Combinatorial Algebraic Geometry
Institute for Computational and Experimental Research in Mathematics (ICERM)
February 1, 2021  May 7, 2021
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Monday, February 1, 2021

9:50  10:00 am ESTWelcomeVirtual
 Brendan Hassett, ICERM/Brown University

11:00 am  12:00 pm EST

12:30  2:00 pm ESTLunch/Free TimeVirtual

3:00  4:00 pm EST

4:30  5:15 pm ESTReceptionVirtual
Tuesday, February 2, 2021

9:00  9:45 am ESTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am ESTBasic notions in cotangent Schubert calculus  Part 1Virtual
 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 oneparameter deformation of the notion "Schubert class", called ChernSchwartzMacPherson (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.

11:00 am  12:00 pm EST

12:00  12:30 pm ESTBasic notions in cotangent Schubert calculus  Part 2Virtual
 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 oneparameter deformation of the notion "Schubert class", called ChernSchwartzMacPherson (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.

12:30  2:00 pm ESTLunch/Free TimeVirtual

2:00  2:45 pm ESTMatroids Part 1Virtual
 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 algebrogeometric models of matroids.

3:00  4:00 pm EST

4:00  4:30 pm ESTMatroids Part 2Virtual
 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 algebrogeometric models of matroids.
Wednesday, February 3, 2021

11:00 am  12:00 pm EST

12:30  2:00 pm ESTLunch/Free TimeVirtual

3:00  4:00 pm EST

4:30  5:15 pm ESTGathertown Afternoon Coffee  Informal Grad Student / Postdoc FocusedCoffee Break  Virtual
Thursday, February 4, 2021

9:00  9:45 am ESTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am ESTIntroduction to Tropical Geometry Through Curves  Part 1Virtual
 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.

11:00 am  12:00 pm EST

12:00  12:30 pm ESTIntroduction to Tropical Geometry Through Curves  Part 2Virtual
 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.

12:30  2:00 pm ESTLunch/Free TimeVirtual

2:00  2:45 pm ESTModuli spaces of tropical curves Part 1Virtual
 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.

3:00  4:00 pm EST

4:00  4:30 pm ESTModuli spaces of tropical curves Part 2Virtual
 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.
Friday, February 5, 2021

10:00  10:45 am ESTCluster structures in commutative rings Part 1Virtual
 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.

11:00 am  12:00 pm EST

12:00  12:30 pm ESTCluster structures in commutative rings Part 2Virtual
 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.

12:30  2:00 pm ESTLunch/Free TimeVirtual

2:00  2:45 pm ESTCluster varieties Part 1Virtual
 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.

3:00  4:00 pm EST

4:00  4:45 pm ESTCluster varieties Part 2Virtual
 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.
Monday, February 8, 2021
Combinatorial Algebraic Geometry

11:00  11:30 am ESTWelcome to ICERMWelcome  Virtual
 Brendan Hassett, ICERM/Brown University

1:30  2:00 pm ESTOrganizer / Directorate MeetingMeeting  Virtual
Tuesday, February 9, 2021
Combinatorial Algebraic Geometry

2:30  3:50 pm ESTBrown Graduate Course: Algebraic GeometryVirtual
 Jeremy Usatine, Brown University
Wednesday, February 10, 2021
Combinatorial Algebraic Geometry

11:00 am  12:00 pm ESTGraduate Students/Postdocs Meeting with DirectorateMeeting  Virtual

3:30  4:30 pm ESTDegeneracy Loci and Schubert PolynomialsVirtual
 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.)
Thursday, February 11, 2021
Combinatorial Algebraic Geometry

2:30  3:50 pm ESTBrown Graduate Course: Algebraic GeometryVirtual
 Jeremy Usatine, Brown University

4:30  5:15 pm ESTHodgeRiemann relations and Lorentzian polynomialsVirtual
 Christopher Eur, Stanford University
Abstract
We introduce the theory of Lorentzian polynomials, motivated by the HodgeRiemann relations and the resulting properties that cohomology rings of smooth complex projective varieties satisfy.
Friday, February 12, 2021
Combinatorial Algebraic Geometry

10:30  11:30 am ESTPost Doc/Graduate Student SeminarVirtual

11:30 am  12:00 pm ESTGraduate Student/postdoc postseminar break in GatherCoffee Break  Virtual

4:30  5:00 pm ESTHodgeRiemann relations and Lorentzian polynomials cont.Virtual
 Christopher Eur, Stanford University
Abstract
We introduce the theory of Lorentzian polynomials, motivated by the HodgeRiemann relations and the resulting properties that cohomology rings of smooth complex projective varieties satisfy.
Monday, February 15, 2021

9:45  10:00 am ESTWelcomeVirtual
 Brendan Hassett, ICERM/Brown University

10:00  10:30 am ESTA brief tour of SageVirtual
 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.

10:45  11:15 am ESTRings and fields in SageVirtual
 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.

