# Webinar 2020

## 斜积流的Sarnak猜想 (刘建亚, 2020-06-04)

Title: 斜积流的Sarnak猜想

Speaker: 刘建亚（山东大学）

Datetime: 2020-06-04 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：965 0633 0103

Abstract

Sarnak猜想预测，Mobius函数与零熵的动力系统正交。本演讲将介绍这个猜想在非正则斜积流情形的一些进展。

## 接触Hamilton动力系统的全局吸引子 (严军, 2020-06-09)

Title: The vanishing discount problem for the system of HJ equations: the full convergence and a counterexample

Speaker: 严军（复旦大学）

Datetime: 2020-06-09 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：971 6180 7575

Abstract

## Viscosity solution of HJ equation: existence, structure of solution set and large time behavior (王林, 2020-06-11)

Title: Viscosity solution of HJ equation: existence, structure of solution set and large time behavior

Speaker: 王林（清华大学）

Datetime: 2020-06-11 15:00 — 16:00 Beijing/Shanghai
Venue: 腾讯会议 APP
Meeting ID：218 308 213

Abstract

I will talk about some new results on the topics listed in the title. It is based on some works joint with X. Shu, K. Wang and J. Yan.

## The vanishing discount problem for the system of HJ equations: the full convergence and a counterexample (Hitoshi Ishii, 2020-06-18)

Title: The vanishing discount problem for the system of HJ equations: the full convergence and a counterexample

Speaker: Hitoshi Ishii（Tsuda University & Waseda University）

Datetime: 2020-06-18 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：963 4455 9881

Abstract

I discuss the vanishing discount problem for the system of HJ equations with the focus on the full convergence of the solutions of the discounted problems as the discount factor tends to zero. I recall joint work with Liang Jin which has established the full convergence result under the convexity, coercivity, and monotonicity hypotheses, and I explain an example of the "non-convex" system, for which the whole family of the solutions of the discounted problems does not converge.

## The Lax-Oleinik representation in non-compact setting (Albert Fathi, 2020-06-18)

Title: The Lax-Oleinik representation in non-compact setting

Speaker: Albert Fathi（Georgia Tech）

Datetime: 2020-06-18 21:00 — 22:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：993 2636 1438

Abstract

We will be interested in viscosity solutions of the evolution Hamilton-Jacobi equation
$\partial _tU+H(x,\partial_xU)=0.$
Here we think of the case where $$U:[0,+\infty[\times M\to\mathbb{R}$$, with $$M$$ is a manifold.

If $$M$$ is compact, as has been known for a long time, the maximum principle yields uniqueness for a given initial condition $$U\vert_{\{0\}\times M}$$. This in turn implies the representation by a Lax-Oleinik type formula. When $$M$$ is not compact, the global maximum principle does not immediately hold. Hitoshi Ishii and his coworkers obtained results about 10 years ago under some restrictions when $$M=\mathbb{R}^n$$. Basically the restrictions are about controlled growth at infinity.

We will explain that under the hypothesis that H is Tonelli, all continuous solutions of the evolution Hamilton-Jacobi equation above satisfy the Lax-Oleinik representation even for non-compact $$M$$. This of course will imply uniqueness for a given initial condition. Moreover, we will also show that if any pointwise finite $$U$$ is given by the Lax-Oleinik representation is automatically continuous and therefore a viscosity solution.

## First Order Mean Field Games: existence and long-time behavior of solutions (王楷植, 2020-06-25)

Title: First Order Mean Field Games--existence and long-time behavior of solutions

Speaker: 王楷植（上海交通大学）

Datetime: 2020-06-25 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：688 9993 3634

Abstract

First I will give a short introduction of the theory of mean field games, which was introduced independently by Lasry and Lions, and Huang, Malhame and Caines to study Nash equilibria for games with a very large number of players. And then I will introduce two long-time behavior results of solutions to evolutionary MFGs system on Rd [P. Cannarsa, W. Cheng, C. Mendico, K. Wang, Dyn. Games Appl., 2020] and with state constraints [P. Cannarsa, W. Cheng, C. Mendico, K. Wang, arXiv: 2004.06505], respectively. At last, I will talk about some new existence results of solutions of discounted stationary MFGs system and of more general (than discounted one) system.

