|Topic:||Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries|
|Affiliation:||Princeton University; Veblen Fellow, School of Mathematics|
|Date:||Monday, November 9|
|Time/Room:||2:00pm - 3:00pm/S-101|
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.