Trends in Arithmetic Geometry

Description and aim

The subject of this workshop is arithmetic geometry, a field in which the interplay between geometry and arithmetic continues to renew itself with surprising connections and spectacular results. The broad scope of this field will be reflected in a rather diverse programme with talks about various new developments. In addition, there will be short lecture series and research talks about two focus areas which are developing rapidly at this moment.

The first focus area is that of twisted sheaves. In a series of papers starting in 2008 Max Lieblich and co-authors have used moduli spaces of twisted sheaves (highly geometric objects whose invention was motivated by string theory) as a powerful tool to study Brauer groups and other arithmetic problems. For example, they made several contributions to the period-index problem for Brauer classes, to the Tate conjecture for K3 surfaces, and to the period-index problem for torsors under abelian varieties.

The second is singularities in characteristic p. Rather than trying to find a general method for resolving resolutions in characteristic p (a difficult open problem), several people have been focussing on using typical characteristic p techniques (Frobenius, Witt vectors) to study the structure of singularities in positive characteristic. One of the surprising results is that classical properties of singularities in characteristic zero correspond to properties of the action of Frobenius on the reduction of the singularity modulo various primes.

With this workshop we aim for two goals. By bringing together specialists we will allow them to share the latest developments, discuss new ideas and start new collaborations. At the same time we will make the topic more accessible to Ph.D. students and young postdocs through lecture series and Q&A sessions.