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Mathematics/CS Colloquium Series Winter 2011
Title: Discrete Volume Computations for Polytopes: An Invitation to Ehrhart Theory
Speaker: Matthias Beck, San Francisco State University
22nd February 2011, Tuesday
Abstract: Our goal is to compute the volume of certain easy (and fun!) geometric objects, called polytopes, which are fundamental in many areas of mathematics. Although polytopes have an easy description, e.g., using a linear system of equalities and inequalities, volume computation is hard even for these basic objects. Our approach is to compute the discrete volume of a polytope P, namely, the number of grid points that lie inside P, given a fixed grid in Euclidean space such as the set of all integer points. A theory initiated by Ehrhart implies that the discrete volume of a polytope has some remarkable properties. We will exemplify Ehrhart theory with the help of several families of polytopes whose discrete volumes are connected with some of our friends in various mathematical areas, such as binomial coefficients, Eulerian, Stirling, and Bernoulli numbers.
This talk will be accessible to anybody who has finished the basic calculus and linear algebra courses. In particular, we will not assume that the audience knows the terms mentioned in this abstract, such as the concept of a polytope.