Main Page/Reading Seminar Stacks (2025): Difference between revisions

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Current Room Information: Fri Feb 28 2025 12:00 pm - 1:30 pm in Ingraham 225 (Repeats every week on Friday through 5/2)
Current Room Information: Van Vleck B129


Disclaimer: Notes from this page were scribed by an audience member. Mistakes, typos, and so forth are possible and likely. No originality is claimed.
Time: Fridays 4:00pm - 5:30pm (we actually have the room until 6:00pm in case people go over).
 
Disclaimer: Any notes from this page were scribed by an audience member and/or written up by the speaker. **Mistakes, typos, and so forth are common and likely.** No originality is claimed.


==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
Line 19: Line 21:
|Kevin D.
|Kevin D.
|Fibred Categories
|Fibred Categories
|Introduction to fibred categories. Describe correspondence between fibred categories /C and presheaves on C. Groupoids in C, fibre products of fibred categories, Yoneda Lemma, and discussion of categories fibred in groupoids. Discuss examples. The following exercises in Olsson are relevant for the future: 3.A, 3.B, 3.C, 3.D, 3.F, and 3.G.
|Introduction to fibred categories. Describe correspondence between fibred categories /C and presheaves on C. Groupoids in C, fibre products of fibred categories, Yoneda Lemma, and discussion of categories fibred in groupoids. Discuss examples.  
The following exercises in Olsson are relevant for the future: 3.A, 3.B, 3.C, 3.D, 3.F, and 3.G.
 
[[File:StacksNotesHairuoKevinScribe.pdf|thumb]]
|-
|-
|03/14/2025
|03/14/2025
|Kevin D.
|Moduli, Representability, and Motivation for the Future
|Representable functors, moduli functors of interest, why the \'etale topology?,  schemes vs. algebraic spaces vs. Deligne-Mumford stacks vs. algebraic stacks. Sketch of future goals.
[[File:ScribeNotesKevinMotivationTalk.pdf|thumb]]
|-
|3/21/2025
|Jeremy N.
|Jeremy N.
|Descent and Stack Conditions
|Descent and Stack Conditions
|Discuss generalities on descent. Explain why fppf descent and fpqc descent work. Applications of descent e.g. closed subschemes, open embeddings, affine morphisms, polarized schemes. Discuss torsors and principal homogeneous spaces. See exercise 4.G for invertible sheaves on sites.
|Discuss generalities on descent. Explain why fppf descent and fpqc descent work. Applications of descent e.g. closed subschemes, open embeddings, affine morphisms, polarized schemes. (Optional: Discuss torsors and principal homogeneous spaces.) Definition of stack using fibred categories and the stackification functor. Examples of stacks.
Definition of stack using fibred categories. Fibred products of stacks, explanation of the stackification functor. Examples of stacks: Olsson's examples are found in the exercises 4.C, 4.E, 4.H. If exercises are too hard, see Laumon-Moret-Bailly.
[[File:ScribeNotesJeremyDescent.pdf|thumb]]
|-
|3/21/2025
|Kevin D. (can give up slot)
|Algebraic Spaces Part 1
|TBD. Olsson's presentation spans three chapters. Alper's notes are really good for this. Note the Stacks project uses the fppf topology instead of the etale topology.
|-
|-
|3/28/2025
|3/28/2025
|Kevin D.  (can give up slot)
|No Speaker
|Algebraic Spaces Part 2
|Spring Break!
|TBD
|
|-
|-
|04/01/2025
|04/01/2025
|Kevin D. (can give up slot)
|Jameson A.
|Algebraic Spaces Part 3
|Algebraic Spaces Part 1
|TBD
|Olsson's presentation spans three chapters. Alper's notes are really good for this. Note the Stacks project uses the fppf topology instead of the etale topology.
[[File:ScribeNotesJamesonAlgSpace.pdf|thumb]]
|-
|-
|04/11/2025
|04/11/2025
|Kevin D. (can give up slot)
|Hairuo X.
|Algebraic Stacks
|Algebraic Spaces Part 2
|TBD
|Define quasicoherent sheaves on algebraic spaces. Discuss more examples of algebraic spaces and where they might arise naturally. Explain why algebraic spaces are not enough for many moduli problems. Notes can be found [https://drive.google.com/file/d/1HXu62XhWF6euTrDDuoeoWt6UNMqPaWfm/view?usp=sharing here].
|-
|-
|04/18/2025
|04/18/2025
|Jeremy N.
|Kevin D.
|Stacks: Motivation and Artin Stacks
|Definition of an algebraic (Artin) stack. Define what is a Deligne-Mumford stack. Define properties of morphisms of stacks (for representable morphisms only). Define M_g, quotient stacks, and classifying stacks as examples (to be verified later). Discuss separation axioms. Theorem: An algebraic stack is Deligne-Mumford iff \Delta:X->X\times_S X is formally unramified.
[[File:NotesAlgebraicStacks1.pdf|thumb]]
|-
|04/25/2025
|No Talk
|Break!
|Break! People are either busy this week or will be traveling so a break right now is great!
|-
|05/02/2025
|Kevin D.
|Expedition: Quot Scheme and Hilbert Scheme
|Expedition: Quot Scheme and Hilbert Scheme
|Introduction to Quot Scheme and Hilbert Scheme. Phrase it using the language introduced thus far.  
|Introduction to Quot Scheme and Hilbert Scheme. Phrase it using the language introduced thus far.  
Indicate application to identifying M_g=[H'/PGL_{5g-5}] for H' a locally closed subscheme of the Hilbert scheme.  
Indicate application to identifying M_g=[H'/PGL_{5g-5}] for H' a locally closed subscheme of the Hilbert scheme. Also indicate proof of algebraicity of the stack Coh_X/S.
 
