## Deformation Theory, Completed Cohomology, Leopoldt Conjecture and K-theory

**Date:**
5 - 9 December 2016

**Location: ** CIRM, Marseille

**Event type:**
Workshop

**Organisers: ** Laurent Clozel (Paris Sud), Francesco Calegari (Chicago)

**Website:**conferences.cirm-math.fr/1676.html

There has been exciting progress recently on two (three ?) connected topics: the extension of the Taylor-Wiles method to situations where the standard (im)parity condition fails, and the relation of deformation theory (starting with Hida’s idea) with the Leopoldt conjecture. The salient points are the paper(s) of Calegari and Geraghty, Calegari’s paper on stable (completed) homology for congruence groups, and the Khare-Thorne paper on Leopoldt; the construction of new Galois representations by Harris-Lan-Thorne-Taylor and Scholze, in particular for torsion classes, is also a crucial advance. Finally, Calegari’s paper reveals new relations with problems in K-theory on which our kowledge does not seem to have improved much since the Batelle meetings.

This workshop will be limited to the researchers involved in these developments. The aim is to obtain an up-to date integrated picture of these developments as well as their interrelations. Connected theories such as completed cohomology should be reviewed, as well as some complements in K-theory.

CMI Enhancement and Partnership Program