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

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)

Partial support is also being provided by...

and...

University of Kentucky  College of Arts and Sciences 

F A L L   2 0 2 5

Fourth meeting:  December 6, 2025 (in-person)

Location: University of Kentucky, Lexington KY

Local Organizing Committee:
Eric Burkholder (UK), Pablo Castilla (UK), Alvaro Cornejo (UK),  Richard Ehrenborg (UK), KT Griffis (UK), Evan Henning (UK), Margaret Readdy  (UK), Doel Rivera Laboy (UK), Williem Rizer (UK), Kumar Vinayak (UK). 
 

Speakers: Bruno Benedetti (University of Miami), Jacob Matherne (NC State University), Dhruv Mubayi (University of Illinois Chicago)

There will also be a Colloquium talk on Friday, December 5, 2025 by Allen Knutson (Cornell University)

Conference Schedule:

Friday, December 5, 2025
03:14 (π time) - 03:50 pm Tea POT 745
04:10 - 05:10 pm Colloquium Talk:  Allen Knutson, "Pipe dreams, Schubert varieties, and the commuting scheme"
06:00 pm Colloquium Dinner:  We will be taking Allen out to dinner at place TBA.  

The Colloquium tea will be held in Patterson Office Tower 745 (POT 745).
The Colloquium talk will be held in Chemistry-Physics 287 (CP 287). 

Saturday, December 6, 2025
All Saturday 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.  

Talk Titles and Abstracts:

Allen Knutson, "Pipe dreams, Schubert varieties, and the commuting scheme"

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} in order 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.

Bruno Benedetti, "Simplicial complexes and decompositions of manifolds"

Every manifold can be decomposed M into balls of the same dimension such that the intersection of each ball with the union of the previous ones has a very specific topology. This is called a ‘handle decomposition of M’.  We sketch a way to characterize this notion combinatorially, that is, in the language of simplicial complexes. This allows us to answer two questions by Gil Kalai and Ed Swartz.

This is joint work with Karim Adiprasito.

Jacob Matherne, "Chow functions for partially ordered sets"

Three decades ago, Stanley and Brenti initiated the study of the Kazhdan–Lusztig–Stanley (KLS) theory.  To each kernel in a graded poset, they associate special functions called KLS polynomials.  This unifies and puts on common ground several important theories in combinatorics and representation theory: (i) the classical Kazhdan–Lusztig polynomial of a Bruhat interval in a Coxeter group, (ii) the toric g-polynomial of a polytope, and (iii) the Kazhdan–Lusztig polynomial of a matroid.

In this talk, I will develop a theory that parallels the KLS theory by introducing a new family of functions, called Chow functions, that encode various deep cohomological aspects of the above-mentioned combinatorial objects.  In the three respective examples above, the Chow function describes (i) a descent-like statistic enumerator for paths in the Bruhat graph, (ii) the enumeration of chains of faces of the polytope, and (iii) the Hilbert series of the matroid Chow ring.  This is joint work with Luis Ferroni and Lorenzo Vecchi.

Dhruv Mubayi, "Randomness and determinism in Ramsey theory"

For many decades randomness proved to be the central paradigm in Ramsey theory.  Recent results suggest, however, that optimal constructions in the field may involve a subtle blend of randomness and determinism. We will illustrate this by surveying many recent results, mainly about graph and hypergraph Ramsey numbers.

One Picture One Theorem Poster Session Presenters:

Amin Bahmanian, TBA

Allan Bickle, "Intersecting families of 3-sets"

Saúl A. Blanco, "Generating the symmetric group by three prefix reversals"

Caroline Boone, "Probabilities of randomly generating ladders: up/down numbers"

Eric Burkholder, TBA

Ce Chen, "Maximum independent sets in the middle two layers of the Boolean lattice"

Hugh Dennin, TBA

Luke Hawranick, "Ramsey numbers of matchings in ordered graphs"

Evan Henning, "The Hopf algebra of juggling"

Omid Khormali, TBA

Bowen Li, TBA

Brian Macdonald, "Using shellings to show the Whitney numbers of threshold graphs are top-heavy"

Russ May, "Alternating sign matrices, bumpless pipe dreams, and quantum knot mosaics"

Hannah Mobley, "Partizan games on integer partitions"

Asia Morgenstern, "Prime juggling patterns with a fixed state"

William Newman, "Chow rings of moduli spaces of genus 0 curves with collisions"

Doel Rivera, TBA

Pranali Sohoni, TBA

Parking:

Parking is in the Cornerstone Garage (PS #5):  Cornerstone Garage is located within Avenue of the Champions (aka Euclid Avenue), South Limestone and South Upper Streets.  It has an entrance and exit on both South Limestone and on Upper Street and is adjacent to "The Cornerstone".  (See red star on above map.). The cost for parking is $2 per hour with a $30 per exit maximum. Visitors must pay to park in the structure on weekends.  There is a possibility that parking will be free on Saturday, December 6th.

