Mathematical Sciences: The Conley Index and Applications to Physical Problems

詹姆斯·赖内克将继续发展 Conley指数和连接矩阵。根本 目标是利用拓扑信息提供动态的 关于流量的信息。康利指数概括了 相对同源序列的过滤， 过去研究过。 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。的 指数理论中的问题是庞加莱定理的证明- Hopf型定理，指标之间关系的研究 在一个不变子空间中的集合和整体的索引 空间，和发展的一个等变理论。连接 矩阵问题包括连接映射和 一组连接的拓扑结构， 描述莫尔斯分解中的连接轨道族 更全面地理解 在连接矩阵中。

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.