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SUMMARY:From mixed multiplicities to combinatorial geometries to Hodge the
 ory (and log-concavity of the Whitney coefficients)
DTSTART:20160322T151500
DTEND:20160322T170000
DTSTAMP:20260512T015111Z
UID:85116d8956e423d1211d768128a8a2a514d6517d12a7dd2dd9410767
CATEGORIES:Conferences - Seminars
DESCRIPTION:Karim Adiprasito\, Hebrew University of Jerusalem\nA conjectur
 e of Read predicts that the coefficients of the chromatic polynomial of an
 y graph form a log-concave\nsequence. A related conjecture of Welsh predic
 ts 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\nassertin
 g the log-concavity of the coefficients of the characteristic polynomial o
 f any matroid. The recent story of these problems starts in 2010\, when Ju
 ne Huh proved Rota's conjecture for the special case hyperplane arrangemen
 ts by identifying the Whitney\ncoefficients with mixed multiplicities of i
 ts Jacobian ideal. It subsequently emerged that virtually all proofs we co
 uld come up with for this case use nontrivial geometric facts about the ar
 rangement and/or Hodge theory for projective\nvarieties\, and the more gen
 eral conjecture of Rota for possibly ``nonrealizable'' configurations/matr
 oids remained open until recently. I will discuss how to extend Hodge theo
 ry beyond the classical setting to general matroids\, starting with the su
 rprising\njoint work with Björner on Lefschetz theorems for Mikhalkin's p
 \,q-groups\, and then discuss the proof of the "Kähler package" for gener
 al matroidal fans\, which proves the Rota and Welsh conjecture in full gen
 erality. All proofs are purely combinatorial\, and\ndo not rely on analyti
 fications or projective algebraic geometry\, although there are some usefu
 l relations I will mention. Joint work with June Huh and Eric Katz
LOCATION:BI A0 448 https://plan.epfl.ch/?room==BI%20A0%20448
STATUS:CONFIRMED
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