Monday, February 1, 2021

• 9:50 - 10:00 am EST

  Welcome

  Virtual

  • Brendan Hassett, ICERM/Brown University
• 10:00 - 10:45 am EST

  Short Introduction to Hodge Structures Part 1

  Virtual

  • Colleen Robles, Duke University

  Abstract

  I will introduce Hodge structures and describe the linear structures underlying the Hard Lefschetz Theorem and Hodge--Riemann Bilinear Relations.

  • Video
  • Slides
• 11:00 am - 12:00 pm EST

  Problem Session 1

  Problem Session - Virtual

  • Slides
• 12:00 - 12:30 pm EST

  Short Introduction to Hodge Structures Part 2

  Virtual

  • Colleen Robles, Duke University

  Abstract

  I will introduce Hodge structures and describe the linear structures underlying the Hard Lefschetz Theorem and Hodge--Riemann Bilinear Relations.

  • Video
  • Slides
• 12:30 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 2:45 pm EST

  Basic Schubert Calculus Part 1

  Virtual

  • Sara Billey, University of Washington

  • Video
  • Slides
• 3:00 - 4:00 pm EST

  Problem Session 2

  Problem Session - Virtual

  • Slides
• 4:00 - 4:30 pm EST

  Basic Schubert Calculus Part 2

  Virtual

  • Sara Billey, University of Washington

  • Video
  • Slides
• 4:30 - 5:15 pm EST

  Reception

  Virtual

Tuesday, February 2, 2021

• 9:00 - 9:45 am EST

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:45 am EST

  Basic notions in cotangent Schubert calculus - Part 1

  Virtual

  • Richard Rimanyi, University of North Carolina at Chapel Hill

  Abstract

  A key notion of Schubert Calculus is the cohomology ring of a homogeneous space, together with a distinguished basis: the collection of Schubert classes. There is a one-parameter deformation of the notion "Schubert class", called Chern-Schwartz-MacPherson (CSM-) class (a.k.a. characteristic cycle class, or cohomological stable envelope). In this lecture/workshop we will define the CSM class, and illustrate many of its properties through examples.

  • Slides
• 11:00 am - 12:00 pm EST

  Problem Session 3

  Problem Session - Virtual

  • Slides
• 12:00 - 12:30 pm EST

  Basic notions in cotangent Schubert calculus - Part 2

  Virtual

  • Richard Rimanyi, University of North Carolina at Chapel Hill

  Abstract

  A key notion of Schubert Calculus is the cohomology ring of a homogeneous space, together with a distinguished basis: the collection of Schubert classes. There is a one-parameter deformation of the notion "Schubert class", called Chern-Schwartz-MacPherson (CSM-) class (a.k.a. characteristic cycle class, or cohomological stable envelope). In this lecture/workshop we will define the CSM class, and illustrate many of its properties through examples.

  • Slides
• 12:30 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 2:45 pm EST

  Matroids Part 1

  Virtual

  • Christopher Eur, Stanford University

  Abstract

  We give an introduction to matroid theory with a view towards its recent interactions with algebraic geometry. In the first half, we study matroids as combinatorial abstractions of hyperplane arrangements, which will lead us to Chow rings of matroids, modeled after the geometry of wonderful compactifications of hyperplane arrangement complements. In the second half, we give a broad survey of some recent developments involving other different but related algebro-geometric models of matroids.

  • Video
  • Slides
  • Notes
• 3:00 - 4:00 pm EST

  Problem Session 4

  Problem Session - Virtual

  • Slides
• 4:00 - 4:30 pm EST

  Matroids Part 2

  Virtual

  • Christopher Eur, Stanford University

  Abstract

  We give an introduction to matroid theory with a view towards its recent interactions with algebraic geometry. In the first half, we study matroids as combinatorial abstractions of hyperplane arrangements, which will lead us to Chow rings of matroids, modeled after the geometry of wonderful compactifications of hyperplane arrangement complements. In the second half, we give a broad survey of some recent developments involving other different but related algebro-geometric models of matroids.

  • Video
  • Slides
  • Notes

Wednesday, February 3, 2021

• 10:00 - 10:45 am EST

  Quantum Cohomology Part 1

  Virtual

  • Nicolas Perrin, Versailles Saint-Quentin-en-Yvelines University

  • Video
  • Slides
• 11:00 am - 12:00 pm EST

  Problem Session 5

  Problem Session - Virtual

  • Slides
• 12:00 - 12:30 pm EST

  Quantum Cohomology Part 2

  Virtual

  • Nicolas Perrin, Versailles Saint-Quentin-en-Yvelines University

  • Video
  • Slides
• 12:30 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 2:45 pm EST

  Affine Schubert Calculus Part 1

  Virtual

  • Mark Shimozono, Virginia Tech

  • Video
  • Slides
• 3:00 - 4:00 pm EST

  Problem Session 6

  Problem Session - Virtual

  • Slides
• 4:00 - 4:30 pm EST

  Affine Schubert Calculus Part 2

  Virtual

  • Mark Shimozono, Virginia Tech

  • Video
  • Slides
• 4:30 - 5:15 pm EST

  Gathertown Afternoon Coffee - Informal Grad Student / Postdoc Focused

  Coffee Break - Virtual

Thursday, February 4, 2021

• 9:00 - 9:45 am EST

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:45 am EST

  Introduction to Tropical Geometry Through Curves - Part 1

  Virtual

  • Madeline Brandt, Brown University

  Abstract

  Tropical geometry equips varieties with a combinatorial counterpart called the tropicalization. In this talk, I will introduce some of the key ideas in tropical geometry by studying curves. This will include definitions and examples of embedded tropicalization for curves in the plane, abstract tropicalization / dual graphs, and Berkovich skeleta.

  • Video
  • Slides
• 11:00 am - 12:00 pm EST

  Problem Session 7

  Problem Session - Virtual

  • Slides
• 12:00 - 12:30 pm EST

  Introduction to Tropical Geometry Through Curves - Part 2

  Virtual

  • Madeline Brandt, Brown University

  Abstract

  Tropical geometry equips varieties with a combinatorial counterpart called the tropicalization. In this talk, I will introduce some of the key ideas in tropical geometry by studying curves. This will include definitions and examples of embedded tropicalization for curves in the plane, abstract tropicalization / dual graphs, and Berkovich skeleta.

  • Video
  • Slides
• 12:30 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 2:45 pm EST

  Moduli spaces of tropical curves Part 1

  Virtual

  • Sam Payne, University of Texas at Austin

  Abstract

  In the first part of this talk, I will introduce the moduli space of stable tropical curves and discuss its combinatorial structure and topology, illustrating with simple examples in low genus. In the problem session, you will compute one further example. And in the second half of the talk, I will explain some general results about the topology of moduli spaces of stable tropical curves, how these relate to the topology of the algebraic moduli spaces M_g and M_{g,n}, and then state some open problems and conjectures that you may find interesting to think about during the semester.

  • Video
  • Slides
• 3:00 - 4:00 pm EST

  Problem Session 8

  Problem Session - Virtual

  • Slides
• 4:00 - 4:30 pm EST

  Moduli spaces of tropical curves Part 2

  Virtual

  • Sam Payne, University of Texas at Austin

  Abstract

  In the first part of this talk, I will introduce the moduli space of stable tropical curves and discuss its combinatorial structure and topology, illustrating with simple examples in low genus. In the problem session, you will compute one further example. And in the second half of the talk, I will explain some general results about the topology of moduli spaces of stable tropical curves, how these relate to the topology of the algebraic moduli spaces M_g and M_{g,n}, and then state some open problems and conjectures that you may find interesting to think about during the semester.

  • Video
  • Slides

Friday, February 5, 2021

• 10:00 - 10:45 am EST

  Cluster structures in commutative rings Part 1

  Virtual

  • Lauren Williams, Harvard University

  Abstract

  In my first talk, I will give a gentle introduction to cluster algebras. In the second talk, I will describe how to identify a commutative ring (such as the coordinate ring of an algebraic variety) with a cluster algebra, and provide several examples. I will also discuss how to describe cluster algebras by generators and relations.

  • Video
  • Slides
• 11:00 am - 12:00 pm EST

  Problem Session 9

  Problem Session - Virtual

  • Slides
• 12:00 - 12:30 pm EST

  Cluster structures in commutative rings Part 2

  Virtual

  • Lauren Williams, Harvard University

  Abstract

  In my first talk, I will give a gentle introduction to cluster algebras. In the second talk, I will describe how to identify a commutative ring (such as the coordinate ring of an algebraic variety) with a cluster algebra, and provide several examples. I will also discuss how to describe cluster algebras by generators and relations.

  • Video
  • Slides
• 12:30 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 2:45 pm EST

  Cluster varieties Part 1

  Virtual

  • David Speyer, University of Michigan

  Abstract

  We will discuss the geometry of the affine algebraic varieties associated to cluster algebras. In the first hour, we will give examples and talk about open sets, smoothness and covering maps; in the second hour, we will talk about mixed Hodge structures.

  • Video
  • Slides
• 3:00 - 4:00 pm EST

  Problem Session 10

  Problem Session - Virtual

  • Slides
• 4:00 - 4:45 pm EST

  Cluster varieties Part 2

  Virtual

  • David Speyer, University of Michigan

  Abstract

  We will discuss the geometry of the affine algebraic varieties associated to cluster algebras. In the first hour, we will give examples and talk about open sets, smoothness and covering maps; in the second hour, we will talk about mixed Hodge structures.

  • Video
  • Slides

Monday, February 8, 2021

• 11:00 - 11:30 am EST

  Welcome to ICERM

  Welcome - Virtual

  • Brendan Hassett, ICERM/Brown University
• 1:30 - 2:00 pm EST

  Organizer / Directorate Meeting

  Meeting - Virtual

Tuesday, February 9, 2021

• 2:30 - 3:50 pm EST

  Brown Graduate Course: Algebraic Geometry

  Virtual

  • Jeremy Usatine, Brown University

Wednesday, February 10, 2021

• 11:00 am - 12:00 pm EST

  Graduate Students/Postdocs Meeting with Directorate

  Meeting - Virtual
• 3:30 - 4:30 pm EST

  Degeneracy Loci and Schubert Polynomials

  Virtual

  • William Fulton, University of Michigan

  Abstract

  This talk emphasizes how finding formulae for degeneracy loci leads naturally to the algebra of Schubert polynomials. We'll see how this illuminates the study of Schubert polynomials in the 19th, 20th, and 21st century. (Joint work with Dave Anderson.)

  • Video

Thursday, February 11, 2021

• 2:30 - 3:50 pm EST

  Brown Graduate Course: Algebraic Geometry

  Virtual

  • Jeremy Usatine, Brown University
• 4:30 - 5:15 pm EST

  Hodge-Riemann relations and Lorentzian polynomials

  Virtual

  • Christopher Eur, Stanford University

  Abstract

  We introduce the theory of Lorentzian polynomials, motivated by the Hodge-Riemann relations and the resulting properties that cohomology rings of smooth complex projective varieties satisfy.