[[File:QuotSchemes.pdf|thumb]]
|-
|-
|04/25/2025
|05/09/2025, 05/16/2025, 05/23/2025, 05/30/2025
|No Talk
|Break!
|
|
|Examples: M_g, Bun(C), Quotient Stacks, Classifying Stack.
|-
|06/05/2025
|Kevin D.
|Introduction to Bun_G
|Introductory talk to some topics regarding Bun_G. Outline main goals for the seminar. Talked about structure groups, defined G-bundle. Formulate the statements of the main theorems.
[[File:BunGTalkNotes1.pdf|thumb]]
|-
|06/18/2025
|Jeremy N.
|Topological Classification and Examples
|Discuss topological classification, and explicit examples of Bun_G e.g. Bun_{G_m}^d(X)\cong Pic^d(X)\times BG_m, describe Bun_GL_2(P^1).
[[File:BunGTalkSlides2.pdf|thumb]]
|-
|06/25/2025
|Jameson A.
|Bun_G is an Artin Stack and Uniformization of Bun_G
|Explain why, for X a smooth algebraic curve of genus g(X) and G a reductive algebraic group that Bun_G(X) is an algebraic stack. Then discuss Weil's Uniformization of Bun_G(X), connected components of Bun_G(X).
|-
|07/09/2025
|Kev D.
|Deformation Theory of Bun_G
|[[File:SmoothDimBunG.pdf|thumb]]Explain why Bun_G(X) is a smooth algebraic stack of dimension (g(X)-1)\dim(G).
|-
|07/16/2025
|Jeremy N.
|Gr_G and its Line Bundles
|Discuss the affine Grassmannian Gr_G and line bundles on Gr_G
[[File:Ind-Schemes, Affine Lie Algebras, and the Affine Grassmannian.pdf|thumb]]
|-
|07/23/2025
|BREAK!
|BREAK!
|Due to various conflicts in schedules, we took a break this week!
|-
|07/30/2025
|Yourong Z.
|Line Bundles on Bun_G
|Discuss line bundles on Bun_G. In particular, work out some examples and outline the ideas of the proof (don't worry about tedious details).
|-
|08/06/2025
|Kevin D.
|Cohomology of Bun_n^d
|[[File:CohomologyRingOfBunrdAndAtiyahBottClasses.pdf|thumb|0x0px]]Discuss the cohomology of Bun_n^d and the answer via Atiyah-Bott Classes.
|-
|08/20/2025
|Kevin D.
|More on Line Bundles on Bun_G
|Clean up some details regarding Line Bundles on Bun_G. Some general facts about Gr_G to be used later if we cover the Geometric Satake Correspondence. Determinant bundles and Pfaffian Bundles and Pic(Bun_G) when G is simple and simply connected of various types. Determine when the determinant bundle or the Pfaffian bundle is a generator. [[File:LineBundlesOnBunG.pdf|thumb]]
|-
|Break
|Break
|Break
|We will skip this time period since there is LSA training and many are traveling.
|-
|09/12/2025
|Break
|Break
