Main Page/Reading Seminar Stacks (2025)
Current Room Information: Van Vleck B321
Time: Wednesdays 1:30pm - 2:30pm
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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| Note | Outline below is tentative. Likely need to spread out the talks because 1 hour will not be enough for some of these talks. | ||
| 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). |
| 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 |
| 08/27/2025 | Break | Break | We will skip this week since many people will be at the LSA training for the Fall. |
| 09/03/2025 | Break | Break | File:OutlineFall2025Seminar.pdfThis is the first week of classes for the Fall Semester. The first talk will be next week. I have attached a sketchy outline of series of talks. |
| 9/10/2025 | Jeremy N. | Semisimple Lie Algebras over C Part I | |
| 9/17/2025 | Jeremy N. | Semisimple Lie Algebras over C Part II | |
| 9/24/2025 | Reductive Algebraic Groups and their Root Datum | ||
| Below is the sketched out schedule. | |||
| Line Bundles on Flag Varieties and Beilinson-Bernstein Localization | |||
| On the Tannakian Reconstruction Theorem | |||
| Introduction to Perverse Sheaves | |||
| Introduction to the Affine Grassmannian (again) | |||
| On the Satake Category | |||
| Construction of the Fibre Functor for the Satake Category | |||
| Why the Satake Category is a Tensor Category | |||
| Identifying \widetilde{G} Part I | We probably need far more than 1 talk but hopefully less than 5. | ||
| Identifying \widetilde{G} Part II | |||
| Identifying \widetilde{G} Part III |
References
- Martin Olsson's Algebraic Spaces and Stacks. See the errata here.
- Laumon-Moret-Bailly Champs Algebriques
- Alper's Stacks and Moduli
- Dan Edidin Notes on the Construction of the Moduli Space of Curves
- Timm Peerenboom's Thesis on the Affine Grassmannian
- Xinwen Zhu's Introduction to affine Grassmannians and the geometric Satake equivalence