  • Video
  • Slides

Friday, February 12, 2021

• 10:30 - 11:30 am EST

  Post Doc/Graduate Student Seminar

  Virtual
• 11:30 am - 12:00 pm EST

  Graduate Student/postdoc post-seminar break in Gather

  Coffee Break - Virtual
• 4:30 - 5:00 pm EST

  Hodge-Riemann relations and Lorentzian polynomials cont.

  Virtual

  • Christopher Eur, Stanford University

  Abstract

  We introduce the theory of Lorentzian polynomials, motivated by the Hodge-Riemann relations and the resulting properties that cohomology rings of smooth complex projective varieties satisfy.

  • Video
  • Slides

Monday, February 15, 2021

• 9:45 - 10:00 am EST

  Welcome

  Virtual

  • Brendan Hassett, ICERM/Brown University
• 10:00 - 10:30 am EST

  A brief tour of Sage

  Virtual

  • Nicolas Thiéry, Université Paris Sud

  Abstract

  I will offer a brief tour of Sage, showcasing some features and use cases, hinting at its development model, pointing to some recent trends, and highlighting how it fits within the larger ecosystem of free computational (mathematics) software.

  • Video
• 10:45 - 11:15 am EST

  Rings and fields in Sage

  Virtual

  • David Roe, Massachusetts Institute of Technology

  Abstract

  I will give an introduction to basic algebraic structures in Sage, with a focus on the coercion model, finite fields and extensions of rings. I will also give an overview of how you can contribute to Sage.

  • Video
  • Slides
• 11:15 - 11:30 am EST

  Coffee Break

  Virtual
• 11:30 am - 12:00 pm EST

  Celestial mechanics via tropical geometry (gfan and Macaulay2)

  Virtual

  • Anton Leykin, Georgia Tech

  • Video
• 12:15 - 12:45 pm EST

  FusionRings in Sage 9.2

  Virtual

  • Daniel Bump, Stanford University

  Abstract

  The FusionRing class implements useful methods for Verlinde Algebras. These are elegant rings similar to WeylCharacterRings (representation rings of Lie groups) except that the fusion categories have only finitely many objects. These rings have applications to conformal field theory, quantum groups, topological quantum computing and knot theory. Most of the methods needed to work with these have been implemented in Sage 9.2. We will review the math and show what the code can do. The FusionRing code is joint work with Guillermo Aboumrad.

  • Video
  • Slides
• 1:00 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 3:00 pm EST

  Gathertown Welcome Reception

  Reception - Virtual
• 3:00 - 4:00 pm EST

  Sage/Oscar Installation Help

  Tutorial - Virtual

Tuesday, February 16, 2021

• 9:00 - 9:45 am EST

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:30 am EST

  OSCAR - The Project

  Virtual

  • Michael Joswig, TU Berlin & MPI Leipzig

  Abstract

  The OSCAR project is a collaborative effort to shape a new computer algebra system, written in Julia. OSCAR is built on top of the four "cornerstone systems" ANTIC (for number theory), GAP (for group and representation theory), polymake (for polyhedral and tropical geometry) and Singular (for commutative algebra and algebraic geometry). We present three examples to showcase the current version 0.5.1. This is joint work with The OSCAR Development Team.

  • Video
  • Slides
  • OSCAR System
• 10:45 - 11:15 am EST

  OSCAR - Selected Features

  Virtual

  • Daniel Schultz, Technische Universität Kaiserslautern

  Abstract

  Introducing OSCAR, a new computer algebra system combining GAP, Polymake, Hecke and Singular.

  • Slides
• 11:15 - 11:30 am EST

  Coffee Break

  Virtual
• 11:30 am - 12:00 pm EST

  Computing the Newton polytope of a large discriminant

  Virtual

  • Lars Kastner, Institute of Mathematics of the Technical University

  Abstract

  The Newton polytope of the discriminant of a quaternary cubic form has 166'104 vertices. One way to obtain these vertices is to enumerate all D-equivalence classes of regular triangulations of the 3- dilated tetrahedron. The only known way to do this is to enumerate all regular triangulations of the 3-dilated tetrahedron and group them into classes in a second step. This talk will focus on the computations carried out to arrive at this result. It involved the use of polymake and mptopcom on large computing clusters in parallel which in turn brought other obstacles. This software can also be used via polymake.jl in OSCAR. Since computer experiments in algebraic geometry are becoming larger and larger, this talks aims at providing insights on how to set up these experiments such that they give reliable results, and how to avoid the pitfalls we encountered. This is joint work with Robert Loewe.

  • Video
  • Slides
• 12:15 - 12:45 pm EST

  Some hybrid symbolic-numeric methods in algebraic geometry

  Virtual

  • Jonathan Hauenstein, University of Notre Dame

  Abstract

  On the theoretical side, algebraic geometry combines aspects of algebra and geometry to provide many tools to prove new results. On the computational side, symbolic computations typically based on algebra and numerical computations typically based on geometry can be combined to provide many new computational tools to study a variety of problems in algebraic geometry. This talk will explore some hybrid symbolic-numeric methods and applications in computational algebraic geometry.

  • Video
• 1:00 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 3:00 pm EST

  Problem Session

  Virtual
• 3:00 - 4:00 pm EST

  Contributing to Sage Tutorial

  Tutorial - Virtual

Wednesday, February 17, 2021

• 9:00 - 9:45 am EST

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:45 am EST

  Parallelization of Triangular Decompositions- Design and implementation with the BPAS library

  Virtual

  • Marc Moreno, University of Western Ontario

  Abstract

  We discuss the parallelization of algorithms for solving polynomial systems by way of triangular decomposition. The "Triangularize" algorithm proceeds through incremental intersections of polynomials to produce the different components of the solution set. Independent components imply the opportunity for concurrency. This "component-level" parallelization of triangular decompositions, our focus here, belongs to the class of dynamic irregular parallelism. Potential parallel speed-up depends only on geometrical properties of the solution set (number of components, their dimensions and degrees); these algorithms do not scale with the number of processors. To manage the irregularities of component-level parallelization we combine different concurrency patterns: map, workpile, producer-consumer, pipeline and fork/join. We report on our implementation in the freely available BPAS library. Comprehensive experimentation with thousands of polynomial systems yields examples with up to 10.8-times speed up on a 12-core machine.
• 11:00 - 11:30 am EST

  Coffee Break

  Virtual
• 11:30 am - 12:00 pm EST

  Rational integrals and periods with Sagemath and Julia

  Virtual

  • Pierre Lairez, INRIA

  Abstract

  Based on symbolic integration and numerical analytic continuation, we can compute to high precision integrals of multivariate rational functions. I will show applications to volume computation and to the study of quartic surfaces. I will emphasize on some software aspects, specific to Sagemath and Julia.
• 12:15 - 12:45 pm EST

  Generalized cohomology quotients of the symmetric functions

  Virtual

  • Darij Grinberg, Drexel University

  • Video
• 1:00 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 2:40 pm EST

  Lightning Talks

  Virtual

  • Adam Afandi, Colorado State University
  • Jose Brox, Centre for Mathematics of the University of Coimbra
  • Juliette Bruce, University of California, Berkeley / MSRI
  • Laura Brustenga i Moncusi, University of Copenhagen
  • Taylor Brysiewicz, Max Planck Institute for Mathematics in the Sciences
  • Papri Dey, University of Missouri
  • Sean Griffin, Brown University
  • Shinyoung KIM, Institute for Basic Science Center for Geometry and Physics

  • Slides
  • Abstracts
• 2:40 - 2:50 pm EST

  Coffee Break

  Virtual
• 2:50 - 3:30 pm EST

  Lightning Talks

  Virtual

  • Lukas Kühne, Max Planck Institute for Mathematics in the Sciences
  • Jianping Pan, University of California, Davis
  • Marta Panizzut, TU Berlin
  • Theodoros Stylianos Papazachariou, University of Essex
  • Colleen Robichaux, University of Illinois at Urbana-Champaign
  • Mahrud Sayrafi, University of Minnesota
  • Weihong Xu, Rutgers

  • Slides
  • Abstracts
• 3:30 - 4:30 pm EST

  Code Demonstrations

  Tutorial - Virtual

Thursday, February 18, 2021

• 9:00 - 9:45 am EST

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:45 am EST

  msolve - A Library for Solving Polynomial Systems

  Virtual

  • Christian Eder, University of Kaiserslautern

  Abstract

  We present a new open source C library msolve dedicated to solve multivariate polynomial systems exactly through computer algebra methods. The core algorithmic framework of msolve relies on Gröbner bases and linear algebra based algorithms for polynomial system solving. It relies on Gröbner basis computation w.r.t. the degree reverse lexicographical order, Gröbner conversion to a lexicographical Gröbner basis and real solving of univariate polynomials. We explain in detail how these three main steps of the solving process are implemented exploiting the computational capabilities of the framework. We compare the practical performance of the different parts of msolve with similar functionalities of leading computer algebra systems such as Magma and Maple on a wide range of polynomial systems with a particular focus on those which have finitely many complex solutions, showing that msolve can tackle systems which were out of reach by the software state-of-the-art. This is joint work with Jérémy Berthomieu, Jean-Charles Faugère and Mohab Safey El Din from the PolSys Team at the Sorbonne Université in Paris.

  • Video
• 11:00 - 11:30 am EST

  Parallelism in Algebraic Geometry - Examples with Singular and GPI-Space

  Virtual

  • Anne Frühbis-Krüger, University of Oldenburg

  Abstract

  I shall illustrate the use of the Singular - GPI-space interplay in some examples including a smoothness test, GIT-fans and desingularization.

  • Slides
• 11:45 am - 12:45 pm EST

  Coffee Break

  Virtual
• 12:45 - 1:15 pm EST

  Computing Galois groups in enumerative geometry

  Virtual

  • Frank Sottile, Texas A&M University

  • Video
• 1:30 - 2:30 pm EST

  Lunch/Free Time

  Virtual
• 2:30 - 3:30 pm EST

  Problem Session

  Virtual
• 3:30 - 4:30 pm EST

  Code Demonstrations

  Tutorial - Virtual

Friday, February 19, 2021

• 9:00 - 9:45 am EST

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:30 am EST

  Factorizations into irreducibles and polytopes

  Virtual

  • Tommy Hofmann, TU Kaiserslautern

  Abstract

  Dedekind domains form a family of commutative rings that plays an important role in algebraic geometry and number theory. While elements of Dedekind domains factor into irreducible elements, such a factorization is in general not unique. We present an algorithm, which for a given element of the ring of integers of a number field, determines all factorizations into irreducible elements. The algorithm makes heavy use of computations with polytopes and is implemented in Oscar. This is joint work with Claus Fieker.
• 10:45 - 11:30 am EST

  Computational challenges for tropical del Pezzo surfaces

  Virtual

  • María Angélica Cueto, Ohio State University

  Abstract

  A smooth degree d del Pezzo surface is obtained by blowing up the projective plane at (9-d) generic points. In this talk, we will discuss how to tropicalize these surfaces for various embeddings as we vary the input points and the computational challenges that arise when doing so.
• 11:15 - 11:30 am EST

  Coffee Break

  Virtual
• 11:30 am - 12:00 pm EST

  Presenting the multipolynomial bases package

  Virtual

  • Viviane Pons, Université Paris Sud

  Abstract

  In this talk, we present an external SageMath package to work on multivariate polynomials seen as an algebra over integer vectors (the exponents). This allows for manipulation of divided differences operators and the definition of many bases of multivariate polynomials such as the Schubert polynomials, Grothendieck, and Demazure Characters.

