Date/Time : 20 Jun 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Sunil Chebolu, Illinois State University

More than 50 years ago, Fuchs asked which abelian groups can be the group of units of a commutative ring? Though progress has been made, the question remains open. I will present an overview of my joint work with Keir Lockridge on this problem. We answered this question for various classes of groups including indecomposable abelian groups, dihedral groups, quaternion groups and some p-groups. This work also gave us several new characterizations of Mersenne primes and Fermat primes.

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Date/Time : 10 May 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Erez Lapid, Weissman Institute of Science

Prehomogeneous vector spaces are ubiquitous in the theory of nilpotent orbits of semisimple Lie algebras. Their study was initiated by M. Sato in the 1960s. Zeta functions attached to them are useful for the asymptotics of arithmetic invariants. They are also important in the geometric side of Arthur’s trace formula. I will discuss some recent work on the analysis of these zeta functions.

Joint work with Tobias Finis.

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Date/Time : 18 Apr 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Teruhisa Koshikawa , Kyoto University

I will talk about some relationships among categorical Langlands, L-packets, A-packets (or rather ABV-packets), and eigensheaves, focusing on the case of GL_2.

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Date/Time : 11 Apr 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Dougal Davis, University of Melbourne

In a seminal paper from 2011, Kari Vilonen and Wilfried Schmid proposed that the deep theory of mixed Hodge modules can be applied to the old problem of determining the unitary representations of a real reductive Lie group. They conjectured that the unitarity of a representation is completely determined by a canonical filtration, the Hodge filtration, which arises naturally from the geometry of the complex flag variety. In this talk, I will explain a recent paper with Vilonen (arXiv:2309.13215) in which we prove this conjecture. A key role (in both the proof of the conjecture and in applications) is played by a Hodge-theoretic refinement of Beilinson-Bernstein localisation: we show that, for mixed Hodge modules on flag varieties, the Hodge filtration is always globally generated and has vanishing higher cohomology.

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Date/Time : 03 Apr 2024 16:30
Venue : S17 #04-06 (SR1)
Speaker : Xuanzhong Dai, RIMS, Kyoto University

It has been suspected for a long time that the Rankin-Cohen brackets of modular forms should be related to vertex operator algebras as initially suggested by W. Eholzer, Y. Manin and D. Zagier. Our attempt involves sheaf constructions known as the chiral de Rham complex, whose cohomology is linked to the Witten genus. For arbitrary congruence subgroup of modular group, the fractional linear transformation induces an action on the global section of chiral de Rham complex on the upper half plane. We consider the invariant sections under the action of the congruence subgroups with suitable cuspidal conditions. It is noteworthy that modular forms appear naturally on invariant sections.

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Date/Time : 21 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jinfeng Song, National University of Singapore

It is well-known that the Drinfeld–Jimbo quantum groups can be viewed as quantized coordinate algebras of the dual Poisson groups. A quantum symmetric pair consists of a quantum group and a coideal subalgebra, called an i-quantum group. In this talk, I will explain that i-quantum groups can be viewed as quantized coordinate algebras of Poisson homogeneous spaces of dual Poisson groups. The consruction fits into the general picture, called the quantum duality principal. The relation with cluster algebras and KZ equations will also be briefly discussed.

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Date/Time : 14 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Le Nhat Hoang, National University of Singapore

The Gan-Gross-Prasad (GGP) conjecture studies a family of restriction problems for classical groups and proposes precise answers to these problems using the local and global Langlands correspondences. It also has a twisted variant in the Fourier-Jacobi case, which is called the twisted Gan-Gross-Prasad conjecture. In this talk, I will report my progress on the local twisted GGP conjecture for tempered representations. The strategy is to adapt Waldspurger and Beuzart-Plessis’s method to develop a local relative trace formula as well as a twisted trace formula and compare their elliptic parts. Although the geometric sides of both trace formulae have not been developed, one can use a partial comparison and an instance of an elliptic L-packet to prove the statement. Moreover, I will propose a geometric multiplicity formula for the local twisted GGP and necessary ingredients to prove it.

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Date/Time : 07 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Ziyang Gao, Leibniz University Hannover

The goal of this talk is to propose a possible framework to study quadratic relations among holomorphic periods of CM abelian varieties. We define, for each Shimura variety and a CM point, a bi-$\overline{\mathbb{Q}}$-structure on the tangent space. Then we explain that in the case of the Siegel moduli variety, the numbers comparing the two $\overline{\mathbb{Q}}$-structures are precisely the products of the holomorphic periods of the CM abelian varieties parametrized by the CM point (up to $2\pi i$). Next we propose a hyperbolic analytic subspace conjecture, which is the analogue of Wüstholz’s analytic subgroup theorem in this context, and explain why it implies the desired consequence on the quadratic relations among these holomorphic CM periods. This is joint work with Emmanuel Ullmo and (partially with) Andrei Yafaev.

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Date/Time : 27 Feb 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Shilin Yu, Xiamen University

Unipotent representations are believed to be the building blocks of unitary dual of semisimple/reductive Lie groups. In their study of special unipotent representations for complex semisimple groups, Barbasch and Vogan constructed a duality map between the nilpotent orbits of $G$ and that of its Langlands dual group $G^\vee$ (also discovered by Lusztig and Spaltenstein), which allows them to describe the unipotent ideals and representations of $G$ in terms of $G^\vee$. Later Sommers and Achar extended this duality map by considering pairs $(\mathbb{O}, C)$ consisting of a nilpotent orbit $\mathbb{O}$ and a conjugacy class $C$ in its Lusztig canonical quotient group.

In the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853), we have extended the Sommers’ duality map in another direction by considering covers of nilpotent orbits, which turns out to be equivalent to Achar’s duality map (in a nontrivial way). This extended duality map allows us to describe the (generalized) unipotent ideals and bimodules attached to nilpotent covers, proposed previously by Ivan Losev and my other two coauthors, in terms of the Langlands dual group.

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