Math 18.708


Math 18.708 will consist of 6 mini-courses, with topics as listed below. Pre-qual students taking the class for credit should submit homework solutions.

Time and location: TTh, 1-2:30 pm, Room 2-143.

Instructors and topics:

  • Pavel Etingof: Tensor categories. February 1, 3, 8, 10, 15.
    • Abstract: I will give an introduction to tensor categories, with a special emphasis on symmetric tensor categories (STC). In particular, I will discuss (super)Tannakian formalism, Deligne's characterization theorem for Tannakian and super-Tannakian categories, Deligne interpolation categories, semisimplification of tensor categories, Frobenius functors and analogs of Deligne's theorem in positive characteristic, applications to modular representation theory of finite groups.
  • Jeremy Hahn: Hochschild homology. February 17, 24, March 1, 3.
    • Abstract: We will discuss various fundamental computations and filtrations on the Hochschild homology of commutative rings, such as the HKR filtration. We will also discuss circle actions on Hochschild homology. We aim to present this material in a way that smoothly generalizes to the setting of topological Hochschild homology, where it is related to work of Bhatt--Morrow--Scholze.
  • Kent Vashaw: Support varieties and tensor triangular geometry. March 8 and 10, and the second half of the lectures on March 15, 17, April 5, 7.
    • Abstract: we will begin by covering the basics of cohomological support varietiess and rank varieties for modular representations of finite groups, as originally studied by Carlson, Alperin, Avrunin-Scott, etc. We will show that such cohomology theories can be used to classify the thick ideals of the stable category of a finite group, along the way discussing Dade's Lemma and the Benson-Carlson-Rickard theory of support varieties for infinitely-generated modules. Lastly, we will generalize these results to finite group schemes via the theory of pi-points (Friedlander-Pevtsova), and show that support varieties can be understood in the general framework of Balmer's tensor triangular geometry.
  • Tony Feng: Simplicial commutative algebra. First half of the lectures on March 15, 17, 29, 31, April 5, 7.
    • Abstract: Just as commutative rings form the basic building blocks of classical algebraic geometry, simplicial commutative rings form the basic building blocks of derived algebraic geometry. This mini-course will introduce simplicial commutative rings, and focus on their more classical applications. The main goal will be to construct the cotangent complex and demonstrate some of its uses.
  • Minh-Tam Trinh: Von Neumann algebras, Jones index. April 12, 14, 19, 21.
    • Abstract: We'll begin by explaining how the operator formalism of quantum mechanics can be expressed in the language of Hiilbert spaces and their bounded operators. The rings of operators that appear here are neatly subsumed by the concept of a von Neumann algebra. After giving a tour of the structure and classification of von Neumann algebras, focusing on the special case of subfactors, we'll present and prove a remarkable theorem due to the late Vaughan Jones: The index of an inclusion of subfactors - a priori, merely a positive real number or infinity - is quantized: If it is less than 4, then it must take the form 4*cos(pi/n)^2 for some integer n at least 3. To conclude, we'll explain how the study of such inclusions of subfactors naturally leads one to discover the Temperley–Lieb algebra. This is a diagrammatic algebra with surprising applications across statistical mechanics, knot theory, and graph theory.
  • Davesh Maulik: FI-modules. April 26, 28, May 3, 5, 10.
    • Abstract: TBA

Problem sets

  • Due Feb 17: Exercises 2.10.6, 8.1.9, 9.9.4, 9.9.9, 9.9.10, 9.11.8, 9.11.13, 9.12.16, in the textbook Tensor Categories (Etingof et al). Solve at least 5 out of these exercises. (post-qual graduate students are exempt).
  • Due March 17: Problem set 2.
  • Due April 7: Problem set 3.
  • Due April 15: Exercises 2.1, 2.2, 2.14, 3.4, 3.7.
  • Due April 28: Problem set 5.
  • Due May 10: Problem set 6.


    • Pavel Etingof: Tensor categories.
      • P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, "Tensor categories": Link, corrections.
      • P. Etingof and A. Kannan, "Lectures on symmetric tensor categories": Link.
    • Jeremy Hahn: Hochschild homology.
      • M. Morrow, "Topological Hochschild homology in arithmetic geometry": Link.
      • A. Krause and T. Nikolaus, "Lectures on topological Hochschild homology and cyclotomic spectra": Link.
      • B. Antieau, "Periodic cyclic homology and derived de Rham cohomology": Link.
    • Kent Vashaw: Support varieties and tensor triangular geometry.
      • D. J. Benson, "Representations and cohomology II: Cohomology of groups and modules".
      • J. Pevtsova, "Representations and cohomology of finite group schemes": Link.
      • P. Balmer, "The spectrum of prime ideals in tensor triangulated categories": Link.
    • Tony Feng: Simplicial commutative rings.
      • Course notes by Tony Feng: PDF.
    • Minh-Tam Trinh: Von Neumann algebras, Jones index.
      • G. B. Folland. Quantum Field Theory: A Tourist Guide for Mathematicians. AMS Mathematical Surveys and Monographs, Vol. 149, AMS (2008).
      • C. K. Fan & R. M. Green. Monomials and Temperley--Lieb Algebras. J. Algebra, 190 (1997), 498-517.
      • V. F. R. Jones. Index for Subfactors. Invent. math., 72 (1983), 1-25.
      • V. F. R. Jones. A Polynomial Invariant for Knots via von Neumann Algebras. Bull. AMS, 12(1) (Jan. 1985), 103-111.
      • V. F. R. Jones. Hecke Algebra Representations of Braid Groups and Link Polynomials. Ann. of Math. (2), 126(2) (Sep., 1987), 335-388.
      • V. F. R. Jones. Von Neumann Algebras. Vanderbilt University. Unpublished notes (November 13, 2015). Link.
      • D. Kazhdan & G. Lusztig. Representations of Coxeter Groups and Hecke Algebras. Invent. math., 53 (1979), 165-184.
    • Davesh Maulik: FI-modules.
      • T. Church, J. S. Ellenberg, and B. Farb, "FI-modules and stability for representations of symmetric groups": Link.
      • T. Church, J. S. Ellenberg, B. Farb, and R. Nagpal, "FI-modules over Noetherian rings": Link.