  • Video
• 12:15 - 12:45 pm EST

  Hasse-Witt matrices and mirror toric pencils

  Virtual

  • Adriana Salerno, Bates College

  • Video
• 1:00 - 2:00 pm EST

  Lunch/Free Time

  Virtual
• 2:00 - 3:00 pm EST

  Gathertown Closing Reception

  Reception - Virtual

Monday, February 22, 2021

• 11:00 am - 12:00 pm EST

  Grad Student/ PostDoc Professional Development: Ethics 1

  Professional Development - Virtual

Wednesday, February 24, 2021

• 3:30 - 4:30 pm EST

  Log-Concavity of Littlewood-Richardson Coefficients

  Virtual

  • Avery St. Dizier, Univeristy of Illinois at Urbana-Champaign

  Abstract

  We describe a log-concavity property of the Littlewood--Richardson numbers and explain its connection to the theory of Lorentzian polynomials.

  • Video
  • Slides

Thursday, February 25, 2021

• 10:00 - 11:00 am EST

  Learning Seminar - Differential forms on tropical moduli spaces

  Seminar - Virtual

  • Melody Chan, Brown University
  • Sam Payne, University of Texas at Austin

  Abstract

  The goal is to understand Francis Brown's recent arXiv preprint Invariant differential forms on complexes of graphs and Feynman integrals. A sub-goal, which we hope takes shape over the course of the seminar, is to draw out the connection between Brown's article and tropical moduli spaces of curves and abelian varieties.
  
  This is not an introductory seminar per se in that the choice of topic is of specialized interest and of particular interest to us vis-a-vis our current work. On the other hand, we will try to develop the machinery without relying on background from tropical geometry, much as Brown's article is able to do.
  
  Resources to get started:
  
  1. Sam's lectures from the introductory Workshop, available from the webpage
  
  2. Draft of a survey articleon classical/tropical moduli spaces by Melody, which outlines the appearance of graph complexes in the study of tropical moduli.
• 11:00 am - 12:00 pm EST

  Semistable Reduction Seminar

  Seminar - Virtual

  • Dan Abramovich, Brown University

Friday, February 26, 2021

• 10:00 - 10:30 am EST

  Graduate Student/ PostDoc pre-seminar break in Gather

  Coffee Break - Virtual
• 10:30 - 11:30 am EST

  CM-regularity

  Post Doc/Graduate Student Seminar - Virtual

  • Matthew Larson, Stanford University
  • Mahrud Sayrafi, University of Minnesota

  Abstract

  The geometry of Castelnuovo-Mumford regularity - Matt Larson, Stanford University
  
  Castelnuovo-Mumford regularity is a tool for proving effective versions of Serre vanishing. I will explain why this is frequently necessary throughout algebraic geometry. I will also define Castelnuovo-Mumford regularity, state its basic properties, and explain its relationship to syzygies of graded module.
  
  Computing the Multigraded Castelnuovo-Mumford Regularity - Mahrud Sayrafi, University of Minnesota
  
  Motivated by toric geometry, Maclagan-Smith defined the multigraded Castelnuovo- Mumford regularity. While this definition reduces to the usual regularity for saturated modules in P^n, some properties of the classical regularity are not yet proven to be true in the multigraded case, while others hold true with subtle differences that illuminate the differences in geometry. In this talk I will focus on the case of a product of projective spaces and describe an algorithm for computing the multigraded regularity for saturated modules implemented in Macaulay2.
• 11:30 am - 12:00 pm EST

  Graduate Student/ PostDoc post-seminar break in Gather

  Coffee Break - Virtual

Monday, March 1, 2021

• 11:00 am - 12:00 pm EST

  Professional Development: Ethics II

  Professional Development - Virtual

Wednesday, March 3, 2021

• 3:00 - 3:30 pm EST

  Gathertown Afternoon Coffee

  Coffee Break - Virtual

• 4:00 - 5:00 pm EST

  Quantum integrability and Grassmannians

  Virtual

  • Paul Zinn-Justin, The University of Melbourne

  Abstract

  We will investigate in the simplest setting, how an``R-matrix'' (the building block of ``quantum integrable systems'') is attached to the equivariant cohomology of Grassmannians. We will compute the R-matrix in the case of CP^1 and discuss how the result generalizes to arbitrary Grassmannians. As an application, we shall derive the AJSBilley formula (restriction of Schubert classes to fixed points).

  • Video

Thursday, March 4, 2021

• 10:00 - 11:00 am EST

  Learning Seminar - Differential forms on tropical moduli spaces

  Seminar - Virtual

  • Melody Chan, Brown University
  • Sam Payne, University of Texas at Austin

  Abstract

  The goal is to understand Francis Brown's recent arXiv preprint Invariant differential forms on complexes of graphs and Feynman integrals. A sub-goal, which we hope takes shape over the course of the seminar, is to draw out the connection between Brown's article and tropical moduli spaces of curves and abelian varieties.
  
  This is not an introductory seminar per se in that the choice of topic is of specialized interest and of particular interest to us vis-a-vis our current work. On the other hand, we will try to develop the machinery without relying on background from tropical geometry, much as Brown's article is able to do.
  
  Resources to get started:
  
  1. Sam's lectures from the introductory Workshop, available from the webpage
  
  2. Draft of a survey articleon classical/tropical moduli spaces by Melody, which outlines the appearance of graph complexes in the study of tropical moduli.
• 11:00 am - 12:00 pm EST

  Semistable Reduction Seminar

  Seminar - Virtual

  • Dan Abramovich, Brown University

Friday, March 5, 2021

• 10:00 - 10:30 am EST

  Graduate Student/ PostDoc pre-seminar break in Gather

  Coffee Break - Virtual
• 10:30 - 11:30 am EST

  Tropical Geometry

  Post Doc/Graduate Student Seminar - Virtual

  • Claudia Yun, Brown University

  Abstract

  The $S_n$-equivariant rational homology of the tropical moduli spaces $\Delta_{2,n}$ Abstract: Fix non-negative integers $g$ and $n$ such that $2g-2+n>0$, the tropical moduli space $\Delta_{g,n}$ is a topological space that parametrizes isomorphism classes of $n$-marked stable tropical curves of genus $g$ with total edge length 1. These spaces are important because their reduced rational homology is identified equivariantly with the top weight cohomology of the algebraic moduli spaces $M_{g,n}$, with respect to the $S_n$-action of permuting marked points. In this talk, we focus on the genus 2 case and compute the characters of $\tilde{H}_i(\Delta_{2,n};\Q)$ as $S_n$-representations for $n$ up to 8. To carry out the computations, we use a cellular chain complex for symmetric $\Delta$-complexes, a technique developed by Chan, Glatius, and Payne. The computation is done in SageMath.
• 11:30 am - 12:00 pm EST

  Graduate Student/ PostDoc post-seminar break in Gather

  Coffee Break - Virtual

Wednesday, March 10, 2021

• 3:00 - 3:30 pm EST

  Gathertown Afternoon Coffee

  Coffee Break - Virtual

• 3:30 - 4:30 pm EST

  Springer fibers and the Delta Conjecture

  Virtual

  • Sean Griffin, Brown University

  Abstract

  Springer fibers are a family of varieties that have remarkable connections to representation theory and combinatorics. Springer constructed an action of the symmetric group on the cohomology ring of a Springer fiber, and used it to geometrically construct the Specht modules (in type A), which are the irreducible representations of the symmetric group. In this talk, I will survey some of the many nice properties of Springer fibers. I will then introduce a new family of varieties generalizing the Springer fibers, and show how they are connected to the (recently proved) Delta Conjecture from algebraic combinatorics. We’ll then use these varieties to geometrically construct the induced Specht modules. This is joint work with Jake Levinson and Alexander Woo.

  • Video

Thursday, March 11, 2021

• 10:00 - 11:00 am EST

  Learning Seminar - Differential forms on tropical moduli spaces

  Seminar - Virtual

  • Melody Chan, Brown University
  • Sam Payne, University of Texas at Austin

  Abstract

  The goal is to understand Francis Brown's recent arXiv preprint Invariant differential forms on complexes of graphs and Feynman integrals. A sub-goal, which we hope takes shape over the course of the seminar, is to draw out the connection between Brown's article and tropical moduli spaces of curves and abelian varieties.
  
  This is not an introductory seminar per se in that the choice of topic is of specialized interest and of particular interest to us vis-a-vis our current work. On the other hand, we will try to develop the machinery without relying on background from tropical geometry, much as Brown's article is able to do.
  
  Resources to get started:
  
  1. Sam's lectures from the introductory Workshop, available from the webpage
  
  2. Draft of a survey articleon classical/tropical moduli spaces by Melody, which outlines the appearance of graph complexes in the study of tropical moduli.
• 11:00 am - 12:00 pm EST

  Semistable Reduction Seminar

  Seminar - Virtual

  • Dan Abramovich, Brown University

Friday, March 12, 2021

• 10:00 - 10:30 am EST

  Graduate Student/ PostDoc pre-seminar break in Gather

  Coffee Break - Virtual
• 10:30 - 11:30 am EST

  Matroids

  Post Doc/Graduate Student Seminar - Virtual

  • Ahmed Umer Ashraf, University of Western Ontario
  • Alex McDonough, Brown University

  Abstract

  Arithmetic Matroids Alex McDonough Alex will introduce matroids and show the relationship between orientable arithmetic matroids and the row lattice of integer matrices. Ahmed will give an understanding of some coefficients of the Tutte polynomial of a matroid as tropical intersection numbers. Tropical Intersection Numbers Ahmed Umer Ashraf
• 11:30 am - 12:00 pm EST

  Graduate Student/ PostDoc post-seminar break in Gather

  Coffee Break - Virtual

Monday, March 15, 2021

• 11:00 am - 12:00 pm EDT

  Professional Development: Job Applications

  Professional Development - Virtual

Wednesday, March 17, 2021

• 3:00 - 3:30 pm EDT

  Gathertown Afternoon Coffee

  Coffee Break - Virtual

• 3:30 - 4:30 pm EDT

  Schubert Products for Permutations with Separated Descents

  Virtual

  • Daoji Huang, Brown University

  Abstract

  We say that two permutations w and v have separated descents at position k if w has no descents before k and v has no descents after k. We give a counting formula in terms of reduced word tableaux for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents. This generalizes previous results by Kogan '00, rediscovered using different methods by Knutson-Yong '04, Lenart '10, and Assaf '17, that solved special cases of this separated descent problem where one of the permutations is required to have a single descent. Our approach uses generalizations of Schutzenberger's jeu de taquin and the Edelman-Greene correspondence via bumpless pipe dreams.

