Math 607 FALL 2021, 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.
  • Tuesday, September 28: Mathematical and physical motivations for TQFT.
  • Reading assignment: try to read foundational paper by Atiyah .
  • Thursday, September 30: TQFT vector spaces are finite dimensional. Definition of categories of cobordisms. Euler's field theory.
  • Reading assignment: read Chapter 1 of Milnor's "Lectures on h-cobordism theorem".
  • Tuesday, October 5: Definition of TQFT. Euler's field theory. Tensor categories. New homework!
  • Thursday, October 7: More of tensor categories.
  • Reading assignment: read Section 2.2 of Tensor Categories book.
  • Tuesday, October 12: Tensor functors and their morphisms.
  • Reading assignment: read Sections 2.4-2.6 of Tensor Categories book. Also look at the solution of Exercise 2.10.15 here.
  • Thursday, October 14: Semisimple tensor categories. Braided and symmetric categories. New homework!
  • Tuesday, October 19: Traces and dimensions. Tannakian categories.
  • Reading assignment: read Section 8.1 of Tensor Categories book about Yang-Baxter equation (Proposition 8.1.10) Also read Section 8.4 for a proof of Joyal-Street theorem.
  • Practice your mathematical French by reading Section 3 of paper by P.Deligne (most importantly, Lemme 3.5 shows that trace behaves well in short exact sequences).
  • Thursday, October 21: Super-Tannakian categories. Negligible morphisms and gligible quotients.
  • Tuesday, October 26: Fibonacci category as subcategory of gligible quotient. Untwisted Dijkgraaf-Witten theory with finite gauge group.
  • Reading assignment: read paper by N.Snyder about Frobenius-Mednykh formula. Also read paper by S.Matveev and V.Turaev about computation of twisted DW invariant in the simplest case.
  • Thursday, October 28: Twisted DW theories. New homework!
  • Tuesday, November 2: One dimensional TQFTs.
  • Reading assignment: read Section 4.4 of Dan Freed's book for a complete argument on 1-dimensional theories. Try to modify it to work in unoriented setup.
  • Sections 1.7-1.9 of paper by P.Deligne is a convenient reference on Karoubian (= pseudo-abelian) envelope construction.
  • Thursday, November 4: Endomorphisms in the category Bord_1(k,t) and Brauer algebras.
  • Tuesday, November 9: Deligne's categories Rep(O_t).
  • Reading assignment: Section 9 of paper by P.Deligne contains more details about categories Rep(O_t); here is paper by H.Wenzl used in Section 9.
  • Read more about Tannakian categories in Chapter 9 (especially Section 9g) of book "Algebraic Groups" by J.S.Milne.
  • Thursday, November 11: No classes in observance of Veteran's Day. New homework!
  • Tuesday, November 16: Deligne's categorie Rep(GL_t). Two dimensional theories and Frobenius algebras.
  • Reading assignment: Section 10 of paper by P.Deligne contain more details about categories Rep(GL_t).
  • Read more about Morse theory and its use in classification of two dimensional theories in Sections 4.1-4.2 of D.Freed's book.
  • Thursday, November 18: More on two dimensional theories.
  • Tuesday, November 23: Two dimensional theories and Rep(S_t).
  • Reading assignment: Section 8 of paper by P.Deligne discusses definition of category Rep(S_t) close to the one discussed in the class. Note that Section 2 contains a quite different definition.
  • Thursday, November 25: No classes in observance of Thanksgiving.
  • Tuesday, November 30: 3 dimensional theories and modular tensor categories.
  • Reading assignment: read more on Reshetikhin-Turaev theories in chapter 4 of Bakalov-Kirillov's book.
  • Read more about Turaev-Viro models in paper by R.Koenig, G.Kuperberg, and B.W.Reichardt.
  • Thursday, December 2: More on modular tensor categories. Turaev-Viro models.
  • THE END