Raju Krishnamoorthy - Flat Connections Seminar

Goal

The goal of this seminar was to study flat connections in positive and mixed characteristic, with a central aim of understanding recent work of Esnault-Groechenig. We surveyed p-curvature, mod p nonabelian Hodge theory, the Hitchin fibration, crystals, and isocrystals. The preliminary program is available as a PDF.

Schedule

Date	Speaker	Topic	Notes
April 24	Raju	Introduction to flat connections, peculiarities in positive characteristic, and p-curvature	Notes
May 8	Raju	The ring of crystalline differential operators and p-curvature	Notes
May 10	Raju	p-curvature continued	Summary
May 15	Raju	Crystals and isocrystals	Notes
May 19	Raju	Crystals, continued	Notes
May 22	Paul	Nonabelian Hodge theory mod p	Notes
May 25	Paul	Nonabelian Hodge theory mod p II
May 30	Josh	Properties of nonabelian Hodge theory mod p and periodicity	Notes
June 12	Raju	The BNR correspondence and its de Rham incarnation in characteristic p	Notes
June 19	Yichen	Spreading out and global nilpotence of flat connections
July 3	Raju	Global nilpotence and periodicity of rigid flat connections	Notes

References

The following are orienting references. For complete bibliographic information, see the seminar program.

[EG20] Esnault and Groechenig, Rigid connections and F-isocrystals.
[E22] Esnault, Lectures on local systems in algebraic-arithmetic geometry.
[G15] Groechenig, Moduli of flat connections in positive characteristic.
[K16] Kedlaya, Notes on isocrystals.
[LSZ15] Lan, Sheng, and Zuo, Nonabelian Hodge theory in positive characteristic via exponential twisting.
[LSZ21] Lan, Sheng, and Zuo, Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups.
[OV07] Ogus and Vologodsky, Nonabelian Hodge theory in characteristic p.
[Oss] Osserman, Connections, curvature, and p-curvature.
[Yun19] Yun, A Simpson correspondence for abelian varieties in positive characteristic.