  • Video

Thursday, March 18, 2021

• 10:00 - 11:00 am EDT

  Learning Seminar - Differential forms on tropical moduli spaces

  Seminar - Virtual

  • Melody Chan, Brown University
  • Sam Payne, University of Texas at Austin

  Abstract

  The goal is to understand Francis Brown's recent arXiv preprint Invariant differential forms on complexes of graphs and Feynman integrals. A sub-goal, which we hope takes shape over the course of the seminar, is to draw out the connection between Brown's article and tropical moduli spaces of curves and abelian varieties.
  
  This is not an introductory seminar per se in that the choice of topic is of specialized interest and of particular interest to us vis-a-vis our current work. On the other hand, we will try to develop the machinery without relying on background from tropical geometry, much as Brown's article is able to do.
  
  Resources to get started:
  
  1. Sam's lectures from the introductory Workshop, available from the webpage
  
  2. Draft of a survey articleon classical/tropical moduli spaces by Melody, which outlines the appearance of graph complexes in the study of tropical moduli.
• 11:00 am - 12:00 pm EDT

  Semistable Reduction Seminar

  Seminar - Virtual

  • Dan Abramovich, Brown University

  • Abstract

Friday, March 19, 2021

• 10:00 - 10:30 am EDT

  Graduate Student/postdoc pre-seminar break in Gather

  Coffee Break - Virtual
• 10:30 - 11:30 am EDT

  Matroids and Algebra

  Post Doc/Graduate Student Seminar - Virtual

  • Lukas Kühne, Max Planck Institute for Mathematics in the Sciences

  Abstract

  A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. I will discuss how matroid theory interacts with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids. A main application is concerned with generalized matroid representations over division rings and matrix rings and their relation to group theory.
• 11:30 am - 12:00 pm EDT

  Graduate Student/postdoc post-seminar break in Gather

  Coffee Break - Virtual

Monday, March 22, 2021

• 10:00 - 10:15 am EDT

  Welcome

  Virtual

  • Brendan Hassett, ICERM/Brown University
• 10:15 - 10:45 am EDT

  Hey Series, Tell Me About the Extended Delta Conjecture

  Virtual

  • Speaker
  • Jennifer Morse, University of Virginia

  • Session Chair
  • Leonardo Mihalcea, Virginia Polytechnic Institute and State University (Virtual)

  Abstract

  The space of diagonal harmonics $DH_n$ is an $S_n$-module in two sets of $n$ variables, which arose from a representation theoretic program to study Macdonald polynomials initiated by Garsia and Haiman. The doubly graded character of $DH_n$ has both a symmetric function and a combinatorial description: Haiman proved it is $\Delta'_{e_{n-1}}e_n$ for an eigenoperator $\Delta'$ of modified Macdonald polynomials, while Carlsson and Mellit established the combinatorial ``Shuffle Conjecture"" of Haiman, Haglund, Loehr, Remmel, and Ulyanov, expressing it as as a sum of LLT polynomials over Dyck paths. An expanded investigation led Haglund, Remmel and Wilson to the Extended Delta Conjecture, a combinatorial prediction for $\Delta_{h_l}\Delta'_{e_k}e_n$, allowing for any $0\leq l,k<n$. Bringing in new results about the action of the elliptic Hall algebra on symmetric functions, we reformulate the conjecture as the polynomial truncation of an identity of infinite series of $GL_l$ characters, expressed in terms of LLT series. The stronger infinite series identity is not difficult to prove using identities on non-symmetric Hall-Littlewood polynomials. This is joint work with Blasiak, Haiman, Pun, and Seelinger.
• 11:00 - 11:15 am EDT

  Break

  Coffee Break - Virtual
• 11:15 - 11:45 am EDT

  Castelnuovo-Mumford regularity of matrix Schubert varieties

  Virtual

  • Speaker
  • Oliver Pechenik, University of Waterloo

  • Session Chair
  • Leonardo Mihalcea, Virginia Polytechnic Institute and State University (Virtual)

  Abstract

  Jenna Rajchgot observed that the Castelnuovo-Mumford regularity of matrix Schubert varieties is computed by the degrees of the corresponding Grothendieck polynomials. We give a formula for these degrees. Indeed, we compute the leading terms of the top degree pieces of Grothendieck polynomials and give a complete description of when two Grothendieck polynomials have the same top degree piece (up to scalars). Our formulas rely on some new facts about major index of permutations.

  • Video
  • Slides
• 11:45 am - 2:00 pm EDT

  Lunch/Free Time

  Virtual
• 2:00 - 2:30 pm EDT

  Conformal blocks: an overview

  Virtual

  • Speaker
  • Chiara Damiolini, Rutgers University

  • Session Chair
  • June Huh, Stanford University (Virtual)

  Abstract

  In this talk I will introduce simple examples of sheaves of conformal blocks arising from representations of Lie algebras and discuss how they can be used to study certain moduli spaces. I will further discuss generalizations of these sheaves which go beyond the representation theory of Lie algebras, and related works in progress.

  • Video
  • Slides
• 2:45 - 3:15 pm EDT

  Quantum geometric Satake at a root of unity and Schubert calculus for G(m,m,n)

  Virtual

  • Speaker
  • Ben Elias, University of Oregon

  • Session Chair
  • June Huh, Stanford University (Virtual)

  Abstract

  There is a q-deformation of the reflection representation of the affine Weyl group in type A which leads to a q-deformation of many common constructions: Demazure operators, Soergel bimodules, geometric Satake, etcetera. When q is set to a root of unity, the action of the affine Weyl group factors through a finite quotient, the complex reflection group G(m,m,n), and a new kind of "Schubert calculus" appears. This talk will demonstrate this unusual construction in the well-understood case of affine A_1 and the mysterious case of affine A_2.

  • Video
  • Slides
• 3:15 - 4:30 pm EDT

  Gathertown Reception

  Reception - Virtual

Tuesday, March 23, 2021

• 9:00 - 9:45 am EDT

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 9:45 - 10:15 am EDT

  Geometry of semi-infinite flag manifolds

  Virtual

  • Speaker
  • Syu Kato, Kyoto University

  • Session Chair
  • Thomas Lam, University of Michigan (Virtual)

  Abstract

  Semi-infinite flag manifolds are variants of affine flag manifolds whose Schubert cells are simultaneously infinite-dimensional and infinite-codimensional. Such objects are introduced by Drinfeld and Lusztig around 1980 (as sets), and its finite-dimensional approximation model was presented by Finkelberg-Mirkovic in 1999. Recently, we described them as an explicit ind-scheme of ind-infinite type that satisfies a certain universal property. It lead us to a description of quantum $K$-groups of partial flag manifolds and some functorial relations between them. In this talk, I will start from a brief review of affine flag manifolds/varieties, define semi-infinite flag manifolds and their natural subschemes, and then explain their combinatorial and algebro-geometric properties (trying to stress the difference from affine flag manifolds/varieties). If time allows, then I will explain how some of these properties are derived as a shadow of the homological properties of affine Lie algebras. This talks is mainly based on arXiv:1810.07106.

  • Video
  • Slides
• 10:30 - 10:45 am EDT

  Break

  Coffee Break - Virtual
• 10:45 - 11:35 am EDT

  Lightning Talks

  Virtual

  • Speakers
  • Aram Bingham, Tulane University
  • Papri Dey, University of Missouri
  • Yifeng Huang, University of Michigan
  • Michael Perlman, Queen's University
  • Semin Yoo, University of Rochester

  • Session Chair
  • David Anderson, Ohio State University (Virtual)

  Abstract

  Combinatorics of quadratic spaces over finite fields
  Semin Yoo, University of Rochester
  Let $\mathbb{F}_q^n$ be the finite field with $q$ elements, where the char$F$ is not $2$. The size of the set of $k$-dimensional subspaces of $\mathbb{F}_q^n$ is called $q$-binomial coefficient, (or Gaussian binomial coefficient.) which has various interesting combinatorial descriptions and related works. Furthermore, the fact that makes $q$-binomial coefficients more interesting is that $q$-binomial coefficients have $q$-analogues. $\mathbf{q}$-\textbf{analogues} of quantities in mathematics involve perturbations of classical quantities using the parameter $q$ and revert to the original quantities when $q$ goes to $1$. In this talk, we add one more structure, called a quadratic form. We are mainly interested in the size of the set of $k$-dimensional subspaces that have orthonormal bases, which can be also written as analogues of binomial coefficients. Various combinatorial properties of this new binomial coefficient will be discussed. We will also show what will happen in the set side if we take the limit when $q$ goes to $1$.
  
  Mixed Hodge structure on local cohomology with support in determinantal varieties
  Michael Perlman, Queen's University
  Given a closed subvariety Z in a smooth variety, the local cohomology sheaves with support in Z are functorially endowed with structures as mixed Hodge modules. This implies that they are equipped with two increasing filtrations: the Hodge filtration and the weight filtration. We will discuss new calculations of these filtrations in the case when Z is a generic determinantal variety. This talk includes joint work with Claudiu Raicu.
  
  Geometric and Combinatorial aspects of Nonlinear Algebra
  Papri Dey, University of Missouri-Columbia
  Nonlinear algebra is an interdisciplinary area and I shall talk about some interaction among algebraic, geometric and combinatorial objects in this field.
  
  Clans, sects, and symmetric space closure orders
  Aram Bingham, Tulane University of Louisiana
  We will give a combinatorial description of the closure order on Borel orbits in symmetric spaces of Hermitian type in terms of parameterizing objects called clans and their projections onto $G/P$ (sects). This description resolves part of a conjecture of Wyser on the restriction of Bruhat orders on these spaces.
  
  A generating function for counting mutually annihilating matrices over a finite field
  Yifeng Huang, University of Michigan
  We count the number of pairs of n x n matrices (A, B) over a finite field such that AB=BA=0. We then give an explicit factorization of a generating function associated to this count, which in particular shows that the function can be meromorphically extended to the entire complex plane. This essentially says that the motivic formula about (the stack of) finite-length coherent sheaves on a nodal singular curve behaves as the geometry predicts.

  • Slides
• 11:35 am - 2:00 pm EDT

  Gathertown with Lightning speakers / Lunch

  Lunch/Free Time - Virtual
• 2:00 - 2:30 pm EDT

  Gröbner Geometry of Schubert Polynomials Through Ice

  Virtual

  • Speaker
  • Anna Weigandt, University of Michigan

  • Session Chair
  • Angela Gibney, Rutgers University, New Brunswick (Virtual)

  Abstract

  The geometric naturality of Schubert polynomials and the related combinatorics of pipe dreams was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of vexillary matrix Schubert varieties. We will discuss general diagonal degenerations, relating them to an older formula of Lascoux (2002) in terms of the 6-vertex ice model. Lascoux's formula was recently rediscovered by Lam, Lee, and Shimozono (2018), as "bumpless pipe dreams." We will explain this connection and discuss conjectures and progress towards understanding diagonal Gröbner degenerations of matrix Schubert varieties.

