Date/Time : 02 Dec 2020 09:00
Venue : via Zoom
Speaker : Yu Zhao, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/85482725152?pwd=em9iM2ZlQk5EZEsyQ3ZIUm9ZVVN3Zz09 Meeting ID: 854 8272 5152 Password: 638381 The quantum toroidal algebra is the affinization of the quantum Heisenberg algebra. Schiffmann-Vasserot, Feigin-Tsymbaliuk and Negut studied the quantum toroidal algebra action on the Grothendieck group of Hilbert schemes of points on surfaces, which generalized the action by Nakajima and Grojnowski in cohomology. In this talk, we will categorify the above quantum toroidal algebra action. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities.

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Date/Time : 18 Nov 2020 09:00
Venue : via Zoom
Speaker : Pablo Alvarez, Institute for Advanced Study

Join Zoom Meeting https://nus-sg.zoom.us/j/89214308777?pwd=VmZBVmZaamF5Z1h5VnU1MERaZ2ZVQT09 Meeting ID: 892 1430 8777 Password: 369675 Abstract:  The question of the center of the quantum group at a generic parameter was understood by Drinfeld in 1989, but the center of this algebra when you specialize at a root of unity remained an open question. By work of Lachowska, Qi and Bezrukavnikov we had some description of the center of the small quantum group. In joint work with Bezrukavnikov, Shen and Vasserot, we prove a more concrete description of the center of the small quantum group, in terms of the cohomology of a certain affine Springer fiber. In joint work with Kivinen and Losev, we link certain questions of the center in Type A with the Hilbert scheme of points on C2.

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Date/Time : 12 Nov 2020 10:00
Venue : via Zoom
Speaker : Chengze Duan, University of Maryland, college park

Join Zoom Meeting https://nus-sg.zoom.us/j/85957375797?pwd=clE4UTk1WDF0VGMzdU5wVldIMnl2UT09 Meeting ID: 859 5737 5797 Password: 456131 Abstract:  n Vinberg’s study of characterization of reductive algebraic monoids and Vinberg monoids, he introduced the notion of perfect submonoids of dominant weights. Given a reductive algebraic monoids, we shall see a perfect submonoids of dominant weights of the group of units. Meanwhile, the perfect submonoids of dominant weights can also appears naturally from the tensor product decomposition in representation theory. We will see some explicit descriptions of the perfect submonoids of dominant weights and compare it with Vinberg’s results on reductive monoids.

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Date/Time : 05 Nov 2020 10:00
Venue : via Zoom
Speaker : Rahul Dalal, University of California, Berkeley

Join Zoom Meeting https://nus-sg.zoom.us/j/94719446250?pwd=bzRVL1Z6blh1RUViMTUzVnFFblllUT09 Meeting ID: 947 1944 6250 Password: 022183 Abstract:  Since Automorphic representations for general groups are very difficult to study individually, they are often studied in families instead. The Arthur-Selberg trace formula lends itself very naturally to answering questions about averages of various parameters of the local components of automorphic representations in so-called harmonic families. In their 2016 work, Shin and Templier realized that, in the special case of representations with discrete series at infinity, the trace formula simplified dramatically enough to compute statistics with good error bounds. These bounds were good enough for applications: an averaged Sato-Tate law analogous to Sato-Tate for families of elliptic curves and computations of the specific random-matrix statistics followed by low-lying zeros of L-functions in the family. Shin-Templier’s work used the invariant trace formula which disallowed families that distinguish representations with infinite component in the same L-packet. However, which member of this L-packet a representation might correspond to determines some important characteristics–whether the representation is holomorphic or quaternionic for example. I will discuss how methods related to the stable trace formula can remove this restriction. The key idea is applying a certain “hyperendoscopy” formulation of stabilization used first by Ferrari, though many technical difficulties come up.

