Mathematical Sciences: Conley Index Theory and Its Applications

研究者正在开发研究孤立不变集结构的方法，这些方法适用于各种各样的问题，例如，反应扩散系统的行波，动态相变，磁流体动力激波，延迟方程，仅举几例。更具体地说，它涉及推导描述不变量集的代数不变量，确定它们如何通过分岔变化，以及确定代数不变量在不变量集上施加哪些几何拓扑结构。这些方法背后的指导思想来自这样一个事实，即许多作为生物、化学或物理系统模型的微分方程都有以下三个特征：(1)它们是从复杂现象的理想化描述中推导出来的，(2)它们是参数化系统，参数的确切值是未知的，(3)它们是非线性的。因此，从应用的角度来看，描述这些微分方程解的全局和鲁棒结构是非常重要的。同时，众所周知，非线性系统会经历极其复杂的分岔（例如混沌动力学的出现）。因此，任何一般理论都必须依赖于对不变集的精细结构漠不关心的工具，例如康利指标理论。

The investigator is developing methods for studying the structure of isolated invariant sets that are applicable to a wide variety of problems, e.g., travelling waves for reaction-diffusion systems, dynamic phase transition, magnetohydrodynamic shock waves, delay equations, to name a few. More specifically, it is concerned with deriving algebraic invariants that describe invariant sets, determining how they can change through bifurcations, and determining which geometric topological structures are imposed on the invariant sets by the algebraic invariants. The guiding philosophy behind these methods comes from the fact that many differential equations that arise as models of biological, chemical, or physical systems share the following three characteristics: (1) they are derived from idealized descriptions of complicated phenomena, (2) they are parameterized systems and exact values of the parameters are not known, (3) they are nonlinear. Thus, from the point of view of applications, describing the global and robust structure of the solutions to these differential equations is of great importance. At the same time, it is well known that nonlinear systems undergo tremendously complicated bifurcations (e.g., the appearance of chaotic dynamics). Therefore, any general theory must rely on tools that are indifferent to the fine structure of the invariant sets, e.g. the Conley index theory.