Arithmetic Topological Field Theory and Relative Langlands Duality
Organisers: Minhyong Kim (ICMS and KIAS) and Sug Woo Shin (UC Berkeley and KIAS)
In this online seminar, we will try to understand portions of a new preprint by David Ben-Zvi, Yiannis Sakellaridis, and Akshay Venkatesh:
https://www.math.ias.edu/~akshay/research/BZSVpaperV1.pdf
If you are interested in participating, contact Minhyong Kim.
Unless there is specific objection from the speakers, the sessions will be recorded.
Schedule
Session 1:
2023/10/31 11:00-13:00 (UK time)
Speaker: Sug Woo Shin
Title: Dual groups of spherical varieties
Abstract: After a general introduction, I will explain the dual group of a spherical variety following Gaitsgory-Nadler and Sakellaridis-Venkatesh.
Recording: shin-1.mp4
Session 2:
2023/11/07 11:00-13:00 (UK time)
Speaker: Sug Woo Shin
Title: Local and global conjectures on automorphic periods
Abstract: In the context of spherical varieties, some local and global conjectures in Sakellaridis-Venkatesh will be discussed. The local part describes the Plancherel decomposition of the L2 space of a spherical variety. The global conjecture, modeled on the Ichino-Ikeda conjecture, will then express a global period in terms of local periods arising from the local conjecture.
Recording: shin-2.mp4
Session 3:
2023/11/14 11:00-13:00 (UK time)
Speaker: Minhyong Kim
Title: A Slice of TQFT for Number Theorists 1
Abstract: This is the first of two talks that will try to familiarise number-theorists with minimal prior knowledge of physics to *some* of the notions that might be useful in understanding the BZSV paper. Obviously, caveat emptor.
Recording: kim-1.mp4
Session 4:
2023/11/21 11:00-13:00 (UK time)
Speaker: Minhyong Kim
Title: A Slice of TQFT for Number Theorists 2
Abstract: This is the second of two talks that will try to familiarise number-theorists with minimal prior knowledge of physics to *some* of the notions that might be useful in understanding the BZSV paper. We will discuss TQFTs and arithmetic analogues.
Recording: kim-2.mp4
Session 5:
2023/11/28 11:00-13:00 (UK time)
Speaker: Dennis Gaitsgory
Title: proof of the geometric Langlands conjecture 1
Abstract: In the course of these two talks we will explain the construction of the Langlands functor (in the context of D-modules), and outline the steps in the proof that it is an equivalence of categories. This is a joint work with Arinkin, Beraldo, Chen, Faergeman, Lin, Raskin and Rozenblyum.
Recording: gaitsgory-1.mp4
Session 6:
2023/12/05 11:00-13:00 (UK time)
Speaker: Dennis Gaitsgory
Title: proof of the geometric Langlands conjecture 2
Abstract: In the course of these two talks we will explain the construction of the Langlands functor (in the context of D-modules), and outline the steps in the proof that it is an equivalence of categories. This is a joint work with Arinkin, Beraldo, Chen, Faergeman, Lin, Raskin and Rozenblyum.
Recording: gaitsgory-2.mp4
Session 7:
2023/12/12 11:00-13:00 (UK time)
Speaker: Jack Thorne
Postponed to 2024/02/06 (below)
Session 8:
2023/12/19 11:00-13:00 (UK time)
Speaker: David Kazhdan
Title: Hecke operators on the space S of half-densities on the groupoid Bun_G(P^1)_{0,∞} over finite and local fields.
Abstract: Let G be a split reductive group over a field k which is either finite or local and Bun_G be the groupoid of G-bundles on P^1 trivialized at 0 and ∞. The goal of this talk is
(1) To define the Schwartz space S of half-densities on BunG and a commutative algebra H_G of Hecke operators acting on S.
(2) To show how to use the algebra H_G for a definition of functions c_{D,ρ} on the space of irreducible cuspidal representations of the group G(k) where D is a divisor on P^1 \ (0, ∞) defined over k and ρ is an algebraic representation of the dual group G^(C).
(3) Provide for finite fields k an interpretation of functions c_{D,ρ} in terms of
the local Langland’s correspondence for the group G(k((t))).
Recording: kazhdan.mp4
Session 9:
2024/01/23 19:00-21:00 (UK time)
Speaker: Pavel Safronov
Title: Global conjecture: period and L-sheaves 1
Abstract: The goal of these two lectures is to state the BZSV relative Langlands duality in the setting of global geometric Langlands which I will recall. I will also explain a connection between Hamiltonian spaces and shifted symplectic structures. This will serve as a motivation for period sheaves constructed using sheaf quantization and L-sheaves constructed using spectral quantization.
Recording: safronov-1.mp4
Slides: safronov-1-slides.pdf
Session 10:
2024/01/30 19:00-21:00 (UK time)
Speaker: Pavel Safronov
Title: Global conjecture: period and L-sheaves 2
Recording: safronov-2.mp4
Slides: safronov-2-slides.pdf
Session 11:
2024/02/06 19:00-21:00 (UK time)
Speaker: Jack Thorne
Title: The dual group and unramified duality
Abstract: I will attempt to give a detailed description of the dual group of a spherical variety (and more generally, of the dual Hamiltonian space of a polarized hyperspecial variety, in the sense of BZSV), along with some examples. Time permitting, I will discuss some aspects of the “local unramified picture” as laid out in BZSV.