Date/Time : 04 Feb 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : David Hansen, National University of Singapore

The Bernstein center of a p-adic reductive group G is a beautiful and explicit commutative ring which acts on “everything” related to the representation theory of G. In recent years, the idea has emerged that this ring contains a canonical subring – the stable Bernstein center – which should be intimately related with the local Langlands correspondence. However, while it is easy to define the stable Bernstein center, it is very difficult to exhibit elements in this subring. On the other hand, recent work of Fargues-Scholze defines another totally canonical subring of the Bernstein center, whose construction uses V. Lafforgue’s theory of excursion operators adapted to the Fargues-Fontaine curve. After reviewing these stories, I’ll sketch a proof that the Fargues-Scholze subring is actually contained in the stable Bernstein center, for all G.

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Date/Time : 22 Jan 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Ma Jiajun, Xiamen University and Xiamen University Malaysia

This mini-course explores the fundamental aspects of local theta correspondence. The series consists of 5 lectures covering the following topics:
1. Models of the Heisenberg-Weil representation and the Metaplectic group 2. Dual pairs and general properties of theta correspondences 3. Theta correspondences over real fields 4. Theta correspondences over finite fields and p-adic fields 5. Geometric aspects of theta correspondence This mini-course is designed for senior undergraduate students and graduate students.

This mini-course will live stream. Here is the zoom ID:
Meeting ID: 848 8485 0855
Passcode: 226224

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Date/Time : 21 Jan 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Hui Gao, Southern University of Science and Technology

Using the Breuil–Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.

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Date/Time : 15 Jan 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Ma Jiajun, Xiamen University and Xiamen University Malaysia

This mini-course focuses on the fundamental aspects of local theta correspondence. In the first lecture, we will examine the Weil representation in detail. We will explore various realizations of the Weil representation and definitions of the Metaplectic group. Additionally, we will discuss the definition of the Weil index and related concepts. The course material is based on Gan-Kudla-Takada’s book The Local Theta Correspondence. This mini-course is designed for senior undergraduate students and graduate students.

This mini-course will live stream.  Here is the zoom ID:
Meeting ID: 848 8485 0855
Passcode: 226224

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Date/Time : 07 Jan 2025 16:00
Venue : S17 #05-12 (SR4)
Speaker : Chen Wan, Rutgers University—Newark

In this talk, I will discuss the geometric expansion of a local (twisted) trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.

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Date/Time : 18 Dec 2024 15:15
Venue : S17 #04-05 (SR2)
Speaker : Matteo Parisi, Harvard University

The amplituhedron is the image of the positive Grassmannian—the region of the Grassmannian where all Plücker coordinates are nonnegative—under a totally positive linear map. It is a far-reaching generalization of cyclic polytopes and hyperplane arrangements, and the positive Grassmannian itself. The “volume” of the amplituhedron encodes probabilities of particle interactions in the quantum field theory N=4 super Yang-Mills, and calculating this volume involves decomposing (or tiling) the amplituhedron into smaller pieces (or tiles) and summing their volumes. This talk will delve into the rich combinatorics of these tiles and tilings, presenting recent results on some of the central conjectures in this area, including the magic number, BCFW tiling, and cluster adjacency conjectures.

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Date/Time : 18 Dec 2024 09:30
Venue : S17 #04-05 (SR2)
Speaker : Kazim Buyukboduk, University College Dublin

I will discuss the factorization of a certain triple product p-adic L-function whose interpolation range is empty. The said factorization reflects, but does not formally follow from due to the absence of an interpolation locus, the Artin formalism for the underlying family of motives (that decompose as the sum of 2 motives of respective degrees 2 and 6). I will explain how this factorization problem can be recast as the comparison of two families of Arithmetic GGP conjectures (and can be proved in some cases using this reduction).

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Date/Time : 18 Dec 2024 10:45
Venue : S17 #04-05 (SR2)
Speaker : Robin Zhang, MIT

In general, local converse theorems say that generic or cuspidal representations can be uniquely determined by their Rankin–Selberg gamma factors. After Jacquet and Langlands first proved such a statement for complex representations of local GL(2), an extensive series of works has established local converse theorems for complex representations of classical groups, some other local quasi-split reductive groups, and finite GL(n). A “naïve local converse theorem” fails for modular representations of finite GL(n) in characteristic ℓ > 0, but I will discuss how to fix this and prove a local converse theorem for finite GL(n) after defining a more general gamma factor with values in certain ℤ[1/p, ζp]-algebras. In the spirit of Henniart’s proof that the local Langlands correspondence for GL(n) is uniquely characterized by the compatibility of gamma factors, one may hope that this can give a “finite field local Langlands in families” determined by gamma factors. This is based on joint work with J. Bakeberg, M. Gerbelli-Gauthier, H. Goodson, A. Iyengar, and G. Moss.

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Date/Time : 18 Dec 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Jiepeng Fang, University of Hong Kong

By using Lusztig sheaves on the framed quiver, we give a categorical realization of the tensor product of integrable highest weight modules of the quantum group such that simple perverse sheaves in the category provide the canonical basis of the tensor product. As a consequence, we obtain the positivity of the canonical basis of the tensor product. This talk is based on a joint work with Yixin Lan.

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