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Discrete Vector Bundles with Connection and the First Chern Class

Presented by: 
Anil Hirani
Date: 
Wednesday 2nd October 2019 - 09:30 to 10:30
Venue: 
INI Seminar Room 1
Abstract: 
The use of differential forms in general relativity requires ingredients like the covariant exterior derivative and curvature. One potential approach to numerical relativity would require discretizations of these ingredients. I will describe a discrete combinatorial theory of vector bundles with connections. The main operator we develop is a discrete covariant exterior derivative that generalizes the coboundary operator and yields a discrete curvature and a discrete Bianchi identity. We test this theory by defining a discrete first Chern class, a topological invariant of vector bundles. This discrete theory is built by generalizing discrete exterior calculus (DEC) which is a discretization of exterior calculus on manifolds for real-valued differential forms. In the first part of the talk I will describe DEC and its applications to the Hodge-Laplace problem and Navier-Stokes equations on surfaces, and then I will develop the discrete covariant exterior derivative and its implications. This is joint work with Daniel Berwick-Evans and Mark Schubel.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons