Masterclass – Stable Homotopy Theory and p-adic Hodge Theory

University of Copenhagen - 26-30 June 2017

Stable Homotopy theory and arithmetic geometry are two very active areas of research today. Our speakers, Thomas Nikolaus and Matthew Morrow represent two researchers at the forefront of these areas, working in different but overlapping fields. The overall motivation for this masterclass is to study the relations between these two areas of research.

In recent years stable homotopy theory has seen unexpected applications to arithmetic geometry. In particular the work of Matthew Morrow (in collaboration with Bhargav Bhatt and Peter Scholze) on integral p-adic Hodge theory was, in part, motivated by calculations of topological Hochschild homology for certain arithmetically important rings. Very recently Lars Hesselholt has used the cyclotomic structure on topological Hochschild homology to define a topological version of periodic cyclic homology and used it to give cohomological interpretations of zeta functions for schemes over finite fields. This was in part motivated by new perspectives and results on cyclotomic structures afforded by work of Thomas Nikolaus (in collaboration with Peter Scholze).

The goal of this masterclass is to study this recent progress. In particular we will focus on the work by Nikolaus and Scholze on cyclotomic spectra and the work by Bhatt, Morrow and Scholze giving integral relations between p-adic cohomology theories. The masterclass will consist of two lecture series by Matthew Morrow and Thomas Nikolaus as well as several discussion sessions.

Lecture series:

  • Matthew Morrow (CNRS , Jussieu--Paris Rive Gauche)
  • Thomas Nikolaus (MPI Bonn)

Additionally there will be discussion and exercise sesssions.


  • Lars Hesselholt (Copenhagen, Nagoya) 
  • Kristian Moi (MPI Bonn) 


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

All lectures will take place in Auditorium 10.

Schedule and abstracts:

The topics of this masterclass will include integral p-adic Hodge theory (following Bhatt-Morrow-Scholze), topological Hochschild homology and cyclotomic spectra.

References, lecture material, and notes:

These will be made available here in due course. 

Exercises: General (updated with exercises from Nikolaus' lectures)

Morrow: Decalage, divided powers, crystalline cohomology and de Rham-Witt

Lecture Notes

Morrow I, Morrow IIMorrow III, Morrow IV, Morrow V, Morrow VI

Nikolaus I, Nikolaus II, Nikolaus III, Nikolaus IV


Moi (Real THH)

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:

Registration online before 20 June 2017. We especially encourage women and underrepresented minorities to apply. 

A list of participants will be published here eventually.

Organised by Ryo Horiuchi, Martin Speirs and Lars Hesselholt.