|[[File:OutlineFall2025Seminar.pdf|thumb]]Attached is an outline for the Fall semester with a list of talk topics. We will not necessarily follow this ''exact'' outline.Volunteer speakers are needed!
|-
|09/19/2025
|Jeremy N.
|Semisimple Lie Algebras over C Part I
|
|
|-
|-
|05/02/2025
|09/26/2025
|Jeremy N.
|Semisimple Lie Algebras over C Part II
|
|
|Quasicoherent Sheaves on Algebraic Stacks
|-
|10/03/2025
|Jameson A.
|Reductive Algebraic Groups and their Root Datum
|
|
|-
|-
|05/09/2025
|10/10/2025
|
|Kevin D. ??
|Coarse Moduli Spaces and more Moduli of Curves
|Line Bundles on Flag Varieties and Beilinson-Bernstein Localization
|
|Note: Someone suggested to split this into two talks and have a talk on Borel-Weil included somewhere.
|-
|10/17/2025
|??
(Kevin cannot make it -- will provide notes to replacement speaker and/or post video lecture)
|On the Tannakian Reconstruction Theorem
|(This might be pushed to a GAGS talk since it is interesting to a broader audience)
|-
|-
|
|10/24/2025
|
|Qianyi Cao
|Gerbes
|Introduction to Perverse Sheaves
|
|
|-
|-
|10/31/2025
|
|
|
|Introduction to the Affine Grassmannian (again)
|Expedition: Coarse Moduli Spaces and Geometric Invariant Theory
|
|
|-
|-
|11/07/2025
|Jeremy N.
|On the Satake Category
|
|
|Kevin (willing to give up slot)
|The Keel-Mori Theorem
|References: Chapter 11 of Olsson has a more general formulation than 4.4.2 of Alper's notes. Applications in next lecture. See also [https://math.stanford.edu/~conrad/papers/coarsespace.pdf Conrad's paper] which has more details and which Olsson follows.
|-
|-
|11/14/2025
|
|
|Construction of the Fibre Functor for the Satake Category
|
|
|Local Structure of Algebraic Stacks
|Apply Keel-Mori Theorem to give local structure of Deligne-Mumford stacks. Discuss the local structure of Artin stacks afterwards.
|-
|-
|
|11/21/2025
|
|Jeremy N.
|Luna's Etale Slice Theorem
|Why the Satake Category is a Tensor Category
|
|
|-
|-
|
|12/03/2025
|
(last talk unfortunately...perhaps not enough time).
|
|Yourong Z. ?
|
|Identifying \widetilde{G} Part I
|We probably need far more than 1 talk but hopefully less than 5.
|-
|-
|
|
|
|Yourong Z. ?
|Moduli of Semistable Bundles
|Identifying \widetilde{G} Part II
|
|
|-
|-
|
|
|
|
|
|Identifying \widetilde{G} Part III
|
|-
|Future Potential Topics
|
|Bun_G (perhaps following [https://www.math.sciences.univ-nantes.fr/~sorger/assets/pdf/trieste.pdf Sorger]), Artin Algebraization, Formal Moduli, etc. We'll figure it out when we get there.
|
|
|- }
|- }
Line 116: Line 206:
==References==
==References==


