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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