## 复动力系统介绍 (王跃飞 , 2020-07-01)

Title: 复动力系统介绍

Speaker: 王跃飞 （中科院数学所）

Datetime: 2020-07-01 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：641 6124 6859

## Interface Profile Near the Contact Line in Electro-Wetting on Dielectric (Weiqing REN, 2020-07-02)

Title: Interface Profile Near the Contact Line in Electro-Wetting on Dielectric

Speaker: Weiqing REN（任维清）（National University of Singapore）

Datetime: 2020-07-02 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：688 9993 3634

Abstract

We consider a charged droplet sitting on a dielectric substrate and study the static profile of the interface near the contact line. We first derive the governing equations using the principle of minimum energy, then discuss the distinguished limit of the model. Analysis of the inner problem, which governs the interface profile near the contact line, shows the existence of a well-defined apparent contact angle. The apparent contact angle depends on the applied voltage and the thickness of the dielectric substrate, and the relation agrees well with the empirical Young-Lippmann equation.

## On the relations between principal eigenvalue and torsional rigidity (Giuseppe Buttazzo, 2020-07-15)

Title: On the relations between principal eigenvalue and torsional rigidity

Speaker: Giuseppe Buttazzo（Università di Pisa）

Datetime: 2020-07-15 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：647 9139 0694

Abstract

The relations between principal eigenvalue of the Laplace operator and torsional rigidity are studied in the class of general domains, convex domains, and domains with a small thickness. This is of help to provide some bounds for the Blasche-Santaló diagram of the two quantities. The results have been obtained in a joint work with Michiel van den Berg (Bristol) and Aldo Pratelli (Pisa).

## Weak KAM and Aubry–Mather theory On graphs (Antonio Siconolfi, 2020-07-21)

Title: Weak KAM and Aubry–Mather theory On graphs

Speaker: Antonio Siconolfi（Università degli Studi di Roma “La Sapienza” ）

Datetime: 2020-07-21 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：654 5189 5068

Abstract

Even if the structure of graphs is very simple, we show that it is possible to build an adaptation of Weak KAM and Aubry Mather theory in this setting, for a suitable class of Hamiltonians. We recover the main properties of the continuous case. One of the applications we have in mind, , still under investigation, is to extend the homogenization result for Hamilton–Jacobi equations on graphs/networks. A further possible application is to define and study a first order time–dependent Mean Field Games system in this setting. Work in collaboration with Alfonso Sorrentino.

## Weak solutions of second order master equations for mean field games with common noise (Chenchen MOU, 2020-07-27)

Title: Weak solutions of second order master equations for mean field games with common noise

Speaker: Chenchen MOU（牟宸晨）（UCLA & City University of Hong Kong）

Datetime: 2020-07-27 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：621 0545 9046

Abstract

In this talk we study master equations arising from mean field game problems, under the crucial monotonicity conditions. Classical solutions of such equations require very strong technical conditions. Moreover, unlike the master equations arising from mean field control problems, the mean field game master equations are non-local and even classical solutions typically do not satisfy the comparison principle, so the standard viscosity solution approach seems infeasible. We shall propose a notion of weak solution for such equations and establish its wellposedness. Our approach relies on a new smooth mollifier for functions of measures, which unfortunately does not keep the monotonicity property, and the stability result of master equations. The talk is based on a joint work with Jianfeng Zhang.

## Equilibrium states which is not Gibbs measure on hereditary subshifts (陈二才, 2020-08-26)

Title: Equilibrium states which is not Gibbs measure on hereditary subshifts

Speaker: 陈二才（南京师范大学）

Datetime: 2020-08-26 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：618 1982 4695

Abstract

We consider consider which kind of invariant measure on hereditary subshifts is not Gibbs measure. For the hereditary closure of a subshift $$(X,S)$$, we prove that in some situation, the invariant measure $$\nu\ast B_{p,1-p}$$ can not be Gibbs measure where $$\nu$$ is any invariant measure on $$(X,S)$$. As an application, we show that for some $$\mathscr{B}$$-free shifts, the equilibrium state $$\nu_{\eta}\ast B_{p,1-p}$$ is not Gibbs measure.

