KOI Combinatorics Lectures
K O I   C O M B I N A T O R I C S   L E C T U R E S

In Japan koi symbolize strength, courage, patience and success through perseverance.

The KOI Combinatorics Lectures is a joint venture between combinatorialists from the Kentucky, Ohio and Indiana area.

The goal of this series of meetings is to foster and build intergenerational friendship and collaboration between researchers broadly defined (graduate students, postdocs, faculty) in the KOI area.

Organizing Committee:
Saúl A. Blanco (IU), Mihai Ciucu (IU), Richard Ehrenborg (UK), Eric Katz (OSU), Margaret Readdy (UK)

Local Organizing Committee:
Bethany Baker (UK), Richard Ehrenborg (UK), Will Gustafson (UK), Chloe Napier (UK), Zach Peterson (UK), Margaret Readdy (UK), Ben Reese (UK), Williem Rizer (UK), Martha Yip (UK)

S P R I N G   2 0 2 3

First meeting: Saturday April 1, 2023 (in-person)

Location: University of Kentucky, 114 White Hall Classroom Building, 140 Patterson Drive, Lexington KY 40506

Parking: We suggest parking in the Cornerstone Garage (Parking Structure #5). It is bounded by South Limestone, Upper Street and Euclid Avenue (also known as Avenue of the Champions). In this area these are one-way streets oriented positively, of course. Cornerstone Garage has an entrance and exit on South Limestone and on Upper Street. It is free and open to the public on weekends, beginning at 7 p.m. Friday through 10 p.m. Sunday. For Friday, the cost for parking is $2 per hour with a $16 per exit maximum.

Lei Xue (University of Michigan), Eric Katz (Ohio State University), Richard Ehrenborg (University of Kentucky).
On Friday March 31st there will also be a Colloquium talk by Mihai Ciucu (Indiana University) followed by dinner.

Conference Schedule:

Friday, March 31, 2023
03:14 (π time) - 03:50 pm Coffee/Tea, 745 Patterson Office Tower
04:00 - 05:00 pm Mihai Ciucu, Colloquium, Cruciform regions and a conjecture of Di Francesco , 214 Whitehall Classroom Bldg
06:00 - 08:00 pm, Colloquium Dinner, Location TBA

Saturday, April 1, 2023
All events held in 114 Whitehall Classroom Building
09:00 - 10:00 am, Arrival/Registration/Meet and Greet
09:59 - 10:00 am Welcome, Welcome speech
10:00 - 11:00 am, Lei Xue, A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds
11:00 - 11:30 am, Coffee Break
11:30 - 12:30 pm, Eric Katz, Models of matroids
12:30 - 02:30 pm, Lunch Break, Lunch List
02:30 - 03:00 pm, Problem Session, run by Saúl A. Blanco
03:00 - 03:30 pm, Tea time and the One Picture/One Theorem Poster Session
04:00 - 05:00 pm, Richard Ehrenborg, Sharing pizza in n dimensions
06:00 - 08:00 pm, Conference Dinner, Location TBA

Registration: Registration form

The nearest hotel is Holiday Inn Express & Suites, 1000 Export Street, Lexington KY 40504; Phone 859-389-6800
Other options include the Hyatt Regency, 401 West High Street, Lexington KY 40507; 855-516-1090, Hilton Lexington/Downtown, 369 W Vine Street, Lexington KY 40507; 859-231-9000, and 21c Museum Hotel Lexington, 167 W Main Street, Lexington KY 40507; 859-899-6800

Talk titles and Abstracts:

Mihai Ciucu, Cruciform regions and a conjecture of Di Francesco
The problem of finding formulas for the number of tilings of lattice regions goes back to the early 1900's, when MacMahon proved (in an equivalent form) that the number of lozenge tilings of a hexagon is given by an elegant product formula. In 1992, Elkies, Kuperberg, Larsen and Propp proved that the Aztec diamond (a certain natural region on the square lattice) of order n has 2n(n+1)/2 domino tilings. A large related body of work developed motivated by a multitude of factors, including symmetries, refinements and connections with other combinatorial objects and statistical physics. It was a model from statistical physics that motivated the conjecture which inspired the regions we will discuss in this talk.

In recent work on the twenty vertex model, Di Francesco was led to a conjecture which states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted Tn, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions Tn, we construct a family of cruciform regions Cm,na,b,c,d generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of the region Tn is a divisor of the number of tilings of the cruciform region C2n-1,2n-1n-1,n,n,n-2, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture.

Richard Ehrenborg, Sharing pizza in n dimensions
We introduce and prove the n-dimensional Pizza Theorem. Let H be a real n-dimensional hyperplane arrangement. If K is a convex set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement H. We prove that if H is a Coxeter arrangement different from A1n such that the group of isometries W generated by the reflections in the hyperplanes of H contains the negative of the identity map, and if K is a translate of a convex set that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement H with |H|-n an even positive integer. This is joint work with Sophie Morel and Margaret Readdy.

Eric Katz, Models of matroids
Matroids are known for having many equivalent cryptomorphic definitions, each offering a different perspective. A similar phenomenon holds in the algebraic geometric approach to them. We will discuss different ways of viewing matroids, motivated by group actions, intersection theory, and K-theory, each of which has been generalized from the realizable case to a purely combinatorial approach.

Lei Xue, A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds
In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s ≤ 2d vertices has at least φk(d+s, d) = {d+1 choose k+1} + {d choose k+1} - {d+1-k \choose k+1} k-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on d-dimensional polytopes with 2d+1 or 2d+2 vertices.

Pre-registered Participants (so far):>
Bethany Baker (U Kentucky)
Jonah Berggren (U Kentucky)
Saúl A. Blanco (Indiana U)
Seok Hyun Byun (Clemson U)
Mihai Ciucu (Indiana U)
Richard Ehrenborg (U Kentucky)
Eric Gottlieb (Rhodes College)*
Will Gustafson (U Kentucky)
Gábor Hetyei (UNC Charlotte)*
JiYoon Jung (Marshall U)*
Eric Katz (Ohio State)
Elizabeth Kelley (UIUC)*
Jeff Lagarias (U Michigan)*
Yi-Lin Lee (Indiana U)
Chloe Napier (U Kentucky)
McCabe Olsen (Rose-Hulman)
Zach Peterson (U Kentucky)
Margaret Readdy (U Kentucky)
Ben Reese (U Kentucky)
Williem Rizer (U Kentucky)
Daniel Skora (Indiana U)
MLE Slone (U Kentucky)
Lei Xue (U Michigan)
Martha Yip (U Kentucky)

* = Online participant

If you wish to be added to the KOI Combinatorics Lectures mailing list, please send an e-mail to: koi.combinatorics@gmail.com.

The KOI Combinatorics Lectures is partially supported by the University of Kentucky Mathematics Department and Richard Ehrenborg's Simons Foundation Collaboration grant.

Last updated March 8, 2023.