  • Video
  • Slides
• 2:45 - 3:00 pm EDT

  Break

  Coffee Break - Virtual
• 3:00 - 3:30 pm EDT

  Rigid local systems and the multiplicative eigenvalue problem

  Virtual

  • Speaker
  • Prakash Belkale, University of North Carolina at Chapel Hill

  • Session Chair
  • Angela Gibney, Rutgers University, New Brunswick (Virtual)

  Abstract

  Local systems are sheaves which describe the behavior of solutions of differential equations. A local system is rigid if local monodromy determines global monodromy. We give a construction which produces irreducible complex rigid local systems on a punctured Riemann sphere via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU(n) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices (which can be inductively determined) of these polytopes give (all) possible unitary irreducible rigid local systems. We note that these polytopes are generalizations of the classical Littlewood-Richardson cones of algebraic combinatorics. Answering a question of Nick Katz, we show that there are no irreducible rigid local systems on a punctured Riemann sphere of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing n, when n is a prime number.

  • Video
  • Slides
• 3:45 - 4:15 pm EDT

  Cyclotomic generating functions

  Virtual

  • Speaker
  • Joshua Swanson, University of California, San Diego

  • Session Chair
  • Angela Gibney, Rutgers University, New Brunswick (Virtual)

  Abstract

  Generating functions are a major theme in mathematics which unify many disciplines. It is remarkably common to find a combinatorial generating function which factors as a ratio of products of $q$-integers. Examples of such quotients arise from enumerative combinatorics (e.g. subset sums and $q$-binomials), representation theory (the $q$-Weyl dimension formula), geometry (iterated $\mathbb{P}^1$-bundles), probability (random variable decompositions), commutative algebra (homogeneous systems of parameters), and more. We call such quotients "cyclotomic generating functions" (CGFs) and initiate their general study. This talk will review some of the many known constructions of CGFs and give asymptotic estimates of their coefficients. We will also highlight a range of conjectures and accessible open problems. Joint work in progress with Sara Billey.

  • Video
  • Slides

Wednesday, March 24, 2021

• 9:00 - 9:45 am EDT

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 9:45 - 10:15 am EDT

  Doing Schubert Calculus with Bumpless Pipe Dreams

  Virtual

  • Speaker
  • Daoji Huang, Brown University

  • Session Chair
  • Laura Escobar, Washington University- St. Louis (Virtual)

  Abstract

  Bumpless pipe dreams were introduced by Lam, Lee, and Shimozono in the context of back stable Schubert calculus. Like ordinary pipe dreams, they compute Schubert and double Schubert polynomials. In this talk, I will give a bijective proof of Monk's rule for Schubert polynomials, and show that the proof extends easily to the proof of Monk's rule for double Schubert polynomials. As an application, I will explain how to biject bumpless pipe dreams and ordinary pipe dreams using the transition and co-transition formulas, which are specializations of Monk's rule. If time permits I will also briefly discuss some work on how bumpless pipe dreams can be used to compute products of certain Schubert polynomials that generalize the Grassmannian case.

  • Video
  • Slides
• 10:30 - 10:45 am EDT

  Break

  Coffee Break - Virtual
• 10:45 - 11:35 am EDT

  Lightning Talks

  Virtual

  • Speakers
  • anastasia chavez, University of California, Davis
  • Theo Douvropoulos, University of Massachusetts, Amherst
  • Brian Hwang, Cornell University
  • Maiko Serizawa, University of Ottawa
  • Weihong Xu, Rutgers

  • Session Chair
  • Leonardo Mihalcea, Virginia Polytechnic Institute and State University (Virtual)

  Abstract

  Volumes of Root Zonotopes via the W-Laplacian
  Theo Douvropoulos, University of Massachusetts, Amherst
  The normalized volume of the classical permutahedron is given by the formula n^{n-2} which, ask any combinatorialist and they will tell you, agrees with the number of trees on n vertices. This coincidence is well understood; the classical permutahedron is a unimodular zonotope, on the set of positive roots of the Symmetric group S_n, and its bases are indexed by trees and hence enumerated by the determinant of a Laplacian matrix. This description of the permutahedron lends itself to a natural generalization: for a Weyl group W, the zonotope associated to the collection of positive roots of W will be called the root zonotope Z_W. These zonotopes Z_W are not unimodular, but their bases can be differentiated with respect to the reflection subgroup they generate (and its connection index). In joint work with Guillaume Chapuy (arxiv.2012.04519) we have introduced for any Weyl group W a (nxn) Laplacian matrix L_W whose spectrum encodes (nontrivially) many enumerative properties of W. In particular we will present new short formulas (ibid: Section 8.3) for the volumes of the root zonotopes Z_W involving only the Coxeter numbers of W and its reflection subgroups. Our approach is uniform and does not rely on the classification.
  
  Effective point-counting for polygons in flag varieties
  Brian Hwang, Cornell University
  There are a number of seemingly similar "big" (e.g. affine, open) spaces that arise in the context of flag varieties. Certain classes of these spaces are known to exhibit special features. For example, open Richardson varieties have a decomposition into a disjoint union of simple components and the coordinate rings of double Bruhat cells are known to be cluster algebras. There are often combinatorial objects that govern their geometry, such as subword complexes for brick manifolds or various kinds of polytopes, especially in the context of toric varieties and related deformations and degenerations. The bestiary is rich and multifarious. Many of these spaces, however, turn out to admit a description as a polygon in a flag variety, that is, a (cyclic) tuple of flags where each flag is a prescribed distance from its immediate neighbors. Here, the distance is taken not with respect to the Euclidean metric, but with a non-symmetric Weyl-group-valued notion of "distance." Such polygons in flag varieties turn out to admit simple decompositions that can be phrased in terms of triangulations of the polygon into certain triangles with an "easy-exit" property. As an illustration of this organizing principle, we show how we can use this to easily count the points of open Richardson varieties over finite fields–––recovering a classic result of Deodhar, as well as its later enhancement by Marsh and Rietsch–––and how this naturally extends to spaces like double Bruhat cells and their amalgamations.
  
  A 'Quantum Equals Classical' Theorem for n-pointed (K-theoretic) Gromov-Witten Invariants of Lines in Homogeneous Spaces
  Weihong Xu, Rutgers University
  'Quantum equals classical' refers to ways of identifying (K-theoretic) Gromov-Witten invariants on a flag variety G/P with (K-theoretic) classical invariants on a possibly different flag variety G/Q. It has been an active research topic in the last 20 years, but by far all the work has been for 3-pointed invariants. We obtain a 'quantum equals classical' theorem for n-pointed, genus 0 (T-equivariant K-theoretic) Gromov-Witten invariants of lines in G/P, which generalizes the 3-pointed result of Leonardo Mihalcea and Changzheng Li from 2013. This talk is based on joint work with Anders Buch, Linda Chen, Angela Gibney, Lauren Heller, Elana Kalashnikov, and Hannah Larson.
  
  Matroids, Positroids, and Combinatorial Characterizations
  Anastasia Chavez, UC Davis
  Positroids are a special class of representable matroids introduced by Postnikov in the study of the nonnegative part of the Grassmannian. Postnikov defined several combinatorial objects that index positroids. In this talk, I'll briefly introduce some of these objects and how they can combinatorially characterize matroid properties.
  
  Twisted quadratic foldings of root systems and combinatorial Schubert calculus
  Maiko Serizawa, University of Ottawa
  This thesis builds on the connection of two widely studied objects in the literature, that is, foldings of finite root systems and the structure algebras of moment graphs associated with finite root systems. Given a finite crystallographic root system <I> whose Dynkin diagram has a non­trivial automorphism, it yields a new root system <l>7 by a so-called classical folding. On the other hand, Lusztig's folding (1983) folds the root system of type Es to type H4 starting from an automorphism of the root lattice of type Es. Lanini-Zainoulline (2018) developed the notion of a twisted quadratic folding of a root system, which describes both the classical foldings and Lusztig's folding on the same footing. Our second key object of study is the structure algebra Z(Q) of the moment graph g associated with a finite root system and its reflection group W. The structure algebra Z(Q) is an algebra over a certain polynomial ring S whose underlying module is free with a distinguished basis { a( w) I w E W} called combinatorial Schubert classes. Each Schubert class a(w) is an S-valued function on W, whose value is explicitly known for any finite reflection group W. Lanini-Zainoulline (2018) showed that a twisted quadratic folding <I> -v--; <l>7 in­duces an embedding of the respective Coxeter groups E : W7 Y W and a ring homomorphism c:* : Z(Q) ➔ Z(Q7 ) between the corresponding structure algebras. This thesis investigates how the induced map c:* relates the Schubert classes of the original structure algebra Z(Q) to those of the folded structure algebra Z(Q7 ). In particular, we will provide a combinatorial criterion for a Schubert class a?u) of Z(Q7 ) to admit a Schubert class a(w) of Z(Q) such that the relation c:*(a(w) ) = c · a?u) holds for some nonzero scalar c. We will also prove that c:* is surjective after an appropriate extension of the coefficient ring.

  • Slides
• 11:35 am - 2:00 pm EDT

  Gathertown with Lightning speakers / Lunch

  Lunch/Free Time - Virtual
• 2:00 - 2:30 pm EDT

  Differential operators for Schur and Schubert polynomials

  Virtual

  • Speaker
  • Gleb Nenashev, Brown University

  • Session Chair
  • Alexander Yong, University of Illinois at Urbana-Champaign (Virtual)

  Abstract

  We study differential operators for Schur and back stable Schubert polynomials. Our operators are based on two operators of degree (-1), which satisfy Leibniz rule. For the case of Schur functions, these two operators fully determine the product of Schur functions, i.e., it is possible to define Littlewood-Richardson coefficients only using these operators. This new point of view on Schur functions gives us an elementary proof of The Giambelli identity and Jacobi-Trudi identities. For the case of Schubert polynomials, we construct a larger class of decreasing operators, which are indexed by Young diagrams. Operators from this family are related to Stanley symmetric functions. In particular, we extend bosonic operators from Schur to Schubert polynomials.