11:15  11:30 am ESTCoffee BreakVirtual

11:30 am  12:00 pm ESTCelestial mechanics via tropical geometry (gfan and Macaulay2)Virtual
 Anton Leykin, Georgia Tech

12:15  12:45 pm ESTFusionRings in Sage 9.2Virtual
 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.

1:00  2:00 pm ESTLunch/Free TimeVirtual

2:00  3:00 pm ESTGathertown Welcome ReceptionReception  Virtual

3:00  4:00 pm ESTSage/Oscar Installation HelpTutorial  Virtual
Tuesday, February 16, 2021

9:00  9:45 am ESTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:30 am ESTOSCAR  The ProjectVirtual
 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.

10:45  11:15 am ESTOSCAR  Selected FeaturesVirtual
 Daniel Schultz, Technische Universität Kaiserslautern
Abstract
Introducing OSCAR, a new computer algebra system combining GAP, Polymake, Hecke and Singular.

11:15  11:30 am ESTCoffee BreakVirtual

11:30 am  12:00 pm ESTComputing the Newton polytope of a large discriminantVirtual
 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 Dequivalence classes of regular triangulations of the 3 dilated tetrahedron. The only known way to do this is to enumerate all regular triangulations of the 3dilated 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.

12:15  12:45 pm ESTSome hybrid symbolicnumeric methods in algebraic geometryVirtual
 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 symbolicnumeric methods and applications in computational algebraic geometry.

1:00  2:00 pm ESTLunch/Free TimeVirtual

2:00  3:00 pm ESTProblem SessionVirtual

3:00  4:00 pm ESTContributing to Sage TutorialTutorial  Virtual
Wednesday, February 17, 2021

9:00  9:45 am ESTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am ESTParallelization of Triangular Decompositions Design and implementation with the BPAS libraryVirtual
 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 "componentlevel" parallelization of triangular decompositions, our focus here, belongs to the class of dynamic irregular parallelism. Potential parallel speedup 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 componentlevel parallelization we combine different concurrency patterns: map, workpile, producerconsumer, 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 10.8times speed up on a 12core machine.

11:00  11:30 am ESTCoffee BreakVirtual

11:30 am  12:00 pm ESTRational integrals and periods with Sagemath and JuliaVirtual
 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 ESTGeneralized cohomology quotients of the symmetric functionsVirtual
 Darij Grinberg, Drexel University

1:00  2:00 pm ESTLunch/Free TimeVirtual

2:00  2:40 pm ESTLightning TalksVirtual
 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

2:40  2:50 pm ESTCoffee BreakVirtual

2:50  3:30 pm ESTLightning TalksVirtual
 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 UrbanaChampaign
 Mahrud Sayrafi, University of Minnesota
 Weihong Xu, Rutgers

3:30  4:30 pm ESTCode DemonstrationsTutorial  Virtual
Thursday, February 18, 2021

9:00  9:45 am ESTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am ESTmsolve  A Library for Solving Polynomial SystemsVirtual
 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 stateoftheart. This is joint work with Jérémy Berthomieu, JeanCharles Faugère and Mohab Safey El Din from the PolSys Team at the Sorbonne Université in Paris.

11:00  11:30 am ESTParallelism in Algebraic Geometry  Examples with Singular and GPISpaceVirtual
 Anne FrühbisKrüger, University of Oldenburg
Abstract
I shall illustrate the use of the Singular  GPIspace interplay in some examples including a smoothness test, GITfans and desingularization.

11:45 am  12:45 pm ESTCoffee BreakVirtual

12:45  1:15 pm EST

1:30  2:30 pm ESTLunch/Free TimeVirtual

2:30  3:30 pm ESTProblem SessionVirtual

3:30  4:30 pm ESTCode DemonstrationsTutorial  Virtual
Friday, February 19, 2021

9:00  9:45 am ESTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:30 am ESTFactorizations into irreducibles and polytopesVirtual
 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 ESTComputational challenges for tropical del Pezzo surfacesVirtual
 María Angélica Cueto, Ohio State University
Abstract
A smooth degree d del Pezzo surface is obtained by blowing up the projective plane at (9d) 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 ESTCoffee BreakVirtual

11:30 am  12:00 pm ESTPresenting the multipolynomial bases packageVirtual
 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.

