Grant Barkley (University of Michigan)
Title: Bruhat orders and applications
Abstract: Coxeter groups are groups that appear throughout algebra and geometry, usually in the form of groups generated by reflections. In this mini-course, we will introduce Coxeter groups, focusing on the most important example: the symmetric group. Our main goal is to understand two important partial orderings on the elements of Coxeter groups: the Bruhat order and the weak order. Bruhat order is closely tied to Schubert calculus and flag varieties, and weak order has applications to root systems and cluster algebras. We will become comfortable thinking about these orders from different perspectives, and discuss some of their applications in algebra and geometry.
Lecture notes and problems (link)
Amanda Burcroff (Massachusetts Institute of Technology)
Title: Cluster algebras
Abstract: This minicourse will explore cluster algebras, which have exploded in importance throughout mathematics and physics since they were introduced in 2002 by Fomin and Zelevinsky. Cluster algebras are commutative rings with a distinguished set of generators obtained via the combinatorial process of mutation. The theory of cluster algebras gives a discrete framework for understanding many important algebraic and geometric spaces, and their beautiful properties unify phenomena across various contexts. We will look at what the fundamental characteristics of a cluster algebra are, where cluster algebras arise, and why they are important. Along the way, we’ll see some foundational constructions of cluster algebras coming from friezes, triangulations of surfaces, total positivity, and more.
Suggested reading in advance of the program: Matthew Pressland, “From frieze patterns to cluster categories” (arXiv:2010.14302), Section 2 (pages 2–7).
Lecture notes and problems (link)
Colleen Robichaux (University of California, Los Angeles)
Title: Schubert calculus
Abstract: In this minicourse we explore the Schubert calculus of the Grassmannian, an area of mathematics which aims to understand the intersection theory of the Grassmannian using tools from several areas such as geometry, topology, and combinatorics. We begin with an introduction to the Grassmannian and Schubert varieties, motivated by Hilbert’s Fifteenth Problem. Then we investigate the underlying connection between Grassmannian Schubert calculus and symmetric function theory to see how these intersection numbers may be computed through purely combinatorial means. Further, these intersection numbers naturally appear in linear algebra and representation theory. In the last lecture we discuss their connection to the Horn eigenvalue problem and the problem’s solution through the Knutson–Tao Saturation Theorem.
Lecture 1 (PDF)
Lecture 2 (PDF)
Lecture 3 (PDF)
Problems 1 (PDF)
Problems 2 (PDF)
Problems 3 (PDF)