## Multiplicity of closed geodesics on Finsler compact non-simply connected manifolds (刘会, 2020-09-02)

Title: Multiplicity of closed geodesics on Finsler compact non-simply connected manifolds

Speaker: 刘会（武汉大学）

Datetime: 2020-09-02 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：672 9144 4139

Abstract

For many years, there seem to be very few works on the multiplicity of closed geodesics on non-simply connected manifolds, the main reason is that the topological structures of the free loop spaces on these manifolds are not well known, so that the classical Morse theory is diﬃcult to be applicable. In recent years, motivated by the studies on simply connected manifolds and closed characteristics on Hamiltonian energy surfaces, we study the topological structure of the contractible component and non-contractible component of the free loop space on Finsler real projective space and compact space form, which are typically non-simply connected manifolds, and then we establish some new resonance identities, which are successfully applied to get many multiplicity results of closed geodesics on these non-simply connected manifolds. In this talk, I will give a survey of our results.

## Restricted independence in displacement function (张伟年, 2020-09-09)

Title: Restricted independence in displacement function

Speaker: 张伟年（四川大学）

Datetime: 2020-09-09 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：651 2716 6946

Abstract

Since the independence of focal values is a sufficient condition to give a number of limit cycles arising from a center-focus equilibrium, in this paper we consider a restricted independence to a parametric curve, which gives a method not only to increase the lower bound for the cyclicity of the center-focus equilibrium but also to be available when those focal values are not independent. We apply the method to a nondegenerate center-focus system and prove that the cyclicity reaches its an upper bound. This is a joint work with Xingwu Chen, Jaume Llibre, Zhaoxia Wang.

## 动力系统中的光滑不变流形和不变叶层问题 (张文萌, 2020-09-09)

Title: 动力系统中的光滑不变流形和不变叶层问题

Speaker: 张文萌（重庆师范大学）

Datetime: 2020-09-09 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：651 2716 6946

Abstract

## On the existence of SRB measures for a class of partially hyperbolic attractors (曹永罗, 2020-09-23)

Title: On the existence of SRB measures for a class of partially hyperbolic attractors

Speaker: 曹永罗（苏州大学）

Datetime: 2020-09-23 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：652 9681 7446

Abstract

In this talk, we consider the existence of SRB measure for partially hyperbolic attractors. If the systems's central direction can be decomposed into one dimension sub-bundles which are dominated splitting, then there exists a SRB measure.

## Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations (秦文新, 2020-09-23)

Title: Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations

Speaker: 秦文新（苏州大学）

Datetime: 2020-09-23 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：652 9681 7446

Abstract

By introducing for monotone recurrence relations pseudo solutions, which are analogues of pseudo orbits of dynamical systems, we show that for general monotone recurrence relations the rotation set is closed, and each element in the rotation set is realized by a Birkhoff orbit. Moreover, if there is an orbit without rotation number, then the system has positive topological entropy, and we can construct orbits shadowing different rotation numbers.

## Regularity and generic divergence of local first integrals for analytic systems (张祥, 2020-09-28)

Title: Regularity and generic divergence of local first integrals for analytic systems

Speaker: 张祥（上海交通大学）

Datetime: 2020-09-28 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：676 5125 5799

Abstract

In this talk we first introduce background on local integrability of analytic differential systems, and then present our recent results about regularity and generic divergence of analytic differential systems near a singularity with one zero eigenvalue and others nonresonant. These results answer the related problems partially existing from 2003.

