Discrete Vector Bundles with Connection and Čech Cohomology

Anil Hirani (U Illinois)

May 21. 2026, 10:45 — 11:30

Since the initial introduction of our framework for discrete vector bundles with connection in 2021, the theory has matured significantly, yielding a clearer overall picture and opening up new applications. In this talk, I will describe the modernized formulation of this theory, which leverages ordered simplicial complexes, subdivision functors, and coarsening operators, alongside a localized perspective using connection 1-forms. One consequence of this setup is an application to Čech cohomology. The vanishing of curvature transforms the sequence of discrete vector bundle-valued cochains into a cochain complex. We identify the cohomology of this complex with the cohomology of a specific local coefficient system, constructed from a discrete vector bundle equipped with a flat discrete connection. By utilizing the open stars of vertices as a good cover, we demonstrate that this discrete complex coincides with the Čech complex for the sheaf cohomology associated with that cover. Joint work with Daniel Berwick-Evans and Mark Schubel.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Differential Complexes: Theory, Discretization, and Applications (Thematic Programme)
Organizer(s):
Andreas Cap (U of Vienna)
Ralf Hiptmair (ETH Zurich)
Kaibo Hu (U of Oxford)
Joachim Schöberl (TU Wien)