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Date/Time : 29 Oct 2020 09:00
Venue : via Zoom
Speaker : Charlotte Chan, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/91964791213?pwd=ZFlvbVdySWNlL2NpNzJXSnlSOUtxQT09 Meeting ID: 919 6479 1213 Password: 006362 Abstract:  Deligne–Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to so-called “depth-zero” supercuspidal representations of p-adic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is partially based on joint work with Alexander Ivanov and joint work with Masao Oi.

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Date/Time : 22 Oct 2020 10:00
Venue : via Zoom
Speaker : Jonathan Wang, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/92431908138?pwd=NGs4eWRENWRERWlJRUZIUDdyVFVOUT09 Meeting ID: 924 3190 8138 Password: 865981 Abstract:  Part of the aim of the relative Langlands program is to produce new integral representations of automorphic L-functions in order to study their analytic properties. The program is still very much in progress, but generalized Ichino-Ikeda formulas of Sakellaridis-Venkatesh relate the global problem to certain problems in local harmonic analysis on spherical varieties. I will explain how we can use techniques from geometric Langlands to compute “new” integrals which give special values of unramified local L-functions over a local function field. Our results also give new integral representations of L-functions (in a right half plane) over global function fields when the integral “unfolds”. This is joint work with Yiannis Sakellaridis.

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Date/Time : 15 Oct 2020 10:00
Venue : via Zoom
Speaker : Zhilin Luo, University of Minnesota--Twin Cities

Join Zoom Meeting https://nus-sg.zoom.us/j/94022303433?pwd=b3hSZWpNTVN2VW9sU21XdThEdlJ0QT09 Meeting ID: 940 2230 3433 Password: 100202 Abstract: The local Gan-Gross-Prasad conjecture studies the restrictioin and branching problems for representations of classical and metaplectic groups . In this talk, I will talk about my proof for the tempered part of the local Gan-Gross-Prasad conejecture (multiplicity one in Vogan packets) for special orthogonal groups over any local fields of characteristic zero, which combines the work of Waldspurger (for the tempered part of the conjecture for special orthogonal groups over p-adic fields) and Beuzart-Plessis (for the tempered part of the conjecture for unitary groups over real fields) in a non-trivial way. In the proof, an indispensable result which is also of independent interest is a formula expressing the regular nilpotent germs of quasi-split reductive Lie algebras over any local fields of characteristic zero via endoscopic invariants, which was previously proved by Shelstad over $p$-adic fields. We also relate the formula with the Kostant’s sections.

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Date/Time : 08 Oct 2020 10:00
Venue : via Zoom
Speaker : Zhilin Luo, University of Minnesota--Twin Cities

Join Zoom Meeting https://nus-sg.zoom.us/j/91356784967?pwd=Q0ZxYlc5VU9UT3NCczJJNUNYTjlOZz09 Meeting ID: 913 5678 4967 Password: 256173 Abstract: I am going to talk about my joint work with D. Jiang and L. Zhang. We develop a new type of harmonic analysis on an extended symplectic groups G=G_m x Sp_2n over p-adic fields. It is associated to the Langlands gamma-functions attached to irreducible admissbile representations of G(F) and the standard representation rho of the dual group. Our work can be viewed as an extension of the work of Godement-Jacquet (which is a generalization of Tate’s thesis). We confirm a series of conjectures in the local theory of the Braverman-Kazhdan proposal in this setting.

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Date/Time : 01 Oct 2020 10:00
Venue : via Zoom
Speaker : Hang Xue, University of Arizona

Join Zoom Meeting https://nus-sg.zoom.us/j/97939482481?pwd=OW10bDJ2T2l0M2VFc0oyek1pZXNmUT09 Meeting ID: 979 3948 2481 Password: 721722 Abstract: A classical branching theorem of Weyl determines the decomposition of an irreducible representation of (compact) U(n+1) when restricted to U(n). The local Gan–Gross–Prasad conjecture puts this in the framework of local Langlands correspondence and extends it to the setting of noncompact unitary groups (and also p-adic unitary groups). We present a very simple proof of this conjecture for tempered packets of real unitary groups based on theta correspondences.  

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