Abstract: Stability thresholds, particularly the \alpha_k and \delta_k invariants, are a fundamental topic in the theory of K-stability, with connections to various different fields such as algebraic geometry, convex geometry, and geometric analysis. In this talk, we investigate their asymptotic behavior, revealing new phenomena in both toric and non-toric settings.

In the toric setting, Ehrhart theory precisely describes the asymptotics via lattice point approximations of the moment polytope. We establish the stabilization of \alpha_k and derive an asymptotic expansion for \delta_k. In the general setting, we demonstrate that \alpha_k may fail to stabilize. To study their asymptotics we analyze the Okounkov body and its discrete approximation, which are a generalization of moment polytopes on toric varieties. Using tools from convex geometry and lattice point enumeration techniques, we prove the first asymptotic result for \delta_k. Based on joint work with Y. Rubinstein and G. Tian.

Brief biography: Chenzi Jin obtained his bachelor’s degree from Peking University in 2017 and his master’s degree from New York University in 2019. He is currently pursuing his PhD at the University of Maryland under the supervision of Prof. Yanir Rubinstein, with an expected graduation in Spring 2025. Following his doctoral studies, he will join Princeton University and the Institute for Advanced Study as a Veblen Research Instructor.

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