This seminar will provide a brief introduction to the fundamental aspects of mean-field stochastic differential equations (SDEs), covering their origins and development, and further comparing the certain differences between mean-field SDEs and classical SDEs in terms of solution properties and dynamical behaviors.
In this talk, we will discuss the quenched invariance principle and Wassertein convergence rate for random Young towers. We will also discuss a new martingale-coboundary decomposition for the random tower map, which provides a good control over sums of squares of the approximating martingale. We apply our results to independent and identically distributed (i.i.d.) translations of LSV maps.
We discuss formal results on the structure of the Hamiltonians of the NLS hierarchy. As a result, we are able to prove the well-posedness of the NLS hierarchy in the setting of nonzero boundary data, and its approximation in the long-wave regime by the KdV hierarchy.
We discuss the definition of the Hamiltonians of the NLS hierarchy via direct scattering theory, and how to prove things about them. This involves working with recurrence relations and their generating functions.
In these two lectures, we will discuss a class of partial differential equations that models a large range of biological and physical phenomena. Some of their solutions are given as a fixed profile traveling with constant speed. We will discuss the long time stability of these specific solutions. Due to translational invariance, the linearized dynamics spectrum touches the imaginary axis. As a consequence, it is necessary to distinguish between shape and phase dynamics. I will present the different tools and approaches that where developed in the past fifty years to separate and control these different dynamics. If time allows, we will discuss possible extensions of these tools to open problems.
In these two lectures, we will discuss a class of partial differential equations that models a large range of biological and physical phenomena. Some of their solutions are given as a fixed profile traveling with constant speed. We will discuss the long time stability of these specific solutions. Due to translational invariance, the linearized dynamics spectrum touches the imaginary axis. As a consequence, it is necessary to distinguish between shape and phase dynamics. I will present the different tools and approaches that where developed in the past fifty years to separate and control these different dynamics. If time allows, we will discuss possible extensions of these tools to open problems.
This talk will present a geometric formulation of GENERIC stochastic differential equations, based on the recent work [Peletier and Seri, arXiv:2509.09566v2]. They propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Their construction builds on the classical GENERIC or metriplectic formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic GENERIC/metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems.
Ergodicity, originally introduced by Boltzmann in 19th century as a hypothesis in thermodynamics, has been extensively studied as a powerful framework for formulating the equidistribution of typical orbits in dynamical systems. Using the argument introduced by Hopf, the so-called "Hopf argument", and developed by Anosov and Sinai, it is well-known that all $C^2$ volume-preserving uniformly hyperbolic systems are ergodic. The famous Stable Ergodicity Conjecture proposed by Pugh-Shub 30-years ago asserts the prevalence of ergodicity among volume-preserving partially hyperbolic systems. I will present our results jointly with Raul Ures on characterizing this prevalence, giving advances to the Hertz-Hertz-Ures Ergodicity Conjecture.
The Kolmogorov-Arnold-Moser (KAM) theory provides a powerful framework for understanding the persistence of quasi-periodic motion in nearly-integrable Hamiltonian systems. However, it requires strong conditions, including high differentiability, proximity to integrability, and Diophantine frequencies. This talk will explore the alternative offered by Aubry-Mather theory via variational method. Aubry-Mather theory originated from the independent research in differential geometry, dynamical systems, and solid state physics. We will contextualize the theory with examples from these fields and explain its fundamental existence theorem, which establishes the presence of quasi-periodic invariant sets for every rotation number under minimal hypotheses.
Abstract: This talk focuses on ergodic theory on horseshoes. It will begin by introducing the horseshoe map, a classic example in differentiable dynamical systems, and establish its relationship with symbolic systems. Then, fundamental concepts in ergodic theory, such as invariant measures, ergodic measures, and entropy, will be discussed. Finally, we will investigate the intermediate value properties and multifractal analysis on horseshoes.
It is well known that deterministic dynamical systems can exhibit some stistical properties if the system is chaotic enough and the observable satisfies some regularity conditions. This two-part talk will present recent results on the weak invariance principle for non-uniformly hyperbolic systems, along with convergence rates in the Wasserstein distance. The first talk will introduce the Wasserstein distance and its fundamental properties. The second talk will then present the main results concerning the weak invariance principle and the convergence rates in this metric.
It is well known that deterministic dynamical systems can exhibit some stistical properties if the system is chaotic enough and the observable satisfies some regularity conditions. This two-part talk will present recent results on the weak invariance principle for non-uniformly hyperbolic systems, along with convergence rates in the Wasserstein distance. The first talk will introduce the Wasserstein distance and its fundamental properties. The second talk will then present the main results concerning the weak invariance principle and the convergence rates in this metric.
简要介绍弹球系统 (主要介绍Birkhoff billiards以及symplectic billiards)和扭转映射，并展示圆形和椭圆形弹球桌面中系统的可积性。
In this talk, I will introduction to the fundamental theory of Hamiltonian systems, including key concepts such as integrable systems, KAM tori, and KAM theory. Following this, I will explore the almost-periodic solutions of the one-dimensional nonlinear Klein-Gordon (NLKG) equation under periodic boundary conditions, with the aim of demonstrating their existence and linear stability in the limit from the relativistic ($𝑐=1$) to the non-relativistic ($𝑐\to\infty$) regime. The research steps are as follows: transform this equation into an infinite-dimensional Hamiltonian system；define the Hamiltonian norm to quantify the perturbation terms； construct the KAM iteration to eliminate specific perturbation terms；prove the convergence of the iteration to establish the existence and linear stability of full-dimensional KAM tori. The methodological foundation of this study is derived from the approach outlined in Bourgain. J. Funct. Anal 229(1) 62–94, and it is a collaborative effort with Cong Hongzi and Wu Xiaoqing.