Abstract: I will start by discussing some basic facts about the semigroup of finite subsets of $\mathbb Z^n.$ This leads us to beautiful results in toric geometry (Bernstein-Kushnirenko theorem on the number of solutions of a system of polynomial equations). I will discuss generalizations to arbitrary varieties/graded algebras and theory of Newton-Okounkov bodies. Beside applications in study of divisors on varieties, this extension introduces stunning new tools and ideas in a number of other areas such local commutative algebra (Hilbert-Samuel multiplicity), symplectic geometry (moment map and integrable systems) and representation theory (flag varieties and Schubert calculus). Most of the talk should be accessible to anybody with just a basic knowledge of algebra and geometry.
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