**Goals**: The topic this quarter is Weil’s Tamagawa Number conjecture for function fields — an extraordinary result of Gaitsgory and Lurie, which beautifully bridges algebraic topology, algebraic geometry, representation theory, and number theory. In previous quarters we have shown that BunG good — in fact, miraculous — but this quarter we are going to tell you how much BunG weighs.

**Setup**: There will be weekly talks ** Wednesdays 4-6pm at Eckhart E203.** There is also an option to attend via Zoom -- contact one of the organizers to receive the Zoom link. Talks will be broken into two periods of 50 minutes each, with a 10 minute break after each half for questions/discussions. As UChicago consists of leading experts in the broad area of BunG, we have invited faculty and postdocs to give (and attend!) a majority of the talks. All talk recordings are uploaded to our YouTube channel.

**Prerequisites**: Basic algebraic geometry: elementary knowledge of projective varieties, line bundles, (quasi-)coherent sheaves, morphisms (smooth, etale, flat), D-modules on varieties. Basic representation theory: finite-dimensional representation theory for reductive or semi-simple Lie groups and Lie algebras.

** Organizers: **
Justin Campbell (cjcampbell AT uchicago.edu), Nikolay Grantcharov (nikolayg AT uchicago.edu), Aaron Slipper (aaronslipper AT uchicago.edu),

Title and Recording | Speaker | Notes | Date |
---|---|---|---|

## An Overview of the Tamagawa Number Theorem for Function FieldsIn this talk, I will introduce the notion of the Tamagawa number as a canonical “volume” of automorphic space. I will discuss the Tamagawa number’s role in classical number theory — in particular, its relevance to the problem of enumerating lattices in R^n (the Weil-Siegel Mass Formula) as well as its relation to special values of L-functions. I will then give a brief discussion of the work of Langlands, Lai, Kottwitz and Chernousov for number fields, and, time permitting, some discussion of the Bloch-Kato Tamagawa number for motives. Finally, I will give an overview of Gaitsgory and Lurie's method for the Tamagawa number problem over function fields. Here, the problem is given a geometric (cohomological!) reinterpretation: as we have already seen, automorphic space is closely related to our hero, BunG, the moduli stack of principal G-bundles over a curve. We may thus reinterpret the question as one about the “stacky” mass of BunG, which can be computed using a suitable “stacky” analogue of the Lefschetz fixed-point theorem for stacks. We will reduce the question to finding a suitable “algebraic” version of Atiyah-Bott’s formula for the global cohomology of BunG. Recording |
Aaron Slipper | Notes | 1/3/24 |

## Higher Algebra and Koszul DualityIn this talk, I'll explain some basic definitions in higher algebra, including (symmetric) monoidal structures on higher categories, associative and commutative algebras, etc., with minimal hand-waving. Then we'll move on to the theory of operads, which provides a convenient framework for all kinds of algebraic structures, notably including Lie algebras. Finally, we'll see how Lie algebras and commutative algebras are related by the operation of Koszul duality, which gives a conceptual and formula-free explanation for why the tangent space of a group has a Lie bracket. Recording |
Justin Campbell | 1/10/24 | |

## Localizing invariants of large categories: general theoryI will explain how to extend the scope of algebraic K-theory (and other localizing invariants) to the context of dualizable (compactly assembled) stable categories. After giving a general introduction to dualizable categories I will define the so-called continuous algebraic K-theory of a dualizable category. It is functorial with respect to functors with colimit-preserving right adjoint. If a category is compactly generated, its continuous K-theory is equivalent to the usual K-theory of the subcategory of compact objects. I will outline the computation of continuous K-theory for categories of sheaves on locally compact Hausdorff spaces. As an application, I will explain an alternative simple proof of a theorem of Kasprowski and Winges about the commutation of K-theory with infinite products. Recording |
Alexander Efimov | 1/17/23 | |

## E_n algebrasIn the first half of the talk, we will go through some of the definitions of E_n-algebras and discuss important basic properties like Dunn additivity. We will spend some time discussing the classical story, but in the end our view will be focused towards infinity-operads, a la Justin's earlier talk this quarter. After the break, we will try to reap the fruits of our labor by discussing connections with Verdier duality and non-abelian Poincare duality. Recording |
Michael Barz | 1/24/23 | |

## Chiral Koszul DualityLast week we saw the definition of topological (Betti) factorization algebras. This week we'll see 2.5 definitions in the holomorphic (de-Rham) setting, and why the last two are in fact the same using the theory of (co)operads. Main reference: Francis paper Recording |
Elchanan Nafcha | Notes | 1/31/24 |

