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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Date/Time : 30 Mar 2022 09:00
Venue : via Zoom
Speaker : Wei Kuo Chen, University of Minnesota

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https://nus-sg.zoom.us/j/81402485618?pwd=WlBSN2NhM2xXaGlTc1NhTFBqRjRIUT09

Meeting ID: 814 0248 5618
Passcode: 266695

Abstract: The Grothendieck L_p problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit L_p ball. The p=2 case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while p=\infty this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general p and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.

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Date/Time : 30 Mar 2022 16:00
Venue : via Zoom
Speaker : Wu Liuzhen,

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract: The continuum function is a key and long-studied object inside set theory. We will survey the study on the behavior of continuum function in presence of strongly compact cardinals. We will also introduce some major research problems in this area. Finally, We discuss our recent work on forcing continuum function of some special pattern.

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Date/Time : 30 Mar 2022 16:00
Venue : via Zoom
Speaker : Zexing Li, University of Cambridge

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https://nus-sg.zoom.us/j/86113239460?pwd=anRLYzEvaGFQVGVLNVF3dGVRM1pOUT09

Meeting ID: 861 1323 9460
Passcode: 220330

Abstract: The Chern-Simons-Schr\”odinger equation is a nonrelativistic quantum model describing the dynamics of a large number of charged particles in the plane interacting both directly and via a self-generated electromagnetic field. Mathematically, it is a gauged $L^2$-critical Schr\”odinger equation has soliton behavior. Considering the equivariant and self-dual case, Liu-Smith has proved below threshold charge wellposedness via concentration compactness method. In this talk, I will present our result on characterizing dynamics at the threshold charge. This is a joint work with Baoping Liu.

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Date/Time : 28 Mar 2022 10:00
Venue : via Zoom
Speaker : Guanglian Li, University of Hong Kong

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https://nus-sg.zoom.us/j/87674182782?pwd=dEtkOWxldGl4RFVFNkJpV2syalJhUT09

Meeting ID: 876 7418 2782
Passcode: 684943

Abstract: I will present in this talk the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm recently proposed in [Li and Hu, J. Comput. Phys., 2021] to efficiently solve subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the advantages of multiscale methods that can effectively deal with heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Compared with the previous work for parabolic problem, the main challenge in both the analysis and simulation arises from the nonlocality of the fractional derivative. To conquer this obstacle, an auxiliary problem is constructed on each coarse temporal subdomain to uncouple the temporal variable completely. In this manner, the approximation properties of the correction operator is proved. In addition, a new summation of exponential sums is derived to generate single-step time stepping scheme, with the number of terms of O(|log τ_f|) independent of final time. Here, τ_f is the fine-scale time step size. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size and the fine-scale time step size. Several numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly.

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Date/Time : 25 Mar 2022 09:00
Venue : via Zoom
Speaker : Chao Li, Columbia University

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https://nus-sg.zoom.us/j/87647484260?pwd=Zys0M211ekFkeE9OZlRCOHcvS0NLZz09

Meeting ID: 876 4748 4260
Passcode: 511096

Abstract: Can an integer n be represented as a sum of two squares n=x^2+y^2? If so, how many different representations are there? We begin with the answers to these classical questions due to Fermat and Jacobi and put it in the modern perspective of the Siegel-Weil formula. After illustrating Kudla’s influential program on geometric and arithmetic generalizations using the example of modular curves, we report on recent theorems establishing arithmetic Siegel-Weil formulas and discuss applications including BSD-type conjectures for higher dimensional varieties.

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Date/Time : 24 Mar 2022 15:00
Venue : via Zoom
Speaker : Patrick Rebeschini, Oxford University

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https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

The local Rademacher complexity framework is one of the most successful general-purpose toolboxes for establishing sharp excess risk bounds for statistical estimators based on the framework of empirical risk minimization. Applying this toolbox typically requires using the Bernstein condition, which often restricts applicability to convex and proper settings. Recent years have witnessed several examples of problems where optimal statistical performance is only achievable via non-convex and improper estimators originating from aggregation theory, including the fundamental problem of model selection. These examples are currently outside of the reach of the classical localization theory.

In this work, we build upon the recent approach to localization via offset Rademacher complexities, for which a general high-probability theory has yet to be established. Our main result is an exponential-tail excess risk bound expressed in terms of the offset Rademacher complexity that yields results at least as sharp as those obtainable via the classical theory. However, our bound applies under an estimator-dependent geometric condition (the “offset condition”) instead of the estimator-independent (but, in general, distribution-dependent) Bernstein condition on which the classical theory relies. Our results apply to improper prediction regimes not directly covered by the classical theory. (Based on joint work with V. Kanade and T. Vaškevičius)

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Date/Time : 23 Mar 2022 16:00
Venue : via Zoom
Speaker : Wu Guohua, Nanyang Technological University

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract:  In this talk, I will review some existing work on splittings of c.e. sets, and then present an ongoing paper on nonsplitting, a joint work with Downey. The main result is the following.

Theorem: There are c.e. sets A and B such that B is strictly Turing reducible to A and for any c.e. sets U, V, if U and V form a set-splitting of A, then one of them is Turing reducible to B.

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Date/Time : 22 Mar 2022 16:00
Venue : via Zoom
Speaker : Takahiro Shibata, National University of Singapore

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https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: In this talk, we deal with finite quotients of abelian varieties (fqav for short), that is, the quotients of abelian varieties by finite group actions. We mainly focus on the positivity of the canonical divisor of such a variety. Those varieties with Q-linearly trivial (resp. anti-ample) canonical divisors are called Q-abelian varieties (resp. Q-Fano fqav). We gives characterizations of those varieties and see that they are building blocks of general fqav’s.

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