Reading Seminar on D-modules (2024S): Difference between revisions

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== Tentative schedule ==
== Tentative schedule ==
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|October 14
|October 14
|'''Available'''
|Kevin (if no one else will volunteer)
|Regular singularities
|Regular singularities
|Connections and holonomic D-modules with regular singularities. Deligne's Riemann-Hilbert correspondence.
|Connections and holonomic D-modules with regular singularities. Deligne's Riemann-Hilbert correspondence.
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|October 28
|October 28
|Hairuo|Hairuo
| Hairuo |Hairuo
|Intro to perverse sheaves
|Intro to perverse sheaves
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|Irregular singularities and the Stokes phenomenon
|Irregular singularities and the Stokes phenomenon
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== References ==
 
If you have other suggestions, please let me know (or just add to this list)!


* J.Bernstein's [https://gauss.math.yale.edu/~il282/Bernstein_D_mod.pdf 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 [https://gauss.math.yale.edu/~il282/Ginzburg_D_mod.pdf notes]
* C.Schnell's course on D-modules with lecture-by-lecture notes ([https://www.math.stonybrook.edu/~cschnell/mat615/ 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 [https://arxiv.org/abs/1702.03608 paper] by M.Kamgarpour and S.Weatherhog.
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Revision as of 05:03, 1 October 2024

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.
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 Kevin (if no one else will volunteer) 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
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