Geometry & Topology

Event Information Generalizations of the Gram relations for polyhedra, volumes of spherical polytopes, and their cone theta functions
16:10 on Monday June 06, 2016
17:00 on Monday June 06, 2016
BA6183, Bahen Center, 40 St. George St.
Sinai Robins
https://sites.google.com/site/sinairobins/home
ICERM and Sao Paulo

How do we extend the 2 -dim'l familiar fact that "the sum of the angles of a triangle equals \pi " , to hight dim'l polyhedra? We begin with a natural discretization of the volume of a spherical polytope, and define certain natural analogues of theta functions, called cone theta functions. These meromorphic functions help us analyze the structure of spherical polytopes. It turns out that we can obtain good asymptotics of cone theta functions, attached to any simplicial polyhedral cone, near a rational 'cusp', and we use these asymptotics to give new extensions of the Gram relations for the solid angles of faces of a simple rational polytope. These new relations involve 'Gauss' sums over parallelepipeds, and we will introduce all objects from 'almost first principles', including lots of pictures.