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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.