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.