Spring 2021

- This semester the Combinatorics Seminar meets on
**Fridays, 3-4pm Central Time (UTC -5 through October; UTC -6 in November).** - All meetings will take place on Zoom: meeting #924 6702 4322, passcode 954485
- Please contact Jeremy Martin if you are interested in speaking or attending.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- Also consider attending the Graduate Online Combinatorics Colloquium.

**Friday 2/5**

Organizational meeting

**Friday 2/12**

No seminar

**Friday 2/19**

Sophie Rehberg (FU Berlin)

*Combinatorial reciprocity theorems for pruned inside-out polytopes and their application to generalized permutahedra and hypergraphs*

Video

__Abstract:__:
Generalized permutahedra are a class of polytopes with many interesting
combinatorial subclasses. We introduce pruned inside-out polytopes, a
generalization of inside-out polytopes introduced by Beck-Zaslavsky
(2006),which have many applications such as recovering the famous
reciprocity result for graph colorings by Stanley. We study the integer
point count of pruned inside-out polytopes by applying classical Ehrhart
polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric
perspective on and a generalization of a combinatorial reciprocity
theorem for generalized permutahedra by Aguiar-Ardila (2017) and
Billera-Jia-Reiner (2009). Applying this reciprocity theorem to
hypergraphic polytopes allows us to give an arguably simpler proof of a
recent combinatorial reciprocity theorem for hypergraph colorings by
Aval-Karaboghossian-Tanasa (2020). Our proof relies, aside from the
reciprocity for generalized permutahedra, only on elementary geometric
and combinatorial properties of hypergraphs and their associated
polytopes. In this talk we will focus on the application to hypergraphs
and their polytopes.

**Friday 2/26**

No seminar

**Friday 3/5**

No seminar

**Friday 3/12**

Jeremy Martin

*What is... Schubert calculus?*

Video

__Abstract:__ I will give an introduction to Schubert calculus on
Grassmannians and flag varieties, assuming that the audience has never
heard of any of these words before. I will use my algebraic
combinatorics lecture notes (sections 11.4 and 11.5).

**Friday 3/19**

Jeremy Martin

*What is... Schubert calculus? (part 2)*

Video

__Comment:__ At the end of the talk, I got asked the question,
"So, what is Schubert calculus?" Here is answer: it is about
using combinatorics (thr symmetric group, partitions, tableaux) to model
and solving geometry and topology problems involving linear subspaces and
configurations of linear subspaces.

Marge Bayer will be speaking at the Graduate
Online Combinatorics Colloquium on **Wednesday, March
24.**

**Friday 3/26**

No seminar

**Friday 4/2**

No seminar

Kevin Marshall will be speaking at the Graduate
Online Combinatorics Colloquium on **Wednesday, April
7.**

**Friday 4/9**

No seminar

**Friday 4/16**

Bryan Gillespie (Colorado State University)

*Convexity in Ordered Matroids and the Generalized External Order*

Video

__Abstract:__ In this talk we will use generalized matroid
activity to construct the "external order" on the independent sets of an
ordered matroid. The poset simultaneously generalizes the active orders
on matroid bases first studied by Björner, as well as the discrete
convexity theory of Las Vergnas and Edelman for oriented matroids. We
will relate the defining operator of the external order to the theory of
anti-exchange closure functions and convex geometries, and we will
discuss a dual view of the order from the perspective of the class of
greedoids called antimatroids. Time permitting, we will characterize
the lattices isomorphic to the external order.

**Friday 4/23**

Julianne Vega (Kennesaw State University)

*Triangulations, Order Polytopes and Generalized Snake Posets*

Video

__Abstract:__ In this talk, we will introduce generalized snake
posets \(Q\) and investigate circuits, flips, and regular triangulations of
the corresponding order polytopes \(O(Q)\). In particular, we will begin
with an introduction and some basic properties of \(O(Q)\) for certain \(Q\). By
the end of the talk we will find ourselves immersed in the secondary
polytope of \(O(Q)\) and considering "twists" which extend to an action on
regular triangulations. This work is joint with the Kentucky Crew:
Matias von Bell, Benjamin Braun, Derek Henely, Khrystyna Serhiyenko,
Andrés Vindas Meléndez, and Martha Yip.

**Friday 4/30**

Byeongsu Yu (Texas A&M)

*When is the quotient of a semigroup ring by a monomial ideal Cohen-Macaulay?*

Video

__Abstract:__ We give a new combinatorial criterion for quotients
of affine semigroup rings by monomial ideals to be Cohen-Macaulay, by
computing the homology of finitely many polyhedral complexes. This
provides a common generalization of well-known criteria for affine
semigroup rings and monomial ideals in polynomial rings. This is joint
work with Laura Matusevich.

**Friday 5/7**

Laura Escobar (Washington University in St. Louis)

*Which Schubert varieties are Hessenberg varieties? *

__Abstract:__ Schubert varieties are subvarieties of the flag
variety parametrized by permutations; they induce an important basis for
the cohomology of the flag variety. Hessenberg varieties are also
subvarieties of the flag variety with connections to both algebraic
combinatorics and representation theory. I will discuss joint work with
Martha Precup and John Shareshian in which we investigate which Schubert
varieties in the full flag variety are Hessenberg varieties.

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

Last updated Sat 5/1/21