Math 583: Grassmannians, clusters algebras, and positroids
Spring, 2023
Prof. Sara Billey
Monday, Wednesday, Friday, time TBD UW Seattle Campus
Summary:
This proposed course is based on an emerging book on the arXiv by
Sergey Fomin, Lauren William, and Andrei Zelevinsky called
"Introduction to cluster algebras". The story starts approximately 70
years ago with totally positive matrices. A matrix is totally
positive if every minor is positive. Similarly, a matrix is totally
nonnegative if every minor is nonnegative. Such matrices appear in a
wide range of mathematical subjects including combinatorics,
probability, stochastic processes, representation theory, and inverse
problems. For example, totally positive matrices characterize certain
invertible electrical networks studied by our emeritus faculty Ed
Curtis, Jim Morrow and several of their REU students. Other examples
come from the work of Fomin and Zelevinsky on stratified spaces,
double Bruhat cells, and cluster algebras. This paper in particular
has become highly cited and inspired a wealth of new research in
combinatorics, algebra, representation theory, algebraic geometry, and
theoretical physics.
In this course, we will introduce cluster algebras and prove many of
their basic properties. We will discuss their connections with several
areas including the positroid cells in the totally nonnegative
Grassmannians and asymmetric exclusion processes. Students will
present material from the text and/or from recent related papers in
this active area of research.
Prerequists:
This is a topics course aimed at advanced math
students. Some coursework in algebraic geometry and combinatorics at
the graduate level is required. Contact the instructor if there are
questions on prereqs.
Course Materials
Resources
- Cluster
Algebras Portal extensive collection of resources maintained by
Sergey Fomin including the book chapters currently on the arXiv by
Sergey Fomin, Lauren William, and Andrei Zelevinsky, "Introduction to
cluster algebras," that we will use as our textbook.
- The Positive
Grassmannian, the Amplituhedron, and Cluster Algebras
by Lauren K. Williams, preprint October 2021. Paper based on the
author's ICM 2022 talk.
- Dimer
Conference 2020 Great collection of online talks on Cluster
Algebras and related topics.
- Edward B. Curtis and James A. Morrow, "Inverse Problems for
Electrical Networks" World Scientific Pub Co; (May 2000).
- S.Fomin and A.Zelevinsky, "Total positivity: tests and
parametrizations", Math. Intelligencer 22 (2000), 23-33.
Syllabus
TBD, see Canvas page.
Sara Billey
Last modified: Mon Sep 26 16:44:35 PDT 2022