Spring 2018

The Combinatorics Seminar meets on **Wednesdays** in Snow 408 from 3-4pm.

Please contact Jeremy Martin if you are interested in speaking.

Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil

Our group project this semester is to read the expository article "Hodge Theory in Combinatorics" by Matt Baker, about the recent wotk of Karim Adiprasito, June Huh and Eric Katz. Some additional resources (provided by Federico Castillo):

- Eric Katz's survey on matroids (contains the representable case of Rota's conjecture): https://arxiv.org/pdf/1409.3503.pdf
- June Huh's recent survey on using HR relations
- Stanley's survey on Hard Lefschetz applications
- 3264 And All That, a relatively accessible introduction to intersection theory by David Eisenbud and Joe Harris

**Friday 1/19**

Organizational Meeting

**Wednesday 1/24**

Joseph Doolittle

*Partition Extenders*

__Abstract:__
In this talk, we will discuss partitions of simplicial complexes, and their relationship to the
\(h\)-vector of the complex. We will show that while not every simplicial complex is partitionable,
every simplicial complex does have a partition extender. This will allow a combinatorial
interpretation of the h-vector of any pure simplicial complex. We further show some bounds on
the size of a partition extender as well as some difficulties that arise when attempting to
construct minimal partion extenders.

**Wednesday 1/31**

Federico Castillo

*Hodge theory in combinatorics: an overview*

__Abstract:__
We take a general look into the recent proof (by
Adiprasito-Huh-Katz) of Rota's conjecture that the absolute value of the
coefficients in the characteristic polynomial of any matroid form a
unimodal sequence. The main point is to explain what all of those word
mean, to give examples, and to mention a thing or two about the proof,
which uses ideas from Hodge theory.

**Wednesday 2/7**

Federico Castillo

*The Chow ring of a matroid*

__Abstract:__ We continue with the definition of the Chow ring \(A(M)\) of a matroid
\(M\). This is modeled on Chow rings of wonderful compactifications and toric varieties of
Bergman fans. However, this admits a completely combinatorial description which is what allows
to extend known results to the non representable case. The key property of this ring are the
Hodge-Riemann relations, a concrete, linear algebra condition.

**Wednesday 2/14**

Jeremy Martin

*The Chow ring of a matroid, II*

__Abstract:__ I'll give some geometric background (focusing more on ideas and less on
technical specs) for what a Chow ring is, then we'll play around with the presentation of
\(A(M)\) for some specific examples.

**Wednesday 2/21**

Ken Duna

~~The Chow ring of a matroid, III~~ **CANCELLED;** will be rescheduled

**Wednesday 2/28**

Bennet Goeckner

Title TBA (*Preliminary Oral Exam for PhD*)

**Wednesday 3/7**

TBA

**Wednesday 3/14**

Bruno Benedetti (University of Miami)

Title TBA

**Wednesday 3/21**

No seminar (Spring Break)

**Wednesday 3/28**

Jose Samper (University of Miami)

Title TBA

**Wednesday 4/4**

TBA

**Wednesday 4/11**

TBA

**Wednesday 4/18**

TBA

**Wednesday 4/25**

TBA

**Wednesday 5/2**

TBA

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated 2/21/18