12:15  12:45 pm EST

1:00  2:00 pm ESTLunch/Free TimeVirtual

2:00  3:00 pm ESTGathertown Closing ReceptionReception  Virtual
Monday, February 22, 2021
Combinatorial Algebraic Geometry

11:00 am  12:00 pm ESTGrad Student/ PostDoc Professional Development: Ethics 1Professional Development  Virtual
Wednesday, February 24, 2021

3:30  4:30 pm EST
Thursday, February 25, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am ESTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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 ESTSemistable Reduction SeminarSeminar  Virtual
 Dan Abramovich, Brown University
Friday, February 26, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am ESTGraduate Student/ PostDoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am ESTCMregularityPost Doc/Graduate Student Seminar  Virtual
 Matthew Larson, Stanford University
 Mahrud Sayrafi, University of Minnesota
Abstract
The geometry of CastelnuovoMumford regularity  Matt Larson, Stanford University
CastelnuovoMumford 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 CastelnuovoMumford regularity, state its basic properties, and explain its relationship to syzygies of graded module.
Computing the Multigraded CastelnuovoMumford Regularity  Mahrud Sayrafi, University of Minnesota
Motivated by toric geometry, MaclaganSmith 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 ESTGraduate Student/ PostDoc postseminar break in GatherCoffee Break  Virtual
Monday, March 1, 2021
Combinatorial Algebraic Geometry

11:00 am  12:00 pm ESTProfessional Development: Ethics IIProfessional Development  Virtual
Wednesday, March 3, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm ESTGathertown Afternoon CoffeeCoffee Break  Virtual

4:00  5:00 pm ESTQuantum integrability and GrassmanniansVirtual
 Paul ZinnJustin, The University of Melbourne
Abstract
We will investigate in the simplest setting, how an``Rmatrix'' (the building block of ``quantum integrable systems'') is attached to the equivariant cohomology of Grassmannians. We will compute the Rmatrix 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).
Thursday, March 4, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am ESTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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 ESTSemistable Reduction SeminarSeminar  Virtual
 Dan Abramovich, Brown University
Friday, March 5, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am ESTGraduate Student/ PostDoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am ESTTropical GeometryPost 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 nonnegative integers $g$ and $n$ such that $2g2+n>0$, 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 ESTGraduate Student/ PostDoc postseminar break in GatherCoffee Break  Virtual
Wednesday, March 10, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm ESTGathertown Afternoon CoffeeCoffee Break  Virtual

3:30  4:30 pm ESTSpringer fibers and the Delta ConjectureVirtual
 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.
Thursday, March 11, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am ESTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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 ESTSemistable Reduction SeminarSeminar  Virtual
 Dan Abramovich, Brown University
Friday, March 12, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am ESTGraduate Student/ PostDoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am ESTMatroidsPost 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 ESTGraduate Student/ PostDoc postseminar break in GatherCoffee Break  Virtual
Monday, March 15, 2021
Combinatorial Algebraic Geometry

11:00 am  12:00 pm EDTProfessional Development: Job ApplicationsProfessional Development  Virtual
Wednesday, March 17, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual

3:30  4:30 pm EDTSchubert Products for Permutations with Separated DescentsVirtual
 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 KnutsonYong '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 EdelmanGreene correspondence via bumpless pipe dreams.
Thursday, March 18, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am EDTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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
Friday, March 19, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am EDTGraduate Student/postdoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am EDTMatroids and AlgebraPost 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 socalled 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 EDTGraduate Student/postdoc postseminar break in GatherCoffee Break  Virtual
Monday, March 22, 2021

10:00  10:15 am EDTWelcomeVirtual
 Brendan Hassett, ICERM/Brown University

10:15  10:45 am EDTHey Series, Tell Me About the Extended Delta ConjectureVirtual
 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 $\Delta'_{e_{n1}}e_n$ 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 nonsymmetric HallLittlewood polynomials. This is joint work with Blasiak, Haiman, Pun, and Seelinger.

11:00  11:15 am EDTBreakCoffee Break  Virtual

11:15  11:45 am EDTCastelnuovoMumford regularity of matrix Schubert varietiesVirtual
 Speaker
 Oliver Pechenik, University of Waterloo
 Session Chair
 Leonardo Mihalcea, Virginia Polytechnic Institute and State University (Virtual)
Abstract
Jenna Rajchgot observed that the CastelnuovoMumford 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.

11:45 am  2:00 pm EDTLunch/Free TimeVirtual

2:00  2:30 pm EDTConformal blocks: an overviewVirtual
 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.

2:45  3:15 pm EDTQuantum 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 qdeformation of the reflection representation of the affine Weyl group in type A which leads to a qdeformation 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 wellunderstood case of affine A_1 and the mysterious case of affine A_2.

3:15  4:30 pm EDTGathertown ReceptionReception  Virtual
Tuesday, March 23, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

9:45  10:15 am EDTGeometry of semiinfinite flag manifoldsVirtual
 Speaker
 Syu Kato, Kyoto University
 Session Chair
 Thomas Lam, University of Michigan (Virtual)
Abstract
Semiinfinite flag manifolds are variants of affine flag manifolds whose Schubert cells are simultaneously infinitedimensional and infinitecodimensional. Such objects are introduced by Drinfeld and Lusztig around 1980 (as sets), and its finitedimensional approximation model was presented by FinkelbergMirkovic in 1999. Recently, we described them as an explicit indscheme of indinfinite 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 semiinfinite flag manifolds and their natural subschemes, and then explain their combinatorial and algebrogeometric 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.

