中科院数学与系统科学研究院

数学研究所

学术报告

代数几何研讨班

报告人：胡佳俊（清华大学）

题  目：Intersection theoretic inequalities via Lorentzian polynomials

时  间：2023.05.27（星期六）14:00-15:30

地  点：腾讯会议（816-542-609）

摘 要：We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to m-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for m-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by m-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Plunnecke-Ruzsa inequalities) from additive combinatorics in our contexts. This is a joint work with Jian Xiao.