## Geometry Seminar

**Date:** January 22, 2018

**Time:** 3:00PM - 4:00PM

**Location:** BLOC 628

**Speaker:** Frank Sottile, TAMU

**Title:** *Newton-Okounkov Bodies for Applications*

**Abstract:** Newton-Okounkov bodies were introduced by Kaveh-Khovanskii and Lazarsfeld-Mustata to extend the theory of Newton polytopes to functions more general than Laurent polynomials. This theory has at least two implications for applications. First is that Newton-Okounkov bodies provide an approach to counting the number of solutions to systems of equations that arise in applications. Another is that when the Newton-Okounkov body is an integer polytope (there is a Khovanskii basis), there is a degeneration to a toric variety which in principal should give a numerical homotopy algorithm for computing the solutions. This talk will sketch both applications.