Date: Monday, March 25, 2013
Location: 3096 East Hall (3:00 PM to 4:00 PM)
Title: How to compute hyperbolic volume
Abstract: From the Gauss-Bonet theorem, we know that the area of a hyperbolic triangle is determined completely by its three angles. In particular the area of a triangle with angles a, b and c is pi-a-b-c. If the triangle is ideal (i.e. all angles are 0) then the triangle has (maximal) area pi. Given a simplex of higher dimension, how do we compute its volume? What is the maximal volume of a hyperbolic n-simplex? It turns out that maximal volume is achieved by the regular ideal simplex (much as in the n=2 case). The upshot of this result is that many manifolds have rather accessible triangulations in terms of ideal simplices. Knowing the volumes of ideal simplices will allow us to calculate the volume of a given manifold. I will attempt to leave the "computation" in this talk to a minimum (and to computers), relying on diagrams and pictures for most of the arguments. I will also provide some classic examples on the computer using the geometry/topology program Snap Pea.
Speaker: David Renardy
Institution: UM
Event Organizer: Tengren Zhang tengren@umich.edu
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