Date/Time : 09 Sep 2020 15:00
Venue : via Zoom
Speaker : Raphaël Beuzart-Plessis, Université d’Aix-Marseille

Join Zoom Meeting https://nus-sg.zoom.us/j/94485648813?pwd=d2pRWVJoVWdybTNhems3L2V4VkQvZz09 Meeting ID: 944 8564 8813 Password: 690359 Abstract: This talk concerns a global conjecture of Gan-Gross-Prasad relating the non-vanishing of certain automorphic periods on unitary groups to the non-vanishing of certain Rankin-Selberg L-functions at their center of symmetry. It also admits a refinement, proposed by Ichino-Ikeda, in the form of an exact identity connecting these two invariants. Following an approach initiated by Jacquet-Rallis of comparing two (relative) trace formulas, W. Zhang and more recently Y. Liu, W.Zhang, X. Zhu and myself have established both of these conjectures for cuspidal representations that are “stable” i.e. whose quadratic base-change is again cuspidal. In this talk, I will explain how to deal with the remaining cuspidal representations, sometimes called “endoscopic”, to which the conjectures apply through a fine spectral expansion of parts of the Jacquet-Rallis trace formula for general linear groups. This is joint work with Pierre-Henri Chaudouard and Michal Zydor..

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Date/Time : 09 Sep 2020 11:00
Venue : via Zoom
Speaker : Hsueh-Yung Lin, Kavli-IPMU, Japan

Join Zoom Meeting https://nus-sg.zoom.us/j/95748660096?pwd=M2Y1eVJvb01iTzQrZ29CNGZuREM4UT09 Meeting ID: 957 4866 0096 Password: 871058 Abstract: Let X be a compact Kähler manifold. The celebrated Kodaira embedding theorem asserts that if the Kähler cone of X contains a rational cohomology class, then X is projective. Instead of considering Kähler classes, we will study the algebraicity of X when X carries a 1-dimensional positive rational Hodge class. We will first discuss and formulate some precise problems (depending on the notion of positivity) then focus on the following results: 1) If the interior of the dual Kähler cone of X contains a rational class, then the Picard variety of X is projective. 2) If the interior of the dual pseudoeffective cone of a compact Kähler threefold X contains a rational class, then X is projective.

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Date/Time : 02 Sep 2020 10:00
Venue : via Zoom
Speaker : Hongtaek Jung , Institute for Basic Science - South Korea

Join Zoom Meeting https://nus-sg.zoom.us/j/94193089627?pwd=SGZ6cUtXNnp6ejFpZUpGWHFkalM5dz09 Meeting ID: 941 9308 9627 Password: 788615 Abstract: It is known that the PSL(n,R)- Hitchin components of closed surfaces of negative Euler characteristic admit global Darboux coordinates with respect to the Goldman symplectic form. I will discuss an analogous result for the “PSL(3,R)-Hitchin components”  or the deformation spaces C(O) of convex projective structures of closed orbifolds O of negative Euler characteristic.  For this, I will present how to construct the Goldman type symplectic form on C(O).  Then I will introduce two ingredients to construct the global Darboux coordinates: a decomposition theorem that splits C(O) into simpler symplectic manifolds and an action-angle principle. This is joint work with Suhyoung Choi.

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Date/Time : 02 Sep 2020 15:00
Venue : via Zoom
Speaker : Sug Woo Shin, University of California, Berkeley

Zoom link: https://nus-sg.zoom.us/j/99468790870?pwd=SHIveFFOalkvdTczNTVteWdJbmMzdz09 Meeting ID: 994 6879 0870 Password: 374233 I will present my joint work with Arno Kret, where we construct a GSpin(2n)-valued ell-adic Galois representation attached to a cuspidal cohomological automorphic representation pi of a suitable quasi-split form of GSO(2n) over a totally real field, under the hypothesis that pi has a Steinberg component at a finite place. This uses input from the cohomology of certain Shimura varieties for GSO(2n); as such we need to take a suitable form of GSO(2n) depending on the parity of n. (We take the split form if and only if n is even.) The relevant preprint is available at https://math.berkeley.edu/~swshin/GSO.pdf.      

