Math 206B Combinatorial Theory: Cluster algebras (Winter 2021)

Course description: Cluster algebras were discovered by Fomin and Zelevinsky in the early 2000s. In the past 20 years, the subject has exploded, and lots of unexpected connections to other fields of mathematics were found. We will review the combinatorial aspects of the theory and will explore some of the connections, including a couple of very recent ones. Many open problems will be mentioned throughout the course. (A subset of) the following topics will be covered:

• Definition, basic properties and examples of cluster algebras.
• Combinatorics of root systems and Coxeter groups.
• Finite type classification and generalized associahedra.
• Periodicity and integrability phenomena.
• Grassmannian cluster algebras and planar bipartite graphs.
• q,t-Catalan numbers and connections to knot theory.

Instructor: Pavel Galashin (udе.аlсu.htаm@nihsаlаg). Please put "206B" into the subject line.

Time and location: No live lectures on Mondays! Wednesdays and Fridays we have live lectures at 1-1:50 on Zoom, but on Mondays I will put out a pre-recorded video lecture.

Zoom link: https://ucla.zoom.us/j/99372195118
Meeting ID: 993 7219 5118
Password: the missing term in the sequence 1, 2, 5, ??, 42, 132, 429. (The password is a number, e.g., "13" as opposed to "thirteen.") Alternatively, see CCLE or feel free to email me.

Recordings: Youtube playlist, see also handwritten lecture notes. The audience will not be recorded.

Grading: 95% homework problem sets, 5% class participation.

Prerequisites: the material will be accessible to first year graduate students and advanced undergraduates.

Office hours: by appointment.