Reading Seminar on D-modules (2024S): Difference between revisions
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(Created page with "We meet on Tuesdays on Zoom (email arinkin@math.wisc.edu for the link if you need it). The first meeting is on Tuesday, June 25th. == Tentative schedule == {| cellpadding="8" !align="left" | date !align="left" | speaker !align="left" | title !align="left" | 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...") |
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In the fall, we are meeting in person on Mondays, 2:20-3:50pm in VV B321. | |||
== Tentative schedule == | == Tentative schedule == | ||
{| cellpadding="8" | {| cellpadding="8" | ||
!align="left" | date | ! align="left" | date | ||
!align="left" | speaker | ! align="left" | speaker | ||
!align="left" | title | ! align="left" | title | ||
!align="left" | topics | ! align="left" | topics | ||
|- | |- | ||
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|- | |- | ||
|July 2 | |July 2 | ||
| | |Jameson | ||
|Left and right D-modules. Inverse images | |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). | |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 | |July 16 | ||
| | |Dima | ||
|Inverse and direct images. Derived category of D-modules | |||
| | |`Naive' definition. Definition in the derived category (examples). | ||
|`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 [https://drive.google.com/file/d/10mi00kI_CrBxm42NnyOgfPhK8eZTvrLI/view?usp=sharing 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 [https://drive.google.com/file/d/1hVj7IDcjZNJGeeoTUUPNGRw6V1lXtBia/view?usp=sharing 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 [https://drive.google.com/file/d/1slQT9mTYh0KTRjG9D_Tf23rk8Doe5L3G/view?usp=sharing 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 | |||
|Regular singularities and Regular Holonomic D-modules | |||
|Connections and holonomic D-modules with regular singularities. Deligne's Riemann-Hilbert correspondence. Here are the [https://drive.google.com/file/d/1rFuQjAcwE9vBpz0-ar6WjuuocrO8jVxN/view?usp=sharing notes]. | |||
|- | |||
|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 | |||
|Intro to perverse sheaves | |||
|- | |||
|November 4 | |||
| | |||
|No meeting | |||
| | |||
|- | |||
|November 11 '''2pm in Sterling 2319''' | |||
|Jeremy | |||
|Perverse sheaves - II | |||
|Intro to perverse sheaves and IC complexes | |||
|- | |- | ||
| | |November 18 | ||
| | |Dima | ||
| | |Irregular singularities and the Stokes phenomenon | ||
| | |Stokes phenomenon via an example. | ||
|- | |||
|November 25 | |||
|Dima | |||
|Irregular singularities and the Stokes phenomenon - II | |||
|Stokes phenomenon in for higher rank. Irregular Riemann-Hilbert Correspondence for Riemann Surfaces. | |||
|- } | |||
|} | |||
== 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. | |||
* N. Katz, An Overview of Deligne's Work on Hilbert's Twenty-First Problem ([https://web.math.princeton.edu/~nmk/old/DeligneXXIHilbert.pdf Online pdf]). |
Latest revision as of 21:45, 25 November 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. 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 | Regular singularities and Regular Holonomic D-modules | Connections and holonomic D-modules with regular singularities. Deligne's Riemann-Hilbert correspondence. Here are the notes. |
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 | Intro to perverse sheaves |
November 4 | No meeting | ||
November 11 2pm in Sterling 2319 | Jeremy | Perverse sheaves - II | Intro to perverse sheaves and IC complexes |
November 18 | Dima | Irregular singularities and the Stokes phenomenon | Stokes phenomenon via an example. |
November 25 | Dima | Irregular singularities and the Stokes phenomenon - II | Stokes phenomenon in for higher rank. Irregular Riemann-Hilbert Correspondence for Riemann Surfaces. |
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.
- N. Katz, An Overview of Deligne's Work on Hilbert's Twenty-First Problem (Online pdf).