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

Lecturer: Lauren Williams (office Evans Hall 913, e-mail williams@math.berkeley.edu)

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.

Grading

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

References for matroids:

• Bjorner, Las Vergnas, Sturmfels, White, Ziegler, Oriented Matroids, Cambridge University Press, Cambridge, 1999.
• Oxley, Matroid theory, Oxford University Press, Oxford, 2011.

References for total positivity:

• Karlin, Total positivity, Volume I, Stanford University Press, Stanford, CA 1968.
• Lusztig, Total positivity in reductive groups, Lie theory and geometry: in honor of B. Kostant, Progress in Math. 123, Birkhauser, 1994, 531--568.
• Fomin and Zelevinsky, Total positivity: tests and parametrizations.

References for cluster algebras:

• Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapters 1-3.
• Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapters 4-5.
• Williams, Cluster algebras: an introduction.
• Fomin and Zelevinsky, Cluster algebras: Notes for the CDM-03 conferences, International Press, 2004.
• Cluster Algebras Portal

References for Grassmannians:

• Postnikov, Total Positivity, Grassmannians, and networks.
• Morales (writeup of lecture notes by Postnikov), Lectures on the positive Grassmannian.
• Postnikov, Speyer, Williams Matching polytopes, toric geometry, and the non-negative part of the Grassmannian.

Other topics:

• Arkani-Hamed and Trnka, The Amplituhedron.
• Karp and Williams, The m=1 amplituhedron and cyclic hyperplane arrangements.
• Ardila, Rincon, Williams, Positively oriented matroids are realizable.
• Kodama, Williams, KP solitons, total positivity, and cluster algebras.

Lectures