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An algebraic-topological approach to dynamical systems and symplectic topology

Abstract

In this project, we propose scientific problems related to the study of dynamical systems via Conley Index Theory and to the study ofFloer Homology via Microlocal Sheaf Theory. In the dynamical setting, we want to explore the dynamical effects of spectral sequence analysis for Morse-Smale flows in order to detect cancellations and reductions of periodic orbits. For a more general flow, with no restrictions on the chain recurrent set, we propose a generalization of the classical Connection Matrix Theory in order to obtain stronger results related to the connections within the flow, as well as, a generalization of the Novikov chain complex to encompass dynamics with richer invariant sets. In the symplectic topology side, we propose to contribute with the confirmation of Kontsevich's conjecture by giving new examples and proving it for 4-dimensional Weinstein manifolds, i.e. the wrapped Fukaya category of them and wrapped microlocal sheaves on their skeleta match. In particular, we want to reformulate Heegard Floer homology using microlocal sheaves. (AU)

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