  • Video
  • Slides
• 2:45 - 3:00 pm EDT

  Break

  Coffee Break - Virtual
• 3:00 - 3:30 pm EDT

  Backstable K-theory Schubert calculus

  Virtual

  • Speaker
  • Mark Shimozono, Virginia Tech

  • Session Chair
  • Alexander Yong, University of Illinois at Urbana-Champaign (Virtual)

  Abstract

  We explain some properties of back-stable limits of double Grothendieck polynomials. We discuss the dual basis of equivariant K-homology of the infinite Grassmannian.
• 3:45 - 4:15 pm EDT

  Bow varieties

  Virtual

  • Speaker
  • Richard Rimanyi, University of North Carolina at Chapel Hill

  • Session Chair
  • Alexander Yong, University of Illinois at Urbana-Champaign (Virtual)

  Abstract

  There are pairs of seemingly unrelated spaces such that Schubert calculus on the two spaces `match’ in some concrete sense. This duality, called 3d mirror symmetry, is best observed in equivariant elliptic cohomology. For example, the dual of the 12-dimensional $T^*Gr(2,5)$ is a certain 4-dimensional Nakajima quiver variety. However, the set of cotangent bundles of homogeneous spaces, or even the set of quiver varieties are not closed for 3d mirror symmetry. In this talk, based on a joint work with Y. Shou, we present a larger pool of spaces: Cherkis bow varieties. Superstring theory predicts that bow varieties are closed for 3d mirror symmetry. The combinatorics necessary to play Schubert calculus on bow varieties includes binary contingency tables and tie diagrams. The existence of an operation (called Hanany-Witten transition) gives bow varieties extra flexibility. We will illustrate the combinatorics and geometry of bow varieties with examples, and we will calculate cohomological and elliptic Schubert classes (rather, `stable envelopes’) to explain the `matching Schubert calculus’ phenomenon.

  • Video
  • Slides

Thursday, March 25, 2021

• 9:00 - 9:45 am EDT

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 9:45 - 10:15 am EDT

  Refined Dubrovin's conjecture for coadjoint varieties

  Virtual

  • Speaker
  • Nicolas Perrin, Versailles Saint-Quentin-en-Yvelines University

  • Session Chair
  • Anders Buch, Rutgers University (Virtual)

  Abstract

  Let X be a Fano variety. Dubrovin’s conjecture predicts, among other things, an equivalence between the semi-simplicity of QH(X) the big quantum cohomology of X and the existence of a full exceptional collection in D(X) the bounded derived category of X. Recently, Kuznetsov and Smirnov proposed a refinement of this conjecture if qH(X) the small quantum cohomology is not semi-simple. I will present this conjecture, discuss it in the case of coadjoint varieties and make connections with roots systems and simple surface singularities. This is a joint work with Maxim Smirnov

  • Video
  • Slides
• 10:30 - 10:45 am EDT

  Break

  Coffee Break - Virtual
• 10:45 - 11:15 am EDT

  motivic Chern classes of Schubert cells and applications

  Virtual

  • Speaker
  • Changjian Su, University of Toronto

  • Session Chair
  • Anders Buch, Rutgers University (Virtual)

  Abstract

  The motivic Chern class in K-theory is a generalization of the Chern-Schwartz-MacPherson class in homology. As the affine Hecke algebra, the motivic Chern classes of Schubert cells have a Langlands dual description. In joint works with P. Aluffi, L. Mihalcea, and J. Schurmann, we use this description to solve some problems about the Casselman basis for the p-adic dual group, and relate the Euler characteristic of the motivic Chern classes to the Iwahori-Whittaker functions. 

  • Video
  • Slides
• 11:30 am - 1:30 pm EDT

  Lunch/Free Time

  Virtual
• 2:00 - 2:30 pm EDT

  Skew-symmetric matrix Schubert varieties

  Virtual

  • Speaker
  • Brendan Pawlowski, University of Southern California

  • Session Chair
  • Thomas Lam, University of Michigan (Virtual)

  Abstract

  Imposing conditions on the ranks of upper-left corners of a skew-symmetric matrix defines a skew-symmetric matrix Schubert variety. We give Gröbner bases for the prime ideals of these varieties, and identify the corresponding initial ideals as the Stanley-Reisner ideals of certain explicit shellable simplicial complexes. These results are analogous to results of Knutson and Miller in the setting of ordinary matrix Schubert varieties, but the techniques are new, and can be used to give new proofs of some of their results. Based on joint work with Eric Marberg.

  • Video
  • Slides
• 2:45 - 3:00 pm EDT

  Break

  Coffee Break - Virtual
• 3:00 - 3:30 pm EDT

  Canonical Bases for Coulomb Branches

  Virtual

  • Speaker
  • Harold Williams, University of Southern California

  • Session Chair
  • Thomas Lam, University of Michigan (Virtual)

  Abstract

  Coulomb branches of 4d N=2 gauge theories are a class of algebraic varieties which, together with their quantizations, appear in a variety of guises in geometry and representation theory. In this talk we outline a construction of canonical bases in the quantized coordinate rings of these varieties. These bases appear as byproducts of a more fundamental construction, a nonstandard t-structure on the dg category of coherent sheaves on the Braverman-Finkelberg-Nakajima space of triples. The heart of this t-structure is a tensor category which is not braided, but admits renormalized r-matrices abstracting those appearing in the finite dimensional representation theory of quantum affine algebras and KLR algebras. It is intended to provide a mathematical model of the category of half-BPS line defects in the relevant gauge theory, and is inspired by earlier work of Kapustin-Saulina and Gaiotto-Moore-Neitzke. On a combinatorial level, the resulting bases are expected to be controlled by specific cluster algebras, which we can confirm in simple examples. This is joint work with Sabin Cautis.

  • Video
• 3:45 - 4:00 pm EDT

  Break

  Coffee Break - Virtual
• 4:00 - 4:30 pm EDT

  Schubert puzzles from Lagrangian correspondences

  Virtual

  • Speaker
  • Allen Knutson, Cornell University

  • Session Chair
  • Thomas Lam, University of Michigan (Virtual)

  Abstract

  Correspondences Z inside M x N induce maps on cohomology H^*(M) -> H^*(N), modulo smoothness and compactness issues. We soup up the diagonal inclusion Gr(k,n) -> Gr(k,n)^2 (whose induced cohomology map is multiplication) to a correspondence of cotangent bundles, which we then factor through T^*(a 2-step flag manifold). Grassmannian Schubert puzzles turn out to be a calculation on this intermediate manifold. As an application of this viewpoint we get a puzzle rule for the Euler characteristic of the intersection of three generically translated Bruhat cells (in up-to-4-step flag manifolds).

  • Slides

Friday, March 26, 2021

• 9:00 - 9:45 am EDT

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 9:45 - 10:15 am EDT

  A combinatorial Chevalley formula for semi-infinite flag manifolds and its applications

  Virtual

  • Speaker
  • Cristian Lenart, University at Albany

  • Session Chair
  • David Anderson, Ohio State University (Virtual)

  Abstract

  I present a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the so-called quantum alcove model. One application is the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture. I also discuss the Chevalley formula for partial flag manifolds G/P. Another application is that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum K-theory of the type A flag manifold. This is joint work with Satoshi Naito and Daisuke Sagaki.

  • Video
  • Slides
• 10:30 - 10:45 am EDT

  Break

  Coffee Break - Virtual
• 10:45 - 11:15 am EDT

  Coxeter-like elements, Schubert geometry, and multiplicity-freeness in algebraic combinatorics

  Virtual

  • Speaker
  • Alexander Yong, University of Illinois at Urbana-Champaign

  • Session Chair
  • David Anderson, Ohio State University (Virtual)

  Abstract

  For a finite Coxeter system, we define “spherical elements” (extending Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Levi-spherical Schubert varieties in G/B. In type A, this connects to key polynomials (Demazure characters), multiplicity-freeness, and “split-symmetry” in algebraic combinatorics. This is joint work (some ongoing) with subsets of David Brewster (UIUC), Yibo Gao (MIT), and Reuven Hodges (UIUC). See arXiv:2007.09238, arXiv:2007.09229, and arXiv:2012.09749.

  • Video
  • Slides
• 11:30 am - 2:00 pm EDT

  Lunch/Free Time

  Virtual
• 2:00 - 2:30 pm EDT

  Positroids, knots, and q,t-Catalan numbers

  Virtual

  • Speaker
  • Pavel Galashin, University of California, Los Angeles

  • Session Chair
  • Angela Gibney, Rutgers University, New Brunswick (Virtual)

  Abstract

  We relate the cohomology of open positroid varieties and their point counts over finite fields to knot homology. In particular, we show that the bigraded Poincaré polynomials of top-dimensional open positroid varieties are given by rational q,t-Catalan numbers. As a consequence of the curious Lefschetz property, we obtain q,t-symmetry and unimodality statements for rational q,t-Catalan numbers. Joint work with Thomas Lam. No special background on the above objects will be assumed.

  • Video
  • Slides
• 2:45 - 3:00 pm EDT

  Break

  Coffee Break - Virtual
• 3:00 - 3:30 pm EDT

  Puzzle rules in cotangent Schubert calculus

  Virtual

  • Speaker
  • Paul Zinn-Justin, The University of Melbourne

  • Session Chair
  • Angela Gibney, Rutgers University, New Brunswick (Virtual)

  Abstract

  We'll present several combinatorial rules (in terms of ``puzzles'') for the expansion of the product of motivic Segre classes (in equivariant K-theory) and Segre-Schwartz-MacPherson classes (in equivariant cohomology) in partial flag varieties. We'll show how to compute such classes and puzzles using the software Macaulay2.

Monday, March 29, 2021

• 11:00 am - 12:00 pm EDT

  Professional Development: Hiring Process

  Professional Development - Virtual

Wednesday, March 31, 2021

• 3:00 - 3:30 pm EDT

  Gathertown Afternoon Coffee

  Coffee Break - Virtual

• 3:30 - 4:30 pm EDT

  The isomorphism problem for Schubert varieties.

  Virtual

  • Edward Richmond, Oklahoma State University

  Abstract

  Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. In this talk, I will discuss recent work with William Slofstra where we give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis. As an application, we show that the isomorphism classes of Schubert varieties in a given flag variety are controlled by graph automorphisms of the Dynkin diagram.

  • Video

Thursday, April 1, 2021

• 10:00 - 11:00 am EDT

  Learning Seminar - Differential forms on tropical moduli spaces

  Seminar - Virtual

  • Melody Chan, Brown University
  • Sam Payne, University of Texas at Austin

  Abstract

  The goal is to understand Francis Brown's recent arXiv preprint Invariant differential forms on complexes of graphs and Feynman integrals. A sub-goal, which we hope takes shape over the course of the seminar, is to draw out the connection between Brown's article and tropical moduli spaces of curves and abelian varieties.
  
  This is not an introductory seminar per se in that the choice of topic is of specialized interest and of particular interest to us vis-a-vis our current work. On the other hand, we will try to develop the machinery without relying on background from tropical geometry, much as Brown's article is able to do.
  
