KU Combinatorics Seminar
Spring 2022

Friday 1/21
Organizational meeting

Friday 1/28
Dania Morales
The character group of a Hopf monoid, Part II

Friday 2/4
Mark Denker
A Hopf monoid on posets

Tuesday 2/8: Kevin Marshall
A Hopf Monoid on Set Families (Ph.D. Oral Comprehensive Exam)

Zoom link (passcode: 2022)

Friday 2/11
Eleonore Bach (FU Berlin)
A concrete construction of the cographic hyperplane arrangement
Zoom link (passcode: 1430)
Zoom recording

Abstract: Geometrically carrying a trove of information about the underlying simple graph, the graphic hyperplane arrangement \(H_G\) yields an interesting mathematical object to study a simple graph \(G\). For example, one proves that the normal vectors of \(H_G\) are linearly independent if and only if they induce forests on \(G\) and the regions of \(H_G\) are in a one-to-one correspondence to the acyclic orientations of \(G\). With the graphic hyperplane arrangement we can associate the graphic matroid whose bases are spanning forests of \(G\). What information do we obtain if we apply duality, i.e., if we start with the dual of the graphic matroid, called the cographic matroid whose bases are complements of spanning forests of \(G\)? In this talk we are going to start answering the above question by constructing the normal vectors of the cographic hyperplane arrangement associated with the cographic matroid for simple, connected and bridgeless graphs.

Friday 2/18
Jeremy Martin
Simplicial Effective Resistance and Tree Enumeration

Abstract: A graph \(G\) can be viewed as an electrical network, whose parameters are the resistances along its edges. Remarkably, this model leads to a way to calculate combinatorial information about \(G\), including generating functions for spanning trees. Also remarkably, my coauthors Woong Kook and Kang-Ju Lee have shown that the same method can be used more generally to enumerate spanning trees of simplicial complexes (which I will explain). In particular, we can recover explicit formulas for shifted complexes (first obtained by Art Duval, Caroline Klivans and myself) and prove a new formula for color-shifted complexes (conjectured by Ghodrat Aalipour and Duval). This is joint work with Duval, Kook, and Lee.

Friday 2/25
Dylan Beck
Some new invariants of the edge ring of a finite simple graph

Abstract: Given any positive integer \(n\), there is a one-to-one correspondence between the collection of finite simple graphs on \(n\) vertices and the collection of squarefree monomial ideals of a polynomial ring in \(n\) indeterminates over a specified field; this bijection is induced by the assignment of an edge \(\{i, j\}\) to the squarefree monomial \(x_i x_j\). Consequently, the quotient of a polynomial ring over a field by a squarefree monomial ideal can be understood by studying the associated finite simple graph and vice-versa. One of the most celebrated examples of this relationship is due to Fröberg, who showed in 1990 that the complement of a finite simple graph is chordal if and only if the corresponding squarefree monomial ideal admits a linear resolution. In joint work with Souvik Dey, we introduce two new invariants of a finite simple graph via the quotient of the complex polynomial ring by the corresponding squarefree monomial ideal. We demonstrate that all though these invariants are often quite subtle, they can be computed or bounded explicitly in several familiar cases. Even more, we show that these invariants behave well with respect to operations such as the join of two finite simple graphs or the wedge of complete graphs. We conclude by stating some questions and possible directions for future work.

Friday 3/4
Sheila Sundaram (Pierrepont School)
Subword Order: Topology, Homology Representations, and Enumeration
Zoom recording

Abstract: We show that the homology module for words of bounded length, over an alphabet of size \(n\), decomposes into a sum of tensor powers of the \(S_n\)-irreducible \(S_{(n-1,1)}\) indexed by the partition \((n-1,1)\), recovering, as a special case, a result of Bj\"orner and Stanley for words of length at most \(k\). For arbitrary ranks we show that the homology is an integer combination of positive tensor powers of the reflection representation \(S_{(n-1,1)}\), and conjecture that this combination is nonnegative. We uncover a curious duality in homology in the case when one rank is deleted.

Our most definitive result describes the homology representation for an arbitrary set of ranks, as being almost a permutation module, modulo one copy of the reflection representation.