## Denjoy subsystems and Horseshoes (Marie-Claude Arnaud, 2020-10-08)

Title: Denjoy subsystems and Horseshoes

Speaker: Marie-Claude Arnaud（Université Paris Diderot）

Datetime: 2020-10-08 14:30 — 15:30 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：685 3634 5303

Abstract

I will describe some connections between two kinds of restricted dynamics for area preserving diffeomorphisms, horseshoes and Denjoy subsystems: this last notion has been introduced by myself and P. Le Calvez and I will explain it. Then I will explain that any horseshoe contains a continuous 1-parameter family of Denjoy subsystems that is parametrized by the rotation number. After that, I will consider the inverse problem and give some partial answer: if an area preserving diffeomorphism $$f$$ has a Denjoy subsystem, does there exist a horseshoe for $$f$$?

## Hamilton-Jacobi方程解的长期行为：从动力系统的观点看 (严军, 2020-10-13)

Title: Hamilton-Jacobi方程解的长期行为：从动力系统的观点看

Speaker: 严军（复旦大学）

Datetime: 2020-10-13 15:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：637 8719 4462

Abstract

## Rigorous theory of viscous and inviscid 1d turbulence (Sergei Kuksin, 2020-10-21)

Title: Rigorous theory of viscous and inviscid 1d turbulence

Speaker: Sergei Kuksin（Université Paris Diderot）

Datetime: 2020-10-21 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：687 5772 3983

Abstract

I will present three main laws from the Kolmogorov theory of turbulence ("the K41 model"), discuss their versions for one-dimensional fluid and will show that the latter may be rigorously justified for the 1d fluid, described by the Burgers equation with small positive or zero viscosity. The proof relies on a qualitative analysis of the dynamical system which the Burgers equation with small positive viscosity defines in Sobolev spaces. The talk is based on my joint book with Alex Boritchev which will appear next year in Publications of AMS.

## Lyapunov optimizing measures and periodic measures for $$C^2$$ expanding maps (黄文, 2020-10-27)

Title: Lyapunov optimizing measures and periodic measures for $$C^2$$ expanding maps

Speaker: 黄文（中国科学技术大学）

Datetime: 2020-10-27 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：693 5063 1465

Abstract

We consider the typical Lyapunov minimizing measures for expanding self-maps on the circle. The main result obtained in this paper is that there exists an open and dense subset $$\mathscr{P}$$ of all $$C^2$$ expanding self-maps such that for each $$T\in\mathscr{P}$$, the Lyapunov minimizing measures of $$T$$ are uniquely supported on a periodic orbit. This answers a conjecture of Jenkinson-Morris positively. This is a joint work with Leiye Xu and Dawei Yang.

## Characteristic Factors in Dynamical Systems (邵松, 2020-10-27)

Title: Characteristic Factors in Dynamical Systems

Speaker: 邵松（中国科学技术大学）

Datetime: 2020-10-27 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：693 5063 1465

Abstract

In this talk, we will discuss characteristic factors in dynamical systems. First we give some examples to show what are characteristic factors. Then we show some recent results about characteristic factors in dynamical systems and their applications. This is a joint work with Wen Huang, Xiangdong Ye etc.

## Ergodic Optimization of a sequence of continuous obervables (赵云, 2020-10-28)

Title: Ergodic Optimization of a sequence of continuous obervables

Speaker: 赵云（苏州大学）

Datetime: 2020-10-28 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：676 0808 8080

Abstract

In this talk, we will review some research progresses in ergodic optimization for a single continuous function, and give some results of the ergodic optimization for a sequence of continuous functions, including subordination principle, constrained ergodic optimization and typical properties.

## Frenkel-Kontorova模型中的有序结构 (王亚南, 2020-10-28)

Title: Frenkel-Kontorova模型中的有序结构

Speaker: 王亚南（南京师范大学）

Datetime: 2020-10-28 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：676 0808 8080

Abstract

Frenkel-Kontorova模型（F-K模型）描述了一列相互作用的粒子在给定势能下的运动规律。系统的有序结构对于系统的稳定性有重要意义。在本报告中，我们主要讨论过阻尼的F-K模型中有序结构性质，主要包括叶状结构存在的判定，层状结构性质及其性质。

## Small amplitude generalized breathers for nonlinear Klein-Gordon equations (Chongchun Zeng, 2020-10-30)

Title: Small amplitude generalized breathers for nonlinear Klein-Gordon equations

Speaker: Chongchun Zeng（Georgia Institute of Technology）

Datetime: 2020-10-30 10:00 — 11:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：613 9977 2969

Abstract

Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states.

