Hodge Theory in Combinatorics

An undergraduate seminar 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 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