动力系统线上论坛
作为探索物理世界的数学前沿，Hamilton动力系统以其广泛的实用性一直活跃在微分方程研究的中心。自牛顿时代以来，每一次新兴科技变革背后都伴随着这一课题重大的理论突破。自上世纪 80年代以来，随着变分法在正定Hamilton系统上的⻓足发展，高维度方程的复杂动力现象层出 不穷的揭示出来。
带有阻尼的Hamilton系统更加契合实际的物理动态，并由稳定吸引子的产生约束全局的动力行为。从变分法的⻆度，此类带有optimal control意味的方程衍生出更加丰富的变分特性，并因其在一阶PDE方程中的重要作用而得到了包括Arnold，Lions等诸多数学家的关注。
该系列报告旨在研究广泛范畴下的Hamilton 系统轨道的拓扑特征和几何刚性，并展示这类方程在近代物理、天文、经济学乃至数值算法、机器学习等领域的研究前景。
时间：2020年7月1日-2020年10月30日 策划：中国科学院数学与系统科学研究院，北京师范大学数学科学学院 顾问：程崇庆 教 授（南京大学数学系） 尚在久 研究员（中科院数学所） 王跃⻜ 研究员（中科院数学所） 组织：苏喜锋（北京师范大学） 张建路（中科院数学所）
腾讯会议: 242 605 286 密码：123456
报告人：张建路（中科院数学所） 摘要：作为保面积映射的主导情形，辛扭转映射因其广泛的应用背景以及良好的拓扑特性成为Hamilton动力学研究最为透彻的客体。变分法的引入也给高维动力学的探索提供了诸多直观依据。我们将着重介绍weak KAM解的全局参数化，并以此来阐述奇性乃至局部转移链的产生机制。
报告人：金亮 （南京理工大学） 摘要：由一阶偏微分方程诱导的特征系统是经典Hamilton系统的一类重要推广。在这个报告中，我们将介绍有关该系统的发展脉络，并解释我们最近关于严格耗散型(激励型)特征系统最大全局吸引子(排斥子)和此类系统整体动力学的工作。我们的结果表明，一大类带有耗散或激励机制的系统可以纳入该类系统的研究框架内。
Zoom会议: 690 776 66313 密码：123456
报告人：赵恺 （复旦大学数学学院） 摘要：线性阻尼Hamilton系统是一种特殊的特征 系统，它在天体力学潮汐牵引、⻛阻效应的物理模型乃至经济学衰减效应上有广泛的应用。由其在一阶偏微分方程上的特殊性，粘性解收敛速度估计一直受到了广泛关注，并应用来获取变分Aubry集的动力学信息。我们就两种最为普遍关注的Aubry集形态(KAM环面、双曲周期轨道)进行收敛速度分析。
报告人：王林 （清华大学丘成桐数学中心） 摘要：基于经典Hamilton系统的Aubry-Mather理论和弱KAM理论，我们对接触Hamilton系统建立 并发展了全局作用量极小方法。在此报告中，我会介绍此方法的基本工具以及利用这些工具在接触 Hamilton系统中发现的区别与经典Hamilton系统的新现象。此报告基于报告人与苏喜锋、王楷植和严军合作的一系列工作。
Zoom会议: 661 689 45483 密码：123456
报告人：严军 （复旦大学数学学院） 摘要：介绍经典Hamilton系统和接触Hamilton系统中的H-J方法，探讨H-J方程经典解和粘性解在动 力系统中的应用。
报告人：王亚南 （南京师范大学数学学院） 摘要：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。
腾讯会议: 826 339 058 密码：123456
报告人：李天虹 （中科院数学所） 摘要：作为一阶偏微分方程的Hamilton-Jacobi 方程， 它的特征系统是Hamilton ODE。我们研究 Hamilton-Jacobi方程解的奇异点集合的拓扑性质和整体结构，以及奇异点集合之外解的正则性。此报告基于报告人与王靖华，温海瑞合作的一系列工作。
报告人：程伟 （南京大学数学系） 摘要：We try to review known and unknown results on singularities and their propagation from Hamilton-Jacobi equations in the past decade. We will emphasize the potential problem and observation for the applications to dynamical systems, geometry, calculus of variation and optimal control and PDE. This talk can be regarded as an extension of the preprint "On and beyond propagation of singularities of viscosity solutions", arXiv:1805.11583, 2018, by Cannarsa and the speaker.
腾讯会议: 581 311 991 密码：123456
报告人：王奕倩 （南京⼤学） 摘要： Quasiperiodic Schrödinger operators (QPSO) is the mathematical model for the conductivity on quasi-crystals which was found by a Nobel prize winner. Several great mathematicians have been captivated by this field. In last decades, various methods have been developed in the study of one-dimensional analytic QPSO, which led to a lot of deep result. However, these methods depend heavily on analytic conditions and are difficult to be extended to smooth situations. Recently a series of sharp results for Sinai's model (QPSO with a cos-type potential and a large coupling) have been obtained. In this talk, we will discuss the roles played by geometric conditions and regularity conditions on the potentials as well as the relationship between them.
