The Birkhoff Conjecture, Spectral Rigidity for Convex Reflecting Particle Systems, and Stochastic Arnold Diffusion

数学家G.D.提出了一种分析在凸域内无摩擦运动并在壁面上弹性反射的粒子运动的方法。1927年的伯克霍夫 这个数学动力学系统可以很好地近似几个重要的物理系统，例如，教堂或大教堂内的声音传播。 这个研究项目的目标是这样的动力系统的经典逆问题，例如：如果我们知道粒子的运动的一些属性，我们可以推断出潜在的域的属性？ 这与一个问题密切相关，“你能听到鼓的形状吗？“也就是说，人们可以仅仅从它产生的声音重建鼓的形状吗？ 事实证明，凸域内声音的性质与同一域内相关反射粒子的可能轨迹的性质密切相关。本计画旨在加深对动力系统分析中相关数学问题的了解，第一部份为研究凸台球的扰动，并推导凸区域的可积性（椭圆扰动）与变形等谱性的必要条件。 前一个条件与回音壁现象和关于可积台球特征的Birkhoff猜想密切相关，而后一个问题与形变谱刚性密切相关。 M. Kac问题可以表示为凸平面区域内的拉普拉斯方程的Dirichlet问题的谱。 由于波迹函数，拉普拉斯谱一般决定了同一域内相关台球的长度谱，并将分析与台球联系起来。 第二部分是证明随机Arnold扩散，即近似可积系统的随机扩散行为和Chirikov猜想。 建立柯克伍德间隙内的随机扩散行为是三体问题的一个具有挑战性的目标。

Analysis of the motion of a particle moving inside of a convex domain without friction and reflecting elastically against the walls was initiated by mathematician G.D. Birkhoff in 1927. This mathematical dynamical system serves as a good approximation to several important physical systems, for instance, sound propagation inside of a chapel or a cathedral. This research project targets classical inverse problems for such dynamical systems, for example: if we know some properties of the particle's motion, can we infer properties of the underlying domain? This is closely related to the question, "Can you hear the shape of a drum?" That is, can one reconstruct the shape of a drum solely from the sound it produces? It turns out that properties of sound inside of a convex domain are closely related to properties of the possible trajectories of the associated reflecting particles inside of the same domain. This project aims to deepen understanding of related mathematical questions in the analysis of dynamical systems.The first part of the project is to study perturbations of convex billiards and to derive necessary conditions for integrability (for perturbation of ellipses) and for isospectrality of deformations of convex domains. The former condition is closely related to the phenomenon of whispering galleries and the Birkhoff conjecture about characterization of integrable billiards, while the latter problem is closely related to deformation spectral rigidity. M. Kac's question can be expressed in terms of the spectrum of the Dirichlet problem for the Laplace equation inside of a convex planar domain. Due to the wave trace function the Laplace spectrum generically determines the length spectrum of the associated billiard inside of the same domain and connects analysis with billiards. The second part of the project is about proving stochastic Arnold diffusion, that is, stochastic diffusing behavior for nearly integrable systems and the Chirikov conjecture. One of challenging goals is to establish stochastic diffusive behavior inside of Kirkwood gaps for the three-body problem.