Program
 Dynamics seminar Student No.： 20 Time： Fri 13:30-15:05 Instructor： Xue Jinxin, Huang Guan, Wang Lin Place： Conference Room 2, Jin Chun Yuan West Bldg. Starting Date： 2018-3-8 Ending Date： 2018-6-14

2018-5-31

Speaker: 吴云辉 Wu Yunhui [YMSC, Tsinghua University]

Title: Iteration of mapping classes and limits of Weil-Petersson geodesics

Abstract: Let $S=S_{g}$ be a closed surface of genus $g$ with $g\geq 2$, $\Mod(S)$ be the mapping class group of $S$ and $\Teich(S)$ be the Teichmuller space of $S$ endowed with the Weil-Petersson metric. Fix $X,Y\in \Teich(S)$. In this paper, we show that for any $\phi \in \Mod(S)$, there exists a positive integer $k$ only depending on $\phi$ such that the sequence of the directions of the geodesics connecting $X$ and $\phi^{kn}\circ{Y}$ is convergent in the visual sphere of $X$ as $n$ goes to infinity. Moreover, we will give geometric descriptions for these limit geodesics.

2018-5-25

Speaker: 杨磊Yang Lei [四川大学Sichuan University]

Abstract: In this talk, we will introduce the notion of algebraically badly approximable vectors which is a natural generalization of badly approximable vectors. We will prove that the set has full Hausdorff dimension. The problem is closely related to the study of badly approximable matrices. We will introduce the Schmidt's problem and Davenport's problem for badly approximable matrices and report our recent progress in solving these problems. This is a joint work with Beresnevich and Velani.

2018-5-17

Speaker: Diogo Gomes [King Abdullah University of Science and Technology]

Title: Mean-field game price formation models

Abstract: Here, we introduce a price-formation model for electricity markets where a large number of small players can store and trade electricity. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition. Under mild conditions, we prove the uniqueness of the solution. Moreover, we establish several estimates for the solutions. Finally, we examine model with finitely many agents and linear-quadratic models that have explicit solutions.

2018-5-10

Title: Unstable elliptic fixed points in Hamiltonian dynamics

Abstract: We introduce a new diffusion mechanism  from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom. Using this mechanism, we obtain the first examples of real analytic Hamiltonians that have a Lyapunov unstable non-resonant elliptic equilibrium.

In the smooth category, and using different techniques, we give the first examples of symplectic diffeomorphisms having a non-resonant elliptic fixed point that attracts an orbit.

2018-4-19

Speaker: 杨佳刚 Yang Jiagang [IMPA, Brasil]

Title: Lyapunov exponents and rigidity of Anosov automorphism and skew products

Abstract: We obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism L with simple real spectrum, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to L.

We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism f0 over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation f of f_0 with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.

2018-4-12

Speaker: 陈秦波 Qinbo Chen [中国科学院晨兴数学中心 Morningside Center of Mathematics, CAS]

Title: Vanishing contact structure problems for contact Hamiltonian systems

Abstract: In this talk, we will clarify some connections between Herglotz’s generalized variational principle and Hamilton-Jacobi equations of contact type. As applications, we will discuss the vanishing contact structure problems which can be regarded as a generalization of the vanishing discount problem in PDE. This is a joint work with Wei Cheng.

2018-3-29

Speaker: Aaaron Brown [University of Chicago]

Title: Groups acting on manifolds: orbit closures and invariant measures

Abstract: Given an action of a discrete group on a manifolds we ask if we can classify all orbit closures and invariant (or stationary) measures. For actions of Z, this is typically impossible. However, recently it has been observed that for certain examples of actions (including affine actions on tori and homogeneous spaces and P-actions on Moduli space) and for actions satisfying certain dynamical hypotheses (including "expanding actions" on surfaces) all orbit closures are "nice" submanifolds and all measures are natural volumes on such manifolds.

I will give an overview of the above setting and state some known results, including my results with F. Rodriguez Hertz on actions on surfaces. I will then give an overview of some work in progress (in very early stages) with Eskin, Rodriguez Hertz, and Filip on general actions in higher-dimensions.

2018-3-22

Speaker: 龚文敏 Gong Wenmin [清华大学 Tsinghua University]

Title: Floer homology for Hamilitonian periodic orbits

Abstract: The existence problem for periodic orbits has a rich history. One of the most prominent cornerstons was Floer's proof in the Arnold conjecture for a certain class of symplectic manifolds by introducing Floer homology theory. In this talk, we will discuss some recent progress about Hamilitonian periodic orbits by Floer theoretical methods. Applications include non-contractible periodic orbits in twisted contangent bundles, Conley conjecture and extending a theorem of Le Calvez and Yoccoz and Franks to higher dimensions.

2018-3-15

Speaker: Patrice Le Calvez  [Institut de Mathematiques de Jussieu, France]

Title: Orbit forcing theory for surface homeomorphisms

Abstract: In a join work with Fabio Tal (Sao Paulo university), we introduce a novel theory of orbit forcing for surface homeomorphims using maximal isotopies and transverse foliations. We get a simple criteria implying the existence of a topological horseshoe. Some application can be deduced.

2018-3-8

Speaker: 章梅荣 Zhang Meirong [清华大学 Tsinghua University]

Title: The completely continuous dependence on the microscopic quantities of the solutions to differential equations and the macroscopic quantities

Abstract: Differential equations in applied math often has finite dimensional parameters as well as infinite dimensional quantities reflecting the macroscopic states such as the density and potential, etc. In general, besides the solutions to the differential equations, people care about all kinds of macroscopic quantities defined through the solutions. In this talk, we will explain using some prototypical problems the completely continuous dependence on the microscopic quantities of the solutions to differential equations and the macroscopic quantities. Namely, the solutions and the macroscopic quantities vary continuously if the parameters vary continuously in the weak topology. Based on this continuity, we will use the variational approach to obtain some optimal upper and lower bounds results of the macroscopic quantities.