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 : 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 : 25 Feb 2022 09:00
Venue : via Zoom
Speaker : Daniel Russo, Columbia University

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https://nus-sg.zoom.us/j/86446962104?pwd=cFFTQzhtOU9OTFd5RUZhOHBPSjcwQT09

Meeting ID: 864 4696 2104
Passcode: 673331

Abstract: Multi-armed bandit algorithms minimize experimentation costs required to converge on optimal behavior. They do so by rapidly adapting experimentation effort away from poorly performing actions as feedback is observed. But this desirable feature makes them sensitive to confounding, which is the primary concern underlying classical randomized controlled trials. We highlight, for instance, that popular bandit algorithms cannot address the problem of identifying the best action when day-of-week effects may confound inferences. In response, this paper proposes deconfounded Thompson sampling, which makes simple, but critical, modifications to the way Thompson sampling is usually applied. Theoretical guarantees suggest the algorithm strikes a delicate balance between adaptivity and robustness to confounding. It attains asymptotic lower bounds on the number of samples required to confidently identify the best action — suggesting optimal adaptivity — but also satisfies strong performance guarantees in the presence of day-of-week effects and delayed observations — suggesting unusual robustness.

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Date/Time : 28 Jan 2022 16:00
Venue : via Zoom
Speaker : Riddhipratim Basu, ICTS-TIFR

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https://nus-sg.zoom.us/j/81804554884?pwd=MEx6eVptVmxYZjNPMjZXZE1IaWxUQT09

Meeting ID: 818 0455 4884
Passcode: 743763

Abstract: First and last passage percolation on the two dimensional Euclidean lattice are canonical models of stochastic planar growth that are generically believed to exhibit universal behaviour in the Kardar-Parisi-Zhang (KPZ) class. This, however, is rigorously known only for a handful of integrable models, e.g., the exponential LPP on the plane. In this talk, I shall describe some results on understanding of the geometry of the geodesics in these exactly solvable models using estimates from integrable probability and illustrate how this geometric understanding, in turn, can be used to obtain more refined information on the passage time landscape.

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Date/Time : 22 Dec 2021 09:00
Venue : via Zoom
Speaker : Jiaoyang Huang, New York University

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https://nus-sg.zoom.us/j/86201788855?pwd=ZW9WQ0pmM2RTOS9SbSsrakdIS2FRQT09

Meeting ID: 862 0178 8855
Passcode: 627300

Abstract: The success of random matrices in modeling physical systems lies in the universality phenomenon of their eigenvalue statistics. The general belief is for systems with many strongly interacting components, we expect to see random matrix statistics. The universality is well understood for Wigner matrices, which are dense. In this talk, I will first discuss some results concerning random matrix statistics in sparse systems—the sparse random graphs. Beyond matrix setting, random matrix statistics are conjectured to govern the asymptotic behavior of various random growth models and interacting particle systems. However, this was only proven for some exactly solvable models. I will discuss a general strategy to prove the universality of random matrix statistics, and our recent result proving that for random lozenge tilings of polygons, the scaling limit of the extreme path is given by random matrix statistics.

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Date/Time : 26 Nov 2021 10:00
Venue : via Zoom
Speaker : Wenyu Pan, University of Chicago

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https://nus-sg.zoom.us/j/82434099670?pwd=VlhicUs1SlZPd0IrTzZ5d0NrMm1oQT09

Meeting ID: 824 3409 9670
Passcode: 772471

Abstract: Let  Hⁿ be the hyperbolic n-space and Γ be a geometrically finite discrete subgroup in Isom + (Hⁿ) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T¹ (Γ\Hⁿ) (resp. the frame bundle F(Γ\Hⁿ)) mixes exponentially. This result has many applications, including spectral theory, orbit counting, equidistribution, prime geodesic theorems, etc. 

I will start with a survey of the past results and related problems on this topic. Then I will present the joint work with Jialun Li in which we show that the geodesic flow mixes exponentially. I will describe some ingredients in the proof. If there is time, I will also discuss the exponential mixing of the frame flow.

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Date/Time : 29 Oct 2021 09:00
Venue : via Zoom
Speaker : Luca Spolaor, University of California San Diego

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https://nus-sg.zoom.us/j/86945333380?pwd=Q0VEM2drc0YrKzFUc3VHbFVqNjRCQT09

Meeting ID: 869 4533 3380
Passcode: 583463

Abstract:  In this talk I will discuss some results about isolated singularities of minimal surfaces arising from minimization and Min-Max. In particular, I will focus on the local description of minimal surfaces at such singularities and on their generic behavior with respect to a change in the ambient metric.

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Date/Time : 24 Sep 2021 09:00
Venue : via Zoom
Speaker : Qianxiao Li, National University of Singapore

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https://nus-sg.zoom.us/j/84334390977?pwd=SXVUWm50R1RNSlY1NnplTXJJQ20xdz09

Meeting ID: 843 3439 0977
Passcode: 186685

Abstract:  In this talk, we discuss some recent work on the connections between machine learning and dynamical systems. These come broadly in three categories, namely machine learning via, for and of dynamical systems. In the first direction, we introduce a dynamical approach to deep learning theory with particular emphasis on its connections with approximation and control theory. In the second direction, we discuss the approximation and optimization theory of learning input-output temporal relationships using recurrent neural networks and variants, with the goal of highlighting key new phenomena that arise in learning in dynamic settings. In the last direction, we discuss some principled methods that learns stable and interpretable dynamical model from data arising in scientific applications. We conclude with some discussion on future directions in this rapidly developing field.

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Date/Time : 27 Aug 2021 09:00
Venue : via Zoom
Speaker : Yufei Zhao, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/87977920058?pwd=R1FxeXJHeGVGeHQ5RGhYR0xwOXVsZz09 Meeting ID: 879 7792 0058 Passcode: 458785 Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity. My talk will discuss these problems and their connections, along with extensions and open problems (e.g., what is the maximum possible second eigenvalue multiplicity of a connected bounded degree graph?). Joint work with Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang.

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