Date/Time : 15 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Yuanqing Cai, Kanazawa University

We discuss the trace formula for the n-fold covering group of SL_2. Along the way, we also revisit the work of Flicker on the trace formula for covering groups of GL(2).

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Date/Time : 08 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Rui Chen, IASM, Zhejiang University

Branching law is a classical topic in representation theory which concerns certain Hom functors. It is natural to consider Ext functors associated to those Hom functors. In this talk we will report some recent progress on the study of these Ext spaces, and show a conjecture of D. Prasad, which asserts that the Ext-analogue of GGP model vanishes for tempered representations. We will also discuss some applications.

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Date/Time : 01 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jeff York Ye, National University of Singapore

The decomposition of tensor products of finite-dimensional irreducible representations of simple Lie groups is a long-studied problem. I will discuss some known results describing the multiplicity using canonical bases for quantum groups and tropicalization. Then I will discuss the related saturation conjecture, which is proved in type A by Knutson and Tao and open for general types.

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Date/Time : 01 Feb 2024 10:00
Venue : S17 #04-06 (SR1)
Speaker : Wang Jialin, National University of Singapore

Fix a finite group G and an algebraically closed field F of characteristic p. For an FG-module M, the complexity of M is the rate of growth of a minimal projective resolution of M which is a cohomological invariant of M. The rank varieties introduced by Carlson are defined for modules for elementary abelian p-groups and can be extended to modules for G by looking at the restriction to elementary abelian subgroups of G. Moreover, the dimension of the rank variety gives the complexity of the module. In this talk, I will discuss some basic properties of rank varieties and complexities and then review some known results on complexities of some simple modules for symmetric groups.

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Date/Time : 25 Jan 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Yifeng Liu, IASM, Zhejiang University

In this talk, I will formulate the anticyclotomic Iwasawa main conjecture for Rankin-Selberg products of automorphic representations of hermitian type, and state our results. I will then explain the main ingredients of the proof: an Ihara type lemma, Abel-Jacobi image of basic locus, and a bipartite Euler system over the Iwasawa algebra. This is a joint work with Yichao Tian and Liang Xiao.

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Date/Time : 24 Jan 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Ziqing Xiang, Southern University of Science and Technology

We introduce a new notion called the quantum wreath product, which produces an algebra $B \wr_Q H(d)$ from a given associative algebra $B$, a positive integer $d$, and a choice $Q = (R, S, \rho, \sigma)$ of parameters. Important examples include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang’s wreath Hecke algebras, Rosso-Savage’s (affine) Frobenius Hecke algebras, Kleshchev-Muth’s affine zigzag algebras, and the Hu algebra that quantizes the wreath product $\Sigma_m \wr \Sigma_2$ between symmetric groups. We will discuss the bases of quantum wreath product algebras, and some of their representations. This is joint work with Daniel Nakano and Chun-Ju Lai.

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Date/Time : 18 Jan 2024 16:30
Venue : S17 #05-12 (SR4)
Speaker : Yongqi Liang, University of Science and Technology of China

Scharaschkin and Skorobogatov conjectured that when smooth projective curves violate the Hasse principle then the violations are explained by the Brauer-Manin obstruction. Colliot-Thelene and Sansuc have similar conjecture for degree 4 del Pezzo surfaces. We use this obstruction to construct explicite algebraic families of degree 4 del Pezzo surfaces and smooth projective curves of genus congruent to 1 modulo 4, such that they violate the Hasse principle. In particular, over every number field,  we find explicite families of elliptic curves whose Tate-Shafarevich groups are non-trivial. This gives an explicite answer to a question of Colliot-Thelene and Poonen asked about 20 years ago, when they proved the existence of such a family of elliptic curves over the field \mathbb{Q} of rational numbers.

This is a joint work with K. Huang. 

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Date/Time : 12 Dec 2023 16:30
Venue : S17 #04-04 (SR3)
Speaker : Lucas Mann, University of Muenster

Based on classical constructions in algebraic topology and algebraic geometry, we introduce abstract 6-functor formalisms, which are a form of “abstract cohomology theories”. It turns out that the abstract theory of 6-functor formalisms is surprisingly rich and has applications in non-geometric fields as well. As an example we provide new tools for studying smooth representations of locally profinite groups — in particular in natural characteristic, where many of the classical representation theoretic methods fail. An application of the theory is a conceptual construction of an anti-involution on the (derived) Iwahori-Hecke algebra, which previously could only be constructed on the associated Ext-algebra using explicit computations.

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Date/Time : 01 Dec 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Chuijia, Technion - Israel Institute of Technology

Kazhdan-Lusztig polynomials play a fundamental role in various Lie theoretical subjects such as Hecke algebras, category O, etc. It is expected that these polynomials only depend on the combinatorial nature of the associated Bruhat intervals. In a recent work of Williamson and his collaborators, they obtain a hypercube decomposition of q derivatives of Kazhdan-Lusztig polynomials via deep geometric machinery, which serves as an important step towards this “combinatorial invariance conjecture”. In this talk, we will try to reinterpret their results from the point of view of parabolic recursions and discuss some applications. This is based on a joint work with Maxim Gurevich.

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