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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Date/Time : 30 Jul 2021 09:00
Venue : via Zoom
Speaker : Hao Shen, University of Wisconsin – Madison

Join Zoom Meeting https://nus-sg.zoom.us/j/86176910318?pwd=NnQ5SFcxTk9BRFc1TklwSFRhYS9nZz09 Meeting ID: 861 7691 0318 Passcode: 016860 Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a “solution”. The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.

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Date/Time : 23 Jul 2021 09:00
Venue : via Zoom
Speaker : Jingrui Cheng, State University of New York at Stony Brook

Join Zoom Meeting https://nus-sg.zoom.us/j/86843946871?pwd=eXk1ZmF6OElwY2s3MjdzM1piV2VQQT09 Meeting ID: 868 4394 6871 Passcode: 398037 I will explain the a priori estimates for the cscK equation on a compact manifold, and how to use these estimates to obtain existence when the associated energy functional is “coercive”. If time permits, I will also explain how we can hope to get existence from a more “algebraic” condition, which might be easier to check in practice.

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Date/Time : 18 Jun 2021 16:00
Venue : via Zoom
Speaker : Yash Lodha, Korea Institute for Advanced Study

Join Zoom Meeting https://nus-sg.zoom.us/j/84001105600?pwd=OUc1ZGtOb21KSGtaRHhjN3VWMHJsdz09 Meeting ID: 840 0110 5600 Passcode: 346800 The classification of finite simple groups is one of the most incredible achievements of 20th century mathematics. So what do we know about infinite, finitely generated (or even finitely presented), simple groups? Starting from a beautiful example of Higman from 1951: the first example of a finitely generated infinite simple group, to the striking examples of Thompson, we shall survey this territory full of beautiful phenomena. We shall also discuss some questions that remain mysterious.

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Date/Time : 21 May 2021 09:00
Venue : via Zoom
Speaker : Yudong Chen, Cornell University

Join Zoom Meeting https://nus-sg.zoom.us/j/82907817525?pwd=bU1tWlNwTXhwbC9sa3R5eDZKOVovQT09 Meeting ID: 829 0781 7525 Passcode: 752172 We consider the problems of learning discrete structures from network data under statistical settings. Popular examples include various planted and block models, Z2 synchronization and mixture models. Semidefinite programming (SDP) relaxation has emerged as a versatile and robust approach to these problems. We show that despite being a relaxation, SDP achieves the optimal Bayes error rate in terms of distance to the target solution. Moreover, SDP relaxation is provably robust under the so-called semirandom model, which frustrates many existing algorithms. Our proof involves a novel primal-dual analysis that establishes what we call the Hidden Integrality property: the SDP relaxation tightly approximates the optimal (yet unimplementable) integer program with oracle information. Joint work with Yingjie Fei (Cornell Ph.D.), who won 2nd place in INFORMS Nicholson Student Paper Competition.

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