Mathematical Sciences: The Conley Index and Applications to Physical Problems
James Reineck Principal Investigator
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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.