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Organizing Committee
Rutgers University
Brown University
University of Michigan
Virginia Polytechnic Institute and State University
Abstract
This introductory workshop in combinatorial algebraic geometry is aimed at early career mathematicians and other mathematicians looking for an entry point into the field. The workshop will feature expository lectures on some of the basic objects of interest, together with "expert'' lectures discussing some current trends in the field. There will also be ample time for problem sessions and discussions.
Image Credit: Benjamin Schroeter
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Workshop Schedule
Monday, February 01, 2021
9:50 - 10:00 AM EST
Welcome
Virtual
Brendan Hassett, ICERM/Brown University
10:00 - 10:45 AM EST
Short Introduction to Hodge Structures Part 1
Virtual
Colleen Robles, Duke University
Abstract
I will introduce Hodge structures and describe the linear structures underlying the Hard Lefschetz Theorem and Hodge--Riemann Bilinear Relations.
11:00 AM - 12:00 PM EST
12:00 - 12:30 PM EST
Short Introduction to Hodge Structures Part 2
Virtual
Colleen Robles, Duke University
Abstract
I will introduce Hodge structures and describe the linear structures underlying the Hard Lefschetz Theorem and Hodge--Riemann Bilinear Relations.
12:30 - 2:00 PM EST
Lunch/Free Time
Virtual
2:00 - 2:45 PM EST
3:00 - 4:00 PM EST
4:00 - 4:30 PM EST
4:30 - 5:15 PM EST
Reception
Virtual
Tuesday, February 02, 2021
9:00 - 9:45 AM EST
Gathertown Morning Coffee
Coffee Break - Virtual
10:00 - 10:45 AM EST
Basic notions in cotangent Schubert calculus - Part 1
Virtual
Richard Rimanyi, University of North Carolina at Chapel Hill
Abstract
A key notion of Schubert Calculus is the cohomology ring of a homogeneous space, together with a distinguished basis: the collection of Schubert classes. There is a one-parameter deformation of the notion "Schubert class", called Chern-Schwartz-MacPherson (CSM-) class (a.k.a. characteristic cycle class, or cohomological stable envelope). In this lecture/workshop we will define the CSM class, and illustrate many of its properties through examples.
11:00 AM - 12:00 PM EST
12:00 - 12:30 PM EST
Basic notions in cotangent Schubert calculus - Part 2
Virtual
Richard Rimanyi, University of North Carolina at Chapel Hill
Abstract
A key notion of Schubert Calculus is the cohomology ring of a homogeneous space, together with a distinguished basis: the collection of Schubert classes. There is a one-parameter deformation of the notion "Schubert class", called Chern-Schwartz-MacPherson (CSM-) class (a.k.a. characteristic cycle class, or cohomological stable envelope). In this lecture/workshop we will define the CSM class, and illustrate many of its properties through examples.
12:30 - 2:00 PM EST
Lunch/Free Time
Virtual
2:00 - 2:45 PM EST
Matroids Part 1
Virtual
Christopher Eur, Stanford University
Abstract
We give an introduction to matroid theory with a view towards its recent interactions with algebraic geometry. In the first half, we study matroids as combinatorial abstractions of hyperplane arrangements, which will lead us to Chow rings of matroids, modeled after the geometry of wonderful compactifications of hyperplane arrangement complements. In the second half, we give a broad survey of some recent developments involving other different but related algebro-geometric models of matroids.
3:00 - 4:00 PM EST
4:00 - 4:30 PM EST
Matroids Part 2
Virtual
Christopher Eur, Stanford University
Abstract
We give an introduction to matroid theory with a view towards its recent interactions with algebraic geometry. In the first half, we study matroids as combinatorial abstractions of hyperplane arrangements, which will lead us to Chow rings of matroids, modeled after the geometry of wonderful compactifications of hyperplane arrangement complements. In the second half, we give a broad survey of some recent developments involving other different but related algebro-geometric models of matroids.
