Mathematical Sciences: The Conley Index and Applications to Physical Problems

James Reineck将继续发展康利指数和连接矩阵理论。基本目标是使用拓扑信息来提供关于流的动态信息。康利指数推广了过去研究过的过滤相对同源序列的概念。Reineck打算将这一理论应用于反应扩散方程、化学网络和生态学中的问题。指标论中的问题是庞加莱-霍普夫型定理的证明，不变子空间中集合的指标与整个空间的指标之间关系的研究，以及等变理论的发展。连接矩阵问题包括连接映射和连接集拓扑之间的关系，在莫尔斯分解中描述连接轨道族的映射的发展，以及对连接矩阵中出现的模糊性的更全面的理解。

James Reineck will continue the development of the theory of the Conley index and the connection matrix. The fundamental objective is to use topological information to provide dynamical information about flows. The Conley index generalizes the notion of relative homology sequence of a filtration which has been studied in the past. Reineck intends to apply this theory to problems in reaction-diffusion equations, chemical networks and ecology. The problems in the theory of the index are the proof of a Poincare- Hopf type theorem, the study of the relation between the indices of a set in an invariant subspace and the index of the whole space, and development of an equivariant theory. Connection matrix problems include a relation between the connection map and the topology of the set of connections, the development of a map to describe families of connecting orbits in Morse decompositions and a more complete understanding of the ambiguity which arises in the connection matrix.