# Newton-Okounkov Bodies and Khovanskii Bases

### From SIAG-AG

## [edit] **Overview**

The theory of Newton-Okounkov bodies generalizes that of Newton Polytopes, giving a conceptual framework for root counts to systems of functions in terms of volumes and mixed volumes of convex Newton-Okounkov bodies. While the asymptotic definition of these bodies is not particularly constructive, when they admit a Khovanskii basis, they are polyhedral. Having a Khovanskii basis enables other methods based on polyhedra to be used.

While Newton-Okounkov bodies arose to solve questions from pure mathematics they have significant potential in applications. These include providing a foundation for root counts for polynomial systems from applications, the use of these root counts and Khovanskii bases for solving, and a host of algorithmic questions involving computing/determining Newton-Okounkov bodies and Khovanskii bases. The purpose of this minisymposium is to explore some of these opportunities and to advertise this to the wider community of applied algebraic geometry.

## [edit] **Organizer**

Frank Sottile

## [edit] **Confirmed Speakers**

Hiraku Abe McMaster University "Volumes of Newton-Okounkov bodies of Hessenberg varieties and their cohomology rings"

Lara Bossinger Universitaet Koeln "Flag varieties of type A, string polytopes, and superpotentials"

Daniel Corey Yale University

Kiumars Kaveh University of Pittsburgh "Introduction to Khovanskii bases, valuations and Newton-Okounkov bodies"

Chris Manon George Mason University "Khovanskii bases and tropical geometry"

Fatemeh Mohammadi Technische Universitaet Berlin "Computing toric degenerations of flag varieties arising from tropical geometry"

Frank Sottile Texas A&M University "Newton-Okounkov Bodies and Khovanskii Bases for Applications"