Discrete Vector Bundles with Connection and the First Chern Class
1 hour 3 mins,
930.31 MB,
MPEG-4 Video
640x360,
30.0 fps,
44100 Hz,
1.96 Mbits/sec
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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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MPEG-4 Video * | 640x360 | 1.96 Mbits/sec | 930.31 MB | View | Download | |
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MP3 | 44100 Hz | 249.89 kbits/sec | 117.14 MB | Listen | Download | |
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