Title: Hyperplane Arrangements, Intersection Lattices, and Koszul Algebras
Abstract:
Fields Medalist June Huh and his collaborators have proved several long-standing conjectures in combinatorics, such as the Top Heavy Conjecture, using tools from algebra and geometry. Two of the tools in these proofs are Chow rings and graded Moebius algebras associated with hyperplane arrangements. It was shown that Chow rings have many of the same properties as the cohomology rings of compact Kahler manifolds. Vladimir Dotsenko conjectured that these rings also have an additional property – that of being a Koszul algebra.
In my talk, I will first discuss some motivating problems from combinatorics, hyperplane arrangements, and their intersection lattices. Then I will introduce Koszul algebras and their properties. I will then describe joint work with Matt Mastroeni in which we confirmed Dotsenko’s conjecture in full generality. Then I will discuss our investigation (joint with Mastroeni, Adam LaClair, and Irena Peeva) of the Koszul property for graded Moebius algebras, along with connections with classical results on Orlik-Solomon algebras and modern results on strongly chordal graphs.