Lorentz Center - Trends in Arithmetic Geometry from 14 Jan 2013 through 18 Jan 2013
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    Trends in Arithmetic Geometry
    from 14 Jan 2013 through 18 Jan 2013


Scientific report



The workshop focussed on recent developments in arithmetic geometry in the broad sense, but with a focussed emphasis on two themes: \twisted sheaves" and \singularities in characteristic p". The aim was twofold: to bring together specialists from all over the world to share the latest developments, discuss new ideas and start new collaborations, and to make the subject more accessible to Ph.D. students and young postdocs through lecture series and Q&A sessions. Here are some of the highlights.


In the broad area of arithmetic geometry we had inspiring lectures by Laurent Moret-Bailly and Johan de Jong on fundamental questions in algebraic geometry. Moret-Bailly discussed recent work with Ofer Gabber and Philippe Gille around the topology of torsors over valued fields. De Jong talked about ongoing work in homological algebra and its implications on the existence of Quot schemes in large generality.

On the more arithmetic side of the subject we had two brilliant talks with Davesh Maulik sharing his results on the Tate conjecture for K3 surfaces and Olivier Wittenberg explaining his work with Esnault and Levine on the index of varieties over local fields.


The focus area of twisted sheaves was mostly covered by Max Lieblich, one of the founders of this area. In a crystal clear, enthusiastic series of 3 lectures he started from the basic theory of twisted sheaves and worked towards the ideas behind some of their spectacular applications. The level and pace were just right, so that the lectures were both useful for the graduate students and postdocs, as well as for the more experienced algebraic geometers in the audience.


The second focus area singularities in characteristic p, saw lectures by Kevin Tucker on F-singularities, and by André Chatzistamatiou & Kay Rülling on Wittrational singularities. Karl Schwede built upon Tucker's lectures to discuss some surprising geometric applications of F-singularities, and Amaury Thuillier showed how one can use Berkovich spaces to understand the combinatorics of resolutions of singularities in arbitrary characteristic.


The unity of the subject of arithmetic geometry was underlined by some of the cross-connections between the various discussed subjects. For example, the lecture of Thuillier formed a bridge between the \singularieties in characteristic p" theme and the lectures on Berkovich spaces by Poineau and Ducros. The \stacky" ideas in Romagny's talk were closely related to the lectures of Lieblich, and the recent developments on rational points over discretely valued fields were central in the talk of Wittenberg and in the results on the period-index problem in the lectures by Lieblich. Frequently, the same problems and objects appeared in di
erent talks throughout the conference, approached from several angles and studied by various techniques. This greatly stimulated discussions between the speakers and participants.


Besides the programmed lectures, we had programmed a rather experimental series of Q&A sessions for the young participants. This was a huge success, in large part thanks to the enthusiasm of both the young mathematicians and the lecturers who took pride and pleasure in answering questions on hugely differing levels. On popular request, an extra session was organized on Friday, for which the students could choose to invite any participating senior mathematician. They decided to nominate Johan de Jong, who gave them a head-start by explaining some of the ideas for his lecture later that day.


The atmosphere was very friendly and stimulating. Both the lectures as the informal discussions were of high quality, as was expressed repeatedly by both senior and junior participants.

Finally, the staff of the Lorentz Center was most helpful and efficient. In particular, when the entire science campus was shut down due to a power incident on the first day of the conference, the Lorentz Center staff managed to organize an alternative conference location (including lunch and coffee) for about 60 participants in a record-breaking 10 minutes. Thanks to their decisive action we could continue the conference with only minor changes to the schedule.




(1) Johannes Nicaise, KU Leuven

(2) Lenny Taelman, Universiteit Leiden


Scientific commitee:

(1) Bas Edixhoven, Universiteit Leiden

(2) Hélène Esnault, Universität Duisburg-Essen

(3) Ben Moonen, Universiteit van Amsterdam

(4) Mircea Mustaţă, University of Michigan