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

这一建议与数学物理中出现的一些偏微分方程的动力学有关。在有限维情形下，关于加速收敛方法、KAM理论、中心流形和规范形的问题是复杂的。在无限维度中，存在着额外的复杂性，例如零因子的出现，而不仅仅是小的因子。动力系统是一种几何严密的方法，用来描述无法显式求解的常微分方程组或偏微分方程组的解的行为。准确描述此类解决方案的行为的能力对于理解我们周围的世界具有重要的意义。

This proposal is concerned with the dynamics of some partial differential equations that arise in mathematical physics. Problems concerning the method of accelerated convergence, KAM theory, center manifolds, and normal forms are complicated in the finite dimensional case. In infinite dimensions there are additional complications such as the appearance of zero divisors rather that just small divisors. Dynamical systems is a geometric rigorous way of describing the behavior of solutions of systems of ordinary or partial differential equations which can not be solved explicitly. The ability to accurately describe the behavior of such solutions has important ramifications in understanding the world around us.