Last updated Mon 10/26/26
Friday 8/28
Organizational meeting
Friday 9/4
Jeremy Martin
Simplicial and cellular spanning trees
Abstract: The number of spanning trees in the complete graph \(K_n\) is given by the beautiful formula \(n^{n-2}\) (misattributed to Cayley). What about counting trees that are simplicial complexes of higher dimension? The first problem is to define what this even means: there are many ways to say what it means for a graph to be a tree, but they do not all generalize to higher dimension in the same way and there are competing notions in the literature. For purposes of enumeration, however, I claim that there is a correct definition, using simplicial homology. It gives rise to a lovely generalization of Cayley's formula, found by Kalai in 1983. Since then, there have followed more enumeration results (e.g., on shifted simplicial complexes and on skeletons of cubes), as well as higher-dimensional generalizations of algebraic graph theory (critical groups, cuts and flows, and resistor networks). I will try to survey this field, with lots of examples and not a lot of proofs.
Friday 9/11
Nate Krause
Rowmotion on the weak order of permutations
Abstract: Rowmotion is a fundamental map in dynamical algebraic combinatorics defined on the order ideals of a poset. By the Fundamental Theorem of Distributive Lattices, rowmotion can also be interpreted as an operation on the elements of a distributive lattice. After I define rowmotion, I will define semidistributive lattices and describe how the definition of rowmotion can also be extended to them.
The weak order on permutations of a fixed length is an important example of a semidistributive lattice. I will give an overview of recent results I obtained this summer jointly with Jessica Striker at North Dakota State University about the dynamics of rowmotion on the weak order. This involves a remarkable connection with a set of noncrossing arc diagrams that Nathan Reading showed to be in bijection with permutations.
Friday 9/18
Nate Lesnevich (Oklahoma State U.)
Shifted Lower Bruhat Intervals are EL-Shellable
Abstract: For \(w,x\) in a Coxeter group \(W\), the shifted lower Bruhat interval \([e,w]x^{-1}\) is ordered by the Bruhat order of \(W\). It is known that Bruhat intervals, and their tilted or twisted counterparts, are EL-shellable. That is, one can label their cover relations in a fashion that is particularly compatible with maximal chains. These shifted lower intervals we consider are neither twisted nor tilted Bruhat intervals. This talk shows that all shifted lower intervals are nonetheless EL-shellable for every Coxeter group.
Friday 9/25
Camilo Vallamil Chalarca (Oklahoma State U.)
Length distributions in Coxeter folding subgroups
Abstract: Let \((W, S)\) be a Coxeter system with length function \(\ell\). Given a set partition \(\hat S\) of \(S\) such that each block \(\alpha\) generates a finite parabolic subgroup of \(W\), let \(r_\alpha\) denote the longest element of the corresponding subgroup, and let \(\hat W\) be the subgroup of \(W\) generated by \(R = \{r_\alpha\colon\alpha\in S\}\). In 1992, B. Mühlherr showed that if \(\hat S\) is an admissible partition of \(S\), then \((\hat W, R)\) is a Coxeter system. This construction, which we call folding, embeds non-simply-laced Coxeter groups into simply-laced ones via a homomorphism \(\phi\colon\hat W\to W\). We study how the elements of a folding subgroup are distributed with respect to the ambient length function by computing the length-generating function \(\displaystyle\sum_{w\in\hat W} q^{\ell(\phi(w))}\). This generating function admits explicit product formulas in terms of \(q\)-integers, exhibiting a factorization phenomenon governed by parabolic compatibility under folding. Finally, we extend this framework to investigate the structure of the quotient space \(W/\phi(\hat W)\). We analyze the Poincaré series associated with its left cosets, giving a uniform formula and describing structural properties of cosets of minimal length one. This is joint work with Edward Richmond.
Friday 10/2
Josh O'Connor
On the quantum \(\mathfrak{sl}_3\) invariant for positive alternating links
Abstract: The \(\mathfrak{sl}_3\) link polynomial is a higher rank relative of the Jones polynomial which can be studied entirely diagrammatically using Kuperberg's web calculus. Building on the work of Harper and Kalfagianni, we construct formulas for the fourth and fifth coefficients of the quantum \(\mathfrak{sl}_3\) link invariant on positive alternating link diagrams in terms of data read from the associated Seifert graph. Then, we give a characterization of the invariant on positive alternating fibered links in terms of elementary symmetric polynomials. We also discuss behavior of our formulas under connected sums, and applications to knot theory. We also provide a characterization of the invariant on 3-braids. This is joint work with Matthew Harper, José Sánchez Menchén, Anthony Stone, and Angie Wu.
Friday 10/9
Andrés Molina
The directions determined by \(n\) points in space
Abstract: Let \(G=(V,E)\) be a finite, undirected graph. A picture of \(G\) is an arrangement of points and lines in space (for instance, \(d\)-dimensional affine or projective space over a field \(K\) that represent the vertices and edges of \(G\), in that they obey containment constraints corresponding to the vertex-edge incidence relations in \(G\). We are interested in the algebraic relations that determine the variety of all such pictures. The theory was introduced by Martin (2003), who focused exclusively on the case \(d=2\). The case of higher-dimensional pictures was considered by Enkosky and Martin (2013). We continue in this last direction, focusing on the complete graph and on pictures where vertices do not coincide, using what is already known from the 2-dimensional case to come up with explicit algebraic relations with combinatorial meaning. This is joint work with Jeremy Martin.
Friday 10/16
No seminar -- Fall Break
Friday 10/23
Jack Chou (U. of Minnesota)
Coxeter and Schubert combinatorics of \(\mu\)-Involutions
Abstract: The variety of complete quadrics is the wonderful compactification of \(GL_n/O_n\) and admits a cell decomposition into Borel orbits indexed by combinatorial objects called \(\mu\)-involutions. We study Coxeter–theoretic properties of \(\mu\)-involutions with results including a combinatorial description for their atoms, an exchange lemma, and transposition-like operators that characterize their Bruhat order. The corresponding orbit closures can be realized inside the flag variety. In this setting, we study the cohomology representatives of these orbits, which are, up to a scalar, the \(\mu\)-involution Schubert polynomials. We expand \(\mu\)-involution Schubert polynomials as a multiplicity-free sum of \(\nu\)-involution Schubert polynomials when \(\nu\) refines \(\mu\) and provide recurrences analogous to Monk's rule for Schubert polynomials. This is joint work with Zach Hamaker.
Friday 10/30
Han Yin (PhD preliminary exam)
Kleber's conjecture and complementary products of symmetric functions
Friday 11/6
Chris Uchizono
Title TBA
Friday 11/13
Saad Awan
Title TBA
Friday 11/20
Reuven Hodges
Title TBA
Friday 11/27
No seminar -- Thanksgiving
Friday 12/4
Marge Bayer
Title TBA
Friday 12/11
No seminar -- Stop Day
For seminars from previous semesters, please see the KU Combinatorics Group page.
Last updated Mon 10/26/26