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Discrete Math

From mixed multiplicities to combinatorial geometries to Hodge theory (and log-concavity of the Whitney coefficients)

Location:  Hill 705
Date & time: Monday, 01 February 2016 at 2:00PM - 2:11PM

Karim Adiprasito, Hebrew University/IAS:

A conjecture of Read predicts that the coefficients of the chromaticpolynomial of any graph form a log-concave sequence. A relatedconjecture of Welsh predicts that the number of linearly independentsubsets of varying sizes form a log-concave sequence for anyconfiguration of vectors in a vector space. Both conjectures are specialcases of the famous Rota conjecture asserting the log-concavity of thecoefficients of the characteristic polynomial of any matroid.

The recent story of these problems starts in 2010, when June Huh provedRota's conjecture for the special case of a hyperplane arrangement byidentifying the Whitney coefficients with mixed multiplicities of itsJacobian ideal. It subsequently emerged that virtually all proofs wecould come up with for this case use nontrivial geometric facts aboutthe arrangement and~or Hodge theory for projective varieties, and themore general conjecture of Rota for possibly
onrealizable 'configurations~matroids remained open until recently.

I will discuss how to extend Hodge theory beyond the classical settingto general matroids, starting with the surprising joint work withBjorner on Lefschetz theorems for Mikhalkin's p,q-groups, and thendiscuss the proof of the 'Kahler package' for general matroidal fans,which proves the Rota and Welsh conjecture in full generality. Allproofs are purely combinatorial, and do not rely on analytifications orprojective algebraic geometry, although there are some useful relationsI will mention.

Based on joint work with Anders Bjorner and with June Huh and Eric Katz'

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Department of Mathematics

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Rutgers University
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