Reading Seminar on D-modules (2024S)
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We meet on Tuesdays at 3pm on Zoom (email arinkin@math.wisc.edu for the link if you need it). The first meeting is on Tuesday, June 25th.
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. |
| TBD | Kevin (Tentative) | 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 |
| TBD | (Another big topic) | Holonomic D-modules | Singular support, Bernstein's inequality. Duality for holonomic D-modules |
| TBD | available | The six functors | Preservation of holonomicity. Functoriality of singular support (?). |
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.