## Grothendieck-Lefschetz and the product formulaI will recall the strategy of proving the Tamagawa number conjecture, and how it breaks down into two different theorems: Grothendieck-Lefschetz formula and a certain "cohomological product formula" for Bun_G. In the first part of the talk, I will outline the proof of the Grothendieck-Lefschetz trace formula for Bun_G due to Kai Behrend. In the second part, I will review basic facts about l-adic étale sheaves on prestacks, and formulate the product formula of Gaitsgory-Lurie which computes étale cohomology of Bun_G as chiral homology of a certain l-adic sheaf on the Ran space. Recording |
Kirill Magidson | 2/7/24 | |

## Non-abelian Poincare dualityNon-abelian Poincare duality is a theorem which relates the l-adic homology of BunG to that of the Beilinson-Drinfeld Grassmannian. We will explain the main ideas in proving this, and show how it reduces to a concrete statement that the space of rational maps from a curve X to a semi-simple group G is homologically contractible. We will give a new, very simple, proof of this due to Drinfeld. Recording |
Nikolay Grantcharov | Notes | 2/14/24 |

## Verdier duality on the Ran spaceWe explain how the formalism of Verdier duality on the Ran space allows one to deduce the "cohomological product formula" from the "non-abelian Poincare duality" of the previous talk. |
Kevin Lin | Notes | 2/21/24 |

## Counting points on BunGI will sketch a proof of the G-L trace formula for Bun_G, proving Conj. 8 of Lecture 3 of Lurie’s course. |
Matt Emerton | 2/28/24 | |

- Gaitsgory, Lurie: Weil's conjecture for Function Fields: PDF
- Lurie's course notes on Tamagawa numbers course website

Organizers: Justin Campbell, Nikolay Grantcharov, Aaron Slipper

Title and Recording | Speaker | Notes | Date |
---|---|---|---|

## Cuspidal Hecke eigensheaves via localizationIn this talk, I'll lay out a roadmap for the rest of the quarter. The plan is to explain Beilinson-Drinfeld's construction of the Hecke eigensheaves using localization of Kac-Moody representations, with the advantage of hindsight and modern technology. Recording |
Justin Campbell | Notes | 10/4/23 |

## Geometric Satake CorrespondenceWe will discuss the equivalence between the symmetric monoidal categories of spherical perverse sheaves on the affine Grassmannian of a reductive group G and the category of representations of the Langlands Dual Group. We will begin by discussing the geometry of the Affine Gassmannian --- the so-called "semi-infinite orbits" in particular. Next we will introduce the category of spherical perverse sheaves on the affine Grassmannian, and show that the global cohomology functor is faithful and exact. Then we will introduce the convolution product on spherical perverse sheaves; we will show that convolution preserves perversity via a semi-smallness argument. Finally, we will introduce the Beilinson-Drinfeld affine Grassmannian, and discuss how the agreement of the "fusion" product with the convolution product yields a commutativity constraint. Together, these imply that the spherical perverse Hecke category is symmetric monoidal, and that global cohomology is a Tannakian fiber functor. Thus the category of G(O)-equivariant perverse sheaves is equivalent to the category of representations of some group; it can be shown that this group is the Lanlgands Dual. Recording |
Aaron Slipper | Notes | 10/11/23 |

## The local and global Hitchin mapWe begin by giving an overview of Hamiltonian reduction in the classical setting of a Harish-Chandra pair (g,K) acting on a smooth scheme Y. Applying this to the BunG framework provides a construction of the local Hitchin map. The main result we prove will be that the image of this local, and hence global, Hitchin map Poisson commutes. Finally, we show that, assuming a global quantization of the Hitchin map exists (to be proved in talk 8), the Hecke eigensheaf associated to an oper is nonzero. Recording |
Nikolay Grantcharov | Notes | 10/18/23 |

## Representation theory of Affine Lie AlgebrasAn overview of the representation theory of affine Lie algebras with a fixed level. The main goal will be to describe the special properties of critical level representations. We'll start by describing the finite dimensional analog: The block decomposition of category O according to the central character using the Harish-Chandra homomorphism. We'll then generalize that to the infinite dimensional case using the Segal-Sugawara operator. We'll prove the triviality of the center for generic levels, and give a first description of the center in critical level using the language of vertex operator algebras. Recording |
Elchanan Nafcha | Notes | 10/25/23 |

