Reading Seminar on D-modules (2024S)

In the fall, we are meeting in person on Mondays, 2:20-3:50pm in VV B321.

Tentative schedule

date	speaker	title	topics
June 25	Josh	Differential operators and filtrations	We'll define the ring of algebraic differential operators together with its order filtration, and discuss some of its implications for modules over rings of differential operators.
July 2	Jameson	Left and right D-modules. Inverse images	Examples of D-modules on a line. Quasicoherent D-modules. Left vs. right D-modules: an equivalence. Inverse images of D-modules. Examples (open embeddings, smooth morphisms, closed embeddings).
July 16	Dima	Inverse and direct images. Derived category of D-modules	`Naive' definition. Definition in the derived category (examples).
July 23	Alex	Kashiwara's Lemma.	Direct image under closed embeddings. Kashiwara's Lemma and applications.
July 30	Kevin	Integral transforms	Tensor product. Integral transforms (=Fourier-Mukai functor). The Fourier-Mukai transform on a line. Here are the notes.
August 6	Jameson	Levelt-Turritin classification	D-modules on punctured formal disk. Regular and irregular singularities. Extra topics: monodromy, the Stokes phenomenon, perhaps some discussion of non-punctured disk
August 13	Kevin	Holonomic D-modules	Singular support, Bernstein's inequality, elementary properties of holonomic D-modules. (Did not get to duality for holonomic D-modules). Here are the notes.
August 20	Alex	The six functors	Preservation of holonomicity. Functoriality of singular support (?).
September 23	Dima	Introduction to the Riemann-Hilbert correspondence (over reals)	Existence and uniqueness for ODEs and PDEs. Monodromy. Correspondence between bundles with connection/local systems/representations of the fundamental group.
September 30	Kevin	Bundles with connection on complex manifolds	Mostly review: (complex) vector bundles and their sheaves of sections, definition of connection (in real/complex case). Cauchy-Riemann equations as (0,1)-connection. Riemann-Hilbert correspondence for vector bundles over complex manifolds. Here are the notes.
October 7	Jameson	Riemann-Hilbert correspondence on non-compact Riemann surfaces	Connections on a disk. Regular singularities and rate of growth of solutions. Riemann-Hilbert on Riemann surfaces. If time permits: Hilbert's 21st problem.
October 14	Kevin (tentative)	Regular singularities	Connections and holonomic D-modules with regular singularities. Deligne's Riemann-Hilbert correspondence.
October 21	Alex	Riemann-Hilbert correspondence for D-modules	Intro to constructible sheaves. Riemann-Hilbert as an equivalence of derived categories
October 28	Hairuo	Intro to perverse sheaves
Optional	Available	Irregular singularities and the Stokes phenomenon

References

If you have other suggestions, please let me know (or just add to this list)!

• J.Bernstein's notes on D-modules. They are quite informal and move very fast.
• R.Hotta, K.Takeuchi, T.Tanisaki, D-modules, perverse sheaves, and representation theory. Very detailed and carefully written book.
• V.Ginzburg's notes
• C.Schnell's course on D-modules with lecture-by-lecture notes (Course page).
• S.C.Coutinho, A primer of algebraic D-modules. The book does go too deep into theory, focusing instead on examples and practical calculation.
• For modern approach to Levelt-Turritin classification, here's a paper by M.Kamgarpour and S.Weatherhog.