Masterclass: Tensor triangular geometry and equivariant stable homotopy theory

University of Copenhagen, 12-16 March 2018

The goal of this Masterclass is to provide a state-of-the-art overview of tensor triangular geometry and to illustrate the abstract theory by an in-depth study of the equivariant stable homotopy category, leading up to the recent progress on the computation of the spectrum of the equivariant stable homotopy category. Another thread that will run through the workshop will be the systematic development and use of modern descent and nilpotency techniques for concrete applications, as for example the computation of Picard groups in various settings. The techniques and results discussed in this Masterclass will, therefore, be of interest to mathematicians working in equivariant and motivic homotopy theory, representation theory, algebraic geometry, and non-commutative geometry. 

The Masterclass will consist of two lecture series by Paul Balmer and Justin Noel, accompanied by several problem sessions, as well as four contributed talks exploring connections to closely related areas of current research.  

The participants

The participants - Click for large photo

Lecture series:

  • Paul Balmer (UCLA): Tensor triangular geometry
  • Justin Noel (Regensburg): Equivariant stable homotopy theory


  • Markus Hausmann (Copenhagen)
  • Niko Naumann (Regensburg)
  • Beren Sanders (EPFL)
  • Vesna Stojanoska (UIUC)


The masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen, with support from the Centre for Symmetry and Deformation. See detailed instructions on how to reach Copenhagen and the conference venue.

Prerequisites and references:

A solid background in algebra as well as some familiarity with the basics of triangulated categories and algebraic topology will be assumed.  

Schedule and abstracts:

A detailed schedule as well as a description of the lectures will become available in due time. 

Lecture notes:

Friday open problems session:

Excercise sessions:

  • Day 1 problem session:
    • Balmer blackboard pictures B 1 B 2 B 3.
    • Noel: Show that Hopf map as a C_2-equivariant map under complex conjugation is neither torsion nor nilpotent.
  • Day 2 problem session:
    • Balmer: 
      • 1. Show that the spectrum of the Verdier quotient K/J can beidentified with a sub-space of the spectrum of K.
      • 2. Show that the closure of a point P in the spectrum of K consists of all the primes contained in P. Explain the reversal of inclusion, withrespect to the Zariski spectrum of a commutative ring.
      • 3. Show that every irreducible (non-empty) closed subset Z of the spectrum is the closure of a unique point.
      • Show that Spc)K= is quasicompact. Lemma: If S is a tensor-multiplicatively closed set of objects , and I a tt-ideal such that I \cap S = \emptyset. Show that there exists a prime P containing I such at I \cap S = \emptyset. Hint look at the ideals with this property, and try to use Zorn's lemma. 
    • Noel:
      • Take G = C_p, and V a real representation of G. Find an explicit G-CW decomposition of the unit sphere S(V) in V.
      • 2) Keep G = C_p, and let S^V denote the one-point compactification of V. Calculate the Bredon cohomology H^*_G(S(V);M)  and H^*_G(S^V;M) where M is either the constant coefficient system Z or M the Burnside ring functor G/H \mapsto A(H)., where A(H) is the Burnside ring of H.
      • Determine the ring structure on A(p^2)
      • Show that the map A(G) -> \prod_(H) Z given by taking a H-fixed-points for all subgroups H is an injective ring homomorphism. (Sometimes called the "mark homomorphism" to the "ghost ring".) Extra: Show that it is not surjective. Extra-extra: What is the cokernel?
  • (Day 3 Wiles -- no exercise session)
  • Day 3: Exercise here
  • Day 4: Exercise blackboard pictures: E 1 E 2 E 3 E 4 E 5 E 6 E 7

Financial support:

Graduate students and other early career researchers can apply for financial support to partly cover local expenses. If you wish to apply for support, please indicate this on the registration form. The support should be roughly DKK 500 per day, at most DKK 2500 in total. 

Registration and Participants:

The deadline for funding applications is February 16th, 2018. If you are not applying for funding, please register by the end of February.

Organised by Tobias Barthel and Jesper Grodal.