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Discrete volumes of polytopes, and Euler-Maclaurin summation formulae with solid angle weights

Algebra & Discrete Mathematics

Speaker: Sinai Robins, University of Sao Paolo
Location: 2112 MSB
Start time: Wed, Mar 13 2019, 12:10PM

There are many types of discrete volumes for polytopes, the most famous being the Ehrhart quasi-polynomial of a rational polytope. We focus here on another discrete volume, called the solid-angle quasi-polynomial of a rational polytope, and give new formulas for all of its coefficients, in terms of certain new number-theoretic functions and series, which arise naturally from the Poisson summation formula. Next, we see how to extend these ideas to sum any function over the lattice points in a polytope, but with the same solid-angle weights as above. This perspective gives a new Euler-Maclaurin summation formula, with solid angle weights for the lattice points.

Much of the work described here is joint with Ricardo Diaz, Nhat Le-Quang, Jamie Pommersheim, and Nick Salter.