Math and Stats colloquium: On Ehrhart positivity

Friday, Oct. 13, 2023, 4–5 p.m.

Speaker: Fu Liu

Title: On Ehrhart positivity

Abstract: The Ehrhart polynomial counts the number of lattice points inside dilations of an integral polytope, that is, a polytope whose vertices are lattice 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 collaborative work with Castillo in which we try to make this relation more explicit in the context of regular permutohedra. Our motivation is to establish Ehrhart positivity for generalized permutohedra. Although it turns out not all generalized permutohedra are Ehrhart positive, we are able to show that the third and fourth Ehrhart coefficients of generalized permutohedra are always
positive.

If time permits, I will also discuss some other related questions.

Mathematics and Statistics Colloquium Series