Prerequisites: Basic operator algebra theory such as MATH 8120 or Kehe Zhu's An Introduction to Operator Algebras, CRC Press, 1993. This includes basic knowledge of von Neumann algebras (Borel functional calculus, von Neumann's bicommutant theorem, comparison of projections, factors and types of factors, group measure space construction etc.).

Suggested Books: There will be no textbook. Some of the material covered in the course can be found in the following books:
1) Claire Anantharaman, Sorin Popa, An introduction to II₁ factor, book, preprint available here.
2) Vaughan Jones, V. Sunder, Introduction to Subfactors, Cambridge University Press, 1997.
3) David Evans, Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford University Press, 1998.
4) Fred Goodman, Pierre de la Harpe, Vaughan Jones, Coxeter Graphs and Towers of Algebras, Springer 1989, reprint 2011.

There are many books with background material on von Neumann algebras, for instance
1) Serban Stratila, Laszlo Zsido, Lectures on Von Neumann Algebras.
2) Jacques Dixmier, Von Neumann Algebras, North Holland, 1981.
3) Masamichi Takesaki, Theory of Operator Algebras I, II, III, Springer-Verlag 2002.

Additional references will be given throughout the course.

Syllabus: The course will begin with a brief review of basic material on von Neumann algebras, in particular the theory of II₁ factors. For instance, I will discuss representations of II₁ factors and the Murray-von Neumann coupling constant. I will then define the Jones index of a subfactor and discuss its basic properties. The highlight of this part of the course will be the proof of Jones' rigidity theorem for indices. The proof involves iterating the basic construction, Markov traces, Temperley-Lieb algebras and other surprising structures that naturally appear in subfactors. I also plan to show that there are hyperfinite subfactors for each allowable Jones index.

In the second part of the course, I will discuss invariants and explicit constructions of subfactors, including principal (or fusion) graphs of subfactors, commuting squares and Ocneanu compactness. If time allows, I will show how Jones' braid group representation that arises from subfactors led Jones to the discovery of his famous knot invariant, the Jones polynomial. Throughout the course, I will point out open problems and try to connect to ongoing research (including my own). This will probably lead to some interesting detours in the theory of II₁ factors and adjacent fields. If time, I will describe some of the algebraic and topological methods that are used to compute with subfactors (e.g. planar algebras, a pictorial way to handle the standard invariant of a subfactor).

The precise contents of the course will depend on the background and interests of the audience.

Grading: The course grade will be based on attendance and a (team) presentation. I will assign optional homework problems. There will be no exams.

COVID-19: Please adhere to Vanderbilt policy and protocols regarding COVID-19. In particular, no one with symptoms that could be due to COVID-19 is to come to class. You should get tested at Student Health and self-isolate while awaiting results. Please see here for more information on COVID-19 testing. If you test positive, please follow the quarantine protocols.

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