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.

• [EG20], Esnault, H.; Gröchenig, M. Rigid connections and F-isocrystals.
• [E22], Esnault, H. Lectures on local systems in algebraic-arithmetic geometry
• [G15], Gröchenig, M. Moduli of flat connections in positive characteristic
• [K16], Kedlaya, K. Notes on isocrystals. If the crash course on crystals/isocrystals was too succinct, Kedlaya's article is a wonderful resource.
• [LSZ15], Lan, G.; Sheng, M.; Zuo, K. Nonabelian Hodge theory in positive characterstic via exponential twisting
• [LSZ21], Lan, G.; Sheng, M.; Zuo, K. Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
• [OV07], Ogus, A.; Vologodsky, V. Nonabelian Hodge theory in characteristic p
• [Oss], Osserman, B. Connections, curvature, and p-curvature. These are notes of Brian Osserman that I found helpful/clarifying, especially for Mochizuki's perspective on p-curvature.
• [Yun19] Hao, Y. A Simpson Correspondence for Abelian Varieties in Positive Characteristic. This is the PhD thesis of Y. Hao, which is a very helpful guide to Gröchenig's perspective on the BNR correspondence.