SPEAKER:  Jacob Matherne  (University of Oregon)

TITLE:  Singular Hodge theory for matroids:  The Top Heavy Conjecture

ABSTRACT: 

A theorem of de Bruijn and Erdős says that a collection of nlines in a projective plane intersect in at least n points.  This is a special case of the more general “Top-Heavy Conjecture” of Dowling and Wilson (1974).  This conjecture was formulated for all matroids and was proven for hyperplane arrangements (realizable matroids) by Huh and Wang in 2017.  A key idea of their proof is to use the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement.  For arbitrary matroids, no such variety exists; nonetheless, I will discuss a proof of the Top-Heavy Conjecture for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory.  This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.

ZOOM LINK:

                           https://mit.zoom.us/j/97367846931

Password: rota