Please do not confuse this parking lot with the nearby employee lot for Dickey and Taylor Ed Buildings.  If you park there, you will be towed.

Hotels in the area:

Participants must make their own hotel reservations.  Lexington is a popular place for sporting events at Rupp Arena and the Central Bank Center (aka the Convention Center).  We advise making your reservation ASAP.

Holiday Inn Express    1000 Export Street, Lexington KY 40504 (0.7 mile from campus).  $145/night with breakfast included

Tru by Hilton University Medical Center (1.5 miles from campus); 535 Waller Avenue, Lexington KY 40504.  $111/night

Hilton Lexington/Downtown 369 W Vine St, Lexington, KY 40507 (1.1 miles from campus).  $121 (Friday), $238 (Saturday)

Hyatt Regency Lexington 401 W High St, Lexington, KY 40507 (1.1 miles from campus).  $179 (Friday), $379 (Saturday)

KOI Lunch List:  pdf 

Registration:  Is now open!  See registration form 

Funding:  The NSF DMS conference grant for the KOI Combinatorics Lectures provides partial funding for participant travel and local costs.  Priority will be placed on funding those individuals to participate who do not have access to other federal support, or who are students, post-doctoral scholars, or from underrepresented groups.  

Funding request form:  here (due November 4, 2025)

Childcare and family care conference resources:  In order to encourage participants whose attendance is contingent in having childcare or family care in place, the NSF has generously provided childcare and family care support.  Please indicate any need when you register.

Pre-registered Participants (so far):

Ayah Almousa (U Kentucky)

Igor Araujo (U Illinois Urbana-Champaign)

Amin Bahmanian (Illinois State U)

Bruno Benedetti (U Miami)

Allan Bickle (Purdue)

Saúl A. Blanco (Indiana U)

Caroline Boone (U Louisville)

Madeline Brandt (Vanderbilt)

Eric Burkholder (U Kentucky)

Pablo Castilla (U Kentucky)

Abhay Chaudhary (Ohio State U)

Ce Chen (U Illinois Urbana-Champaign)

Mihai Ciucu (Indiana U)

Alvaro Cornejo (U Kentucky)

Hugh Dennin (Ohio State U)

Anakin Dey (Ohio State U)

Richard Ehrenborg (U Kentucky)

Shanzhen Gao (Virginia State U)

William Gustafson (U Minnesota)

Luke Hawranick (U South Carolina)

Evan Henning (U Kentucky)

Tong Jin (Vanderbilt)

Eric Katz (Ohio State U)

Gary Kennedy (Ohio State U)

Omid Khormali (U Evansville)

Allen Knutson (Cornell)

Bowen Li (U Illinois Urbana-Champaign)

Brian Macdonald (U Kentucky)

Luke Martin (U Kentucky)

Jacob Matherne (NC State)

Rus May (Morehead State U)

Hannah Mobley (U Kentucky)

Asia Morgenstern (U Kentucky)

Dhruv Mubayi (U Illinois at Chicago)

Sina Nadi (Purdue)

William Newman (Ohio State U)

Prakod Ngamlamai (Ohio State U)

Antwon Park (U Kentucky)

Margaret Readdy (U Kentucky)

Doel Rivera Laboy (U Kentucky)

Williem Rizer (U Kentucky)

Pranali Sohoni (Ohio State U)

Kumar Vinayak (U Kentucky)

Minh-Quan Vo (U Illinois Urbana-Champaign)

Jacob Vogelpohl (U Kentucky)

Non-discrimination statement and conduct:  The KOI Combinatorics Lectures promotes and encourages a welcoming environment for individuals from all backgrounds.  We actively promote participation from all career stages and from all backgrounds.

Our goal is to foster a wonderful learning community and exchange of ideas.  As per Federal regulations, there is a zero-tolerance policy for discrimination on the basis of race, ethnicity, color, religion, age, national origin, veteran or service status, disability, any other protected category, or identity.  We expect all participants to conduct themselves conscientiously and respectfully. We will not allow harassment or inappropriate behavior. Conference organizers have full discretion to enforce this policy.

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

Last updated:  November 30, 2025