Math 607 WINTER 2024, List of lectures
On this page I will post content of all lectures with reference to the
book. All handouts also will be posted here.
Thursday, January 18: The collection of all representations of a group as a mathematical object. Axioms for semigroup and monoidal categories.
Reading assignment: read about various equivalent definitions of the unit object in Section 2.2 of Tensor Categories book; see also
pages 98-100 of [KS].
Tuesday, January 23: More on unit object. Examples of monoidal categories.
Monoidal functors and their morphisms.
New homework!
Thursday, January 25: Pointed categories and monoidal functors between them;
description in terms of cohomology.
Reading assignment: read Notes by Agustina Czenky on pointed categories associated with finite cyclic groups.
Tuesday, January 30: Duality (= rigidity). Pivotal and spherical structures.
Braided structures.
Reading assignment: read about Maclane's strictness and coherence theorems in Sections 2.8 and 2.9 of Tensor Categories book.
Thursday, February 1: Braided pointed categories: theorem of Joyal-Street.
Reading assignment: look at examples of graphical calculus in Chapter XIV of Christian Kassel's book "Quantum Groups".
Tuesday, February 6: Proofs of Eilenberg-Maclane and Joyal-Street theorems.
Here are some Projects for this class.
Thursday, February 8: Non-pointed categories and their Grothendieck rings.
Tuesday, February 13: Frobenius-Perron dimension less than 2.
New homework!
Thursday, February 8: Temperley-Lieb category.
Tuesday, February 20: Jones polynomial. Karoubian envelopes and idempotents.
Reading assignment: read more about invariants of framed links in Section X.8 of Christian Kassel's book "Quantum Groups".
Thursday, February 22: Karoubian envelope of the Temperley-Lieb category.
Tuesday, February 27: Semisimplification of spherical categories.
Reading assignment: read notes by Scott Morrison about Jones-Wenzl projections
Thursday, February 29: More on semisimplification. Temperley-Lieb category is "nice".
Tuesday, March 5: Fusion categories from Temperley-Lieb categories. Algebras in tensor categories.
Thursday, March 7: Morita theory in fusion categories.
Tuesday, March 12: Dual categories and Drinfeld center.
Thursday, March 14: Modular Tensor Categories.
Reading assignment: read a proof of Anderson-Moore-Vafa theorem in Section 8.18 of Tensor Categories book.
Tuesday, March 19, 1pm, meet in Fenton next to Math office: Symmetric tensor categories.
Reading assignment: read paper by P. Etingof and S. Gelaki about isocategorical groups
Reading assignment: read more about Tannakian duality in Chapter 9 of
book "Algebraic Groups" by J. Milne.
Thursday, March 21, 10am, meet in Fenton: Examples.