Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Ehrhart positivity and McMullen's formula

Seminar | November 27 | 3:45-4:45 p.m. | 939 Evans Hall

 Fu Liu, UC Davis

 Department of Mathematics

The Ehrhart polynomial counts the number of integral points inside dilations of an integral polytope, that is, a polytope whose vertices are integral points. We say a polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients. In the literature, different families of polytopes have been shown to be Ehrhart positive using different techniques. We will survey these results in the first part of the talk, after giving a brief introduction to polytopes and Ehrhart polynomials. Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients relating to the normalized volumes of faces. In the second part of the talk, I will discuss joint work with Castillo in which we try to make this relation more explicit in the case of regular permutohedra. The motivation is to prove Ehrhart positivity for generalized permutohedra. This turns out to be related to formulas for Todd classes of a certain family of toric varieties.