Math 18.708

About

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, 2, 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:
  • Kent Vashaw: Support varieties and tensor triangular geometry. March 8, 10, 15, 17.
    • 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 rings. March 29, 31, April 5, 7.
    • Abstract:
  • 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: TBA. May 3, 5, 10.
    • Abstract:

Notes

  • Notes for mini-courses will be added here.

Problem sets

  • Problem sets will be added here.

References

    • 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.
    • 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.
    • Minh-Tam Trinh: Von Neumann algebras, Jones index.
    • Davesh Maulik: TBA.