9:00–-10：00 薛金鑫（清华大学）
题目： Closing lemma and KAM normal form
摘要：We develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original setting of Poincar\'e and confirm several old and new conjectures with suitable formulations. First, for Poincar\'e's original setting of nearly integrable systems, we prove that typically periodic orbits are asymptotically dense as the size of perturbation tends to zero. Second, we prove that typical smooth perturbation of the geodesic flow on the flat torus has asymptotically dense periodic orbits, which partially solves an open problem since Pugh-Robinson's $C^1$-closing lemma. Third, we prove that for typical Hamiltonian or contact perturbation of the geodesic flows of the ellipsoid has asymptotically dense orbit on the energy level, which greatly enhances the recent researches on strong closing lemma, and also confirms partially a conjecture of Fish-Hofer in this setting. We also show how the problem of closing lemma is related to many-body localization.
10:00—-11:00 陈秦波 （南京大学）
题目：On local rigidity of certain non-ergodic partially hyperbolic actions
摘要：For a $\mathbb{Z}^k$, $k\geq 2$ smooth action where each (non-trivial) element is an isometric toral extension of an ergodic toral automorphism, we will establish the smooth local rigidity under a topological condition. This condition is necessary and sufficient. Our proof uses a KAM type iterative scheme. This talk is based on joint work with D. Damjanovic.
14:00—-15:00 王林 （北京理工大学）
题目：Quantitative destruction of invariant circles
摘要：For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency $\omega$ of an integrable system by a trigonometric polynomial of degree $N$ perturbation $R_N$ with $\|R_N\|_{C^r}<\epsilon$. We obtain a relation among $N$, $r$, $\epsilon$ and the arithmetic property of $\omega$, for which the area-preserving map admit no invariant circles with $\omega$.
15:00—-16:00 苏喜锋 （北京师范大学）
题目：KAM approach for FK type models and related topics
摘要：We consider several models of Frenkel-Kontorova type in both Classical and Quantum mechanics. According to the arithmetic and analytical properties of the rotation numbers and potential respectively, we review and develop different arguments to show the existence of the associated quasi(almost)-periodic equilibria using Nash-Moser iterative method.
