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 : 27 May 2022 09:00
Venue : via Zoom
Speaker : Alan Chang, Princeton University

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https://nus-sg.zoom.us/j/85408531758?pwd=VDVXRXVFQ09RNjQ3QWFNeC9wUDlLdz09

Meeting ID: 854 0853 1758
Passcode: 633851

Abstract: The Kakeya needle problem asks the following: What is the smallest area in the plane in which a unit line segment can be rotated 180 degrees? This question, which can be solved with elementary geometry, has surprising connections to many other topics (e.g., harmonic analysis, PDEs, analytic number theory, theoretical computer science). In this talk, we will give a survey of this classical problem, and then we will discuss a variant where the line segment has been replaced by a C^1 curve (or more generally, a 1-rectifiable set). This is joint work with Marianna Csörnyei.

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Date/Time : 13 May 2022 16:00
Venue : via Zoom
Speaker : Jan S Hesthaven, EPFL

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https://nus-sg.zoom.us/j/82587724360?pwd=dGJ4RTZVb25rK1dGRFI5Y0FyTUJiUT09

Meeting ID: 825 8772 4360
Passcode: 015672

Abstract: The development of reduced order models for complex applications promises rapid and accurate evaluation of the output of complex models under parameterized variation with applications to problems which require many evaluations, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed. However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large scale applications, in particular for nonlinear and/or time-dependent problems.

After a brief introduction to reduced order models, we discuss the development of methods which seek to conserve chosen invariants for nonlinear time-dependent problems. We develop structure-preserving reduced basis methods for a broad class of Hamiltonian dynamical systems, including canonical problems and port-Hamiltonian problems, before considering the more complex situation of Hamiltonian problems endowed with a general Poisson manifold structure which encodes the physical properties, symmetries and conservation laws of the dynamics.

Time permitting we also discuss the extension of structure preserving models within a framework for nonlinear reduced order models in which a local basis allows to maintain a small basis even for problems with a slowly decaying Kolmogorov n-width such a transport dominated problems. We shall demonstrate the efficiency of such techniques for nonlinear transport dominated problems, including problems in kinetic plasma physics,

This work is done in collaboration with B, Maboudi (DTU), C. Pagliantini (TU/e), N. Ripamonti (EPFL).

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Date/Time : 13 May 2022 09:00 - 15:00
Venue : via Zoom
Speaker : Caucher Birkar (Tsinghua), Christopher Hacon (Utah), Kenneth Ascher (UC Irvine), Yuji Odaka (Kyoto)

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https://nus-sg.zoom.us/j/89998054586?pwd=aURNRUZQWDBQSEJtNW9lOW5oa1cyQT09

Meeting ID: 899 9805 4586
Passcode: 123123

Abstract: In this lecture, Professor Birk ar (FRS, Fields Medallist 2018) will focus on parametrising algebraic varieties, a very old subject that has seen significant progress in recent years. He will pay particular attention to questions regarding boundedness and moduli of certain classes of varieties. In particular he will discuss Fano and Calabi-Yau varieties as well as minimal varieties of arbitrary Kodaira dimension.

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Date/Time : 12 May 2022 16:00
Venue : via Zoom
Speaker : Vlad-Dumitru Margarint, NYU Shanghai

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https://nus-sg.zoom.us/j/81881463944?pwd=M2Y0OVpiSUsxRGdnWG9DWm9CY0o1UT09

Meeting ID: 818 8146 3944
Passcode: 177270

Abstract: In this talk I will introduce Schramm-Loewner Evolutions (SLE) theory and I will discuss some PDEs that appear naturally in this study. I will spend most of the time on a specific PDE that models a fundamental quantity in SLE theory: the density of the law of the SLE tip. I will present a series of recent results on this object based on the analysis of this PDE, obtained jointly with Yizheng Yuan and Oleg Butkovksy, as well as some open problems. I will also describe the progress of the SLE community in the study of a different PDE and describe some open problems as an invitation to the community for further discussions!

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Date/Time : 22 Apr 2022 14:00
Venue : via Zoom
Speaker : Zhou Pan, Sea Limited

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

Meeting ID: 977 560 6531
Passcode: 123321

Abstract:  To train networks, lookahead algorithm updates its fast weights k times via an inner-loop optimizer before updating its slow weights once by using the latest fast weights. Any optimizer, e.g. SGD, can serve as the inner-loop optimizer, and the derived lookahead generally enjoys remarkable test performance improvement over the vanilla optimizer. But theoretical understandings on the test performance improvement of lookahead remain absent yet. To solve this issue, we theoretically justify the advantages of lookahead in terms of the excess risk error which measures the test performance. Specifically, we prove that lookahead using SGD as its inner-loop optimizer can better balance the optimization error and generalization error to achieve a smaller excess risk error than vanilla SGD on (strongly) convex problems and nonconvex problems with Polyak-Łojasiewicz condition which has been observed/proved in neural networks. Moreover, we show the stagewise optimization strategy which decays the learning rate several times during training can also benefit lookahead in improving its optimization and generalization errors on strongly convex problems. Finally, we propose a stagewise locally-regularized lookahead (SLRLA) algorithm which sums up the vanilla objective and a local regularizer to minimize at each stage and provably enjoys optimization and generalization improvement over the conventional (stagewise) lookahead. Experimental results on CIFAR10/100 and ImageNet testify its advantages.

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Date/Time : 22 Apr 2022 09:00
Venue : via Zoom
Speaker : Yuansi Chen, Duke University

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https://nus-sg.zoom.us/j/85706793159?pwd=bTlhVVZJeVE3Z2dDU0w1bDZmUGxXQT09

Meeting ID: 857 0679 3159
Passcode: 533414

Abstract: Kannan, Lovász, and Simonovits (KLS) conjectured in 1995 that the Cheeger isoperimetric coefficient of any log-concave density or any convex body is achieved by half-spaces up to a universal constant factor. This conjecture now plays a central role in the field of convex geometry, unifying or implying older conjectures. In particular, it implies Bourgain’s slicing conjecture (1986) and the thin-shell conjecture (2003). The previous best bound with dimension dependency $d^{1/4}$ was established by Lee and Vempala in 2017, which also matches the best dimension dependency Klartag obtained in 2006 for Bourgain’s slicing conjecture.

In recent work, we improve the Eldan’s stochastic localization proof technique deployed in Lee and Vempala (2007) to prove an almost constant Cheeger isoperimetric coefficient in the KLS conjecture with dimension dependency $d^{o(1)}$.

In this talk, first we briefly survey the origin and the main consequences of these conjectures. Then we present the development and the refinement of Eldan’s stochastic localization scheme. Finally, we explain a few proof details which result in the current best bound of the Cheeger isoperimetric coefficient in the KLS conjecture.

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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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