Colloquium Speaker:  Allen Knutson, Cornell University

"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.

This colloquium talk is part of the KOI Combinatorics Lectures.

It is supported in part by the National Science Foundation.

Colloquium Speaker:  Allen Knutson, Cornell University

"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.

This colloquium talk is part of the KOI Combinatorics Lectures.

It is supported in part by the National Science Foundation.

Saturday, Dec. 6, 2025
All Saturday lectures are held in Chem-Physics 139 (CP 139)

09:00 - 09:59 a.m.  Arrival/Registration/Meet and Greet/Poster set up time
09:59 - 10:00 a.m.  Welcome, Welcome speech
10:00 - 11:00 a.m.  Bruno Benedetti, "Simplicial complexes and decompositions of manifolds"
11:00 - 11:30 a.m.  Coffee Break
11:30 - 12:30 p.m.  Jacob Matherne, "Chow functions for partially ordered sets"
12:30 - 02:30 p.m.  Lunch Break
02:30 - 03:14 p.m.  The Koi Pond Panel
03:14 (π time) - 04:09 p.m.  Tea Time and the One Picture/One Theorem Poster Session
04:10 - 05:10 p.m.  Dhruv Mubayi, "Randomness and determinism in Ramsey theory"
05:11 - 05:20 p.m.  Conference Photo
 

Refer to the conference website for titles and abstracts.

Please note there is also a colloquium by Allen Knutson on Friday, Dec. 5, 2025.

     https://sites.google.com/view/koicombinatorics/

Saturday, Dec. 6, 2025
All Saturday lectures are held in Chem-Physics 139 (CP 139)

09:00 - 09:59 a.m.  Arrival/Registration/Meet and Greet/Poster set up time
09:59 - 10:00 a.m.  Welcome, Welcome speech
10:00 - 11:00 a.m.  Bruno Benedetti, "Simplicial complexes and decompositions of manifolds"
11:00 - 11:30 a.m.  Coffee Break
11:30 - 12:30 p.m.  Jacob Matherne, "Chow functions for partially ordered sets"
12:30 - 02:30 p.m.  Lunch Break
02:30 - 03:14 p.m.  The Koi Pond Panel
03:14 (π time) - 04:09 p.m.  Tea Time and the One Picture/One Theorem Poster Session
04:10 - 05:10 p.m.  Dhruv Mubayi, "Randomness and determinism in Ramsey theory"
05:11 - 05:20 p.m.  Conference Photo
 

Refer to the conference website for titles and abstracts.

Please note there is also a colloquium by Allen Knutson on Friday, Dec. 5, 2025.

     https://sites.google.com/view/koicombinatorics/

See schedule

https://www.ms.uky.edu/~readdy/KOI/

See schedule

https://www.ms.uky.edu/~readdy/KOI/

See schedule https://www.ms.uky.edu/~readdy/KOI/

See schedule https://www.ms.uky.edu/~readdy/KOI/

KOI Combinatorics Lectures

Speaker:  Richard Ehrenborg, University of Kentucky

Title:         Sharing pizza in n dimensions

Abstract:

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.

https://www.ms.uky.edu/~readdy/KOI/

KOI Combinatorics Lectures

Speaker:  Richard Ehrenborg, University of Kentucky

Title:         Sharing pizza in n dimensions

Abstract:

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.

https://www.ms.uky.edu/~readdy/KOI/