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INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH (IISER) PUNE
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An Autonomous Institution, Ministry of Human Resource Development, Govt. of India
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Seminars and Colloquia

Mathematics

Discretizations of Exterior Calculus and Differential Geometry 
 
Tue, Jan 07, 2020,   04:00 PM to 05:00 PM at Madhava Hall

Prof. Anil Nirmal Hirani
University of Illinois Urbana Champaign

Exterior calculus on manifolds generalizes vector calculus to smooth manifolds. Real-valued differential forms are the main objects. Vector bundles with connections are a further generalization, with vector bundle valued forms as the main objects. Exterior calculus is suitable for most partial differential equations of engineering physics and vector bundles with connections are suitable for differential geometry and most differential equations of mathematical physics. In computations, manifolds can be approximated by piecewise linear simplicial approximations. I will describe a discrete combinatorial theory of discrete exterior calculus (DEC) and its recent generalization to discrete vector bundles with connections. The main operator we develop is a discrete covariant exterior derivative that generalizes the discrete exterior calculus operator of DEC and yields a discrete curvature and a discrete Bianchi identity. 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. DEC is joint work with several collaborators over the years. The newer work on discrete vector bundles with connection is joint work with Daniel Berwick-Evans and Mark Schubel.
 

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