Math 251Y - Topics in combinatorics: Cluster algebras, positivity, Grassmannians, and beyond -- Fall 2021

Professor: Lauren Williams (Science Center 510, e-mail williams@math.harvard.edu)

Office Hours: After class.

Course assistant: Charles Wang (email cmwang@math.harvard.edu)

Course assistant's office hours: By appointment.

Harvard's covid guidance: Masks are required indoors. All participants must be vaccinated and free of covid symptoms (please stay home if you have any symptoms).

Course description

This course will survey one of the most exciting recent developments in algebraic combinatorics, namely, Fomin and Zelevinsky's theory of cluster algebras. Cluster algebras are a class of combinatorially defined commutative rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. Introduced in 2001, cluster algebras have already been shown to be related to a host of other fields of math, such as quiver representations, Teichmuller theory, Poisson geometry, and total positivity. Cluster structures in Grassmannians have in particular been linked to integrable systems and physics. In the first part of the course I will cover the basics of cluster algebras and total positivity. In the second part of the class I will discuss recent developments and applications of the theory, including Postnikov's positroid decomposition of the positive Grassmannian, Arkani-Hamed--Trnka's notion of amplituhedron, and KP solitons.

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

Grading will be based on some problem sets together with 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 submit it through Canvas.) Please discuss with me the topic of your final project, no later than mid-October, to make sure I approve.

Extra credit will be given to people who find typos in our book ``Introduction to cluster algebras.''

References

References for cluster algebras:

• Fomin, Total positivity and cluster algebras.
• Fomin and Zelevinsky, Cluster algebras: Notes for the CDM-03 conferences, International Press, 2004.
• Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapters 1-3.
• Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapters 4-5.
• Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapter 6: cluster structures in commutative rings.
• Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapter 7: plabic graphs.
• Williams, Cluster algebras: an introduction.
• Cluster Algebras Portal

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 Grassmannians:

• Postnikov, Total Positivity, Grassmannians, and networks.
• Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapter 7: plabic graphs.

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 amplituhedra:

• Arkani-Hamed and Trnka, The Amplituhedron.
• Arkani-Hamed, Thomas, and Trnka, Unwinding the amplituhedron in binary.
• Karp and Williams, The m=1 amplituhedron and cyclic hyperplane arrangements.
• Parisi, Sherman-Bennett, and Williams, The m=2 amplituhedron and the hypersimplex: clusters, triangulations, Eulerian numbers.

Other possible topics:

• Ardila, Rincon, Williams, Positively oriented matroids are realizable.
• Kodama, Williams, KP solitons, total positivity, and cluster algebras.

Lectures