Applied Topology Seminar

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Past events in this series
21 May 2021
15:00
Wan Zhengchao
Abstract

In this work we present a thorough study of the theoretical properties and devise efficient algorithms for the persistent Laplacian, an extension of the standard combinatorial Laplacian to the setting of simplicial pairs: pairs of simplicial complexes related by an inclusion, which was recently introduced by Wang, Nguyen, and Wei. 

In analogy with the non-persistent case, we establish that the nullity of the q-th persistent Laplacian equals the q-th persistent Betti number of any given simplicial pair which provides an interesting connection between spectral graph theory and TDA. 

We further exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix. This relation permits us to uncover a link with the notion of effective resistance from network circuit theory and leads to a persistent version of the Cheeger inequality.

This relationship also leads to a novel and fundamentally different algorithm for computing the persistent Betti number for a pair of simplicial complexes which can be significantly more efficient than standard algorithms. 

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  • Applied Topology Seminar
28 May 2021
15:00
Abstract

 In this work we systematically introduce relations to topological data analysis (TDA) in the categories of sets, simplicial complexes and vector spaces to characterize and study the general dynamical behaviors in a consistent way. The proposed framework not only offers new insights to the classical TDA methodologies, but also motivates new approaches to interesting applications of TDA in dynamical metric spaces, dynamical coverings, etc. The associated algorithm which produces barcode invariants, and relations in more general categories will also be discussed.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Applied Topology Seminar
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