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 log-concave sequence. Log-concavity 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 log-concave.

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 algebro-geometric 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:

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.
12/05
George Drimba.
Log-concavity conjectures.