Long Range Behavior in Dynamical Systems and Partial Differential Equations

我们想研究动力系统和偏微分方程中的几个项目。我们还将考虑与两者共享属性的多粒子系统/一个统一的线索是，我们想要理解变分方法和更多几何方法之间的关系。特别是，我们想用几何和变分方法来证明扩散，并将变分方法得到的动力学结果推广到偏微分方程。通常双曲流形理论是几何方法的一个重要工具，我们想把它推广到偏微分方程和无限粒子系统。有时，短期的小原因可能会产生长期的大影响。例如，小的周期性力可能引起能量的大变化。材料局部性质的一些微小变化可能导致覆盖大距离的图案的出现。然而，在其他情况下，局部效应只是平均的。我们想要建立一个广泛的方法阵列（包括数值研究和几何技术），可以用来决定是建立还是平均发生。我们还想特别注意一些在技术应用中出现的具体模型。

We want to investigate several projects in Dynamical Systems and in Partial Differential equations. We will also consider multi-particle systems that share properties with both/One unifying thread is that we would like to understand the relation between variational approaches and more geometric ones. In particular, we would like to produce proofs of diffusion that use geometric and variational methods and to extend results in dynamics obtained by variational methods to partial differential equations. An important tool for geometric methods is the theory of normally hyperbolic manifolds and we would like to extend it to Partial differentialequations and infinite particle systems.Sometimes, small short range causes may have large long term effects. For example, small periodic forces may build up large changes in energy. Some small changes in the local properties of a material may lead to the emergence of patterns that cover large distances. In other situations, however, local effects just average out.We would like to device a broad based array of methods (including numerical studies and geometric techniques)that can be used to decide whether build up or averaging occurs. We would also like to pay special attention tosome concrete models appearing in technological applications.