In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work with O. Gomide, M. Guardia, and T. Seara.

## On a Hamiltonian approach towards hydrodynamic limit for non-interacting deterministic particles (Jin FENG, 2020-11-04)

Title: On a Hamiltonian approach towards hydrodynamic limit for non-interacting deterministic particles

Speaker: Jin FENG（University of Kansas）

Datetime: 2020-11-04 20:00 — 21:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：624 1270 2131

Abstract

In the context of hydrodynamic limit for $$N$$ non-interacting particles following Hamiltonian dynamics, we identify a type of scaling where we can avoid explicit use of micro-canonical, canonical ensembles and even arguments about equivalence/non-equivalence of ensembles. Replacing them is a deterministic averaging problem studied in the finite dimensional weak KAM theory. As a consequence, we derive a continuum level effective-Hamiltonian as limit from the particle Hamiltonians.

We formulate the above program using well-posedness and multi-scale convergence of first order Hamilton-Jacobi PDEs in the space of probability measures, in a viscosity sense. An extended half-relaxed limit theory and a comparison principle for the Hamilton-Jacobi can be developed exploring two inter-related abstract aspects of the Wasserstein space: One, it is a limit of quotient space of $$N$$-products of Euclidean spaces modeling individual particles; Two, it is an Alexandrov metric spaces with curvature bounded from below.

This is work in progress with Toshio Mikami in Tsuda University, Japan.

## Weak KAM Theory for Sub-Riemannian control systems (Cristian Mendico, 2020-11-05)

Title: Weak KAM Theory for Sub-Riemannian control systems

Speaker: Cristian Mendico（GSSI-Gran Sasso Science Institute）

Datetime: 2020-11-05 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：665 8976 3941

Abstract

The ergodic behavior of control systems is an open and challenging problem since such a systems give rise to non-coercive and not strictly convex Hamiltonian even if the original Lagrangian is of Tonelli type. I will analyze the class of Sub-Riemannian control systems, i.e. system of the form $\dot\gamma(t)=\sum_{i=1}^{m}{f_{i}(\gamma(t))u_{i}(t)}$ for a given set of vector fields $$\{f_{i}\}_{i = 1, \dots m}$$. The assumptions on the model that will play a crucial role are the Chow's condition and the non-existence of singular control for the convergence result and for the Aubry-Mather theory, respectively.

For a Lagrangian $$L$$ and a time horizon $$T>0$$ we, first, obtain that time-average value function converge to a real constant $$c(L)$$ as $$T \to +\infty$$. Then, we show the ergodic Hamilton-Jacobi equation $H(x, Du(x))=c(L), \quad x \in \mathbb{R}^{d}$ has a continuous viscosity solution and moreover, we provide a representation formula for such a constant, in the spirit of Mather's results, by using closed measures.

In conclusion we define the (projected) Mather set and the Aubry set associated with this systems and we study the regularity of a solution to the ergodic equation on these sets.

## KAM theory and quasi-periodic attractors for conformally symplectic systems (Alessandra Celletti, 2020-11-13)

Title: KAM theory and quasi-periodic attractors for conformally symplectic systems

Speaker: Alessandra Celletti（University of Rome Tor Vergata）

Datetime: 2020-11-13 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：686 3847 7527

Abstract

We present results on the existence of quasi-periodic attractors of conformally symplectic systems in non-perturbative regimes. Conformally symplectic systems are characterized by the property that they transform the symplectic form into a multiple of itself. Finding the solution of such systems requires to add a drift parameter. We provide an explicit quantitative theorem in an a-posteriori format. Precisely, assuming the existence of an approximate solution, satisfying the invariance equation up to an error term - small enough with respect to explicit condition numbers, - then we can prove the existence of a solution nearby.