10:30  10:45 am EDTBreakCoffee Break  Virtual

10:45  11:35 am EDTLightning TalksVirtual
 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 MissouriColumbia
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) finitelength coherent sheaves on a nodal singular curve behaves as the geometry predicts. 
11:35 am  2:00 pm EDTGathertown with Lightning speakers / LunchLunch/Free Time  Virtual

2:00  2:30 pm EDTGröbner Geometry of Schubert Polynomials Through IceVirtual
 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 6vertex 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.

2:45  3:00 pm EDTBreakCoffee Break  Virtual

3:00  3:30 pm EDTRigid local systems and the multiplicative eigenvalue problemVirtual
 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 LittlewoodRichardson 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.

3:45  4:15 pm EDTCyclotomic generating functionsVirtual
 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.
Wednesday, March 24, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

9:45  10:15 am EDTDoing Schubert Calculus with Bumpless Pipe DreamsVirtual
 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 cotransition 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.

10:30  10:45 am EDTBreakCoffee Break  Virtual

10:45  11:35 am EDTLightning TalksVirtual
 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 WLaplacian
Theo Douvropoulos, University of Massachusetts, Amherst
The normalized volume of the classical permutahedron is given by the formula n^{n2} 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 pointcounting 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 nonsymmetric Weylgroupvalued 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 "easyexit" 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 npointed (Ktheoretic) GromovWitten Invariants of Lines in Homogeneous Spaces
Weihong Xu, Rutgers University
'Quantum equals classical' refers to ways of identifying (Ktheoretic) GromovWitten invariants on a flag variety G/P with (Ktheoretic) 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 3pointed invariants. We obtain a 'quantum equals classical' theorem for npointed, genus 0 (Tequivariant Ktheoretic) GromovWitten invariants of lines in G/P, which generalizes the 3pointed 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 socalled 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. LaniniZainoulline (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 Svalued function on W, whose value is explicitly known for any finite reflection group W. LaniniZainoulline (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. 
11:35 am  2:00 pm EDTGathertown with Lightning speakers / LunchLunch/Free Time  Virtual

2:00  2:30 pm EDTDifferential operators for Schur and Schubert polynomialsVirtual
 Speaker
 Gleb Nenashev, Brown University
 Session Chair
 Alexander Yong, University of Illinois at UrbanaChampaign (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 LittlewoodRichardson coefficients only using these operators. This new point of view on Schur functions gives us an elementary proof of The Giambelli identity and JacobiTrudi 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.

2:45  3:00 pm EDTBreakCoffee Break  Virtual

3:00  3:30 pm EDTBackstable Ktheory Schubert calculusVirtual
 Speaker
 Mark Shimozono, Virginia Tech
 Session Chair
 Alexander Yong, University of Illinois at UrbanaChampaign (Virtual)
Abstract
We explain some properties of backstable limits of double Grothendieck polynomials. We discuss the dual basis of equivariant Khomology of the infinite Grassmannian.

3:45  4:15 pm EDTBow varietiesVirtual
 Speaker
 Richard Rimanyi, University of North Carolina at Chapel Hill
 Session Chair
 Alexander Yong, University of Illinois at UrbanaChampaign (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 12dimensional $T^*Gr(2,5)$ is a certain 4dimensional 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 HananyWitten 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.
Thursday, March 25, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

9:45  10:15 am EDTRefined Dubrovin's conjecture for coadjoint varietiesVirtual
 Speaker
 Nicolas Perrin, Versailles SaintQuentinenYvelines 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 semisimplicity 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 semisimple. 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

10:30  10:45 am EDTBreakCoffee Break  Virtual

10:45  11:15 am EDTmotivic Chern classes of Schubert cells and applicationsVirtual
 Speaker
 Changjian Su, University of Toronto
 Session Chair
 Anders Buch, Rutgers University (Virtual)
Abstract
The motivic Chern class in Ktheory is a generalization of the ChernSchwartzMacPherson 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 padic dual group, and relate the Euler characteristic of the motivic Chern classes to the IwahoriWhittaker functions.

11:30 am  1:30 pm EDTLunch/Free TimeVirtual

2:00  2:30 pm EDTSkewsymmetric matrix Schubert varietiesVirtual
 Speaker
 Brendan Pawlowski, University of Southern California
 Session Chair
 Thomas Lam, University of Michigan (Virtual)
Abstract
Imposing conditions on the ranks of upperleft corners of a skewsymmetric matrix defines a skewsymmetric matrix Schubert variety. We give Gröbner bases for the prime ideals of these varieties, and identify the corresponding initial ideals as the StanleyReisner 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.

2:45  3:00 pm EDTBreakCoffee Break  Virtual

3:00  3:30 pm EDTCanonical Bases for Coulomb BranchesVirtual
 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 tstructure on the dg category of coherent sheaves on the BravermanFinkelbergNakajima space of triples. The heart of this tstructure is a tensor category which is not braided, but admits renormalized rmatrices 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 halfBPS line defects in the relevant gauge theory, and is inspired by earlier work of KapustinSaulina and GaiottoMooreNeitzke. 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.

