VIRTUAL ONLY: Introductory Workshop: Combinatorial Algebraic Geometry
Institute for Computational and Experimental Research in Mathematics (ICERM)
February 1, 2021  February 5, 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.
All event times are listed in ICERM local time in Providence, RI (Eastern Standard Time / UTC5).
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