Discrete Vector Bundles with Connection and the First Chern Class
1 hour 3 mins,
244.61 MB,
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480x270,
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44100 Hz,
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About this item
| Description: |
Hirani, A
Wednesday 2nd October 2019 - 09:30 to 10:30 |
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| Created: | 2019-10-02 10:46 |
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| Collection: | Geometry, compatibility and structure preservation in computational differential equations |
| Publisher: | Isaac Newton Institute |
| Copyright: | Hirani, A |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
|
| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| 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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