报告人：魏巧玲 （⾸都师范⼤学） 摘要：听⾳辨⿎，即是否能通过两张⿎⾯的声⾳，辨别出它们是否形状⼀样（M.Kac 1966）。数学上对应反谱问题：⼀个平⾯区域，是否可由其带Dirichlet边界的Laplace算⼦的特征值构成的算⼦谱唯⼀决定。另⼀⽅⾯，⼀个平⾯区域也可给出⼀个动⼒系统——弹球系统，弹球系统的所有周期轨的⻓度构成区域的⻓度谱。Laplace算⼦谱和⻓度谱紧密相关。我们将介绍反谱问题的历史和研究进展，重点介绍弹球系统⻓度谱的相关研究⽅法。
腾讯会议: 384 498 320 密码：123456
报告人：崔⼩军 （南京⼤学） 摘要： 我们将主要回顾总结⼏何（黎曼和洛伦兹）框架中的Aubry-Mather理论和弱KAM理论⽅⾯的主要结果，也将讨论⼀些这⽅⾯的理论和⼏何测度论，刚性，Wasserstein空间的测地动⼒学等领域的联系。
报告人：许璐 （吉林⼤学） 摘要：My talk is about the quasi-periodic motions in multi-scaled Hamiltonian systems. It consists of four part.
I will also simply introduce the Technical ingredients of our work.
腾讯会议: 970 257 901 密码：123456
报告人：程旭 （中科院数学所） 指导老师： 尚在久 研究员 摘要： 函数的优化问题由来已久，从一阶梯度算法 到二阶加速梯度算法 ，从凸优化到非凸优化。近年来，越来越多研究者从动力系统的⻆度来理解加速现象。对于凸优化，将离散算 法与连续的动力系统结合起来，Nesterov加速梯度法的极限情形就是一个二阶ODE，通过分析这个连续动力系统的耗散可以解释加速现象和震荡现象。此后，结合变分法，连续系统的收敛速度可以达 到任意多项式次 ；结合哈密尔顿系统，设计辛算法也可以得到更好的优化算法。对于非凸优化，从稳定流形理论出发，可以得到梯度算法以概率1逃离严格鞍点，由此通过加随机扰动得到 一系列逃离鞍点的随机算法（PGD, PAGD, PSGD）;类似的这些离散算法通过连续化，也可以用随机微分方程SDE逼近。从优化中提取动力系统可以理解优化算法中的深刻内涵，进一步的，往往通过离散和连续的对比，应用动力系统理论，可以启发出新的算法。
腾讯会议: 923 270 635 密码：123456
报告人：程崇庆 （南京大学） 摘要： In this talk, I shall introduce some techniques we invented for the construction of global connecting orbits. Some open problems shall be also proposed for further study.
2020-09-03 Zoom: 646 976 07037 密码：123456
报告人：Diogo Gomes （KAUST） 摘要： Here, consider a constrained mean-field game where the price is determined by a supply vs. demand balance condition. We begin by examining problems with a deterministic supply. In this case, we establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well-defined, and it is a Lipschitz function of time. Then, we study linear-quadratic models that can be solved explicitly. Finally, we discuss the case where the supply is a random process and in the case of linear-quadratic models discuss how to solve the problem.
2020-09-04 Zoom: 664 863 69095 密码：123456
报告人：赵磊 （德国 奥格斯堡大学） 摘要： 我们将回顾与评述关于太阳系稳定性，尤其是其非线性KAM稳定性的一些数学理论与进展。
腾讯会议: 803 712 343 密码：123456
报告人：顾险峰 （State University of New York at Stony Brook） 摘要： This work introduces an optimal transportation (OT) view of generative adversarial networks (GANs). Natural datasets have intrinsic patterns, which can be summarized as the manifold distribution principle: the distribution of a class of data is close to a low-dimensional manifold. GANs mainly accomplish two tasks: manifold learning and probability distribution transformation. The latter can be carried out using the classical OT method. From the OT perspective, the generator computes the OT map, while the discriminator computes the Wasserstein distance between the generated data distribution and the real data distribution; both can be reduced to a convex geometric optimization process. Furthermore, OT theory discovers the intrinsic collaborative—instead of competitive—relation between the generator and the discriminator, and the fundamental reason for mode collapse. We also propose a novel generative model, which uses an autoencoder (AE) for manifold learning and OT map for probability distribution transformation. This AE–OT model improves the theoretical rigor and transparency, as well as the computational stability and efficiency; in particular, it eliminates the mode collapse. The experimental results validate our hypothesis, and demonstrate the advantages of our proposed model.
Zoom: 659 865 44977 密码：123456
报告人：张智元 （CNRS, Universite Paris 13） 摘要： We show that a topologically mixing Anosov flow on a 3-dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential. This is a joint work with Masato Tsujii.
