Date/Time : 22 Jan 2021 09:00
Venue : via Zoom
Speaker : Amol Aggarwal, Columbia University

Join Zoom Meeting Meeting ID: 896 0780 2118 Password: 815869 Abstract: Lévy matrices are symmetric matrices whose entries are random variables with infinite variance; they are governed by a parameter \alpha \in (0, 2) dictating the power law decay of their entries. For \alpha < 1, they are believed to serve as one of the few examples of a matrix model exhibiting a mobility edge, also called an Anderson transition, that separates chaotic (GOE) eigenvalue spacing statistics from ordered (Poisson) ones. In this talk we describe results concerning the statistics for the eigenvalue spacings and eigenvector entries of Lévy matrices. In particular, for \alpha \in (1, 2) their eigenvalue statistics asymptotically follow those of the GOE throughout the spectrum, and for \alpha < 1 the same statement holds around small eigenvalues. These describe joint works with Patrick Lopatto, Jake Marcinek, and Horng-Tzer Yau.

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Date/Time : 21 Jan 2021 09:00
Venue : via Zoom
Speaker : Chao Li, Columbia University

Join Zoom Meeting https://nus-sg.zoom.us/j/84339085667?pwd=ZDZvZkZPMEJiSW1HcDNNNnVIbHlSQT09 Meeting ID: 843 3908 5667 Password: 816448 Abstract: For certain automorphic representations π on unitary groups, we show that if L(s, π) vanishes to order one at the center s=1/2, then the associated π-localized Chow group of a unitary Shimura variety is nontrivial. This proves part of the Beilinson-Bloch conjecture for unitary Shimura varieties, which generalizes the BSD conjecture. Assuming the modularity of Kudla’s generating series of special cycles, we further prove a precise height formula for L'(1/2, π). This proves the conjectural arithmetic inner product formula, which generalizes the Gross-Zagier formula to Shimura varieties of higher dimension. We will motivate these conjectures and discuss some aspects of the proof. This is joint work with Yifeng Liu.

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Date/Time : 20 Jan 2021 16:00
Venue : via Zoom
Speaker : Marko Weber, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82724882064?pwd=RFdxUWF4REdhblJKWi9vazJzUW9TQT09&from=addon Meeting ID: 827 2488 2064 Password: 167307 Abstract: We explicitly solve an equilibrium model in which several long-lived agents with heterogenous absolute risk-aversion and discount rates make consumption and investment decisions, trading an asset that pays a dividend whose growth rate randomly fluctuates over time. We analyze equilibrium consumption shares, stock price, risk-free rate, and optimal trading strategies. We introduce a novel technique to study the market equilibrium in an incomplete model.

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Date/Time : 20 Jan 2021 16:00
Venue : via Zoom
Speaker : Frédéric Naud, Institut de Mathématiques de Jussieu - Paris 6

Join Zoom Meeting https://nus-sg.zoom.us/j/85799518051?pwd=N0JDU2pGbXAwcThKZVE3UEh5MmJXZz09 Meeting ID: 857 9951 8051 Password: 336911 We will define a notion of random covers of hyperbolic surfaces (compact or non- compact) through random representations of their fundamental groups. We will then show how some tools from group combinatorics can be used to study, in a probabilistic sense, the “low spectrum” of these surfaces in the large euler characteristic limit. Joint work with Michael Magee and Doron Puder.

