Reading Seminar on Crystalline Cohomology and Gauss-Manin Connections

We will meet on Mondays from 1:00PM to 2:30PM at 2-255, for a 1 hour talk followed by a 30 minute question and discussion session. 

The goal of this reading seminar is to understand the work of Petrov-Vaintrob-Vologodsky on the Getzler-Gauss-Manin connection on the periodic cyclic homology of a dg algebra in positive characteristic. Their work involved a noncommutative generalization of the celebrated result of Katz relating the p-curvature of the Gauss-Manin connection with the Kodaira-Spencer operator, which led to a proof of Grothendieck's p-curvature conjecture for Picard-Fuchs equations.

Tentative topics of this seminar include: different constructions of the Getzler-Gauss-Manin connection on cyclic homology, Crystalline cohomology, Katz's p-curvature formula and Petrov-Vaintrob-Vologodsky's generalization to the noncommutative setting. Depending on audience interests, some potential additional topics include: homotopy theoretic perspective on crystals, the recent work of Pomerleano-Seidel proving that the quantum connection is of unramified exponential type.

Some references:

Katz, Algebraic Solutions to Differential Equations (link)

Petrov-Vaintrob-Vologodsy, The Gauss-Manin Connection on the Periodic Cyclic Homology (link)

Getzler, Cartan Homotopy Formula and the Gauss-Manin Connection in Cyclic Homology (link)

Pomerleano-Seidel, The Quantum Connection, Fourier-Laplace Transform, and Families of A_infty Categories (link)

Berthelot-Ogus, Notes on Crystalline Cohomology (link)

Schedule

Week 1 (Feb 5) Overview (Zihong Chen) Notes

Week 2 (Feb 12) Gauss-Manin Connection, Kodaira-Spencer Class, Hodge Filtration (Yonghwan Kim) Notes

(no talk Feb 19)

Week 3 (Feb 26) Katz's formula for the p-curvature of the Gauss-Manin connection  (Jae Hee Lee)    Notes 

Week 4 (Mar 4) Guest Lecture (Vadim Vologodsky): Fontaine-Laffaille structure on the periodic cyclic homology   Notes

(Harvard Spring Break)

Week 5 (Mar 18) Crystals, Crystalline cohomology, Relations to connections in characteristic p  (Daniel Hu)   Notes

(MIT Spring Break)

Week 6 (Apr 1) Digression - Modern Homotopy-theoretic viewpoint for Crystalline Cohomology (Sanath Devalapurkar) Notes

Week 7 (Apr 8) Examples of Crystals: Tate Cohomology   (Zihong Chen) Notes

(MIT Holiday : Patriots' Day)

Week 8 (Apr 22) Petrov-Vaintrob-Vologodsky - Main results, and the local monodromy theorem (Yonghwan Kim) Notes

Week 9      (Apr 29) Application to the quantum connection in symplectic topology  (Dan Pomerleano) Notes