## Feigin-Frenkel IsomorphismIn the previous talk, Elchanan explained to us the famous theorem of B.Feigin and E.Frenkel: the algebra of endomorphisms of the vacuum module over an affine Lie algebra at the critical level is a polynomial algebra on infinitely many generators. This computation depends on a choice of coordinate on the formal disc. I will explain a coordinate free formulation of this theorem due to Beilinson-Drinfeld describing endomorphisms of the vacuum module as functions on the space of opers, and outline a geometric proof of this theorem due to S.Raskin. Recording |
Kirill Magidson | Notes | 11/1/23 |

## Localization of Kac--Moody representationsIn the song "New Math," T. Lehrer describes addition in base eight as "so very simple, that only a child could do it!" V. Arnold famously proclaimed that he was able to teach group theory to Moscow schoolchildren by rejecting the definition of groups as sets with a binary and unary operation satisfying some obscure identities. In a similar spirit, we shall construct the Kazhdan--Lusztig category KL(G) and the localization functor KL(G) --> D(Bun_G) in terms that are so very simple (the theory of ind-coherent sheaves on infinite type stacks), that V. Arnold could do it. |
Kevin Lin | Notes | 11/8/23 |

## Chiral/Factorization AlgebraI will introduce the notion of chiral and factorization algebra, which are equivalent to each other. Examples of such objects will arise as the chiral envelope of Lie-* algebras. We will end with a commutativity theorem whose proof uses the Eckmann-Hilton argument. Recording |
Lie Qian | Notes | 11/20/23 |

## Hecke eigensheaves via chiral homologyI will explain how the machinery of chiral homology, when applied to the Feigin-Frenkel isomorphism, gives rise to the quantized global Hitchin fibration. A key ingredient is the interpretation of Kac-Moody localization as an enhanced version of chiral homology. Time permitting, I will also sketch how the Fundamental Local Equivalence at critical level allows us to construct arbitrary cuspidal Hecke eigensheaves, including those whose eigenvalue does not admit a global oper structure. Recording |
Justin Campbell | Notes | 11/29/23 |

- A. Beilinson and V.Drinfeld "Quantization of the Hitchin integrable system and Hecke eigensheaves"

Organizers: Justin Campbell, Nikolay Grantcharov, Aaron Slipper

Title and Recording | Speaker | Notes | Date |
---|---|---|---|

## \(\text{Bun}_G\) at infinity\(\text{Bun}_G\) is not quasi-compact, and its behavior at infinity is controlled by the deeper strata of the Harder-Narasimhan stratification. I will discuss the remarkable geometric contractiveness property of this stratification. As will be explained later in the semester, the contractiveness implies that the category of D-modules on \(\text{Bun}_G\) behaves as if \(\text{Bun}_G\) would be quasi-compact. The talk follows the work of V.Drinfeld and D.Gaitsgory "compact generation of the category of D-modules on the stack of G-bundles on a curve", Drinfeld-Gaitsgory Recording |
Sasha Beilinson | 3/22/23 | |

## Infinitesimal Structure of \(\text{Bun}_G\)Given a principal G-bundle P over a smooth projective curve X, one may consider the jet spaces of germs of regular functions on Bun_G(X) over P. Dually, one considers the fiber of the sheaf of differential operators on BunG over P. Following Beilinson-Drinfeld and Beilinson-Ginzburg, we will give several explicit descriptions of these jet spaces, starting with a concrete realization of the 1st order jet = cotangent space \((T_P(Bun_G))^*\). Then upgrading this realization to higher order jets, we will see the notion of “conformal blocks for Kac-Moody algebras” arises naturally. A proof of the main theorem describing jets, using a residue pairing, will be sketched. Recording |
Nikolay Grantcharov | 3/29/23 | |

## A recap on algebraic D-modulesI will review basics on algebraic D-modules following the notes of J. Bernstein. Recording |
Zhilin Luo | 4/5/23 | |

Justin Campbell | 4/12/23 | ||

## Derived Categories of Sheaves on SchemesI'll wrap up the discussion of higher categories from last week, then outline the six-functor formalism for derived categories of sheaves on schemes, and how it extends to more general prestacks. The contrast between D-modules and other flavors of sheaf theory will be emphasized. Recording |
Justin Campbell | 4/19/23 | |

## Compact Generation of D-mod(BunG)I will speak about compact generation of the category of D-modules on Bun_G. This statement is due to Drinfeld and Gaitsgory and uses the stratification of Bun_G by contractive strata from Beilinson's talk. In this talk, I will review some generalities on compactly generated DG categories together with classical examples from algebraic geometry, such as quasi- and Ind-coherent sheaves, and D-modules on sufficiently nice schemes and stacks, and will show how this machinery applies to D-mod(Bun_G). Reference: Drinfeld-Gaitsgory Recording |
Kirill Magidson | 4/26/23 | |

