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 total positivity or cluster algebras.


If you are taking this class for a grade, you need to write a final paper. This can be on a topic of your choice, provided it is related to the class, and should be 5 to 10 pages in length. The paper is due on December 8; no late papers will be accepted. You can email it to me. Please discuss with me (in person or in email) the topic of your final project, no later than mid-October, to make sure I approve.


References for matroids:

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): Matroid polytopes and their characterization by Gelfand-Goresky-MacPherson-Serganova.
  • 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.
  • Lecture 6 (Oct 12): Combinatorics of the tree amplituhedron, 9:30am at MSRI (!) -- see Geometric and topological combinatorics: modern techniques and methods.
  • Lecture 7 (Oct. 19): Reduced plabic graphs and positivity tests for Grassmannians.
  • Lecture 8 (Oct. 26): Guest lecture by Pavel Galashin: plabic graphs and zonotopal tilings.
  • Lecture 9 (Nov. 2): Cluster algebras, the Laurent phenomenon, and positivity.
  • Lecture 10 (Nov. 9): The starfish lemma and the coordinate ring of the (affine cone over the) Grassmannian.
  • Lecture 11 (Nov. 16): The coordinate ring of the Grassmannian (continued).
  • Lecture 12 (Nov. 30): Network charts, cluster charts, and the twist map.
  • Lecture 13 (Dec. 7): Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians.