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

1927年，数学家G·D·伯克霍夫首次提出了对粒子在凸域内无摩擦、弹性反射到壁上的运动的分析。这个数学动力系统很好地近似于几个重要的物理系统，例如，声音在礼拜堂或大教堂内的传播。这个研究项目针对这样的动力系统的经典逆问题，例如：如果我们知道粒子运动的一些性质，我们能推断出潜在区域的性质吗？这与“你能听到鼓的形状吗？”这个问题密切相关。也就是说，仅仅从鼓发出的声音就能重建鼓的形状吗？结果表明，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。这个项目的目的是加深对动力系统分析中相关数学问题的理解。项目的第一部分是研究凸台球的扰动，并推导出凸域变形的可积性(椭圆扰动)和等谱的必要条件。前者与暗廊现象和关于可积台球刻画的Birkhoff猜想密切相关，而后者与形变谱刚性密切相关。M.Kac的问题可以用凸平面区域内Laplace方程的Dirichlet问题的谱来表示。由于波迹函数，拉普拉斯谱一般确定同一区域内相关台球的长度谱，并将分析与台球联系起来。项目的第二部分是关于证明随机Arnold扩散，即近可积系统的随机扩散行为和Chirikov猜想。具有挑战性的目标之一是建立三体问题柯克伍德缝隙内的随机扩散行为。

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.