3:45  4:00 pm EDTBreakCoffee Break  Virtual

4:00  4:30 pm EDTSchubert puzzles from Lagrangian correspondencesVirtual
 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 2step 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 upto4step flag manifolds).
Friday, March 26, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

9:45  10:15 am EDTA combinatorial Chevalley formula for semiinfinite flag manifolds and its applicationsVirtual
 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 torusequivariant Kgroup of semiinfinite flag manifolds, which is expressed in terms of the socalled quantum alcove model. One application is the Chevalley formula for antidominant fundamental weights in the (small) torusequivariant quantum Ktheory 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 socalled quantum Grothendieck polynomials indeed represent Schubert classes in the (nonequivariant) quantum Ktheory of the type A flag manifold. This is joint work with Satoshi Naito and Daisuke Sagaki.

10:30  10:45 am EDTBreakCoffee Break  Virtual

10:45  11:15 am EDTCoxeterlike elements, Schubert geometry, and multiplicityfreeness in algebraic combinatoricsVirtual
 Speaker
 Alexander Yong, University of Illinois at UrbanaChampaign
 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 Levispherical Schubert varieties in G/B. In type A, this connects to key polynomials (Demazure characters), multiplicityfreeness, and “splitsymmetry” 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.

11:30 am  2:00 pm EDTLunch/Free TimeVirtual

2:00  2:30 pm EDTPositroids, knots, and q,tCatalan numbersVirtual
 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 topdimensional open positroid varieties are given by rational q,tCatalan numbers. As a consequence of the curious Lefschetz property, we obtain q,tsymmetry and unimodality statements for rational q,tCatalan numbers. Joint work with Thomas Lam. No special background on the above objects will be assumed.

2:45  3:00 pm EDTBreakCoffee Break  Virtual

3:00  3:30 pm EDTPuzzle rules in cotangent Schubert calculusVirtual
 Speaker
 Paul ZinnJustin, 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 Ktheory) and SegreSchwartzMacPherson 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
Combinatorial Algebraic Geometry

11:00 am  12:00 pm EDTProfessional Development: Hiring ProcessProfessional Development  Virtual
Wednesday, March 31, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual

3:30  4:30 pm EDTThe isomorphism problem for Schubert varieties.Virtual
 Edward Richmond, Oklahoma State University
Abstract
Schubert varieties in the full flag variety of KacMoody 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.
Thursday, April 1, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am EDTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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

3:30  4:40 pm EDTA trinity in affine Schubert calculusVirtual
 Syu Kato, Kyoto University
Abstract
One of the most famous results in affine Schubert calculus is Peterson's quantum = affine theorem (proved by LamShimozono 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 LamLiMihalceaShimozono. In this talk, we add the semiinfinite 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 semiinfinite flag manifold. This establishes the above conjecture of LamLiMihalceaShimozono. 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 semiinfinite flag manifold and its equivariant $K$group, please refer to the slide of my talk on March 23rd.
Friday, April 2, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am EDTGraduate Student/Postdoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am EDTCluster Algebras and Total PositivityPost 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 kpositivity, a generalization of total positivity. We will explore how much of the cluster algebraic structure of totally positive matrices extends to kpositive matrices.

11:30 am  12:00 pm EDTGraduate Student/Postdoc postseminar break in GatherCoffee Break  Virtual
Monday, April 5, 2021
Combinatorial Algebraic Geometry

11:00 am  12:00 pm EDTDirectorate CheckIn with Grads and PostdocsMeeting  Virtual
Wednesday, April 7, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual

2:00  3:00 pm EDTThe unramified affine springer fiber and the nabla operatorVirtual
 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 Schuberttype 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
Combinatorial Algebraic Geometry

10:00  11:00 am EDTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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
Friday, April 9, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am EDTGraduate Student/Postdoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am EDTSchubert CalculusPost 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 MurnaghanNakayama rule, Giambelli identity, and JacobiTrudi identities. They also fully determine LittlewoodRichardson 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 Ktheory 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, edgelabeled 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 EDTGraduate Student/Postdoc postseminar break in GatherCoffee Break  Virtual
Monday, April 12, 2021

9:45  10:00 am EDTWelcomeVirtual
 Brendan Hassett, ICERM/Brown University

10:00  10:45 am EDTThe Foundation of a MatroidVirtual
 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.