# Martin Olsson's Algebraic Spaces and Stacks
# Martin Olsson's Algebraic Spaces and Stacks.  See the errata [https://drive.google.com/file/d/1iuC8Yr293xusJHP9WIPMqvV_USeqEjSh/view here].
# Laumon-Moret-Bailly Champs Algebriques
# Laumon-Moret-Bailly Champs Algebriques
# Alper's [https://sites.math.washington.edu/~jarod/moduli.pdf Stacks and Moduli]
# Alper's [https://sites.math.washington.edu/~jarod/moduli.pdf Stacks and Moduli]
# Dan Edidin [https://arxiv.org/abs/math/9805101 Notes on the Construction of the Moduli Space of Curves]
# Timm Peerenboom's [https://www.math.uni-bonn.de/ag/stroppel/thesis_timm_peerenboom.pdf Thesis on the Affine Grassmannian]
# Xinwen Zhu's [https://arxiv.org/abs/1603.05593 Introduction to affine Grassmannians and the geometric Satake equivalence]
# Gaitsgory's [https://arxiv.org/pdf/math/9912074 Construction of Central Elements...]
# Consider looking [https://people.mpim-bonn.mpg.de/gaitsgde/grad_2009/ here] and the two talks on Affine Grassmannians as well

Latest revision as of 14:15, 13 September 2025

Current Room Information: Van Vleck B129

Time: Fridays 4:00pm - 5:30pm (we actually have the room until 6:00pm in case people go over).

Disclaimer: Any notes from this page were scribed by an audience member and/or written up by the speaker. **Mistakes, typos, and so forth are common and likely.** No originality is claimed.

Tentative schedule

date speaker title topics
02/28/2025 Hairuo X. Grothendieck Topologies / Sites Introduction to Grothendieck Toplogies /sites. More information can be found in Notes on a Seminar by Michael Artin. If one wishes to present more on the \'etale site, Milne's Lecture Notes has far more details.

File:StacksNotesHairuoScribe.pdf

03/07/2025 Kevin D. Fibred Categories Introduction to fibred categories. Describe correspondence between fibred categories /C and presheaves on C. Groupoids in C, fibre products of fibred categories, Yoneda Lemma, and discussion of categories fibred in groupoids. Discuss examples.

The following exercises in Olsson are relevant for the future: 3.A, 3.B, 3.C, 3.D, 3.F, and 3.G.

File:StacksNotesHairuoKevinScribe.pdf

03/14/2025 Kevin D. Moduli, Representability, and Motivation for the Future Representable functors, moduli functors of interest, why the \'etale topology?, schemes vs. algebraic spaces vs. Deligne-Mumford stacks vs. algebraic stacks. Sketch of future goals.

File:ScribeNotesKevinMotivationTalk.pdf

3/21/2025 Jeremy N. Descent and Stack Conditions Discuss generalities on descent. Explain why fppf descent and fpqc descent work. Applications of descent e.g. closed subschemes, open embeddings, affine morphisms, polarized schemes. (Optional: Discuss torsors and principal homogeneous spaces.) Definition of stack using fibred categories and the stackification functor. Examples of stacks.

File:ScribeNotesJeremyDescent.pdf

3/28/2025 No Speaker Spring Break!
04/01/2025 Jameson A. Algebraic Spaces Part 1 Olsson's presentation spans three chapters. Alper's notes are really good for this. Note the Stacks project uses the fppf topology instead of the etale topology.

File:ScribeNotesJamesonAlgSpace.pdf

04/11/2025 Hairuo X. Algebraic Spaces Part 2 Define quasicoherent sheaves on algebraic spaces. Discuss more examples of algebraic spaces and where they might arise naturally. Explain why algebraic spaces are not enough for many moduli problems. Notes can be found here.
04/18/2025 Kevin D. Stacks: Motivation and Artin Stacks Definition of an algebraic (Artin) stack. Define what is a Deligne-Mumford stack. Define properties of morphisms of stacks (for representable morphisms only). Define M_g, quotient stacks, and classifying stacks as examples (to be verified later). Discuss separation axioms. Theorem: An algebraic stack is Deligne-Mumford iff \Delta:X->X\times_S X is formally unramified.

File:NotesAlgebraicStacks1.pdf

04/25/2025 No Talk Break! Break! People are either busy this week or will be traveling so a break right now is great!
05/02/2025 Kevin D. Expedition: Quot Scheme and Hilbert Scheme Introduction to Quot Scheme and Hilbert Scheme. Phrase it using the language introduced thus far.

Indicate application to identifying M_g=[H'/PGL_{5g-5}] for H' a locally closed subscheme of the Hilbert scheme. Also indicate proof of algebraicity of the stack Coh_X/S.

File:QuotSchemes.pdf

05/09/2025, 05/16/2025, 05/23/2025, 05/30/2025 No Talk Break!
06/05/2025 Kevin D. Introduction to Bun_G Introductory talk to some topics regarding Bun_G. Outline main goals for the seminar. Talked about structure groups, defined G-bundle. Formulate the statements of the main theorems.

