** **

**2016 Seminar on Hamiltonian Dynamics**

Our seminar on Hamiltonian dynamics consists of a series of courses and several lectures. It is based on latest developments in dynamical instability of Hamiltonian systems, viscosity solutions of Hamilton-Jacobi equations and other related fields located at the intersection of dynamical system and calculus of variation. The seminar is hosted by Prof. Cheng Chongqing from Nanjing University and will last from January 7, 2016 to January 30, 2016.

**About the courses**

Our courses, given by Prof. Cheng, will discuss some of the complete proof of existence and genericity of Arnold diffusion in nearly integrable positive-definite Hamiltonian systems with arbitrary degrees of freedom.

**Time:** 15:00-17:00, every working day from January 7, 2016 to January 22, 2016.

**Organization:**

1. We will discuss some preliminaries about Hamiltonian systems, differential forms and symplectic topology, define the Tonelli Lagrangian systems and prove the equivalence of Lagrangian and Hamiltonian formulation of a system in this case.

2. We will state and prove Weierstrass Lemma and Tonelli Theorem, which form the foundation of the so-called Aubry-Mather theory.

3. We will construct minimal measures, introduce Mather’s α and β function. Then we will prove some basic properties of these functions, discuss the relationship between the differentiability of these functions and dynamics in low dimensional case. Finally, we will compute the α function of a concrete example.

4. We will define static and semi-static curves and several minimal sets. Then we will prove Mather’s Lipschitz graph theorem and the symplectic invariance of Aubry and Mane sets. Finally, we will review some results in monotone twist maps from the above theorems.

5. We will introduce weak KAM solutions of Hamilton-Jacobi equation, prove its existence by using Lax-Oleinik operator. We will define calibrated curves and prove some properties of weak KAM solution concerning its dynamical meaning. Finally, we will establish the semi-concavity (semi-convexity) of backward (forward) weak KAM solutions and discuss the dynamical meaning of their generalized gradients.

6. We will introduce the elementary weak KAM for one static class and barrier function for two different static classes (both of them are special weak KAM solutions). We will show any weak KAM could be represented by finite many elementary weak KAM in generic cases, discuss the dynamical meaning of minimal points of barrier functions.

7. We will define the viscosity solutions for Hamilton-Jacobi equations and prove the equivalence of viscosity solutions and weak KAM solutions in the Tonelli case. We will present a classical comparison theorem for viscosity solutions of Hamilton-Jacobi equations of certain type that will be used later.

8. We will discuss the upper semi-continuity of global minimizing curves and semi-static curves w.r.t. Lagrangians for time-periodic cases. We will construct local connecting orbits of type-c for time-periodic Tonelli Lagrangian system by using the topological sparsity of Mane sets.

9. We will discuss the dynamical transitivity of static classes if there are only finite of them. We will construct local connecting orbits of type-h for time-periodic Tonelli Lagrangian system by using the topological sparsity of minimal points of barrier functions.

10. We will present the corresponding formulation of the construction of type-c connecting orbits in the autonomous case.

11. We will present the corresponding formulation of the construction of type-h connecting orbits in the autonomous case.

12. We will discuss the locally minimal property of local connecting orbits of type-h we constructed in previous courses in autonomous and time-periodic cases.

**Notation: **All contents of these courses come from a book that the speaker is writing and anyone attend could ask for a copyright.

**About the lectures**

Our lectures concentrate on recent developments in Aubry-Mather theory, weak KAM theory and viscosity solutions for different types of systems. These lectures are given by three speakers: Chengwei, Cuixiaojun from Nanjing University and Yanjun from Fudan University.

**Time:** TBA

**Organization:**

1. Prof. Cheng Wei will present some new results on the global propagation of singularities of weak KAM solutions for Tonelli Lagrangian systems on compact manifolds or Euclidean spaces.

2. Prof. Yanjun will present some new results on weak KAM theory of nonlinear first order PDEs of Tonelli type (i.e. H(x, u, p)=0, H is convex and super-linear for p, p=dxu) and Aubry-Mather theory for the corresponding contact systems.

3. Prof. Cuixiaojun will present some new results on the application of Aubry-Mather theory to geometry: calibrations and laminations of higher dimensions. This is a joint work with V. Bangert; global viscosity solutions to the eikonal equations in the setting of Riemannian and Lorentzian geometry. This is a joint work with L. Jin.

No. 55, Zhongguancun East Road, Haidian District, Beijing 100190

Phone: 86-10-82541165 Email: mcmadmin@math.ac.cn