Wednesday, February 03, 2021
10:00 - 10:45 AM EST
Quantum Cohomology Part 1
Virtual
Nicolas Perrin, Versailles Saint-Quentin-en-Yvelines University
11:00 AM - 12:00 PM EST
12:00 - 12:30 PM EST
Quantum Cohomology Part 2
Virtual
Nicolas Perrin, Versailles Saint-Quentin-en-Yvelines University
12:30 - 2:00 PM EST
Lunch/Free Time
Virtual
2:00 - 2:45 PM EST
3:00 - 4:00 PM EST
4:00 - 4:30 PM EST
4:30 - 5:15 PM EST
Gathertown Afternoon Coffee - Informal Grad Student / Postdoc Focused
Coffee Break - Virtual
Thursday, February 04, 2021
9:00 - 9:45 AM EST
Gathertown Morning Coffee
Coffee Break - Virtual
10:00 - 10:45 AM EST
Introduction to Tropical Geometry Through Curves - Part 1
Virtual
Madeline Brandt, Brown University
Abstract
Tropical geometry equips varieties with a combinatorial counterpart called the tropicalization. In this talk, I will introduce some of the key ideas in tropical geometry by studying curves. This will include definitions and examples of embedded tropicalization for curves in the plane, abstract tropicalization / dual graphs, and Berkovich skeleta.
11:00 AM - 12:00 PM EST
12:00 - 12:30 PM EST
Introduction to Tropical Geometry Through Curves - Part 2
Virtual
Madeline Brandt, Brown University
Abstract
Tropical geometry equips varieties with a combinatorial counterpart called the tropicalization. In this talk, I will introduce some of the key ideas in tropical geometry by studying curves. This will include definitions and examples of embedded tropicalization for curves in the plane, abstract tropicalization / dual graphs, and Berkovich skeleta.
12:30 - 2:00 PM EST
Lunch/Free Time
Virtual
2:00 - 2:45 PM EST
Moduli spaces of tropical curves Part 1
Virtual
Sam Payne, University of Texas at Austin
Abstract
In the first part of this talk, I will introduce the moduli space of stable tropical curves and discuss its combinatorial structure and topology, illustrating with simple examples in low genus. In the problem session, you will compute one further example. And in the second half of the talk, I will explain some general results about the topology of moduli spaces of stable tropical curves, how these relate to the topology of the algebraic moduli spaces M_g and M_{g,n}, and then state some open problems and conjectures that you may find interesting to think about during the semester.
3:00 - 4:00 PM EST
4:00 - 4:30 PM EST
Moduli spaces of tropical curves Part 2
Virtual
Sam Payne, University of Texas at Austin
Abstract
In the first part of this talk, I will introduce the moduli space of stable tropical curves and discuss its combinatorial structure and topology, illustrating with simple examples in low genus. In the problem session, you will compute one further example. And in the second half of the talk, I will explain some general results about the topology of moduli spaces of stable tropical curves, how these relate to the topology of the algebraic moduli spaces M_g and M_{g,n}, and then state some open problems and conjectures that you may find interesting to think about during the semester.
Friday, February 05, 2021
10:00 - 10:45 AM EST
Cluster structures in commutative rings Part 1
Virtual
Lauren Williams, Harvard University
Abstract
In my first talk, I will give a gentle introduction to cluster algebras. In the second talk, I will describe how to identify a commutative ring (such as the coordinate ring of an algebraic variety) with a cluster algebra, and provide several examples. I will also discuss how to describe cluster algebras by generators and relations.
11:00 AM - 12:00 PM EST
12:00 - 12:30 PM EST
Cluster structures in commutative rings Part 2
Virtual
Lauren Williams, Harvard University
Abstract
In my first talk, I will give a gentle introduction to cluster algebras. In the second talk, I will describe how to identify a commutative ring (such as the coordinate ring of an algebraic variety) with a cluster algebra, and provide several examples. I will also discuss how to describe cluster algebras by generators and relations.
12:30 - 2:00 PM EST
Lunch/Free Time
Virtual
2:00 - 2:45 PM EST
Cluster varieties Part 1
Virtual
David Speyer, University of Michigan
Abstract
We will discuss the geometry of the affine algebraic varieties associated to cluster algebras. In the first hour, we will give examples and talk about open sets, smoothness and covering maps; in the second hour, we will talk about mixed Hodge structures.
3:00 - 4:00 PM EST
4:00 - 4:45 PM EST
Cluster varieties Part 2
Virtual
David Speyer, University of Michigan
Abstract
We will discuss the geometry of the affine algebraic varieties associated to cluster algebras. In the first hour, we will give examples and talk about open sets, smoothness and covering maps; in the second hour, we will talk about mixed Hodge structures.
