Reading Seminar on D-modules (2024S)
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In the fall, we are meeting in person on Mondays, 2:20-3:50pm in VV B321.
Tentative schedule
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.
| 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. |
| 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). |
| 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 |
| 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 | Available | Regular singularities | Connections and holonomic D-modules with regular singularities. Deligne's Riemann-Hilbert correspondence. |
| October 21 | Vischer | 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 |