Zoom: 668 0410 3984 密码：123456
报告人：Enrico Valdinoci （University of Western Australia） 摘要： In this talk we consider a nonlocal equation driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This result regarding symbolic dynamics in a fractional framework is part of a study of Peierls-Nabarro model for crystal dislocations. The associated evolution equation can be studied in the mesoscopic and macroscopic limits. Namely, the dislocation function has the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior potential, which can be either repulsive or attractive, depending on the relative orientations of the dislocations. For opposite orientations, collisions occur, after which the system relaxes exponentially fast.
Zoom: 650 0806 2145 密码：123456
报告人：Stefan Suhr（Ruhr-University Bochum） 摘要： The talk will explain the geometric framework of Aubry-Mather theory in Lorentzian geometry and review the main results. If time permits possible future directions and open problems will be discussed.
腾讯会议: 812 805 679 密码：123456
报告人：⻩冠 （清华大学） 摘要： We consider a damped/driven nonlinear Schrödinger equation in , where is arbitrary, under odd periodic boundary conditions. Here is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy uniformly in and . In this work we prove that for small and any initial data, with large probability the Sobolev norms with become large at least to the order of with , on time intervals of order . It proves that solutions of the equation develop short space-scale of order to a positive degree, and rigorously establishes the (direct) cascade of energy for the equation. This is a joint work with Sergei Kuksin.
腾讯会议: 439 739 834 密码：123456
报告人：薛金鑫 （清华大学） 摘要： 我们考虑黑洞背景下的粒子的测地运动，并用哈密顿动力系统的方法进行研究。我们借助黑 洞的光圈得到Arnold扩散的轨道。Arnold扩散是近可积哈密顿动力系统的一个典型的不稳定现象。 这种轨道有明显的物理意义并可以观测。在远离视界的区域，我们可以类比牛顿三体问题，得到振 荡轨道的存在性。最后我们证明，扭转映射的理论可以用于研究光圈的动力学，以及拟周期振荡。
Zoom: 696 196 91647 密码：123456
报告人：Chen Kuo-Chang （台湾清华大学） 摘要： The Newtonian 2-body problem is also known as the Kepler problem in honor of Johannes Kepler (1571-1630) for his discovery of three laws of planetary motion, based on which Newton deduced in 1687 the celebrated law of universal gravitation. It is commonly considered a well-understood problem, as solving it with given initial data and proving Kepler's three laws require nothing more than tools from elementary calculus. In this talk I will briefly describe its history, outline recent discoveries from variational perspectives, and show some progresses regarding singularities.
Zoom: 620 998 77902 密码：123456
报告人：Marie-Claude Arnaud （IMJ-PRG, Université Paris-Diderot） 摘要： This is a joint work with Maxime Zavidovique. For Hamiltonian systems, the well-known Arnol’d-Liouville theorem tells us that if the system has enough independent integrals, then the space is foliated by invariant Lagrangian submanifolds on which the Dynamics is conjugated to a rotation. We will consider a situation with weaker hypothesis: assume that a symplectic twist map of the annulus has an invariant foliation into continuous curve. What can be said on the this foliation and the Dynamics? After explaining some classical and less classical results in the Hamiltonian case, we will explain recent results on twist maps, e.g. that the invariant foliation is Holder, that with some other hypothesis the restricted dynamics to invariant curve is conjugate to a rotation.
2020-11-16，16:00-17:00 Zoom: 622 098 20563 密码：123456
报告人：Alex Haro(University of Barcelona) 摘要： We present a method to find nontwist KAM tori. These are tori for which the twist condition fails. Our method also leads to a natural classification of KAM tori which is based on Singularity Theory. This talk aims to illustrate the main ideas of our approach, going from rigorous results to numerical computations up to the verge of breakdown. This a joint project with Rafael de la Llave and Alejandra González.
2020-11-20，09：00-11：00 Zoom: 687 466 26905 密码：123456
报告人：Renato Calleja (National Autonomous University of Mexico) 摘要：Many physical problems are described by conformally symplectic systems. We study the existence of whiskered tori in a family of conformally symplectic maps depending on parameters . Whiskered tori are tori on which the motion is a rotation but having as many contracting/expanding directions as allowed by the preservation of the geometric structure.
Our main result is formulated in an a-posteriori format. Given an approximately invariant embedding of the torus for a parameter value with an approximately invariant splitting, there is an invariant embedding and invariant splittings for new parameters.
Using the results of formal expansions as the starting point for the a-posteriori method, we study the domains of analiticity of parameterizations of whiskered tori in perturbations of Hamiltonian Systems with dissipation. The proofs of the results lead to efficient algorithms that are quite practical to implement.
Joint work with A. Celletti and R. de la Llave, A.P. Bustamante.
报告人：Rodrigo Treviño (University of Maryland) 摘要：The world of aperiodic tilings is a meeting point of many disciplines such as mathematical physics, dynamical systems, operator algebras, discrete geometry, to name a few. In this talk I will give go over the origins of the topic, some important areas of research, recent results and open questions. I will emphasize the important role that tools from dynamical systems and ergodic theory play in all of it.