Okounkov bodies are a beautiful generalization of moment polytopes from Hamiltonian dynamics on toric varieties. They were introduced in 2008 by Lazarsfeld–Mustata and Kaveh–Khovanskii to study volumes of big line bundles over a variety. In the toric setting, Ehrhart theory provides a precise understanding of how the moment polytope is approximated by discrete collections of lattice points associated to holomorphic sections. What can be said in the general setting?   

It turns out there is a natural notion of discrete Okounkov bodies on any projective variety and their study leads to many interesting results and problems. In this talk I will introduce discrete Okounkov bodies and explain how to use them to answer some open questions in algberaic/complex geometry. For instance, we prove the first asymptotic result for K-stability thresholds. In another direction, I will introduce volume quantiles that generalize the notion of volume and show how they lead to a study of "collapsing" (discrete) Okounkov bodies that leads to the first asymptotic result for global log canonical thresholds. Surprisingly, the latter can be viewed as a higher-dimensional Weierstrass gap theorem. Based on joint work with Y.A. Rubinstein and G. Tian.