# Introduction to the Theory of Newton{Okounkov Bodies

Speaker: Professor Askold Khovanskii

Affiliation: University of Toronto

## Abstract

Newton{Okounkov bodies relate Algebraic geometry with Convex geometry via semigroups of integral

points in the lattice Zn. In general, sub-semigroups of the lattice Zn are very complicated objects. It turns

out that asymptotic behavior of such semigroups is simple enough. It has a nice description via Convex

geometry (via its convex Newton{Okounkov cone and Newton{Okounkov bodies).

Let me present a high school problem, which provides a simplest example of asymptotic behaviour of a

semigroup.

Problem. Let G Z be an additive semigroup generated by 3; 5 2 Z. Prove that G contains all natural

numbers n 8.

One can construct a Zn valued valuation on the multiplicative group of nonzero rational functions on any

irreducible n-dimensional algebraic variety. Such valuation relates algebraic geometry with subsemigroups

in Zn.

Newton{Okounkov bodies allow to provide an elementary proof of isoperimetric type inequalities in Al-

gebraic geometry.

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