Goal

The goal of this seminar is to study flat connections in positive and mixed characteristic. One of the main goals is to understand spectacular recent work of Esnault-Gröchenig. To this end, we will survey the theory of p-curvature, mod p nonabelian Hodge theory, the Hitchin fibration, and (iso)crystals. Time/enthusiasm permitting, we may discuss applications to characteristic 0 algebraic geometry using these techniques, in particular the work of Arapura and Langer. A preliminary program may be found here.

Schedule

April 24 Raju Introduction to flat connections, peculiarities in positive characteristic, p-curvature Notes.
May 8 Raju the ring of crystalline differential operators and p-curvature Notes.
May 10 (online) Raju p-curvature continued See notes from May 8 for details and here for a summary. ([p. 949-955, BMR08] and [G15] may also be helpful.)
May 15 Raju Crystals and Isocrystals Notes. The lecture notes [MHN] are a well-motivated introduction to some of this material.
May 19 (online) Raju Crystals, continued Notes. For a much more complete summary, please see [K16]. The article [Ill15] also has helpful comments on the notion of a crystal, the Taylor formula, etc.
May 22 Paul Non-abelian Hodge theory mod p Notes
May 25 (online) Paul Non-abelian Hodge theory mod p II
May 30 Josh Properties of non-abelian Hodge theory mod p, periodicity Notes
June 12th Raju The BNR correspondence and its de Rham incarnation in characteristic p Notes
June 19th Yichen Spreading out and global nilpotence of flat connections
July 3rd Raju Global nilpotence and periodicity of rigid flat connections Notes


References

The following are orienting references. For a complete list of references (with full bibliographic information), please see the seminar program.