Fall 2021

- We are back in person! This semester the Combinatorics Seminar will meet on
**Fridays, 3-4pm**, in**Snow 306**, - 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 8/27**

Organizational meeting

**Friday 9/3**

Mark Denker

*Creating and shelling cut complexes*

__Abstract:__
Given a graph \(G\), we construct a series of simplicial complexes using
its separating sets called cut complexes. A result of Frö berg states
that the 2-cut complex is shellable if and only if \(G\) is chordal, in
this talk we explain what a simplicial complex is, how we construct the
\(k\)-cut complex, what it means for a complex to be shellable, and how
we can understand shellability of the k-cut complex from properties of
\(G\).

**Friday 9/10**

Kyle Maddox

*Affine semigroups in commutative algebra*

__Abstract:__
An affine semigroup is a set of points inside \(\mathbb{Z}^n\) which is
closed under addition. These simple but combinatorially rich objects
also give rise to interesting commutative algebra by assigning points in
the semigroup to monomials in a ring, typically a subring of a
polynomial ring over a field. In this example-heavy talk, we will give
several interesting examples of affine semigroup rings and discuss the
basics of Hochster's theorem on normality of semigroups.

**Friday 9/17**

Jeremy Martin

*Unbounded matroids*

__Abstract:__
A matroid \(M\) on ground set \(E=[n]\) gives rise to a polytope
\(P\subseteq\mathbb{R}^n\) whose vertices are the characteristic vectors
of the bases of \(M\). Each polytope arising in this way is a
*generalized permutahedron*: its edges are all parallel to
differences of standard basis vectors, or equivalently its normal fan
coarsens the braid fan. By a theorem of Gel'fand, Goresky, Macpherson
and Serganova, matroid polytopes are *exactly* the generalized
permutahedra whose vertices are all 0,1-vectors. What can be said about
0,1-generalized permutahedra that are not assumed to be bounded? On the
combinatorial side, these polyhedra correspond to a special case of the
*submodular systems* studied by Fujishige and others. We describe
the exact correspondence between combinatorial and geometric objects and
show, for example, that every unbounded matroid has a canonical
extension to a matroid.

**Friday 9/24**

Kevin Marshall

*A Hopf Monoid On Set Families*

__Abstract:__
We study a Hopf monoid **SetFam** whose underlying objects are
arbitrary set families, with product given by join and coproduct defined
by a general notion of restriction and contraction, analogous to that
for matroids. Of particular interest is the Hopf submonoid **LOI**
spanned by families \(J(P)\) of order ideals of finite posets \(P\). We
summarize a topological approach, inspired by the work of Aguiar and
Ardila on generalized permutahedra, for a cancellation-free formula for
the antipode in **LOI**.

**Friday 10/1**

Jennifer Elder (Rockhurst University)

*Graphs of Sets of Reduced Words*

__Abstract:__
Any permutation in the finite symmetric group can be written as a
product of adjacent transpositions. For a fixed permutation \(\sigma \in
\mathfrak{S}_n\) we call products of minimal length reduced words, and
the collection of all such reduced words is denoted \(R(\sigma)\). Any
reduced word of \(\sigma\) can be transformed into any other by a
sequence of commutation moves or long braid moves. We are interested in
the congruence classes defined by using only braid moves or only
commutation moves.

The set \(R(\sigma)\) can be represented as a graph \(G(\sigma)\) where the edges denote whether two reduced words are related by a commutation relation, or a braid relation. In this talk, we will present work on induced subgraphs of \(G(\sigma)\), and how understanding these subgraphs will allow us to count the number of edges of a particular type in \(G(\sigma)\). We will also discuss how these subgraphs can be used to give us a better understanding of the sets of congruence classes in \(R(\sigma)\).

**Friday 10/8**

Mark Denker

*Enumerating Triangular and Three-Dimensional Floorplans*

__Abstract:__
A triangular floorplan is a subdivision of a triangle into triangular
rooms. A slicing floorplan is constructed inductively one wall at a
time. In this talk, I will provide an enumeration of slicing triangular
floorplans with a specified number of rooms, as well as bijections to
separable permutation pairs and three-dimensional rectangular
floorplans.

**Friday 10/15**

Marge Bayer

*Matching Complexes of Planar Graphs*

__Abstract:__
The matching complex of a graph is a simplicial complex representing
the matchings (sets of independent edges) of the graph. Matching
complexes of certain classes of graphs have been much studied, with an
array of topological results, particularly on complete graphs and
complete bipartite graphs. This talk will focus on matching complexes
of planar graphs, particularly graphs of certain finite tilings by
polygons. This is based on joint work with Julianne Vega and Marija
Jelic-Milutinovic.

**Friday 10/22**

Jeremy Martin

*Ehrhart theory for paving matroids*

**Friday 10/29**

Andrew Moorhead (EECS)

*Introduction to the universal algebra commutator*

__Abstract:__
Universal algebra is the study of general algebraic structures, which
are nonempty sets equipped with some operations. In this talk we will
give a survey of some main results of the area and then discuss how the
commutator theories for groups, rings, Lie algebras, and more can be
viewed as specializations of a general commutator theory applicable to
any algebraic structure.

**Friday 11/5**

Matias von Bell (University of Kentucky)

*The nu-associahedron via flow polytopes*

__Abstract:__
The nu-associahedron introduced by Ceballos, Pardol, and Sarmiento in
2019 is a polytopal complex which generalizes the classical
associahedron. We will first familiarize ourselves with the
nu-associahedron before introducing the nu-caracol flow polytope. We
will then show how the nu-associahedron can be obtained as the dual of a
triangulation of the nu-caracol flow polytope. As a consequence,
Mészáros' subdivision algebra can be used to obtain the
nu-Tamari complex, answering a question of Ceballos, Padrol, and
Sarmiento. Parts of this talk are based on joint work with Rafael
Gonzáles D'León, Francisco Mayorga Cetina, and Martha Yip.

**Friday 11/12**

Enrique Salcido

*The chromatic symmetric function of a graph*

__Abstract:__
The chromatic symmetric function of a graph \(G\) is a proper extension
of the chromatic polynomial of G that encodes more than the number of
proper vertex colorings of a graph. We will show multiple ways of
representing the chromatic symmetric function in terms of different
bases for the space of symmetric functions and explore what the
coefficients of these representations encode. One of the representations
of interest will give the chromatic symmetric function as a sum over the
broken circuit complex of \(G\).

**Friday 11/19**

Marge Bayer

*Convex Polytopes from a Combinatorial Perspective*

__Abstract:__
This is a basic survey talk on the combinatorics of convex polytopes,
with an emphasis on numbers of faces. I will discuss a few theorems and
an open question about 3-dimensional polytopes, and then look at what is
known and, mostly, not known in higher dimensions.

**Friday 11/26**

No seminar (Thanksgiving)

**Friday 12/3**

Dania Morales

*The character group of a Hopf monoid, Part I*

**Friday 12/10**

No seminar (Stop Day)

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

Last updated Thu 1/27/22