  Resources to get started:
  
  1. Sam's lectures from the introductory Workshop, available from the webpage
  
  2. Draft of a survey articleon classical/tropical moduli spaces by Melody, which outlines the appearance of graph complexes in the study of tropical moduli.
• 11:00 am - 12:00 pm EDT

  Semistable Reduction Seminar

  Seminar - Virtual

  • Dan Abramovich, Brown University

  • Abstract
• 3:30 - 4:40 pm EDT

  A trinity in affine Schubert calculus

  Virtual

  • Syu Kato, Kyoto University

  Abstract

  One of the most famous results in affine Schubert calculus is Peterson's quantum = affine theorem (proved by Lam-Shimozono based on a work of Mihalcea). It states that the homology of affine Grassmanian of a simple algebraic group $G$ is equivalent to the quantum cohomology of the flag manifold of $G$ as based rings. Such an equivalence has a $K$-theoretic counterpart conjectured by Lam-Li-Mihalcea-Shimozono. In this talk, we add the semi-infinite flag manifold of $G$ into this picture. In particular, the $K$-theoretic analogue of the both sides of the Peterson theorem are related to the equivariant $K$-group of the semi-infinite flag manifold. This establishes the above conjecture of Lam-Li-Mihalcea-Shimozono. Arguably, our picture provides the first explanation as to why Peterson's isomorphism should respect the basis. In addition, it has a wide range of implications including the finiteness of the quantum multiplication for the full flag manifold. This talk is based on arXiv:1805.01718. For the semi-infinite flag manifold and its equivariant $K$-group, please refer to the slide of my talk on March 23rd.

  • Slides

Friday, April 2, 2021

• 10:00 - 10:30 am EDT

  Graduate Student/Postdoc pre-seminar break in Gather

  Coffee Break - Virtual
• 10:30 - 11:30 am EDT

  Cluster Algebras and Total Positivity

  Post Doc/Graduate Student Seminar - Virtual

  • Sunita Chepuri, University of Michigan

  Abstract

  We will first define cluster algebras and explain their historical motivation from total positivity. The research portion of the talk will move to the setting of k-positivity, a generalization of total positivity. We will explore how much of the cluster algebraic structure of totally positive matrices extends to k-positive matrices.
• 11:30 am - 12:00 pm EDT

  Graduate Student/Postdoc post-seminar break in Gather

  Coffee Break - Virtual

Monday, April 5, 2021

• 11:00 am - 12:00 pm EDT

  Directorate Check-In with Grads and Postdocs

  Meeting - Virtual

Wednesday, April 7, 2021

• 3:00 - 3:30 pm EDT

  Gathertown Afternoon Coffee

  Coffee Break - Virtual

• 2:00 - 3:00 pm EDT

  The unramified affine springer fiber and the nabla operator

  Virtual

  • Erik Carlsson, UC Davis

  Abstract

  I'll present a new result with A. Mellit, which gives a combinatorial formula for a diagonalizing operator for the modified Macdonald polynomials, known as the nabla operator. This formula was discovered by searching for a Schubert-type basis of a certain explicit module from Haiman's polygraph theory, which is identified with both the matrix elements of this operator, and the equivariant homology of the unramified affine Springer fiber studied by Goresky, Kottwitz, and Macpherson.

Thursday, April 8, 2021

• 10:00 - 11:00 am EDT

  Learning Seminar - Differential forms on tropical moduli spaces

  Seminar - Virtual

  • Melody Chan, Brown University
  • Sam Payne, University of Texas at Austin

  Abstract

  The goal is to understand Francis Brown's recent arXiv preprint Invariant differential forms on complexes of graphs and Feynman integrals. A sub-goal, which we hope takes shape over the course of the seminar, is to draw out the connection between Brown's article and tropical moduli spaces of curves and abelian varieties.
  
  This is not an introductory seminar per se in that the choice of topic is of specialized interest and of particular interest to us vis-a-vis our current work. On the other hand, we will try to develop the machinery without relying on background from tropical geometry, much as Brown's article is able to do.
  
  Resources to get started:
  
  1. Sam's lectures from the introductory Workshop, available from the webpage
  
  2. Draft of a survey articleon classical/tropical moduli spaces by Melody, which outlines the appearance of graph complexes in the study of tropical moduli.
• 11:00 am - 12:00 pm EDT

  Semistable Reduction Seminar

  Seminar - Virtual

  • Dan Abramovich, Brown University

  • Abstract

Friday, April 9, 2021

• 10:00 - 10:30 am EDT

  Graduate Student/Postdoc pre-seminar break in Gather

  Coffee Break - Virtual
• 10:30 - 11:30 am EDT

  Schubert Calculus

  Post Doc/Graduate Student Seminar - Virtual

  • Gleb Nenashev, Brown University
  • Jianping Pan, University of California, Davis

  Abstract

  Decreasing operators for Schur functions.
  Gleb Nenashev
  I will discuss decreasing operators for Young diagrams. These operators give elementary proofs of some famous properties of Schur functions, e.g. for Murnaghan-Nakayama rule, Giambelli identity, and Jacobi-Trudi identities. They also fully determine Littlewood-Richardson coefficients. I will present some of these proofs. Furthermore, the decreasing operators are well defined for Back stable Schubert polynomials, but I will not discuss it.
  
  Crystal combinatorics with stable Grothendieck polynomials
  Jianping Pan
  Grothendieck polynomials are representatives of Schubert varieties in the K-theory of the flag manifold. Taking stable limit, we obtain symmetric functions called the stable Grothendieck polynomials. I will talk about a combinatorial construction of it, developed by Fomin and Kirillov, which turns out to have some nice crystal structure (directed, edge-labeled graphs from representation theory). There are also several properties associated with it, including some insertion algorithms. This talk is based on arXiv:1911.08732.
• 11:30 am - 12:00 pm EDT

  Graduate Student/Postdoc post-seminar break in Gather

  Coffee Break - Virtual

Monday, April 12, 2021

• 9:45 - 10:00 am EDT

  Welcome

  Virtual

  • Brendan Hassett, ICERM/Brown University
• 10:00 - 10:45 am EDT

  The Foundation of a Matroid

  Virtual

  • Speaker
  • Matthew Baker, Georgia Institute of Technology

  • Session Chair
  • Lauren Williams, Harvard University (Virtual)

  Abstract

  Matroid theorists are typically interested in questions concerning representability of matroids over fields. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid, which governs the representations of over all partial fields. Unfortunately, almost all matroids are not representable over any partial field, and in this case, the universal partial field gives no information. Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for matroids having no minor isomorphic to U(2,5) or U(3,5). Among other things, our classification provides a short conceptual proof of the 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic.

  • Video
• 11:00 - 11:15 am EDT

  Break

  Coffee Break
• 11:15 am - 12:00 pm EDT

  Lagrangian geometry of matroids

  Virtual

  • Speaker
  • Graham Denham, University of Western Ontario

  • Session Chair
  • Lauren Williams, Harvard University (Virtual)

  Abstract

  In joint work with Federico Ardila and June Huh, we introduce the conormal fan of a matroid, which is the Lagrangian analogue of the Bergman fan. We use it to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of a matroid. We develop tools for tropical Hodge theory to show that the conormal fan satisfies Poincaré duality, the Hard Lefschetz property, and the Hodge--Riemann relations. Together, these imply conjectures of Brylawski and Dawson about the log-concavity of the h-vectors of the broken circuit complex and independence complex of a matroid.

  • Video
• 12:00 - 1:30 pm EDT

  Lunch/Free Time
• 1:30 - 2:15 pm EDT

  Tautological classes of matroids

  Virtual

  • Speaker
  • Christopher Eur, Stanford University

  • Session Chair
  • Sam Payne, University of Texas at Austin

  Abstract

  We introduce certain torus-equivariant classes on permutohedral varieties which we call ``tautological classes of matroids'' as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory. This is joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

  • Video
• 2:30 - 2:45 pm EDT

  Break

  Coffee Break
• 2:45 - 3:30 pm EDT

  Kazhdan-Lusztig theory and singular Hodge theory for matroids

  Virtual

  • Speaker
  • June Huh, Stanford University

  • Session Chair
  • Sam Payne, University of Texas at Austin

  Abstract

  There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or point configuration) from the perspective of combinatorial cohomology theories. I will give an overview of the similarity and report on recent my joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang on singular Hodge theory for combinatorial geometries: https://arxiv.org/abs/2010.06088

  • Video
• 3:45 - 4:45 pm EDT

  Gathertown Reception

  Reception - Virtual

Tuesday, April 13, 2021

• 9:00 - 9:45 am EDT

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:45 am EDT

  Real phase structures on matroid fans

  Virtual

  • Speaker
  • Kristin Shaw, University of Oslo

  • Session Chair
  • Melody Chan, Brown University (Virtual)

  Abstract

  In this talk, I will propose a definition of real phase structures on polyhedral complexes. I’ll explain that in the case of matroid fans, specifying a real phase structure is cryptomorphic to providing an orientation of the underlying matroid. Then I’ll define the real part of a polyhedral complex with a real phase structure. This determines a closed chain in the real part of a toric variety. In the case when the polyhedral complex is a non-singular tropical variety, the real part is a PL-manifold. Moreover, for a non-singular tropical variety with a real phase structures we can apply the same spectral sequence for tropical hypersurfaces, obtained by Renaudineau and myself, to bound the Betti numbers of the real part by the dimensions of the tropical homology groups. This is joint work in progress with Johannes Rau and Arthur Renaudineau.
• 11:00 - 11:15 am EDT

  Break

  Coffee Break
• 11:15 am - 12:00 pm EDT

  Combinatorics and real lifts of bitangents to tropical quartic curves

  Virtual

  • Speaker
  • María Angélica Cueto, Ohio State University

  • Session Chair
  • Melody Chan, Brown University (Virtual)

  Abstract

  Smooth algebraic plane quartics over algebraically closed fields have 28 bitangent lines. By contrast, their tropical counterparts have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic curve. In this talk, I will discuss recent work joint with Hannah Markwig (arXiv:2004.10891) on the combinatorics of these bitangent classes and its connection to the number of real bitangents to real smooth quartic curves characterized by Pluecker. We will see that they are tropically convex sets and they come in 41 symmetry classes. The classical bitangents map to specific vertices of these polyhedral complexes, and each tropical bitangent class captures four of the 28 bitangents. We will discuss the situation over the reals and show that each tropical bitangent class has either zero or four lifts to classical bitangent defined over the reals, in agreement with Pluecker's classification.

  • Video
  • Slides
• 12:00 - 1:30 pm EDT

  Lunch/Free Time
• 1:30 - 2:15 pm EDT

  Tropical psi classes

  Virtual

  • Speaker
  • Renzo Cavalieri, Colorado State University

  • Session Chair
  • Dhruv Ranganathan, University of Cambridge (Virtual)

  Abstract

  We introduce a tropical geometric framework that allows us to define $\psi$ classes for moduli spaces of tropical curves of arbitrary genus. We prove correspondence theorems between algebraic and tropical $\psi$ classes for some one-dimensional families of genus-one tropical curves.

  • Video
• 2:30 - 2:45 pm EDT

  Break

  Coffee Break
• 2:45 - 3:30 pm EDT

  When are multidegrees positive?

