KU Combinatorics Seminar
Fall 2021

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