Title: 大数据应用案例分析
Speaker: 张晓明（北京应用数学研究院）
Datetime: 2021-01-21 10:00 — 11:30 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：682 0506 6340
Password：034847
Abstract
大数据人工智能是现在和未来相当长一段时期内的重要研究领域，本讲座通过实战案例简要介绍了数学在解决实际问题中的作用、以及未来发展的一些方向。本讲座的四个案例，分别是：电影票房预测和影院排片；应用基因组学数据提前预测重大疾病的发生；应用实际生产流程数据推荐和优化工艺配方；阿里巴巴千里马大数据竞赛《蒸汽量预测》题解。
Title: Long time behaviour of solutions to Hamilton-Jacobi equations for sub-Riemannian control systems
Speaker: Piermarco Cannarsa（University of Rome Tor Vergata）
Datetime: 2021-01-21 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：690 2910 8040
Password：197423
Abstract
Sub-Riemannian control systems are an important class of dynamical systems, with linear dependence on controls (but nonlinear on state variables). Controllability properties for such systems are derived by the so-called Lie Algebra rank condition on the associated family of vector fields, called Hörmander vector fields.
We will discuss the long-time average behaviour of the value function of optimal control problems for sub-Riemannian systems, which cannot be addressed by the classical week KAM theory as the Hamiltonian fails to be coercive in the momentum variable.
Nevertheless, by a dynamical approach, we will show how to prove the existence of a unique critical constant such that the ergodic Hamilton-Jacobi equation admits solutions. Then, we will use such a constant to obtain the limit profile of solutions to the Cauchy problem as the time horizon goes to infinity. We will also provide a representation formula for the critical constant and study the associated Aubry set, giving conditions for such a set to be compact. Finally, we will prove horizontal differentiability for critical solutions to the ergodic Hamilton-Jacobi equation on the Aubry set.
Title: The Hamilton–Jacobi equation on networks: weak KAM and Aubry–Mather theories
Speaker: Alfonso Sorrentino（University of Rome Tor Vergata）
Datetime: 2021-01-29 16:00 — 17:00 Beijing/Shanghai
Venue: Zoom APP
Meeting ID：662 5511 4145
Password：609617
Abstract
Over the last years there has been an increasing interest in the study of the Hamilton–Jacobi Equation on networks and related questions. These problems, in fact, involve a number of subtle theoretical issues and have a great impact in the applications in various fields, for example to data transmission, traffic management problems, etc… While locally — i.e., on each branch of the network (arcs) —, the study reduces to the analysis of 1-dimensional problems, the main difficulties arise in matching together the information converging at the juncture of two or more arcs, and relating the local analysis at a juncture with the global structure/topology of the network.
In this talk I shall discuss several results related to the global analysis of this problem, obtained in collaboration with Antonio Siconolfi (Univ. of Rome La Sapienza); more specifically, we developed analogues of the so-called Weak KAM theory and Aubry–Mather theory in this setting. The salient point of our approach is to associate to the network an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on the graph.
Title: Hamilton-Jacobi equations and viscosity solutions
Speaker: Hitoshi Ishii（Waseda University & Tsuda University）
Datetime:
2021-04-06 14:00 — 15:00 Beijing/Shanghai
2021-04-09 14:00 — 15:00 Beijing/Shanghai
Totally ten times, the same meeting room and times every week (Tuesday and Friday)
Venue: Umeet APP
Meeting ID：183 691 7895
Password：992585
Abstract
Title: Recent progress in sub-Riemannian geometry
Speaker: Ludovic Rifford（Université Côte d'Azur）
Datetime: 2021-05-13 16:00 — 17:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：184 709 9625
Password：349386
Abstract
After a short introduction to sub-Riemannian geometry, we will present the main open problems in the theory and give an overview of recent progress.
Title: On locally linearized billiard maps
Speaker: Dmitry V. Treschev（Steklov Mathematical Institute & Lomonosov Moscow State University）
Datetime: 2021-05-13 17:20 — 18:20 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：189 342 3534
Password：784002
Abstract
Can a billiard map be locally conjugated to a rigid rotation? The answer to this question is positive in the category of formal series that determine the billiard boundary. We present numerical evidence that for good rotation angles the answer is also positive in analytic category. Numerical results suggest several other conjectures including a multidimensional case.
Title: A survey on a critical Coagulation-Fragmentation equation
Speaker: Hung Vinh Tran（University of Wisconsin Madison）
Datetime: 2021-05-27 11:00 — 12:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：157 645 5237
Password：075354
Abstract
We give a survey on our new method to study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of mass-conserving solutions to the Coagulation-Fragmentation equation. These solve partly some long standing open problems in the field. Based on joint works with Hiroyoshi Mitake (University of Tokyo) and Truong-Son Van (CMU).
Title: Viscosity solutions of the Hamilton-Jacobi equations on non-compact manifolds
Speaker: Albert Fathi（Georgia Institute of Technology）
Datetime:
2021-05-31 14:00 — 16:00 Beijing/Shanghai
2021-06-03 14:00 — 16:00 Beijing/Shanghai
2021-06-07 14:00 — 16:00 Beijing/Shanghai
2021-06-11 14:00 — 16:00 Beijing/Shanghai
2021-06-14 14:00 — 16:00 Beijing/Shanghai
Totally five times, the same meeting room
Venue: Umeet APP
Meeting ID：151 270 4553
Password：495123
Abstract
The purpose of this course is to study the properties of viscosity solutions of the Hamilton-Jacobi equations on non-compact manifolds, in the spirit of what was done for the case of compact manifolds in
Albert Fathi, Weak KAM from a PDE point of view: viscosity solutions of the Hamilton- Jacobi equation and Aubry set, Proc. Roy. Soc. Edinburgh Sect. A, 120 (2012) 1193-1236
We will be mainly 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. We will show how to obtain the Lax-Oleinik formula and the uniqueness result. We will consider the pointwise finiteness of the Lax-Oleinik formula for general initial conditions. results.
