Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries

Members' Seminar
Topic:Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries
Speaker:June Huh
Affiliation:Princeton University; Veblen Fellow, School of Mathematics
Date:Monday, November 9
Time/Room:2:00pm - 3:00pm/S-101
Video Link:https://video.ias.edu/membsem/2015/1109-Huh

A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This implies the above mentioned conjectures and their generalization to arbitrary matroids. Joint work with Karim Adiprasito and Eric Katz.