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U N I V E R S I T Y OF K E N T U C K Y
WHERE CATS = COMBINATORICS, ALGEBRA, TOPOLOGY & STATISTICS!
WHERE CATS = COMBINATORICS, ALGEBRA, TOPOLOGY & STATISTICS!
745 PATTERSON OFFICE TOWER
Fall 2025
Fall 2025
Unless otherwise noted, seminar meets at 2:00 pm Mondays in 745 POT.
Sept 8
Richard Ehrenborg
University of Kentucky
Spanning trees and period polynomials
We determine the number of spanning trees in several graphs which are defined using number theoretic means. Our main example is a generalization of the Paley graph in the case when e = 2. The vertex set is the finite field F_q, where two elements are connected by an edge if their difference belongs to a certain collection of cosets of the eth power subgroup. Using results of Myerson on period polynomials, we obtain explicit formulas when the power e is 3 and 4. This is joint work with David Leep.
Sept 29
KOI Organizational Meeting
Oct 13
Maxwell Hosler
University of Kentucky
An order on circular permutations
Masters Exam
Discussing a paper by Abram, Chapelier-Laget, and Reutenauer, we examine a family a lattices with three isomorphic expressions; first, as a lattice of circular permutations, second, as a lattice of natural-valued functions called 'admitted vectors,' and third, as an interval in the weak order on the affine symmetric group. This family turns out to have strong analogies with the weak order on the symmetric group, despite not being a weak order. Amongst other things, admitted vectors act as 'inversion sets with multiplicity' for these permutations, and the Hasse diagram can be labelled by transpositions in a way reminiscent of how the same can be done for the weak order. We end by proving the fact that, in some sense, the 'limit' of this family of posets is Young's lattice.
Oct 20
Evan Henning
University of Kentucky
The incidence Hopf algebra of the non-crossing partition lattice
Masters Exam
First studied in the literature by H.W Becker as planar rhyme schemes, non-crossing partitions have long been a combinatorial object of interest. It is well known that the set of non-crossing partitions of [n] inherit a lattice structure as a sublattice of the partition lattice ordered by refinement. Simion and Ullman showed that this lattice is self-dual. Moreover, intervals in the non-crossing partition lattice factor nicely hence the incidence Hopf algebra on the family of intervals of the non-crossing partition lattice has a nice structure. In this talk we will discuss Hillary Einziger's contributions to the study of the structure of this incidence Hopf algebra. In doing so we will find multiple bases, various formulas for the antipode, and show a bijection between this Hopf algebra and that of the symmetric functions.
Oct 27
No meeting
FALL BREAK
Nov 10
KOI Local Organizing Committee Meeting
Nov 17
KOI Local Organizing Committee Meeting
Nov 24
KOI Local Organizing Committee Meeting
Dec 1
KOI Local Organizing Committee Meeting
CP 139 (Note location)
Dec 5
Allen Knutson
Cornell University
Pipe dreams, Schubert varieties and the commuting scheme
Colloquium
Schubert considered the space of kxn matrices whose Gaussian elimination has fixed pivot columns. The "volume" of this space, in some sense, is a Schur polynomial, with many combinatorial interpretations. Pipe dreams were introduced in 1993 in [Bergeron-Billey] to give a pictorial calculus for "Schubert polynomials," the corresponding volumes of a more general class of Schubert varieties.
In 2005, Miller and I gave a geometric retrodiction of pipe dreams based on Gröbner degeneration. In the same year, I introduced the "lower-upper scheme'' {(X,Y): XY lower triangular, YX upper} to study the scheme of pairs of commuting matrices. I'll explain a (much more natural) pipe dream theory for the lower-upper scheme, use it to rederive the old one (also Lam-Lee-Shimozono's "bumpless pipe dreams'') and give a formula for the degree of the commuting scheme. This is joint with Paul Zinn-Justin.
Colloquium tea is at 3:14 pm (= π time) in 745 POT.
Colloquium talk is at 4:10 pm in Chem-Physics 287.
This colloquium talk is part of the KOI Combinatorics Lectures.
The KOI Combinatorics Lectures are funded by a grant from the National Science Foundation. Partial support provided by the UK Department of Mathematics and the UK College of Arts and Sciences.
Dec 6
See KOI Combinatorics Lectures for further details.
All lectures are held in Chem-Physics 139 (CP 139)
09:00 - 09:59 am Arrival/Registration/Meet and Greet/Poster set up time
09:59 - 10:00 am Welcome, Welcome speech
10:00 - 11:00 am Bruno Benedetti, "Simplicial complexes and decompositions of manifolds"
11:00 - 11:30 am Coffee Break
11:30 - 12:30 pm Jacob Matherne, "Chow functions for partially ordered sets"
12:30 - 02:30 pm Lunch Break
02:30 - 03:14 pm The Koi Pond Panel
03:14 (π time) - 04:09 pm Tea Time and the One Picture/One Theorem Poster Session
04:10 - 05:10 pm Dhruv Mubayi, "Randomness and determinism in Ramsey theory"
05:11 - 05:20 pm Conference Photo
06:00 pm - Conference Dinner, CP 114.
The KOI Combinatorics Lectures are funded by a grant from the National Science Foundation. Partial support provided by the UK Department of Mathematics and the UK College of Arts and Sciences.
Dec 8
Stephen Lacina
Truman State University
Posets that are CC-shellable but are not CL-shellable
We present two perhaps surprisingly small posets, one graded and one non-graded, that are CC-shellable in the sense of Kozlov, but not CL-shellable in the sense of Björner and Wachs. In the spirit of Björner and Wachs’ Recursive Atom Orders (RAO) and Hersh and Stadnyk’s Generalized Recursive Atom Orders (GRAO), we also introduce two new recursive techniques for proving that a bounded poset is shellable, one of which is equivalent to CC-shellability.
If you wish to be added to the Discrete CATS mailing list, please send an e-mail to: margaret.readdy@uky.edu
Last updated: December 4, 2025
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