# 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 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

- K. Adiprasito, J. Huh, E. Katz, Hodge Theory for Combinatorial Geometries.

See also their *Notices* survey

- K. Adiprasito, J. Huh, E. Katz, Hodge Theory of Matroids,
*Notices of the AMS*,*64*(2017), 26-30.

Matt Baker wrote a blog article here and a survey

- M. Baker, Hodge Theory in Combinatorics.

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