Spring 2019

- The Combinatorics Seminar meets on Fridays in
~~Snow 408~~**Snow 306**from 4-5pm. - Please contact Jeremy Martin or Federico Castillo 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
- A good general resource (which we may use this semester):
*A Combinatorial Miscellany*by Anders Björner and Richard Stanley

**Friday 1/25**

Pam Harris (Williams College)

*Kostant's Partition Function*

__Abstract:__ In this talk we introduce Kostant's partition
function which counts the number of ways to represent a particular
weight (vector) as a nonnegative integral sum of positive roots of a Lie
algebra. We provide two fundamental uses for this function. The first is
associated with the computation of weight multiplicities in
finite-dimensional irreducible representations of classical Lie algebras
and the second is in the computation of volumes of flow polytopes. We
provide some recent results in the representation theory setting and
state a direction of ongoing research related to the computation of the
volume of a new flow polytope associated to a Caracol diagram.

**Friday 2/1**

Organizational meeting

**Friday 2/8**

Jeremy Martin

*Report from GRWC 2018: Parking Functions and Elser's Invariant*

__Abstract:__ I will report on a project from the 2018
Graduate Research Workshop in Combinatorics,
about enumeration for completions of parking functions and potential consequences
for Pitman-Stanley polytopes.

~~Friday 2/15~~

**Seminar cancelled** due to inclement weather

**Friday 2/22**

Kevin Marshall

*What Is A Modulus On Graphs?*

__Abstract:__ We discuss the notion of modulus on graphs as first developed by Nathan
Albin et al. in, e.g., arXiv:1401.7640 and
arXiv:1805.10112.
The concept of modulus has its origins in analysis dealing with families of curves. Modulus on
graphs can be thought of as the discrete analogue to this idea. We will cover some basic facts
about modulus on graphs and in particular focus on using modulus with walks and trees of a
graph.

**Friday 3/1**

Mark Denker

*What is Polya's Theorem?*

**Friday 3/8**

Federico Castillo

*Deformations of Coxeter permutahedra and Coxeter submodular functions*

__Abstract:__
One way to decompose a polytope is to represent it as a Minkowski sum of two
other polytopes. These smaller pieces are naturally called summands.
Starting from a polytope we want to explore the set of all summands.
This set can be parametrized by a polyhedral cone, called deformation
cone, in a suitable real vector space. We focus on the case where the
starting polytope is a Coxeter permutahedron, which is a polytope
naturally associated with a root system. This generalizes the type A
case, which corresponds to generalized permutohedra. This is joint work
with Federico Ardila, Chris Eur, and Alexander Postnikov.

**Friday 3/15**

No seminar (Spring Break)

**Friday 3/22**

Justin Lyle

*Rank Selection and Depth Conditions for Balanced Simplicial Complexes*

__Abstract:__ A simplicial complex is said to be balanced if its
vertices can be colored so that none of its faces contain more than one
vertex of a given color. We focus on rank selected subcomplexes of
balanced simplicial complexes, which are formed by excluding all
vertices of a given set of colors. In particular, we are interested in
the depth properties of these subcomplexes and how they connect with
those of the original complex. By passing to a barycentric subdivision,
we discuss how our results can be applied to any simplicial complex.
This is joint work with Brent Holmes.

**Friday 3/29**

TBA

**Friday 4/5**

Carl Lee (University of Kentucky)

*The a,b,c,d,C,D,f,g,h of Polytopes*

__Abstract:__ I will sample some different methods in the
enumeration of chains of faces of convex polytopes, and discuss their
relationships.

**Friday 4/12**

Casey Pinckney (Colorado State University)

*Independence Complexes of Finite Groups*

__Abstract:__ Understanding generating sets for groups is of
interest in group theory, as certain problems involving groups are made
easier by appropriate choices of generators. Our goal is to describe
minimal generating sets for certain finite groups in a visual way. We
will explore some interesting combinatorics that arises from this view.
More specifically, let \(G\) be a finite group. We define an independent
set of \(G\) to be a collection of group elements that is a minimal
generating set for some subgroup of \(G\). These independent sets form a
simplicial complex (called the Independence Complex) whose vertices are
elements of \(G\) and whose faces of size \(k\) correspond to
independent sets with \(k\) generators. We describe the structure and
combinatorics of the resulting simplicial complex.

**Friday 4/19**

Alex McDonough (Brown University)

*Chip-Firing on Simplicial Complexes and Matroids*

__Abstract:__ The sandpile group and its relation to spanning
trees has been studied extensively in the graphical setting in the
context of chip-firing. A more recent area of research has been to
generalize chip-firing results to a larger class of objects. In this
talk, we explore chip-firing on simplicial complexes as well as regular
matroids. Through combining these topics, we produce a bijective proof
for Duval-Klivans-Martin's Simplicial Matrix Tree Theorem on a
particular class of simplicial complexes. This bijection provides
representatives for the elements of the sandpile groups for these
complexes that depend only on a vertex ordering. This talk does not
assume any prior knowledge of chip-firing or matroids.

**Friday 4/26**

Souvik Dey

*Generalized Flag Varieties: An Introduction*

__Abstract:__ Classically, a full flag in a vector space of dimension n is
defined to be an increasing sequence of vector subspaces of length n, starting
with dimension 0 and with increment 1. The collection of all such (full) flags,
called the full flag variety, can be viewed as a Zariski closed subset of a
product of Grassmaninan varieties, from which the collection obtains a variety
structure, so that it is called a full flag variety. It can be seen that the
general linear group acts transitively on this variety, the invariant subgroup
of the action being the group of upper-triangular matrices, also called the
Borel subgroup of the general linear group, so that the variety can be
parametrized by the co-sets of this Borel subgroup. All these concepts can
actually be generalized in the context of algebraic groups, where one can define
a notion of Borel subgroups, and the homogeneous space of that is called a
generalized full (sometimes also called complete) flag variety. And in general,
a generalized flag-variety is the quotient space of a parabolic subgroup. In
this talk, we will try to make sense of these concepts of Borel subgroups,
parabolic subgroups and flag varieties in the general context of algebraic
groups and mention in what way they are special and try to give some results
explaining their geometry.

**Friday 5/3**

McCabe Olsen (Ohio State)

*Birkhoff polytopes of different type and the orthant-lattice property*

__Abstract:__ Given a \(d\)-dimensional lattice polytope \(P\), we say that
\(P\) has the orthant-lattice property (OLP) if the subpolytope obtained by
restriction to any orthant is a lattice polytope. While this property feels somewhat
contrived, it can actually be quite useful in verification of discrete geometric
properties of \(P\). In this talk, we will discuss a number of results for the
existence of triangulation and the integer decomposition property for reflexive OLP
polytopes. One such polytope which fits into the program is a type-B analogue of the
Birkhoff polytope and its dual polytope, the investigation of which led to interest
in this property.

**Friday 5/10**

No seminar (Stop Day)

Here's our potential list of speakers and "What is..." topics (left over from last fall, and subject to change):

- Dylan - universal algebra
- Jeremy - root systems, \(E_8\) (want the full story? Take Math 996!)
- Ken - Cayley graphs
- Kevin - Borel-fixed ideals
- Mark - Polya's theorem

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

Last updated Wed 4/24/19