  Virtual

  • Speaker
  • Federico Castillo, University of Kansas

  • Session Chair
  • Dhruv Ranganathan, University of Cambridge (Virtual)

  Abstract

  The notion of multidegree for multiprojective varieties extends that of degree for projective varieties. They can be defined in geometric terms, using intersection theory, or alternatively in algebraic terms, via multigraded hilbert polynomial. We study the problem of their positivity and establish a combinatorial description using polyhedral geometry. We will show applications for Schubert polynomials and mixed volumes. This is joint work with Y.Cid-Ruiz, B.Li, J.Montano, and N.Zhang.

Wednesday, April 14, 2021

• 9:00 - 9:45 am EDT

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:45 am EDT

  Wall-crossing phenomenon for Newton-Okounkov bodies

  Virtual

  • Speaker
  • Laura Escobar, Washington University- St. Louis

  • Session Chair
  • Lara Bossinger, Mathematics Institute UNAM, Oaxaca (Virtual)

  Abstract

  A Newton-Okounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes and the correspondence between polytopes and projective toric varieties. Work of Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. As an example, we describe wall-crossing formula in the case of the Grassmannian Gr(2,m). This is joint work with Megumi Harada.

  • Slides
• 11:00 - 11:15 am EDT

  Break

  Coffee Break - Virtual
• 11:15 am - 12:00 pm EDT

  On combinatorics of Arthur's trace formula, convex polytopes, and toric varieties

  Virtual

  • Speaker
  • Kiumars Kaveh, University of Pittsburgh

  • Session Chair
  • Lara Bossinger, Mathematics Institute UNAM, Oaxaca (Virtual)

  Abstract

  I start by discussing two beautiful well-known theorems about decomposing a convex polytope into an signed sum of cones, namely the classical Brianchon-Gram theorem and Lawrence-Varchenko theorem. I will then explain a generalization of the Brianchon-Gram which can be summerized as ""truncating a function on the Euclidean space with respect to a polytope"". This is an extraction of the combinatorial ingredients of Arthur's ''convergence'' and ''polynomiality'' results in his famous trace formula. Arthur's trace formula concerns the trace of left action of a reductive group $G$ on the space $L^2(G / \Gamma)$ where $\Gamma$ is a discrete (arithmetic) subgroup. The combinatorics involved is closely related to compactifications of ''locally summetric spaces'' (which btw are hyperbolic manifolds). Our ''combinatorial truncation'' can be thought of as an analogue of Arthur's truncation over a toric variety (in place of a compactification of a locally symmetric space). If there is time, I will briefly sketch geometric interpretations of our combinatorial truncation as a measure and a Lefschetz number on a toric variety respectively. This is a joint work in progress with Mahdi Asgari (Oklahoma State).

  • Video
• 12:00 - 1:30 pm EDT

  Lunch/Free Time
• 1:30 - 2:30 pm EDT

  Poster Session

  Virtual

  Abstract

  Ideal Preserving Operations on Chemical Reaction Networks
  Mark Curiel, University of Hawaii at Manoa
  Under the assumption of mass action kinetics, the associated dynamical system of a reaction network is polynomial. We consider the ideals generated by these polynomials, which are called steady-state ideals. Steady-state ideals appear in multiple contexts within the chemical reaction network literature, however they have yet to be systematically studied. To begin such a study, we ask and partially answer the following question: when do two reaction networks give rise to the same steady-state ideal? In particular, our main results describe three operations on the reaction graph that preserve the steady-state ideal. Furthermore, since the motivation for this work is the classification of steady-state ideals, monomials play a primary role. To this end, combinatorial conditions are given to identify monomials in a steady-state ideal, and we give a sufficient condition for a steady-state ideal to be monomial.
  
  Construction and properties of Kanev surfaces in toric 3-folds
  Julius Giesler, University of Tuebingen
  In this poster Kanev surfaces, which are surfaces of general type, are considered, that arise as nondegenerate hypersurfaces in toric 3-folds. First such an hypersurface might have singularities but we show how to construct a minimal and a canonical model with toric methods. After this construction we consider nondegenerate hypersurfaces with fixed Newton polytope, thus obtaining a family of Kanev surfaces, and we both compute their number of moduli and check whether the infinitesimal Torelli theorem holds for such a family. The results constitute part of the author's doctoral thesis.
  
  On the shifted Littlewood-Richardson coefficients and the Littlewood-Richardson coefficients
  Nguyen Khanh, Institut Camille Jordan
  We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{\lambda\mu}^\nu$ ($\lambda,\mu,\nu$ are strict partitions). The coefficients $g_{\lambda\mu}$ which appear in the decomposition of Schur $Q$-function $Q_\lambda$ into the sum of Schur functions $Q_\lambda = 2^{l(\lambda)}\sum\limits_{\mu}g_{\lambda\mu}s_\mu$ can be considered as a special case of $f_{\lambda\mu}^\nu$ (here $\lambda$ is a strict partition of length $l(\lambda)$). We also give another description for $g_{\lambda\mu}$ as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients $c_{\mu^t\mu}^{\tilde{\lambda}}$. This new point of view allows us to establish connections between $g_{\lambda\mu}$ and $c_{\mu^t \mu}^{\tilde{\lambda}}$. More precisely, we prove that $g_{\lambda\mu}=g_{\lambda\mu^t}$, and $g_{\lambda\mu} \leq c_{\mu^t\mu}^{\tilde{\lambda}}$. We conjecture that $g_{\lambda\mu}^2 \leq c^{\tilde{\lambda}}_{\mu^t\mu}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.
  
  Computational complexity, Newton polytopes, and Schubert polynomials
  Colleen Robichaux, University of Illinois at Urbana-Champaign
  Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The nonvanishing problem asks if a coefficient of a Schubert polynomial is nonzero. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the nxn grid. This is joint work with Anshul Adve and Alexander Yong.
  
  Enumeration of algebraic and tropical singular hypersurfaces
  Uriel Sinichkin, Tel Aviv University
  We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $ \delta $ dimensional linear space of degree $ d $ real hypersurfaces containing $ \frac{1}{\delta!}(\gamma_nd^n)^{\delta}+O(d^{n\delta-1}) $ hypersurfaces with $ \delta $ real nodes, where $ \gamma_n $ are positive and given by a recursive formula.
  This is asymptotically comparable to the number $ \frac{1}{\delta!} \left( (n+1)(d-1)^n \right)^{\delta}+O\left(d^{n(\delta-1)} \right) $ of complex hypersurfaces having $ \delta $ nodes in a $ \delta $ dimensional linear space. In the case $ \delta=1 $ we give a slightly better leading term.
• 2:30 - 3:15 pm EDT

  On Newton-Okounkov bodies associated to Grassmannians

  Virtual

  • Speaker
  • Alfredo Nájera Chávez, Mathematics Institute UNAM, Oaxaca

  • Session Chair
  • Travis Mandel, University of Oklahoma (Virtual)

  Abstract

  In this talk I will elaborate on a certain class of Newton-Okounkov bodies that one can associate to "nice" compactifications of cluster varieties. In particular, I will explain how this approach recovers Rietsch--Williams' construction of Newton--Okounkov bodies for Grassmannians. In order to make the precise connection it will be necessary to explain how the Marsh--Rietsch potential and the Gross--Hacking--Keel--Kontsevich potential for Grassmannians are related. Finally, I will draw some consequences from this relation such as an isomorphism of the toric degenerations obtained by Rietsch-Willimas and the toric degenerations obtained by the celebrated "principal coefficient" construction. Time permitting, I will briefly elaborate on the interpretation of these results from the viewpoint of the representation theory of the associated dimer algebra.

  • Video
  • Slides
• 3:30 - 3:45 pm EDT

  Break

  Coffee Break
• 3:45 - 4:30 pm EDT

  Broken line convexity

  Virtual

  • Speaker
  • Timothy Magee, University of Birmingham

  • Session Chair
  • Daping Weng, Michigan State University (Virtual)

  Abstract

  In this talk, I'll give an overview of how convex polytopes generalizes from the toric world to the cluster world, where the "polytopes" live in a tropical space rather than a vector space. In this setting, "broken line convex polytopes" define projective compactifications of cluster varieties. After this overview, I'll focus on two exciting applications of this more general notion of convexity: 1) an intrinsic version of Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. The overview is based on joint work with Mandy Cheung and Alfredo Nájera Chávez, and the applications are based on ongoing joints works with Mandy, Alfredo, Lara Bossinger, and Bosco Frías Medina.

Thursday, April 15, 2021

• 9:00 - 9:45 am EDT

  Gathertown Morning Coffee

  Coffee Break - Virtual
• 10:00 - 10:45 am EDT

  Toric vector bundles -- an overview

  Virtual

  • Speaker
  • Milena Hering, The University of Edinburgh

  • Session Chair
  • Linda Chen, Swarthmore College (Virtual)

  Abstract

  I will give a brief introduction to toric vector bundles, an overview of what we know about them so far, and explain some more recent developments on the defining equations of embeddings of their projectivisations

  • Video
• 11:00 - 11:15 am EDT

  Break

  Coffee Break
• 11:15 am - 12:00 pm EDT

  The Fulton-MacPherson compactification is not a Mori dream space

  Virtual

  • Speaker
  • José González, University of California, Riverside

  • Session Chair
  • Linda Chen, Swarthmore College (Virtual)

  Abstract

  We show that the Fulton-MacPherson compactification of the configuration space of n distinct labeled points in certain varieties of arbitrary dimension d, including projective space, is not a Mori dream space for n greater than or equal to d+9.
• 12:00 - 1:30 pm EDT

  Lunch/Free Time
• 1:30 - 2:15 pm EDT

  Schubert polynomials from a polytopal point of view

  Virtual

  • Speaker
  • Karola Meszaros, Cornell University

  • Session Chair
  • Laura Escobar, Washington University- St. Louis (Virtual)

  Abstract

  Schubert polynomials are multivariate polynomials representing cohomology classes on the flag manifold. Despite the beautiful formulas developed for them over the past three decades, the coefficients of these polynomials remained mysterious. I will explain Schubert polynomials from a polytopal point of view, answering, at least partially, the questions: Which coefficients are nonzero? How do the coefficients compare to each other in size? Are the Newton polytopes of these polynomials saturated? Are their coefficients log-concave along lines? Is there a polytope whose integer point transform specializes to Schubert polynomials? As the questions themselves suggest, we will find that polytopes play an outsized role in our understanding. The talk is based on joint works with Alex Fink, June Huh, Ricky Liu, Jacob Matherne and Avery St. Dizier.
• 2:30 - 2:45 pm EDT

  Break

  Coffee Break
• 2:45 - 3:30 pm EDT

  Families of Gröbner degenerations

  Virtual

  • Speaker
  • Lara Bossinger, Mathematics Institute UNAM, Oaxaca

  • Session Chair
  • Laura Escobar, Washington University- St. Louis (Virtual)

  Abstract

  Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over affine m-space that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh--Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base the toric variety associated to cone C along its universal torsor. We apply this construction to the Grassmannians of planes with their Plücker embeddings and the Grassmannian Gr(3,6) with its cluster embedding. In each case there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley--Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. This talk is based on joint work with F. Mohammadi and A. Nájera Chávez, arxiv:2007.14972.

  • Video
  • Slides