This method can be also used to get different results: (i) prove the existence of whiskered tori for conformally symplectic systems, (ii) give a characterization of the analyticity domains of the quasi-periodic attractors in the symplectic limit, (iii) provide a very efficient algorithm to generate the solution, which can be implemented successfully in model problems and physically meaningful examples.

The content of this talk refers to works in collaboration with R. Calleja and R. de la Llave.

## On the vanishing discount problem from the negative direction (Andrea Davini, 2020-11-17)

Title: On the vanishing discount problem from the negative direction

Speaker: Andrea Davini（University of Rome La Sapienza）

Datetime: 2020-11-17 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：650 6594 4606

Abstract

It has been proved in A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Invent. Math. (2016) that the unique viscosity solution of
\label{abs}\tag{*}
\lambda u_\lambda+H(x,d_x u_\lambda)=c(H)\qquad\hbox{in $$M$$},

uniformly converges, for $$\lambda\rightarrow 0^+$$, to a specific solution $$u_0$$ of the critical equation
$H(x,d_x u)=c(H)\qquad\hbox{in M},$
where $$M$$ is a closed and connected Riemannian manifold and $$c(H)$$ is the critical value.

In this seminar, we will consider the same problem for $$\lambda\rightarrow 0^-$$. In this case, viscosity solutions of equation \eqref{abs} are not unique, in general, so we focus on the asymptotics of the minimal solution $$u_\lambda^-$$ of \eqref{abs}. Under the assumption that constant functions are subsolutions of the critical equation, we prove that the $$u_\lambda^-$$ also converges to $$u_0$$ as $$\lambda\rightarrow 0^-$$. Furthermore, we exhibit an example of $$H$$ for which equation \eqref{abs} admits a unique solution for $$\lambda<0$$ as well. The talk is based on a joint work with Lin Wang (Tsinghua University).

## Euclidean distance function in the presence of an obstacle (Piermarco Cannarsa, 2020-11-19)

Title: Euclidean distance function in the presence of an obstacle

Speaker: Piermarco Cannarsa（University of Rome Tor Vergata）

Datetime: 2020-11-19 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：680 9256 2983

Abstract

The obstacle problem is a classical topic in analysis, which may take different forms depending on the quantities you observe. In this talk, we are interested in the regularity of the Euclidean distance function from a given point in the presence of a compact obstacle with smooth boundary. First, we will show that the distance is semiconcave with a fractional modulus and that, near the obstacle, such a regularity is optimal. Then, we will show that the distance function is everywhere differentiable (except for the point target) if and only if no obstacle is present. Finally, we will study the propagating structure of the singular set of the distance both at `interior points' and on the boundary of the obstacle. For such an analysis, we will use recent results on the extension of semiconcave functions defined on a closed domain. This is joint work with Paolo Albano and Vincenzo Basco.

## Generic Behavior of smooth monotone systems with respect to $$k$$-cones (王毅, 2020-11-24)

Title: Generic Behavior of smooth monotone systems with respect to $$k$$-cones

Speaker: 王毅（中国科学技术大学）

Datetime: 2020-11-24 14:00 — 15:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：640 5432 5516

Abstract

In this talk, we consider a smooth flow which is ”strongly monotone“ with respect to a $$k$$-cone, a closed set that contains a linear subspace of dimension $$k$$ and no linear subspaces of higher dimension. We will show that orbits with initial data from an open and dense subset of the phase space are either pseudo-ordered or convergent to equilibria. This covers the celebrated Hirsch's Generic Convergence Theorem in the case $$k=1$$, and yields a generic Poincaré-Bendixson Theorem for the case $$k=2$$.

## Spreading speeds of nonlocal diffusion KPP equations (梁兴, 2020-11-24)

Title: Spreading speeds of nonlocal diffusion KPP equations

Speaker: 梁兴（中国科学技术大学）

Datetime: 2020-11-24 15:00 — 16:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：640 5432 5516

Abstract

In this talk I will introduce our work on the dynamics of KPP nonlocal diffusion equation on one-dimensional space. We will discuss the relation between the the existence of the spreading speed and the irreducibility of the diffusion kernel.

