Hodge Theory in Combinatorics
An undergraduate seminar led by Raymond Cheng during the fall of 2017.
The goal of this seminar is to understand the recent proof of
Rota–Welsh Conjecture due to Adiprasito–Huh–Katz.
Briefly, the Rota–Welsh Conjecture is a combinatorial conjecture
that posits that the coefficients of a certain
polynomial
associated with certain combinatorial objects, called
matroids,
form a
logconcave sequence.
Logconcavity of a sequence is a slight strengthening of unimodality
of a sequence, i.e. if the sequence first increases then decreases.
The basic case to keep in mind is that the coefficients of the
chromatic polynomial
of a graph are logconcave.
The recent work of Adiprasito–Huh–Katz establishes the
Rota–Welsh Conjecture by developing a combinatorial
Hodge theory
for matroids in general. Their methods, though inspired by algebraic
geometric considerations, are elementary in the sense that they do
not need to directly invoke the algebrogeometric muse.
References
The primary reference for this seminar is the original paper
See also their
Notices survey
Matt Baker wrote a blog article
here
and a survey
At some point, we will need to discuss matroids. Some references include:
 J. G. Oxley,
Matroid theory.
Oxford Science Publications.
The Clarendon Press, Oxford University Press, New York, 1992.
 E. Katz,
Matroid theory for algebraic geometers,
Simons Symposium Proceedings.
Schedule
We meet every Tuesday in Mathematics Room 507 between 8:30PM and 10:30PM.
 09/07 –

Organization.
 09/19 –

Raymond Cheng,
Motivation: Chromatic polynomials, matroids, and Chow rings of matroids.
 09/26 –

Shiv Gupta,
Brief on Matroids.
 09/26 –

Zach Davis,
Cryptomorphisms of Matroids.
 10/03 –

Joseph Lap,
Matroidal Chow rings.
 10/03 –

George Drimba,
Matroidal flips and pullback homomorphisms.
 10/10 –

Michael Tong,
Gysin homomorphisms and statement of Poincaré duality for matroidal Chow rings.
 10/10 –

Adam Block,
Surjectivity of Gysin ⊕ pullback homomorphisms.
 10/17 –

Ulysses Kim,
Complete the proof of Poincaré duality.
 10/17 –

Zach Davis,
Poincaré duality algebras, hard Lefschetz, and Hodge–Riemann relations.
 10/24 –

Shiv Gupta,
Hodge–Riemann package for subspaces.
 10/24 –

Michael Tong,
Stability of Hodge–Riemann under tensor products.
 10/31 –
 No Seminar. Happy Halloween!
 11/07 –
 No Seminar. Go vote!
 11/14 –

Ulysses Kim.
Combinatorial ample cones and local reductions.
 11/14 –

Adam Block.
Combinatorial ampleness under matroidal flips.
 11/21 –
 No Seminar. Happy Thanksgiving!
 11/28 –

Joseph Lap.
Limits of ample classes and Hodge Theory for combinatorial geometries.
 11/28 –
 12/05 –

George Drimba.
Logconcavity conjectures.