11:00  11:15 am EDTBreakCoffee Break

11:15 am  12:00 pm EDTLagrangian geometry of matroidsVirtual
 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 ChernSchwartzMacPherson 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 HodgeRiemann relations. Together, these imply conjectures of Brylawski and Dawson about the logconcavity of the hvectors of the broken circuit complex and independence complex of a matroid.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTTautological classes of matroidsVirtual
 Speaker
 Christopher Eur, Stanford University
 Session Chair
 Sam Payne, University of Texas at Austin
Abstract
We introduce certain torusequivariant 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 Chowtheoretic description and a logconcavity property for a 4variable transformation of the Tutte polynomial, and by establishing an exceptional HirzebruchRiemannRochtype formula for permutohedral varieties that translates between Ktheory and Chow theory. This is joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

2:30  2:45 pm EDTBreakCoffee Break

2:45  3:30 pm EDTKazhdanLusztig theory and singular Hodge theory for matroidsVirtual
 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

3:45  4:45 pm EDTGathertown ReceptionReception  Virtual
Tuesday, April 13, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTReal phase structures on matroid fansVirtual
 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 nonsingular tropical variety, the real part is a PLmanifold. Moreover, for a nonsingular 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 EDTBreakCoffee Break

11:15 am  12:00 pm EDTCombinatorics and real lifts of bitangents to tropical quartic curvesVirtual
 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.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTTropical psi classesVirtual
 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 onedimensional families of genusone tropical curves.

2:30  2:45 pm EDTBreakCoffee Break

2:45  3:30 pm EDTWhen 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.CidRuiz, B.Li, J.Montano, and N.Zhang.
Wednesday, April 14, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTWallcrossing phenomenon for NewtonOkounkov bodiesVirtual
 Speaker
 Laura Escobar, Washington University St. Louis
 Session Chair
 Lara Bossinger, Mathematics Institute UNAM, Oaxaca (Virtual)
Abstract
A NewtonOkounkov 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 KavehManon gives an explicit link between tropical geometry and NewtonOkounkov bodies. We use this link to describe a wallcrossing phenomenon for NewtonOkounkov bodies. As an example, we describe wallcrossing formula in the case of the Grassmannian Gr(2,m). This is joint work with Megumi Harada.

11:00  11:15 am EDTBreakCoffee Break  Virtual

11:15 am  12:00 pm EDTOn combinatorics of Arthur's trace formula, convex polytopes, and toric varietiesVirtual
 Speaker
 Kiumars Kaveh, University of Pittsburgh
 Session Chair
 Lara Bossinger, Mathematics Institute UNAM, Oaxaca (Virtual)
Abstract
I start by discussing two beautiful wellknown theorems about decomposing a convex polytope into an signed sum of cones, namely the classical BrianchonGram theorem and LawrenceVarchenko theorem. I will then explain a generalization of the BrianchonGram 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).

12:00  1:30 pm EDTLunch/Free Time

1:30  2:30 pm EDTPoster SessionVirtual
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 steadystate ideals. Steadystate 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 steadystate ideal? In particular, our main results describe three operations on the reaction graph that preserve the steadystate ideal. Furthermore, since the motivation for this work is the classification of steadystate ideals, monomials play a primary role. To this end, combinatorial conditions are given to identify monomials in a steadystate ideal, and we give a sufficient condition for a steadystate ideal to be monomial.
Construction and properties of Kanev surfaces in toric 3folds
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 3folds. 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 LittlewoodRichardson coefficients and the LittlewoodRichardson coefficients
Nguyen Khanh, Institut Camille Jordan
We give a new interpretation of the shifted LittlewoodRichardson 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 LittlewoodRichardson 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 UrbanaChampaign
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\delta1}) $ 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)(d1)^n \right)^{\delta}+O\left(d^{n(\delta1)} \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 EDTOn NewtonOkounkov bodies associated to GrassmanniansVirtual
 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 NewtonOkounkov bodies that one can associate to "nice" compactifications of cluster varieties. In particular, I will explain how this approach recovers RietschWilliams' construction of NewtonOkounkov bodies for Grassmannians. In order to make the precise connection it will be necessary to explain how the MarshRietsch potential and the GrossHackingKeelKontsevich potential for Grassmannians are related. Finally, I will draw some consequences from this relation such as an isomorphism of the toric degenerations obtained by RietschWillimas 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.

3:30  3:45 pm EDTBreakCoffee Break

3:45  4:30 pm EDTBroken line convexityVirtual
 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 NewtonOkounkov 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 EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTToric vector bundles  an overviewVirtual
 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

11:00  11:15 am EDTBreakCoffee Break

11:15 am  12:00 pm EDTThe FultonMacPherson compactification is not a Mori dream spaceVirtual
 Speaker
 José González, University of California, Riverside
 Session Chair
 Linda Chen, Swarthmore College (Virtual)
Abstract
We show that the FultonMacPherson 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 EDTLunch/Free Time

1:30  2:15 pm EDTSchubert polynomials from a polytopal point of viewVirtual
 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 logconcave 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 EDTBreakCoffee Break