We will also discuss the results obtained with Piermarco Cannarsa and Wei Cheng on the topology of the set of singularities of such solutions and give applications to Riemannian Geometry.
Some materials on the background (You can only use the materials for this course, all of them will be deleted after the course):
Slides for the course: Lecture 1 Lecture 2
Title: 旋转p-Laplacian周期特征值的结构
Speaker: 章梅荣（清华大学）
Datetime: 2021-06-01 16:00 — 17:00 Beijing/Shanghai
Venue: Tencent Meeting APP
Meeting ID：666 117 203
Password：123456
Abstract
给定介于1与无穷之间的指标\(p\)，考虑d维欧式空间的1-周期运动，旋转\(p\)-Laplacian的1-周期特征值问题是指周期运动在\(p\)-次势能约束下的\(p\)-次动能的临界点和临界值（特征函数和特征值）。当\(p=2\)时，其特征值与维数无关且是简谐振动的周期特征值。对于一般的\(p\)， Manasevich-Mawhin在20多年前观测到2维空间中有两列比较明显的特征值，并试图说明是否这个问题仅有这两列特征值。在这个报告中，我们将看到，对于任何不为2的\(p\)，该问题一定包含有无穷多列不同的特征值。这一结果离旋转\(p\)-Laplacian的周期特征值的全貌还有很大的差距。我们的构造和证明是基于可积哈密顿系统的动力学分析。
Title: Spatial Propagation of Nonlocal Dispersal Equations
Speaker: 李万同（兰州大学）
Datetime: 2021-06-04 14:00 — 15:00 Beijing/Shanghai
Venue: Tencent Meeting APP
Meeting ID：561 661 015
Password：123456
Abstract
In this talk, I will report the spatial propagation of nonlocal dispersal equations. It consists of five parts. I first will present some relations between local (random) and nonlocal dispersal problems and then I will report our recent results on the spatial propagation (traveling waves and entire solutions) of nonlocal dispersal equations. Part III is concerned with acceleration propagation for nonlocal dispersal systems. Part IV is concerned with free boundary problems on nonocal dispersal equations. In Part IV, I list some problems on nonlocal dispersal equations.
Title: Introduction to deterministic Mean Field Games
Speaker: Piermarco Cannarsa（University of Rome Tor Vergata）
Datetime:
2021-06-15 14:00 — 16:00 Beijing/Shanghai
2021-06-17 14:00 — 16:00 Beijing/Shanghai
2021-06-21 15:00 — 17:00 Beijing/Shanghai
2021-06-24 15:00 — 17:00 Beijing/Shanghai
2021-06-29 15:00 — 17:00 Beijing/Shanghai
Totally five times, the same meeting room
Venue: Umeet APP
Meeting ID：159 311 9665
Password：460996
Abstract
The theory of Mean Field Games (MFG) has been developed in the last two decades by economists, engineers, and mathematicians in order to study decision making in very large populations of “small" interacting agents. The approach by Lasry and Lions leads to a system of nonlinear partial differential equations, the solution of which can be used to approximate the limit of an \(N\)-player Nash equilibrium as \(N\) tends to infinity.
This course will be mainly focused on deterministic models in Euclidean space. These problems are associated with a first order PDE system, which couples a Hamilton-Jacobi equation (depending on the distribution of players) with a continuity equation (driving such a distribution by the optimal feedback provided by the first equation). We will first prove the existence of solutions to the MFG system by a fixed point argument. Then, we will discuss uniqueness issues under some monotonicity condition for the coupling functions. Finally, we will study the long time behavior of solutions following the approach of weak KAM theory.
Prerequisites: basics of optimal control theory, viscosity solutions of Hamilton-Jacobi equations, properties of semiconcave functions.
Title: On simultaneous linearization of certain commuting nearly-integrable diffeomorphisms
Speaker: 陈秦波（KTH Royal Institute of Technology）
Datetime: 2021-09-23 15:00 — 16:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：136 416 6292
Password：199711
Abstract
The question of linearization has been one of the central themes in dynamical systems. This talk discusses two typical smooth nearly-integrable maps of the cylinder which commute with each other. Under suitable conditions (including the symplectomorphisms), we use the KAM iterative scheme for the group action to show that the commuting maps considered in this talk are simultaneously \(C^\infty\) linearizable. As a consequence, we get local rigidity of certain class of \(\mathbb{Z}^2\)-actions generated by commuting twist maps.
Title: Some recent progress on non-collision singularities in Newtonian N-body problems
Speaker: 黄冠（清华大学丘成桐数学科学中心）
Datetime: 2021-11-18 16:00 — 17:00 Beijing/Shanghai
Venue: Tencent Meeting APP
Meeting ID：827 244 037
Password：123456
Abstract
In Newtonian N-body problems, the existence of non-collision singularities, which means there are orbits of the system exhibiting the peculiar feature that some bodies would escape to infinity in finite time with velocities of infinite large without the occurrence of collisions (two or more bodies occupying the same point in the physical space), has been long speculated for \(N\geqslant 4\). This was commonly known as the Painlevé conjecture and has been recently resolved by J. Xue [Acta Math 2020]. In this talk we would review several existing models that allow the existence of non-collision singularities and introduce a new model in a planar 4-body problem. Using the method developed by Xue in [Acta Math 2020], with numeric assistant checking certain non-degeneracy properties of the system, we show the existence of non-collision singularities in this new configuration.