## The limit distribution of inhomogeneous Markov processes and Kolmogorov's problem (柳振鑫, 2020-11-27)

Title: The limit distribution of inhomogeneous Markov processes and Kolmogorov's problem

Speaker: 柳振鑫（大连理工大学）

Datetime: 2020-11-27 14:00 — 15:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：643 9526 4495

Abstract

In this talk, we will talk about the limit distribution of inhomogeneous Markov processes generated by SDEs. Meantime, we will also discuss the recent progress in Kolmogorov's problem on the limit behavior of stationary distributions of diffusion processes as the diffusion tends to zero.

## From homogenization to dynamical system (Yifeng Yu, 2020-12-04, 2020-12-11, 2020-12-18)

Title: From homogenization to dynamical system

Speaker: Yifeng Yu（University of California, Irvine）

Datetime:

• 2020-12-04 11:00 — 12:00 Beijing/Shanghai
• 2020-12-11 11:00 — 12:00 Beijing/Shanghai
• 2020-12-18 11:00 — 12:00 Beijing/Shanghai

Venue: Zoom APP
Meeting ID：632 3890 1470

Abstract

In these three talks, I will (1) go over the basic theory of homogenization of Hamilton-Jacobi equations; and (2) talk about how to use Aubry-Mather theory to solve some fundamental problems in homogenization theory that can not be approached by standard PDE tools.

This mini-course is composed of three parts:

1. Introduction of basic theory in Homogenization
2. Optimal convergence rate in the homogenization of Hamilton-Jacobi equation
3. Properties of effective Hamiltonian

## On the regularity of length-minimizers in sub-Riemannian geometry (Davide Barilari, 2020-12-04)

Title: On the regularity of length-minimizers in sub-Riemannian geometry

Speaker: Davide Barilari（Università degli Studi di Padova）

Datetime: 2020-12-04 19:00 — 20:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：632 8456 8140

Abstract

The regularity issue for length-minimizers is one of the main open problem in sub-Riemannian geometry. In this talk, after presenting the question and giving a survey of the known results, we present a recent result on the $$C^1$$ regularity for a class of length-minimizers in rank 2 sub-Riemannian structures. (Joint with Yacine Chitour, Frédéric Jean, Dario Prandi, Mario Sigalotti)

## Noise Impacts on Finite Dimensional Dynamical Systems (Yingfei Yi, 2020-12-08)

Title: Noise Impacts on Finite Dimensional Dynamical Systems

Speaker: Yingfei Yi（University of Alberta and Jilin University）

Datetime: 2020-12-08 14:00 — 15:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：627 0593 7107

Abstract

Dynamical systems are often subjected to noise perturbations either from external sources or from their own intrinsic uncertainties. While it is well believed that noises can have dramatic effects on the stability of a deterministic system at both local and global levels, mechanisms behind noise surviving or robust dynamics have not been well understood especially from distribution perspectives. This talk attempts to outline a mathematical theory for making a fundamental understanding of these mechanisms in white noise perturbed systems of ordinary differential equations, based on the study of stationary measures of the corresponding Fokker-Planck equations. New existence and non-existence results of stationary measures will be presented by relaxing the notion of Lyapunov functions. Limiting behaviors of stationary measures as noises vanish will be discussed in connection to important issues such as stochastic stability and bifurcations.

## Noise Smooth Conjugacy for Random Dynamical Systems (kening Lu, 2020-12-11)

Title: Smooth Conjugacy for Random Dynamical Systems

Speaker: kening Lu（Brigham Young University）

Datetime: 2020-12-11 09:30 — 11:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：641 3668 9354

Abstract

In this talk, we study the smooth conjugacy problems for random dynamical systems when the Lyapunov exponents satisfy various conditions such as Diophantine conditions, nonresonant conditions.