[Topology.beta] Online talk by Clemens Bannwart

[Rot, T.O. (Thomas)]

Dear all,

Tomorrow at 11:00 we have a remote seminar in the CTAAAG. The topic might be of interest to some dynamicists, which is why I’m crossposting.

https://vu-live.zoom.us/j/96375042161?pwd=TGENLNSw7D3Ff2sb9YSaiDSjHNr6k2.1

Meeting ID: 963 7504 2161
Passcode: 541276

We will set up a viewing room in Maryam.

Best,
Thomas

Speaker: Clemens Bannwart

Title: Chain complexes as a topological summary of Morse-Smale vector fields

Abstract: The Morse complex is a chain complex that can be assigned to a gradient-like Morse-Smale vector field on a smooth manifold. It consists of vector spaces generated by the fixed points of the vector field and the differential is defined by counting the flow lines between points of adjacent indices. The homology of this chain complex is isomorphic to the singular homology of the manifold, while the acyclic parts describe the topology of the vector field. In this talk we explore how we can go beyond the gradient-like case. After recalling the relevant definitions, we present a method by Franks to replace a closed orbit by a pair of fixed points and explain why there are different, non-equivalent ways to follow this procedure. Then, we present a pipeline to construct a chain complex also in the presence of closed orbits. To this end, we consider a filtration of the manifold by unstable manifolds first described by Smale and the resulting spectral sequence in Čech homology. The first page of this spectral sequence can be endowed with canonical bases, where every fixed point corresponds to one basis element and every closed orbit corresponds to two basis elements. We show how the algebraic information of this spectral sequence can be rearranged into a chain complex whose homology is isomorphic to the singular homology of the underlying manifold. If time permits, we go back to the gradient-like case and describe a method to decompose the Morse complex in order to obtain an invariant which is related to the persistence barcode. This is joint work with Claudia Landi.


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