2:45  3:30 pm EDTFamilies of Gröbner degenerationsVirtual
 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 mspace that assembles the Gröbner degenerations of V associated with all faces of C. This is a multiparameter generalization of the classical oneparameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from KavehManon's recent work on the classification of toric flat families over toric varieties: it is the pullback 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 StanleyReisner 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.
Friday, April 16, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTThe logarithmic Hilbert scheme of curvesVirtual
 Speaker
 Dhruv Ranganathan, University of Cambridge
 Session Chair
 Federico Ardila, San Francisco State University (Virtual)
Abstract
Within the Hilbert scheme of curves in projective space is a subscheme of curves that are "tropical" in the sense of Tevelev: they interact well with the coordinate subspaces. I will explain why, from the point of view of tropical and logarithmic geometry, this locus ought to be the principal open cell in another moduli space, of which the Hilbert scheme is only an approximation. This "logarithmic Hilbert scheme" was recently constructed in work with Davesh Maulik (MIT) and is the core of a new theory of logarithmic DonaldsonThomas invariants. The story touches another major character in the story of polyhedral and algebraic geometry: the secondary polytope of Gel'fandKapranovZelevinsky. I'll try to give some sense for why.

11:00  11:15 am EDTBreakCoffee Break

11:15 am  12:00 pm EDTInitial degenerations of GrassmanniansVirtual
 Speaker
 Daniel Corey, University of Wisconsin, Madison
 Session Chair
 Federico Ardila, San Francisco State University (Virtual)
Abstract
We construct closed immersions from initial degenerations of Gr_0(d,n)the open cell in the Grassmannian Gr(d,n) given by the nonvanishing of all Plücker coordinatesto limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms in many cases, including (d,n) equal to (2,n), (3,6) and (3,7). As an application, Gr_0(3,7) is schön, and the Chow quotient of Gr(3,7) by the maximal torus in PGL(7) is the log canonical compactification of the moduli space of 7 points in P^2 in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev. Time permitting, I will discuss recent work on extending these results to the Lietype D setting.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTOn the topweight rational cohomology of A_gVirtual
 Speaker
 Melody Chan, Brown University
 Session Chair
 Yoav Len, University of St Andrews (Virtual)
Abstract
I'll report on recent work using tropical techniques to find new rational cohomology classes in moduli spaces A_g of abelian varieties, building on previous joint work with Galatius and Payne on M_g. Joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

2:30  2:45 pm EDTBreakCoffee Break

2:45  3:30 pm EDTTropical Flag VarietiesVirtual
 Speaker
 Madeline Brandt, Brown University
 Session Chair
 Yoav Len, University of St Andrews (Virtual)
Abstract
Flag matroids are combinatorial abstractions of flags of linear subspaces, just as matroids are of linear subspaces. We introduce the flag Dressian as a tropical analogue of the partial flag variety, and give a correspondence between: (a) points on the flag Dressian, (b) valuated flag matroids, (c) flags of projective tropical linear spaces, and (d) coherent flag matroidal subdivisions of the flag matroid polytope. The ideas presented in this talk will be brought to life through examples.
Monday, April 19, 2021
Combinatorial Algebraic Geometry

11:00 am  12:00 pm EDTPapers and JournalsProfessional Development  Virtual
Wednesday, April 21, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual

3:30  4:30 pm EDTSchubert Calculus via bosonic operatorsVirtual
 Gleb Nenashev, Brown University
Abstract
I will present a definition and some important properties of the bosonic operators for backstable Schubert polynomials. The operators act on the left weak Bruhat order (divided difference and Monk’s rule use the right side action on permutations in my notations). These operators with an extra condition give sufficiently enough linear equations for the structure constants of flag varieties. In particular, they provide a recurrent formula for the structure constants. In some special cases it is easy to check the positivity of the structure constants using this formula, examples will be presented. One of the advantages of our method is that we do not need to use formulas for Schubert polynomials and backstable Schubert polynomials. Nevertheless if time permits, I will also show how to establish the pipe dreams formula using these operators.
Thursday, April 22, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am EDTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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

3:30  4:40 pm EDTBow VarietiesCombinatorics, Geometry, and Characteristic ClassesVirtual
 Speaker
 Yiyan Shou, UNC  Chapel Hill
 Session Chair
 Anders Buch, Rutgers University
Abstract
Cherkis bow varieties are believed to be a natural setting for the study of 3d (N=4) mirror symmetry from the perspective of characteristic classes. This talk will review the "quiver" definition of bow varieties, due to Nakajima and Takayama, and introduce a combinatorial framework for their study. This framework includes brane diagrams and related combinatorial operations and constructions. Using this combinatorial framework, we will analyze the tangent bundle, torus fixed points, and invariant curves, key ingredients for defining characteristic classes. Finally, examples of cohomological stable envelopes will be calculated. This talk is based on a joint work with Richárd Rimányi.
Friday, April 23, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am EDTGraduate Student/postdoc postseminar break in GatherCoffee Break  Virtual

10:30  11:00 am EDTSlack realization spaces and realizability of polytopesPost Doc/Graduate Student Seminar  Virtual
 Antonio Macchia, Freie Universität Berlin

11:00  11:30 am EDTIntroduction to Hessenberg varietiesPost Doc/Graduate Student Seminar  Virtual
 Sean Griffin, Brown University
Abstract
Hessenberg varieties are a family of subvarieties of the complete flag variety that have many wonderful combinatorial properties. I'll survey some of these combinatorial properties, including an amazing connection to chromatic quasisymmetric functions, which was conjectured by Shareshian and Wachs and proven by Brosnan and Chow.