File:BunGTalkNotes1.pdf

06/18/2025 Jeremy N. Topological Classification and Examples Discuss topological classification, and explicit examples of Bun_G e.g. Bun_{G_m}^d(X)\cong Pic^d(X)\times BG_m, describe Bun_GL_2(P^1).

File:BunGTalkSlides2.pdf

06/25/2025 Jameson A. Bun_G is an Artin Stack and Uniformization of Bun_G Explain why, for X a smooth algebraic curve of genus g(X) and G a reductive algebraic group that Bun_G(X) is an algebraic stack. Then discuss Weil's Uniformization of Bun_G(X), connected components of Bun_G(X).
07/09/2025 Kev D. Deformation Theory of Bun_G File:SmoothDimBunG.pdfExplain why Bun_G(X) is a smooth algebraic stack of dimension (g(X)-1)\dim(G).
07/16/2025 Jeremy N. Gr_G and its Line Bundles Discuss the affine Grassmannian Gr_G and line bundles on Gr_G

File:Ind-Schemes, Affine Lie Algebras, and the Affine Grassmannian.pdf

07/23/2025 BREAK! BREAK! Due to various conflicts in schedules, we took a break this week!
07/30/2025 Yourong Z. Line Bundles on Bun_G Discuss line bundles on Bun_G. In particular, work out some examples and outline the ideas of the proof (don't worry about tedious details).
08/06/2025 Kevin D. Cohomology of Bun_n^d File:CohomologyRingOfBunrdAndAtiyahBottClasses.pdfDiscuss the cohomology of Bun_n^d and the answer via Atiyah-Bott Classes.
08/20/2025 Kevin D. More on Line Bundles on Bun_G Clean up some details regarding Line Bundles on Bun_G. Some general facts about Gr_G to be used later if we cover the Geometric Satake Correspondence. Determinant bundles and Pfaffian Bundles and Pic(Bun_G) when G is simple and simply connected of various types. Determine when the determinant bundle or the Pfaffian bundle is a generator. File:LineBundlesOnBunG.pdf
Break Break Break We will skip this time period since there is LSA training and many are traveling.
09/12/2025 Break Break File:OutlineFall2025Seminar.pdfAttached is an outline for the Fall semester with a list of talk topics. We will not necessarily follow this exact outline.Volunteer speakers are needed!
09/19/2025 Jeremy N. Semisimple Lie Algebras over C Part I
09/26/2025 Jeremy N. Semisimple Lie Algebras over C Part II
10/03/2025 Jameson A. Reductive Algebraic Groups and their Root Datum
10/10/2025 Kevin D. ?? Line Bundles on Flag Varieties and Beilinson-Bernstein Localization Note: Someone suggested to split this into two talks and have a talk on Borel-Weil included somewhere.
10/17/2025 ??

(Kevin cannot make it -- will provide notes to replacement speaker and/or post video lecture)

On the Tannakian Reconstruction Theorem (This might be pushed to a GAGS talk since it is interesting to a broader audience)
10/24/2025 Qianyi Cao Introduction to Perverse Sheaves
10/31/2025 Introduction to the Affine Grassmannian (again)
11/07/2025 Jeremy N. On the Satake Category
11/14/2025 Construction of the Fibre Functor for the Satake Category
11/21/2025 Jeremy N. Why the Satake Category is a Tensor Category
12/03/2025

(last talk unfortunately...perhaps not enough time).

Yourong Z. ? Identifying \widetilde{G} Part I We probably need far more than 1 talk but hopefully less than 5.
Yourong Z. ? Identifying \widetilde{G} Part II
Identifying \widetilde{G} Part III

References

  1. Martin Olsson's Algebraic Spaces and Stacks. See the errata here.
  2. Laumon-Moret-Bailly Champs Algebriques
  3. Alper's Stacks and Moduli
  4. Dan Edidin Notes on the Construction of the Moduli Space of Curves
  5. Timm Peerenboom's Thesis on the Affine Grassmannian
  6. Xinwen Zhu's Introduction to affine Grassmannians and the geometric Satake equivalence
  7. Gaitsgory's Construction of Central Elements...
  8. Consider looking here and the two talks on Affine Grassmannians as well
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