Lattice polygons and finite generation of certain valuation semigroups

The main theme of the talk is the combinatorics of lattice polygons and its relationship to the geometry of the associated toric surfaces. Our point of view is to measure the complexity of lattice polygons via the complexity of geometric objects to which they give rise. For the latter, we will focus on convex geometric finiteness properties such as the polyhedrality of the cone of curves or the finite generation of valuation semigroups coming from Newton-Okounkov theory. The latter is a central (and wide open)  question in combinatorial algebraic geometry with strong ties to representation theory. Although the talk is in algebraic geometry, various parts of it will be understandable without much specialized knowledge from algebraic geometry. This is an account of joint work with Klaus Altmann, Christian Haase, Karin Schaller, and Lena Walter.