This talk will be self-contained. I will describe the techniques used to determine the topology and the homology representations via the Hopf trace formula. I will also present some open enumerative conjectures involving Stirling numbers of both kinds.

Friday 3/11
No seminar - Spring Break

Friday 3/18
No seminar - Spring Break

Friday 3/25
Derek Hanely (University of Kentucky)
Triangulations of Flow Polytopes, Ample Framings, and Path Algebras
Zoom recording

Abstract: The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. In this talk, we will characterize the DAGs that admit ample framings and enumerate the number of ample framings for a fixed DAG. Moreover, we will establish a connection between maximal simplices in DKK triangulations and \(\tau\)-tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart \(h^\ast\)-polynomials. This is joint work with Matias von Bell, Benjamin Braun, Kaitlin Bruegge, Zachery Peterson, Khrystyna Serhiyenko, and Martha Yip.

Friday 4/1
Ritika Nair
Weak Lefschetz Property for quotients by monomial ideals containing squares of variables

Abstract: Let \(\Delta\) be a simplicial complex on \(n\) vertices. The corresponding Stanley-Reisner ring is obtained from \(S = \Bbbk[x_1, \ldots, x_n]\) by quotienting out \(I_{\Delta} = \langle x_{i_1} \cdots x_{i_m} : \{i_1, \ldots, i_m \} \notin \Delta \rangle\), where \(\Bbbk\) is a field of characteristic \(0\). Let \(J_{\Delta}\) be the ideal generated by \(I_{\Delta}\) and the squares of the variables. \(A(\Delta) = S/J_{\Delta}\) is a graded artinian algebra. For a general linear form \(\ell\), \(A(\Delta)\) has the Weak Lefschetz Property (WLP) if the homomorphism induced by multiplication by \(\ell\), \(\mu_i : A(\Delta)_i \to A(\Delta)_{i+1}\) has maximal rank for all \(i\). In joint work with Hailong Dao, we characterize the Weak Lefschetz Property of \(A(\Delta)\) in terms of the simplicial complex \(\Delta\). In particular, we see a complete characterization of WLP in degree \(1\) in terms of the edge graph of \(\Delta\). We shall also look at pseudomanifolds of small dimension and if time permits, some interesting corollaries including relating WLP with face 2-colorability of certain triangulations.

Friday 4/8
Geoffrey Critzer
Enumeration of Finite Semigroups via the Symbolic Method

Abstract: We introduce the symbolic method as developed in the seminal work Analytic Combinatorics by Phillipe Flajolet and Robert Sedgewick. We employ the method to derive generating functions that enumerate some interesting statistics relating properties of the full transformation semigroup \(\mathcal{T}_n\), the semi-group \(\mathcal{PT}_n\) of partial transformations, the symmetric inverse semigroup \(\mathcal{IS}_n\), and the semigroup \(\mathcal{B}_n\) of all relations on \([n]\). In particular, we count the number of nilpotent, idempotent, and group elements in each semigroup. We classify these semigroups according to various properties, including but not limited to: index, connected components, number of recurrent, fixed, defective, and undefined points. We derive probability generating functions for some distributions associated with the non-functional points of a random partial function.

Friday 4/15
Kyle Maddox
Semigroups and commutative algebra

Abstract: Semigroups are simple combinatorial objects which provide a wealth of interesting and beautiful examples in commutative algebra by forming what's known as an affine semigroup ring. In this talk, we will recap some of the basic definitions surrounding semigroups and discuss recent joint work with Vaibhav Pandey about affine semigroup rings.

Friday 4/22
Kyle Maddox
Commutative algebra and semigroups

Abstract: In this follow-up talk from last week, I will focus a bit more on the algebraic side. We will cover a tiny bit of necessary homological algebra, then demonstrate how the simple idea of "\(p\)-nilpotent embeddings" of affine semigroups can help give interesting and powerful information about otherwise mysterious local cohomology modules of the affine semigroup ring. Kevin Marshall
Title TBA

Friday 4/29
No seminar; attend the Great Plains Combinatorics Conference instead!

Friday 5/6
No seminar (Stop Day)

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

Last updated Mon 5/9/22