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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Date/Time : 25 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Tomasz Przebinda, University of Oklahoma

We describe an integral kernel operator which should map a Harish-Chandra character of a representation of one member of a dual pair to a distribution on the other member of the dual pair. A well founded hope is that this distribution coincides with the character of the big theta lift. In part this is a joint work with Hung Yean Loke.

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Date/Time : 14 Nov 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Finn Wiersig, National University of Singapore

The Riemann-Hilbert correspondence establishes a profound connection between certain differential equations on a given complex analytic manifold and its topology. This result had numerous interesting applications; for example, it was applied by Beilinson-Bernstein and Brylinski-Kashiwara in the 1980s to study the representation theory of semisimple algebraic groups. This talk explores ongoing research leveraging these concepts to investigate representation theory appearing in the p-adic local Langlands program, focusing on progress towards a p-adic analytic Riemann-Hilbert correspondence.

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Date/Time : 07 Nov 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Euiyong Park, University of Seoul

Let $C$ be a generalized Cartan matrix and $U_q(\mathfrak{g})$ the quantum group associated with $C$. In this talk, I am going to talk about the PBW theory for the bosonic extension $\widehat{A}_{\mathfrak{g}}$ of the negative half $U_q^-(\mathfrak{g})$. When $U_q(\mathfrak{g})$ is of finite simply-laced type, the algebra $\widehat{A}_{\mathfrak{g}}$ is isomorphic to the quantum Grothendieck ring of the Hernandez-Leclerc category over a quantum affine algebra. The PBW theory allows us to define a new family of subalgebras, denoted by $\widehat{A}_{\mathfrak{g}}(b)$, for any element $b$ in the (generalized) Braid group associated with $C$. I will explain why this algebra $\widehat{A}_{\mathfrak{g}}(b)$ can be viewed as a natural extension of the quantum unipotent coordinate ring $A_q(n(w))$ for a Weyl group element $w$. This talk is based on arXiv:2401.04878 (with Se-jin Oh) and arXiv:2408.07312 (with Masaki Kashiwara, Myungho Kim and Se-jin Oh).

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