# From mixed multiplicities to combinatorial geometries to Hodge
theory

(and log-concavity of the Whitney coefficients)

## Karim Adiprasito: Hebrew University/IAS

## February 2, 2016

A conjecture of Read predicts that the coefficients of the chromatic
polynomial of any graph form a log-concave sequence. 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. Both conjectures are
special cases of the famous Rota conjecture asserting the
log-concavity of the coefficients of the characteristic polynomial of
any matroid.
The recent story of these problems starts in 2010, when June Huh
proved Rota's conjecture for the special case of hyperplane arrangements
by identifying the Whitney coefficients with mixed multiplicities of
its Jacobian ideal. It subsequently emerged that virtually all proofs
we could come up with for this case use nontrivial geometric facts
about the arrangement and/or Hodge theory for projective varieties,
and the more general conjecture of Rota for possibly "nonrealizable"
configurations/matroids remained open until recently.
I will discuss how to extend Hodge theory beyond the classical setting
to general matroids, starting with the surprising joint work with
Björner on Lefschetz theorems for Mikhalkin's p,q-groups, and then
discuss the proof of the "Kähler package" for general matroidal fans,
which proves the Rota and Welsh conjecture in full generality. All
proofs are purely combinatorial, and do not rely on analytifications
or projective algebraic geometry, although there are some useful
relations I will mention.

Based on joint work with June Huh and Eric Katz.