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On discrete gradient vector fields and Laplacians of simplicial complexes

Andrew Tawfeek, Amherst College
Wednesday, June 2, 2021 - 3:30pm to 5:00pm
via Zoom

Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.

Join Zoom Meeting:
Meeting ID: 915 4733 5974

Discrete Morse theory, a cell complex-analog to smooth Morse theory, has been developed over the past few decades since its original formulation by Robin Forman in 1998, bringing along a wide range of applications. In particular, discrete gradient vector fields on simplicial complexes capture important features of discrete Morse functions. We prove that the characteristic polynomials of the Laplacian matrices of a simplicial complex are generating functions for discrete gradient vector fields when the complex is a triangulation of an orientable manifold. Furthermore, we provide a full characterization of the correspondence between rooted forests in higher dimensions and discrete gradient vector fields. This is based on joint work with Ivan Contreras and Alejandro Morales.