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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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.
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
|'''available'''
|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 9
|July 16
|'''available'''
|Dima  
(This is not explained well in references, so only take this if you are comfortable; otherwise I'll do it. Dima.)
|Inverse and direct images. Derived category of D-modules
|Direct images. Derived category of D-modules.
|`Naive' definition. Definition in the derived category (examples).
|`Naive' definition. Definition in the derived category (examples). Time permitting - Kashiwara's Lemma.  
|-
|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
|-
|-
|July 16
|November 18
|'''available'''
|Dima
|Integral transforms.
|Irregular singularities and the Stokes phenomenon
|Tensor product. Integral transforms (=Fourier-Mukai functor). The Fourier-Mukai transform on a line.
|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).