist maps, we study the dynamics of invariant graphs and the threshold for their existence and breakdown. This talk is based on some work joint with Qi Li and Alfonso Sorrentino.
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
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.
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.
Xu Zhang (Sichuan University, China)
In this talk, I will present some progresses and open problems on control theory for stochastic partial differential equations. I will explain the new phenomena and difficulties in the study of controllability and optimal control problems for these systems. In particular, I will show by some examples that both the formulation of corresponding stochastic control problems and the tools to solve them may differ considerably from their deterministic/finite-dimensional counterparts.