11:30 am  12:00 pm EDTGraduate Student/postdoc postseminar break in GatherCoffee Break  Virtual
Monday, April 26, 2021
Combinatorial Algebraic Geometry

11:00 am  12:00 pm EDTGrant ProposalsProfessional Development  Virtual
Wednesday, April 28, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual

3:30  4:30 pm EDTThe Abelian/nonAbelian correspondence and mirror symmetryVirtual
 Elana Kalashnikov, Harvard University
Abstract
The Abelian/nonAbelian correspondence is a powerful tool that can be used to study GIT quotients V//G, where V is a vector space. Such GIT quotients include type A flag varieties and quiver flag varieties. The principle of the Abelian/nonAbelian correspondence is that a GIT quotient V//G can be studied by considering the much simpler Abelian GIT quotient V//T, where T is maximal torus of G. I'll discuss applications of the Abelian/nonAbelian correspondence to quantum cohomology and mirror symmetry of type A flag varieties and quiver flag varieties, focusing on rimhook removal rules and Plücker coordinate mirrors. Part of this talk will report on joint work with Wei Gu.
Thursday, April 29, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am EDTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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
Friday, April 30, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am EDTGraduate Student/postdoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am EDTToric GeometryPost Doc/Graduate Student Seminar  Virtual
 Sean Griffin, Brown University
 Mariel Supina, University of California, Berkeley
Abstract
An introduction to toric geometry
Sean Griffin, Brown University
I'll introduce the basics of toric geometry, including the construction of a toric variety from a rational convex polytope. We'll do lots of examples to illustrate the ideas.
The equivariant Ehrhart theory of the permutahedron
Mariel Supina, University of California, Berkeley
Ehrhart theory is a topic in geometric combinatorics which involves counting the lattice points inside of lattice polytopes. Stapledon (2010) introduced equivariant Ehrhart theory and conjectured a relationship between the equivariant H*series of a polytope and invariant hypersurfaces in its corresponding toric variety. In this talk, I will describe Stapledon's conjecture and discuss joint work with Ardila and Vindas Meléndez (2020) on verifying his conjecture in this special case. 
11:30 am  12:00 pm EDTGraduate Student/postdoc postseminar break in GatherCoffee Break  Virtual
Wednesday, May 5, 2021
Combinatorial Algebraic Geometry

3:00  3:30 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual

3:30  4:30 pm EDTEquivariant Schubert Calculus of Peterson VarietiesVirtual
 Rahul Singh, Virginia Polytechnic Institute and State University
Abstract
Peterson varieties are certain singular subvarieties of flag manifolds, naturally admitting onedimensional torus action. Starting with a natural basis for the equivariant homology of a Peterson variety, we construct a dual basis in cohomology and show that the structure constants of the cohomology ring are positive with respect to this basis. We also discuss the sense in which the fundamental classes of the Peterson varieties exhibit a stability analogous to the stability of Schubert classes, and how this can be used to streamline various calculations in the Schubert calculus of Peterson varieties. This is joint work with Rebecca Goldin and Leonardo Mihalcea.
Thursday, May 6, 2021
Combinatorial Algebraic Geometry

10:00  11:00 am EDTLearning Seminar  Differential forms on tropical moduli spacesSeminar  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 subgoal, 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 visavis 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

12:00  1:00 pm EDTSocially Distanced Walk with DirectorateSocially Distanced Walk  121 South Main Street Main Entrance
Friday, May 7, 2021
Combinatorial Algebraic Geometry

10:00  10:30 am EDTGraduate Student/postdoc preseminar break in GatherCoffee Break  Virtual

10:30  11:30 am EDTKacMoody Schubert calculusPost Doc/Graduate Student Seminar  Virtual
 Shuai Jiang, Virginia Tech
Abstract
Equivariant homology of infinite Grassmannian
Let G be a semisimple algebraic group over C, B ⊂ G a Borel subgroup and T a maximal torus. Peterson discovered the ”quantum equals affine” phenomena and Lam and Shimozono proved that the equivariant Schubert homology structure constants of affine Grassmannian are the Schubert structure constants for the Tequivariant quantum cohomology rings QHT(G/B). Later, Lam and Shimozono found the equivariant homology Chevalley formula and more recently Lam, Lee and Shimozono use this formula to derive the jbasis coefficients of infinity Grassmannian for a hook. In the first part of this talk, I will discuss the jbasis in equivariant homology of affine Grassmannian. Then, I will explain their hookformula. 
11:30 am  12:00 pm EDTGraduate Student/postdoc postseminar break in GatherCoffee Break  Virtual
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