Mathematical Sciences: Dynamical Systems with Infinitely Many Degrees of Freedom in Mathematical Physics

该项目将通过Kolmogorov-Arnold-Moser理论、不变流形定理和Liapunov指数等技术研究数学物理中出现的具有无限多个自由度的动力系统。首先，该项目将重点关注Kolmogorov-Arnold-Moser理论等技术与加速收敛方法之间的关系，这些方法用于研究具有随机或准周期势的薛定谔算子的局部态；并试图利用这些关系来证明偏微分方程中周期轨道的存在性。其次，它将研究如何将中心流形定理等思想与哈密顿结构相结合，以研究像Boussinesq方程和其他与流体动力学相关的方程这样的偏微分方程。最后，它将研究统计力学中出现的无限维动力系统的李雅普诺夫指数等量的存在性。对这些量的更好的理解将使我们得到平衡统计力学中热力学极限的动力学类比。

This project will study dynamical systems with infinitely many degrees of freedom which arise in mathematical physics through techniques such as the Kolmogorov-Arnold-Moser theory, invariant manifold theorems, and Liapunov exponents. First, the project will focus on the relationship between techniques like the Kolmogorov-Arnold-Moser theory, and the accelerated convergence methods developed to study localized states in Schrodinger operators with random or quasi-periodic potentials, and will attempt to exploit these relationships to prove the existence of periodic orbits in partial differential equations. Second, it will investigate how ideas like center manifold theorem can be combined with a Hamiltonian structure to study partial differential equations like the Boussinesq equation and other equations related to hydrodynamics. Finally, it will study the existence of quantities like the Liapunov exponents for infinite dimensional dynamical systems arising in statistical mechanics. A better understanding of such quantities would give one a dynamical analogue of the thermodynamic limit in equilibrium statistical mechanics.