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| In the fall, we are meeting in person on Mondays, 12:00-1:00pm in VV 307.
| | We are meeting in person on Mondays, 12:00-1:00pm in VV 307. |
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| == Tentative schedule == | | == Tentative schedule == |
| {| cellpadding="8"
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| ! align="left" | date
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| ! align="left" | speaker
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| ! align="left" | title
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| ! align="left" | topics
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| | {| class="wikitable" |
| | |+ |
| | !Date |
| | !Speaker |
| | ! |
| | ! |
| |- | | |- |
| |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.
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| |- | | |- |
| |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
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| |Dima
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| |Inverse and direct images. Derived category of D-modules
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| |`Naive' definition. Definition in the derived category (examples).
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| |-
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| |July 23
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| |Alex
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| |Kashiwara's Lemma.
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| |Direct image under closed embeddings. Kashiwara's Lemma and applications.
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| |-
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| |July 30
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| |Kevin
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| |Integral transforms
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| |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].
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| |-
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| |August 6
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| |Jameson
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| |Levelt-Turritin classification
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| |D-modules on punctured formal disk. Regular and irregular singularities. Extra topics: monodromy, the Stokes phenomenon, perhaps some discussion of non-punctured disk
| |
| |-
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| |August 13
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| |Kevin
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| |Holonomic D-modules
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| |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
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| |Alex
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| |The six functors
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| |Preservation of holonomicity. Functoriality of singular support (?).
| |
| |-
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| |September 23
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| |Dima
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| |Introduction to the Riemann-Hilbert correspondence (over reals)
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| |Existence and uniqueness for ODEs and PDEs. Monodromy. Correspondence between bundles with connection/local systems/representations of the fundamental group.
| |
| |-
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| |September 30
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| |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
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| |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
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| |Kevin
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| |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
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| |Alex
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| |Riemann-Hilbert correspondence for D-modules
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| |Intro to constructible sheaves. Riemann-Hilbert as an equivalence of derived categories
| |
| |-
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| |October 28
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| |Hairuo
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| |Intro to perverse sheaves
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| |Intro to perverse sheaves
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| |-
| |
| |November 4
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| | | | | |
| |No meeting
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| | | | | |
| |- | | | |
| |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
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| |Dima
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| |Irregular singularities and the Stokes phenomenon - II
| |
| |Stokes phenomenon in for higher rank. Irregular Riemann-Hilbert Correspondence for Riemann Surfaces.
| |
| |- }
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| | |
| |} | | |} |
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| == References == | | == References == |
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| If you have other suggestions, please let me know (or just add to this list)!
| | L Maxim. Intersection Homology and Perverse sheaves |
We are meeting in person on Mondays, 12:00-1:00pm in VV 307.
Tentative schedule
References
L Maxim. Intersection Homology and Perverse sheaves