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

*Decompositions of Simplicial Complexes* (__Preliminary Oral Exam for PhD__)

**Wednesday 3/7**

Ken Duna

*Rank functions of positroids*

**Wednesday 3/14**

Bruno Benedetti (University of Miami)

*Some metric and algebraic approaches to look at polytope graphs*

__Abstract:__ We give an invitation to the study of polytope graphs, with
particular focus on the diameter and the connectivity. The plan is to
sketch a metric approach (joint with Karim Adiprasito) and an approach via
commutative algebra that lifts to hyperplane arrangements (joint with
Matteo Varbaro, Michela Di Marca).

**Wednesday 3/21**

No seminar (Spring Break)

**Wednesday 3/28**

Jose Samper (University of Miami)

*Is the \(h\)-vector of a matroid a pure \(O\)-sequence?*

__Abstract:__ We discuss an old conjecture of Stanley asserting that the \(h\)-vector of a matroid independence complex is a pure \(O\)-sequence. We will discuss how ordering of the groundset of the matroid leads to a refined count of the \(h\)-vector and suggests a strategy to tackle the conjecture inductively. We then discuss a few refinements of Stanley's conjecture, how they reduce the problem to a finite problem in fixed rank. After that we discuss a few explicit large examples that elucidate some of the difficulties of the conjecture.

**Wednesday 4/4**

No seminar

**Wednesday 4/11**

Alex Schaefer

*The dimension of the negative cycle vectors of a signed graph*

__Abstract:__ A *signed graph* is a graph \(\Gamma\) where the edges are
assigned sign labels, either \(+\) or \(-\). The sign of a cycle is the product of
the signs of its edges. Let \(\textrm{Spec}(\Gamma)\) denote the list of lengths of cycles in \(\Gamma\). We
equip each signed graph with a vector whose entries are the numbers of negative
\(k\)-cycles for \(k\in\textrm{Spec}(\Gamma)\). These vectors generate a subspace of
\(\mathbb{R}^{|\textrm{\Spec}(\Gamma)|}\). Using matchings with a strong permutability property, we provide
lower bounds on the dimension of this space; in particular, we show for complete graphs,
complete bipartite graphs, and a few other graphs that this space is all of
\(\mathbb{R}^{|\textrm{\Spec}(\Gamma)|}\).

**Wednesday 4/18**

Ken Duna

*Go with the Flow: Combinatorial Applications of A Dual Linear Programming Theorem*

**Wednesday 4/25**

Hailong Dao
*On a conjecture by Conforti and Cornuéjols*

__Abstract:__
I will discuss the conjecture in the title (which still carries a cash prize of $5,000),
which relates the max-flow min-cut property to a combinatorial statement. Also, I will
describe a stronger version, which can be proved for graphs using some simple translation
to commutative algebra. A more detailed description can be found at MathOverflow.

**Wednesday 5/2**

TBA

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

Last updated 4/19/18