Algecom-XXV

Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the Midwest and beyond, with interests in algebra, geometry and combinatorics (widely interpreted). 

Algecom usually happens twice a year.

AlGeCom Committee:

 Alexander Yong  David Speyer Uli Walther Saugata Basu  Evgeny Mukhin Peter Tingley  Chris Drupieski Laura Escobar Kyle Petersen Vic Reiner Anna Weigandt

AlGeCom XXV Date:   Saturday April 5, 2025

Location: Vincent Hall, University of Minnesota, Minneapolis

All talks are in Vincent Hall 16, and registration, coffee and posters in Vincent Hall lobby.

Campus map: https://campusmaps.umn.edu/vincent-hall

The speakers will be

• Tom Braden (UMass-- Amherst)

• Sarah Brauner (Brown University)

• Colleen Robichaux (UCLA)

• Vasu Tewari (Univ. of Toronto)

Local Organizers (questions related to Algecom XXV):

Vic Reiner Anna Weigandt 

ALGECOM is partially funded by a grant from the National Science Foundation, which provides limited support for students and postdocs to offset the cost of attendance.  Everyone should apply by filling out the registration form below and indicating their interest in financial support (and we appreciate early applications). Funding decisions will be made shortly thereafter. 

Registration is free, but please use the registration link  here.

There will also be a poster fair; you can sign up to present a poster!

Recommended Parking:

 Washington Ave Ramp

 Church Street Garage

Lodging:

A block of rooms reserved at the Graduate Hotel, and here is a booking link, with a cutoff date of March 18, 2025

Guests can also call the hotel at (612) 379-8888, ask for reservations, and request the UM ALGECOM 2025 room block.

Schedule:

(All times are in the Central Time Zone)

Talks and coffee are in Vincent Hall. 

9:00-9:30am Coffee and Registration

9:30-10:30am: Tom Braden

Title: The intersection cohomology module of a matroid

Abstract: 

  The graded Möbius algebra H(M) of a matroid M is a finite-dimensional Q-algebra which encodes the structure of its lattice of flats.  The intersection cohomology module of M is a special H(M)-module IH(M) which played a key role in the proof of the Dowling-Wilson top-heaviness conjecture for geometric lattices.  The original construction of IH(M) was as a direct summand of the augmented Chow ring using a complicated inductive process.   I will present a simple module-theoretic characterization of IH(M), which leads to a more elementary construction.  This construction also generalizes IH(M) from rational coefficients to coefficients in any field k.  When k has positive characteristic, new phenomena arise.  (joint work with J. Huh, J. Matherne, N. Proudfoot and B. Wang)  

10:30-11:00am Coffee Break

11:00-12:00am: Sarah Brauner

Title: Spectrum of random-to-random shuffling in the Hecke algebra

Abstract:

    The eigenvalues of a Markov chain determine its mixing time. In this talk, I will describe a Markov chain on the symmetric group called random-to-random shuffling, whose eigenvalues have surprisingly elegant—though mysterious—formulas. In particular, these eigenvalues were shown to be non-negative integers by Dieker and Saliola in 2017, resolving an almost 20 year conjecture.

In recent work with Axelrod-Freed, Chiang, Commins and Lang, we generalize random-to-random shuffling to a Markov chain on the Type A Iwahori Hecke algebra, and prove its eigenvalues are polynomials in q with non-negative integer coefficients.. Our methods simplify the existing proof for q=1 by drawing novel connections between random-to-random shuffling and the Jucys-Murphy elements of the Hecke algebra.

12:00-1:30pm: Lunch

1:30-2:30pm: Colleen Robichaux

Title: Positivity of Schubert coefficients

Abstract:

  Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e., they are in #P. In this talk we discuss the closely related problem of the positivity of Schubert coefficients. We prove this problem has a positive rule based on two standard assumptions.

This is joint work with Igor Pak.

2:30-3:00pm Coffee Break 

3:00-4:00pm: Vasu Tewari

Title: Equivariant quasisymmetry and noncrossing partitions

Abstract::

     Quasisymmetric polynomials are a well-studied subfamily of symmetric polynomials, obtained by replacing usual symmetry by a weakened form called quasisymmetry.  Schubert polynomials comprise a distinguished basis of the polynomial ring generalizing the basis of the ring of symmetric polynomials given by Schur polynomials. 

I will present a new formula for double Schubert polynomials, which represent Schubert classes in the equivariant cohomology of the flag variety, ditching symmetry for quasisymmetry. This change in perspective allows us to build a theory of ''double forest polynomials'' exhibiting a strong parallel to that of Schubert polynomials, with permutations replaced by noncrossing partitions.  This new family of polynomials decomposes Schubert polynomials Graham-nonnegatively, contains as a subfamily a double analogue of fundamental quasisymmetric polynomials, and has an AJS-Billey-type formula for evaluations at noncrossing permutations. Time permitting, I will discuss some underlying geometric ideas.

Joint work with Nantel Bergeron (York), Lucas Gagnon (York), Philippe Nadeau (CNRS & Lyon), and Hunter Spink (Univ. of Toronto).

4:00-5:30pm Poster Fair (probably with coffee)

6:00pm-??? Dinner at Tea House .