Applied math seminar: Anil Hirani (UIUC)

December 12, 2023 @ 3:30 pm – 4:30 pm
Keller 302

Title: Wedge Product and Naturality in Discrete Exterior Calculus

Abstract: In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and the antisymmetrized cup product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson’s cochain product defined using Whitney and de Rham maps. This may lead to existence of more accurate combinatorial wedge products that are associative in the limit. In any case, the combinatorial product may be useful in creating an A-infinity or C-infinity algebra. Joint work with Mark Schubel and Daniel Berwick-Evans.