Organizing Committee
Abstract

The aim of this reunion meeting is to bring together the participants from the spring 2021 program “Combinatorial Algebraic Geometry” bringing together experts in both pure and applied parts of mathematics as well mathematical programmers, all working at the confluence of discrete mathematics and algebraic geometry, with the aim of creating an environment conducive to interdisciplinary collaboration.

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Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

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Monday, July 31, 2023
  • 10:00 am - 3:30 pm EDT
    Check In
  • 3:45 - 4:00 pm EDT
    Welcome
    11th Floor Lecture Hall
    • Brendan Hassett, ICERM/Brown University
  • 4:00 - 5:30 pm EDT
    Reception
    11th Floor Collaborative Space
Tuesday, August 1, 2023
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, August 2, 2023
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, August 3, 2023
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, August 4, 2023
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, August 7, 2023
  • 2:00 - 2:45 pm EDT
    Signed permutohedra
    10th Floor Classroom
    • Matthew Larson, Stanford University
    Abstract
    Postnikov has shown that generalized permutohedra, polytopes whose edges are parallel to vectors of the form e_i - e_j, have remarkable formulas for their volumes and lattice point counts. Additionally, generalized permutohedra can always be decomposed into certain easy-to-understand polytopes. I will discuss a generalization of these results to signed generalized permutohedra, polytopes whose edges are parallel to vectors of the form e_i - e_j, e_i + e_j, or e_i. Joint work with Chris Eur, Alex Fink, and Hunter Spink.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, August 8, 2023
  • 2:00 - 2:45 pm EDT
    Cross-ratio degrees and perfect matchings
    10th Floor Classroom
    • Rob Silversmith, University of Warwick
    Abstract
    Cross-ratio degrees are intersection numbers on M_{0,n}-bar indexed by certain bipartite graphs. I'll discuss several angles on cross-ratio degrees, and give a combinatorial upper bound involving perfect matchings, with a geometric proof. Time permitting, I'll give a connection to tropical intersection theory.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, August 9, 2023
  • 2:00 - 2:45 pm EDT
    Singular matroid realization spaces
    10th Floor Classroom
    • Daniel Corey, University of Wisconsin, Madison
    Abstract
    By Mnëv's universality theorem, every singularity type appears in the realization spaces of a rank 3 matroid. While the proofs of this theorem are constructive, the resulting matroids have large ground sets even for the simplest singularities. I will present recent joint work with Dante Luber (TU Berlin) where we show that, over the complex numbers, the realization spaces of rank 3 matroids on ≤ 11 elements are all smooth, but there are singular realization spaces for matroids on ≥ 12 elements. We use this result to show that the open Grassmannian is not schön in the sense of Tevelev.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, August 10, 2023
  • 11:00 - 11:45 am EDT
    Schubert calculus and bosonic operators
    10th Floor Classroom
    • Gleb Nenashev, St Petersburg University
    Abstract
    In this talk I will present a new point of view on Schubert polynomials via bosonic operators. These operators help to define everything correctly and make it possible to work in a purely combinatorial way. The main advantage of this point of view is to avoid “polynomials” and divided difference operator.
  • 11:55 am - 12:00 pm EDT
    Group Photo (Immediately After Talk)
    10th Floor Classroom
  • 2:00 - 2:45 pm EDT
    Standard monomials and Gröbner bases for positroid varieties
    10th Floor Classroom
    • Daoji Huang, University of Minnesota
    Abstract
    We give a simple and explicit characterization of the standard monomials of positroid varieties with respect to the Hodge degeneration, and give a Gröbner basis for positroid varieties. As an application, we relate promotion on rectangular-shaped semi-standard tableaux to cyclic shifts of positroid varieties.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, August 11, 2023
  • 11:00 - 11:45 am EDT
    Letter-linking - an explicit bridge between combinatorial group theory and cohomology
    10th Floor Classroom
    • Nir Gadish, The University of Michigan
    Abstract
    How can we tell if a group element is a k-fold commutator? A computable invariant of words in groups that does not vanish on k-fold commutators will help. For free groups this is achieved by "free differential calculus" due to Fox, and there is a generalization to RAAGS such as braid groups, but beyond that little is known. We introduce a complete collection of such invariant for any group, starting from the 1-st cohomology of a space, using (higher) linking numbers of letters in words due to Monroe and Sinha. This approach gives a simple algorithm of combinatorial flavor that measures "distance" between group elements, and exhibits a perfect pairing between relations in groups and higher Massey products of H^1 classes.
  • 2:00 - 2:45 pm EDT
    Vanishing theorems for matroids
    10th Floor Classroom
    • Christopher Eur, Harvard University
    Abstract
    In this informal talk, I will describe my recent work on cohomologies of tautological bundles of matroids, with an emphasis on various open questions and future directions.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, August 14, 2023
  • 2:00 - 2:45 pm EDT
    Partial resolutions of the nilpotent cone and Schur-positivity of the Delta Conjecture
    10th Floor Classroom
    • Sean Griffin, UC Davis
    Abstract
    The Delta Conjecture of Haglund, Remmel and Wilson gives two combinatorial formulas for a symmetric function in parameters q,t coming from Macdonald Theory (one of which has been proven). In this talk, we explain how the geometry of partial resolutions of the nilpotent cone developed by Borho and MacPherson can be applied to give a new simple formula for the t=0 slice of this symmetric function. Motivated by this formula, we give conjectural Schur-positive formulas for the t^1 and t^2 coefficients. A Schur expansion for the entire Delta Conjecture function remains elusive, but we give an outline of a possible geometric approach toward proving Schur positivity. Based on joint work with Maria Gillespie.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, August 15, 2023
  • 2:00 - 2:45 pm EDT
    The virtual Euler characteristic for binary matroids
    10th Floor Classroom
    • Madeline Brandt, Brown University
    Abstract
    Inspired by Kontsevich's graphic orbifold Euler characteristic we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank r. Our main result provides a simple formula for the virtual Euler characteristic for the set of isomorphism classes of binary matroids of rank r. This is joint work with Juliette Bruce and Dan Corey.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, August 16, 2023
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, August 17, 2023
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, August 18, 2023
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space

All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).

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