VIRTUAL ONLY: Introductory Workshop: Combinatorial Algebraic Geometry

Institute for Computational and Experimental Research in Mathematics (ICERM)

https://icerm.brown.edu/programs/sp-s21/w1/
February 1, 2021 - February 5, 2021
Monday, February 1, 2021
  • 9:50 - 10:00 am EST
    Welcome
    Virtual
    • Brendan Hassett, ICERM/Brown University
  • 10:00 - 10:45 am EST
    Short Introduction to Hodge Structures Part 1
    Virtual
    • Colleen Robles, Duke University
    Abstract
    I will introduce Hodge structures and describe the linear structures underlying the Hard Lefschetz Theorem and Hodge--Riemann Bilinear Relations.
  • 11:00 am - 12:00 pm EST
    Problem Session 1
    Problem Session - Virtual
  • 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
    Basic Schubert Calculus Part 1
    Virtual
    • Sara Billey, University of Washington
  • 3:00 - 4:00 pm EST
    Problem Session 2
    Problem Session - Virtual
  • 4:00 - 4:30 pm EST
    Basic Schubert Calculus Part 2
    Virtual
    • Sara Billey, University of Washington
  • 4:30 - 5:15 pm EST
    Reception
    Virtual
Tuesday, February 2, 2021
  • 9:00 - 9:45 am EST
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EST
    Basic notions in cotangent Schubert calculus - Part 1
    Virtual
    • Richard Rimanyi, University of North Carolina at Chapel Hill
    Abstract
    A key notion of Schubert Calculus is the cohomology ring of a homogeneous space, together with a distinguished basis: the collection of Schubert classes. There is a one-parameter deformation of the notion "Schubert class", called Chern-Schwartz-MacPherson (CSM-) class (a.k.a. characteristic cycle class, or cohomological stable envelope). In this lecture/workshop we will define the CSM class, and illustrate many of its properties through examples.
  • 11:00 am - 12:00 pm EST
    Problem Session 3
    Problem Session - Virtual
  • 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
    Problem Session 4
    Problem Session - Virtual
  • 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 3, 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
    Problem Session 5
    Problem Session - Virtual
  • 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
    Affine Schubert Calculus Part 1
    Virtual
    • Mark Shimozono, Virginia Tech
  • 3:00 - 4:00 pm EST
    Problem Session 6
    Problem Session - Virtual
  • 4:00 - 4:30 pm EST
    Affine Schubert Calculus Part 2
    Virtual
    • Mark Shimozono, Virginia Tech
  • 4:30 - 5:15 pm EST
    Gathertown Afternoon Coffee - Informal Grad Student / Postdoc Focused
    Coffee Break - Virtual
Thursday, February 4, 2021
  • 9:00 - 9:45 am EST
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EST
    Introduction to Tropical Geometry Through Curves - Part 1
    Virtual
    • Madeline Brandt, Brown University
    Abstract
    Tropical geometry equips varieties with a combinatorial counterpart called the tropicalization. In this talk, I will introduce some of the key ideas in tropical geometry by studying curves. This will include definitions and examples of embedded tropicalization for curves in the plane, abstract tropicalization / dual graphs, and Berkovich skeleta.
  • 11:00 am - 12:00 pm EST
    Problem Session 7
    Problem Session - Virtual
  • 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
    Problem Session 8
    Problem Session - Virtual
  • 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 5, 2021
  • 10:00 - 10:45 am EST
    Cluster structures in commutative rings Part 1
    Virtual
    • Lauren Williams, Harvard University
    Abstract
    In my first talk, I will give a gentle introduction to cluster algebras. In the second talk, I will describe how to identify a commutative ring (such as the coordinate ring of an algebraic variety) with a cluster algebra, and provide several examples. I will also discuss how to describe cluster algebras by generators and relations.
  • 11:00 am - 12:00 pm EST
    Problem Session 9
    Problem Session - Virtual
  • 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
    Problem Session 10
    Problem Session - Virtual
  • 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.

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

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