Math 270 - Hot topics in algebra: Grassmannians, matroids, positivity, clusters, and beyond -- Fall 2017

Lectures: Thursdays 9:30am-11am, 2 Evans, plus a few Tuesdays (maybe Oct 24 and Nov 21, to be confirmed) 9:30am-11am, 2 Evans.

Lecturer: Lauren Williams (office Evans Hall 913, e-mail

Office Hours: by appointment

Course description

The classical theory of total positivity studied totally positive matrices, matrices with all minors positive. The theory was pioneered by Gantmacher, Krein, Schoenberg, Whitney, and others in the first half of the 20th century, and subsequently generalized in the setting of Lie theory by Lusztig in the 1990's, who had observed that his canonical bases had surprising positivity properties. Meanwhile in 2000, Fomin and Zelevinsky introduced cluster algebas, a class of combinatorially defined commutative rings, which provide a unifying structure for phenomena in total positivity, and turned out to be connected to a host of other fields, including quiver representations, Teichmuller theory, Poisson geometry, etc. In this course I'll survey these developments, with a particular focus on the totally nonnegative Grassmannian, whose combinatorial structure was first developed by Postnikov. If time permits I may discuss recent developments in the field, including positroids, amplituhedra, KP solitons, and Newton-Okounkov bodies.

I will assume that people have some familiarity with combinatorics. Familiarity with root systems would also be helpful. I will not assume prior knowledge of cluster algebras.

References for total positivity:

References for cluster algebras:

References for Grassmannians:

Other topics:


  • Lecture 1 (Aug 24): The Grassmannian and its positive part; overview of the course.
  • August 31: MSRI Connections for women workshop: geometric and topological combinatorics.
  • September 7: MSRI Introductory workshop: geometric and topological combinatorics.
  • Lecture 2: (September 14): Matroids, the matroid stratification, and matroid polytopes.
  • Lecture 3 (Sept. 21): The positroid stratification of the totally nonnegative Grassmannian.
  • Lecture 4 (Sept. 28): Positroids, noncrossing partitions, and realizability.
  • Lecture 5 (Oct. 5): Sign variation and the amplituhedron.
  • October 12: MSRI Geometric and topological combinatorics: modern techniques and methods.
  • Lecture 6 (Oct. 19): Reduced plabic graphs: positivity tests and quiver mutation.
  • Lecture 7 (Oct. 26): Cluster algebras, the Laurent phenomenon, and positivity.
  • Lecture 8 (Nov. 2): The starfish lemma and the coordinate ring of the (affine cone over the) Grassmannian.
  • Lecture 9 (Nov. 9): Network charts, cluster charts, and the twist map.
  • Lecture 10 (Nov. 16):
  • Lecture 11 (Nov. 30):
  • Lecture 12 (Dec. 7):