## Conformal Blocks and Bun_GA remarkable fact about Bun_G is that its geometry is closely related to 2d conformal field theory. I will briefly explain the theory of conformal blocks (a key concept in 2d conformal field theory) and its derived version. I will then specialize to the case of a Kac-Moody Lie algebra and its maximal integrable quotient (the so-called WZW model) and relate their conformal blocks to the geometry of Bun_G. I will explain that the space of conformal blocks of the WZW model is given by the cohomology of the line bundle of "non-abelian theta functions" on Bun_G. A key ingredient will be a refined version of the uniformization of Bun_G. Reference: Rozenblyum Recording |
Nick Rozenblyum | 5/3/23 | |

## Geometric Eisenstein Series and Constant Term FunctorsIn this talk we will introduce the geometric Eisenstein series and constant term functors, along with some motivation for their construction in the geometric Langlands program. The first part of the discussion will involve defining the functors and a brief overview of some foundational results of early papers by Braverman-Gaitsgory. In the second part of the talk, we’ll discuss Drinfeld-Gaitsgory’s formulation of Braden’s theorem and its application which relates constant term functors for opposite parabolic subgroups. Reference: Gaistgory Recording |
Andreas Hayash | 5/10/23 | |

## Miraculous DualityAs we have seen a number of times, one of the technical complications of working with the stack Bun_G is that it is not quasi-compact. Nevertheless, for very nontrivial reasons, it behaves in many ways as if it were. For instance, the category of D modules on Bun_G is compactly generated. In particular, it is dualizable. We will show that it is in fact, self-dual with self-duality given by "miraculous duality" and describe some applications to Eisenstein series. Reference: Gaistgory Recording |
Nick Rozenblyum | 5/17/23 |

- V.Drinfeld and D.Gaitsgory "Compact generation of the category of D-modules on the stack of G-bundles on a curve",
- D. Gaitsgory, "A strange functional equation for Eisenstein series and miraculous duality on BunG" Gaistgory
- Chapter 1 of Gaitsgory-Rozenblyum's DAG book

Organizers: Nikolay Grantcharov, Aaron Slipper

Title and Recording | Speaker | Notes | Date |
---|---|---|---|

## Introduction to stacks and \(\text{Bun}_G\)The moduli stack of principal G-bundles on an algebraic curve is an important object in numerous contexts. I will give a motivated introduction to the theory of algebraic stacks in general and BunG in particular. Of particular interest will be relations with geometric representation theory. Recording |
Nick Rozenblyum | Notes | 1/4/23 |

## Functor of Points and DescentAs a warm-up, I'll define schemes from the functor of points perspective. After upgrading to functors valued in groupoids, stacks will emerge as a natural 2-categorical upgrade of sheaves. We'll see some examples that demonstrate why the Zariski topology is too coarse for many purposes, and should be replaced by the étale or fppf topology. Finally, I'll discuss a couple of approaches to defining Artin or algebraic stacks, which are locally isomorphic to affine schemes in a suitable sense. Recording |
Justin Campbell | Notes | 1/11/23 |

## Algebraicity of \(\text{Bun}_G\)We will first review the definition of stacks and algebraic stacks, then introduce the classifying stack BG and show that it is an algebraic stack. We will also introduce the stack \(\text{Bun}_{G,X}\) and focus on the case when \(G = GL_n\) and X is a smooth projective curve over an algebraically closed field k. We will recall Weil’s automorphic interpretation of \(\text{Bun}_{n,X}(k)\). Finally, we will sketch a proof for the algebraicity of \(\text{Bun}_{n,X}\). Recording |
Zhilin Luo | Notes | 1/18/23 |

## \(\text{Bun}_G\) and Higgs BundlesWe will start with a discussion of moduli stacks that parametrize objects of a (sheaf of) categories and how the stack Bun_G fits into this general framework. Next, we define the tangent, cotangent, and also the inertia stack of a stack X. We interpret these stacks as "vector bundle stacks" on X. I'll give a description of the tangent/cotangent stack of a moduli stack and show that the cotangent stack of \(\text{Bun}_G\) is isomorphic to the stack of Higgs bundles. Time permitting, I'll also discuss some more concrete results related to the geometry of Higgs bundles, in particular, the 'global nilpotent cone' that plays an important role in Geometric Langlands. Recording |
Victor Ginzburg | Notes | 1/25/23 |