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Date/Time : 02 Sep 2020 16:00
Venue : via Zoom
Speaker : Sonoda Sho, RIKEN AIP, Japan

Join Zoom Meeting https://nus-sg.zoom.us/j/99186202215?pwd=ZFp5S21NVGVJclphVU82cm54TlpGQT09 Meeting ID: 991 8620 2215 Password: 026907 Abstract: Neural networks are known to be a black box due to the high-dimensionality, non-linearity, and non-convexity of parameters. In the so-called `integral representation’, at the cost of lifting up the parameters to infinite-dimensional function space, the parameter becomes linear and convex. The presenter aims to turn the black box into a white box by using integral representation. In this talk, he describes the integral representation first, then introduces recent application results for analyzing the parameters acquired through several learning algorithms.

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Date/Time : 28 Aug 2020 09:00
Venue : via Zoom
Speaker : Ying Cui , University of Minnesota

Join Zoom Meeting Meeting ID: 972 8455 8347 Password: 733806 Many modern statistical estimation problems are defined by  three major components: a statistical model that postulates the dependence of an output variable on the input features; a loss function measuring the error between the observed output and the model predicted output; and a regularizer that controls the overfitting and/or variable selection in the model.  We study the sampled version of this generic statistical estimation problem where the model parameters are estimated by empirical risk minimization, which involves the minimization of the empirical average of the loss function at the data points weighted by the model regularizer.  In our setup we allow all three component functions to be of the difference-of-convex type and illustrate them with a host of commonly used examples, including those in continuous piecewise affine regression and in deep learning with piecewise affine activation functions.  We describe a non-monotone majorization-minimization (MM) algorithm for solving the unified nonconvex, nondifferentiable optimization problem which is formulated as a specially structured composite dc program of the pointwise max type.  Numerical results are presented to demonstrate the effectiveness of the proposed algorithm and the superiority of continuous piecewise affine regression over the standard linear model. 

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Date/Time : 26 Aug 2020 15:00
Venue : via Zoom
Speaker : Lu Ming, Sichuan University

Join Zoom Meeting https://nus-sg.zoom.us/j/99335000776?pwd=UFVJNnN1ZTJMK1MzQlpFWnVSckwrQT09 Meeting ID: 993 3500 0776 Password: 359589 Abstract: A quantum symmetric pair consists of a quantum group and its coideal subalgebra (called an i-quantum group). A quantum group can be viewed as an example of i-quantum groups associated to symmetric pairs of diagonal type. In this talk, we present a new Hall algebra construction of i-quantum groups. Our approach leads to monomial bases, PBW bases, and braid group actions for i-quantum groups. In case of symmetric pairs of diagonal type, our work reduces to a reformulation of Bridgeland’s Hall algebra realization of a quantum group, which in turn was a generalization of earlier constructions of Ringel and Lusztig for half a quantum group. This is joint work with Weiqiang Wang.

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Date/Time : 26 Aug 2020 10:00
Venue : via Zoom
Speaker : Li Qiongling , Chern Institute of Mathematics, Nankai University

Join Zoom Meeting https://nus-sg.zoom.us/j/91329086826?pwd=RFNXQzROaDhicTdKQkxkeWxpL3lhUT09 Meeting ID: 913 2908 6826 Password: 648142 We start this talk with some basic linear algebras. Consider m(A)=[A,A*] on sl(n,C). Ness studied the (normalized) square of norm |m|^2 and showed some nice properties of |m|^2 which characterize the unitary orbits inside the nilpotent orbits of sl(n,C). We generalize Ness’ results to general orbits by modifying |m|^2, motivated by the curvature formula of symmetric spaces. As geometric applications, we show some domination results in n-Fuchsian fibers in the Hitchin fibration of the moduli space of Higgs bundles over a Riemann surface. And we also show some existence and uniqueness results of equivariant minimal/maximal surfaces in certain product spaces. This is joint work with Song Dai.    

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Date/Time : 26 Aug 2020 16:00
Venue : via Zoom
Speaker : Tong Xin, NUS

Join Zoom Meeting https://nus-sg.zoom.us/j/92076578514?pwd=aWk4Nm1Cc3htS1hGWDVpTlZ2cUgrdz09 Meeting ID: 920 7657 8514 Password: 459566 Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minima. On the other hand, Langevin dynamics (LD) can explore the state space and find global minima, but in order to give accurate estimates, LD needs to run with a small discretization step size and weak stochastic force, which in general slows down its convergence. This paper shows that these two algorithms can “collaborate” through a simple exchange mechanism, in which they swap their current positions if LD yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. We further verify our theoretical results through some numerical experiments, and observe superior performance of the proposed algorithm over running GD or LD alone. We will further explain how does replica exchange method improve LD’s spectral gap using Poincare type of inequality.

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