Date/Time : 20 Jun 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Yap Jit Wu, Harvard University

Abstract: For a rational function on P^1, it is known that the set of periodic points of period n converges weakly to a measure supported on its Julia set, known as the equilibrium measure, as n tends to infinity. In the 1980s, Szpiro-Ullmo-Zhang proved an arithmetic version of this for abelian varieties, where one is allowed to take just the Galois orbit of a torsion point, instead of all torsion points of order n. Since then, much more general arithmetic equidistribution theorems have been proven, along with the discovery of various applications to number theory.

In this talk, we will explain how having a stronger notion called logarithmic equidistribution will allow us to prove finiteness of S-integral points in certain situations and state some results in this direction.

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Date/Time : 16 Jun 2022 11:00
Venue : S17 #04-06 (SR1)
Speaker : Dinesh Thakur, University of Rochester

Abstract: We will explain some basic congruences of number theory, questions, conjectures and results about them and connections with Zeta values.

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Date/Time : 09 Jun 2022 11:00
Venue : S17 #04-06 (SR1)
Speaker : Dinesh Thakur, University of Rochester

Abstract: We will explain some basic congruences of number theory, questions, conjectures and results about them and connections with arithmetic derivatives.

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Date/Time : 01 Jun 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Edmund Karasiewicz, University of Utah

Abstract: The Fourier coefficients of automorphic forms are important objects of study due to their connection to L-functions. In the adelic framework, constructions of L-functions involving Fourier coefficients (e.g. Langland-Shahidi and Rankin-Selberg methods) naturally lead to spherical Whittaker functions on p-adic groups. Thus we would like to study these spherical Whittaker functions to better understand L-functions. Casselman-Shalika determined a formula for the spherical Whittaker functions, and basic algebraic manipulations reveal that their formula can be succinctly expressed in terms of characters of the Langlands dual group. We will describe a new approach to the Casselman-Shalika formula that provides a conceptual explanation of the appearance of characters. This is joint work with Nadya Gurevich.

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Date/Time : 19 Apr 2022 15:00
Venue : via Zoom
Speaker : Xiaoyu Zhang, University of Duisburg-Essen.

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https://nus-sg.zoom.us/j/88995853527?pwd=MHZNbFVGTy93QVJEMGFtdEtLdGJGdz09

Meeting ID: 889 9585 3527
Passcode: 388541

Abstract: Theta correspondence is a powerful tool in the theory of automorphic representations. Many fundamental problems in the complex coefficient case have been resolved thanks to the work of many people and have led to very deep applications in Langlands program and other problems in number theory. In this talk, we consider theta correspondence of orthogonal-symplectic type. Given the integral structures of the spaces of automorphic forms, we will show that if an automorphic form on orthogonal group is non-zero mod p, so is its theta lift. The proof involves a comparison of the Bessel period of the theta lift and the toric period of the original automorphic form and is reduced to an equidistribution result of CM points on the orthogonal group. If time permits, we will also discuss some problems related to these results.

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Date/Time : 12 Apr 2022 14:00
Venue : via Zoom
Speaker : Yu Jun, Peking University

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https://nus-sg.zoom.us/j/89178679746?pwd=NUNjUVdTcWhLZ082di9tM0I1cDRXQT09

Meeting ID: 891 7867 9746
Passcode: 548511

Abstract: Given a quasisplit reductive linear algebraic group $\mathbb{G}$ over a number field $F$, this talk concerns the determination of the functional source group $H_{\pi}$ of an automorphic representation $\pi$ of Ramanujan type of $\mathbb{G}(\mathbb{A}_{F})$. Assuming that $\mathbb{G}$ is split, a unitary group or an orthogonal group and $\pi$ neither comes from the Eisenstein series nor comes from endoscopy, we show that $H_{\pi}$ is determined up to element-conjugacy by its dimension datum alone. We also show results concerning linear independent of dimension data of such $H_{\pi}$.

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Date/Time : 07 Apr 2022 14:00
Venue : S17 #04-05 (SR2)
Speaker : Weiqiang Wang, University of Virginia

Abstract: Symmetric pairs over a local field admit a quantum deformation known as quantum symmetric pairs of affine type, (U, U^i). Here U is a Drinfeld-Jimbo affine quantum group (a quantum deformation of a loop algebra), and it admits a Drinfeld (loop) presentation. In this talk, we give a Drinfeld presentation for the affine i-quantum group U^i. Along the way, we shall construct various root vectors in U^i and describe their classical limits. This is based on joint work with Ming Lu (Sichuan) and Weinan Zhang (Virginia).

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Date/Time : 05 Apr 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Huanchen Bao, National university of Singapore

Abstract:  The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.

We show that the J-totally nonnegative flag variety has a cellular decomposition into totally positive J-Richardson varieties. Moreover, each totally positive J-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.

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Date/Time : 31 Mar 2022 09:00
Venue : via Zoom
Speaker : Stefan Dawydiak, University of Toronto

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https://nus-sg.zoom.us/j/89752886605?pwd=Z1V3N2lPbEtodjViZXVmMXRCNGFpZz09

Meeting ID: 897 5288 6605
Passcode: 343140

Abstract:  In the early 1980s Lusztig defined a ring J that serves as a limit of an affine Hecke algebra as the parameter v tends to 0, and showed that J can be realized as a subalgebra of a completion of the affine Hecke algebra, with elements of J being infinite linear combinations of elements of the affine Hecke algebra with Laurent series coefficients. Using work of Braverman-Kazhdan and the Plancherel formula for p-adic groups, we prove that the above coefficients are in fact rational functions whose denominators all divide a fixed polynomial that depends only on the affine Weyl group. We conjecture, and prove in the case that the p-adic group is GL_n, that this polynomial is precisely the Poincare polynomial of the finite Weyl group of G. We will explain a possible application to the representation theory of affine Hecke algebras at root of unity.

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