## Harder-Narasimhan StratificationThe stability condition for vector bundles was first introduced by D. Mumford using a mysterious notion called "slope". Though seemingly lacking in intuition, this notion is nonetheless very important in GIT and more general moduli problems, due to being the condition to obtain a "good" moduli space in "linearizable situations". Stemming from stability is the Harder-Narasimhan filtration, which provides a canonical way to describe any vector bundle on a curve as successive extensions of stable bundles. Such theory is generalized to any reductive group G by A. Ramananthan and later also by K. Behrend using so-called "complementary polyhedra". We will go through the main results of this theory and use GL2 over P^1 as an example to see the equivalence between Ramananthan-Behrend stability and slope stability. If time allows, we will also briefly state S. Schieder's "slope map" as a third description of stability which also makes the equivalence between the previous two stabilities in GLn case more transparent. Recording |
Griffin Wang | Notes | 2/1/23 |

## Moduli of G-bundles as a quotient by the gauge groupFirstly we will provide a complex geometric description of holomorphic G-bundles as principal bundles together with a connection. We will use this to interpret the moduli space of G-bundles, of a fixed topological type, with a quotient by the (infinite dimensional) group of gauge transformations. We will reinterpret this as a (infinite dimensional) symplectic quotient in an analogue of the Kempf--Ness Theorem. The Yang--Mills functional appears as the norm of the moment map. The Harder--Narasimhan filtration can be reinterpreted in terms of the equivariant Morse theory of this functional in an analogue of the HKKN stratification. Time permitting, we will describe how Atiyah--Bott used this to compute the cohomology of the moduli space of semistable G-bundles. Recording |
Benedict Morrissey | 2/8/23 | |

## Affine GrassmannianI will explain the construction of the affine Grassmannian and its basic geometric properties. Recording |
Ngô Bảo Châu | Zhu's Notes | 2/15/23 |

## Uniformization of Principal BundlesGiven a simply-connected, complex semisimple algebraic group G, a complex smooth projective curve X, and a choice of basepoint on X, there exists a smooth surjective map—called the uniformization map—from the affine Grassmannian of G onto the moduli stack of principal G-bundles over X. The proof has two steps. The first uses a result of Beauville–Laszlo to replace the affine Grassmannian with a global analogue. The second uses a theorem of Drinfeld–Simpson to ensure that any principal G-bundle over the complement of the basepoint admits a Borel reduction, and hence, admits a trivialization over the complement. The force of these statements is that they hold not just on complex points, but on R-points for any C-algebra R. Recording |
Minh-Tâm Trinh | Notes | 2/22/23 |

## A Mosaic of Examples and Basic Properties of BunGThis talk will explore in detail many of the aspects of the geometry of BunG that we have encountered so far this quarter. But we shall try to accomplish this through explicit examination of concrete examples. We hope to make this talk ostentatiously low-tech, utilizing what Shreeram Abhyankar calls “high-school algebra” (ie, explicit polynomial computations) when possible. (But don’t worry — cohomology will still make an appearance!) Of particular interest will be the case where the underlying curve is P^1, since here the coarse moduli space is essentially trivial and all the geometry is quintessentially “stacky.” Some of the things we hope to cover include Grothendieck’s theory of vector bundles over P^1; Grothendieck-Harder’s classification of general reductive G-torsors over P^1; and Ramanathan’s description of the deformations and automorphisms of reductive G-torsors over P^1. This will give us a fairly complete picture of what the (stacky) k-points of BunG(P^1) "look like” and how they are topologized. Along the way, we will hope to offer a number of digressions relating these basic examples to the general properties we have encountered/will encounter in future quarters; including, potentially, the general theory of the connected components of BunG, the Harder-Narasimhan filtration, Tamagawa Numbers, one-point and Adelic uniformization, the Tannakian formalism, and the Narasimhan-Seshadri theorem. If time permits, we will discuss some geometry in the case where the underlying curve is elliptic. Recording |
Aaron Slipper | 3/1/23 |

- Gaitsgory, 2009 seminar on geometric representation theory
- Sorger, Lectures on moduli of principal G-bundles over algebraic curves
- Faltings, Vector Bundles on Curves
- Alper, Stacks and Moduli
- Halpern-Leistner, Moduli Theory
- PCMI: Geometry of Moduli Spaces and Representation Theory (see Xinwen Zhu's lecture)
- Schmitt: Affine Flag Manifolds and Principal Bundles (see Görtz’s lecture)