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Date/Time : 18 Jan 2021 09:00
Venue : via Zoom
Speaker : Sourav Sarkar, University of Toronto

Join Zoom Meeting https://nus-sg.zoom.us/j/85233351827?pwd=U2dESEhORVN4eHJwZlhNZEt0MEE3Zz09 Meeting ID: 852 3335 1827 Password: 149940 Abstract: Universality in disordered systems has always played a central role in the direction of research in Probability and Mathematical Physics, a classical example being the Gaussian universality class (the central limit theorem). In this talk, I will describe a different universality class for random growth models, called the KPZ universality class. Since Kardar, Parisi and Zhang introduced the KPZ equation in their seminal paper in 1986, the equation has made appearances everywhere from bacterial growth, fire front, coffee stain to the top edge of a randomized game of Tetris; and this field has become a subject of intense research interest in Mathematics and Physics for the last 15 to 20 years. The random growth processes that are expected to have the same scaling and asymptotic fluctuations as the KPZ equation and converge to the universal limiting object called the KPZ fixed point, are said to lie in the KPZ universality class, though this KPZ universality conjecture has been rigorously proved for only a handful of models till now. Here, I will talk about some universal geometric properties of the KPZ fixed point and show that the KPZ equation and exclusion processes converge to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ universality conjecture for these models, which were long-standing open problems in this field.

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Date/Time : 13 Jan 2021 09:00
Venue : via Zoom
Speaker : Lars Ruthotto, Emory university

Join Zoom Meeting https://nus-sg.zoom.us/j/83743056445?pwd=QTFIeE9RZVh1M3FvNDhNNENlcFh0Zz09 Meeting ID: 837 4305 6445 Password: 384731 Abstract: In recent years, OT and ML have become increasingly intertwined. This talk presents new avenues for solving high-dimensional optimal transport (OT) problems using machine learning (ML). The first part of the talk shows how neural networks can be used to efficiently approximate the optimal transport map between two densities in high dimensions. To avoid the curse-of-dimensionality, we combine Lagrangian and Eulerian viewpoints and employ neural networks to solve the underlying Hamilton-Jacobi-Bellman equation. Our approach avoids any space discretization and can be implemented in existing machine learning frameworks. We present numerical results for OT in up to 100 dimensions and validate our solver in a two-dimensional setting. The second part of the talk shows how optimal transport theory can improve the efficiency of training generative models and density estimators, which are critical in machine learning. We consider continuous normalizing flows (CNF) that have emerged as one of the most promising approaches for variational inference in the ML community. Our numerical implementation is a discretize-optimize method whose forward problem relies on manually derived gradients and Laplacian of the neural network and uses automatic differentiation in the optimization. In common benchmark challenges, our method outperforms state-of-the-art CNF approaches by 1-2 orders of magnitude during training and inference.

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Date/Time : 08 Jan 2021 09:00
Venue : via Zoom
Speaker : Wei Zhang , MIT

Join Zoom Meeting Meeting ID: 871 9433 1381 Password: 878103 Abstract: This will be an introductory talk to conjectures on algebraic cycles on varieties over number fields and their relation to L-functions, viewed as a version of the “local-to-global principle”. We will focus on “modular varieties” whose L-functions are known to be related to automorphic representations. They often admit a natural collection of algebraic cycles (e.g., arising from the theory of quadratic forms and hermitian forms over various integer rings). We will indicate a few applications to arithmetic and geometric questions on elliptic curves and K3 surfaces parameterized by modular functions on suitable classical hermitian symmetric domains.

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Date/Time : 06 Jan 2021 10:00
Venue : via Zoom
Speaker : Jingjun Han, Johns Hopkins University

Join Zoom Meeting https://nus-sg.zoom.us/j/83092658434?pwd=dDlHclhrMmNvQjZkUFFVeEpOdEhDUT09 Meeting ID: 830 9265 8434 Password: 664940 Abstract: Kawamata log terminal (klt) singularities form an important class of singularities due to its fundamental roles in MMP, Kähler-Einstein geometry, and K-stability. Recently, Chi Li introduced a new invariant called the local volume of a klt singularity which encodes lots of interesting geometric and topological information. A folklore conjecture predicts that local volumes satisfy the ascending chain condition (ACC) when the coefficients of the boundary divisors belong to a DCC set. In this talk, we will show the ACC conjecture for local volumes holds when the ambient variety is fixed. We will also explore the relation between log canonical thresholds, local volumes, and delta-plt blow-ups. This is based on ongoing joint work with Yuchen Liu and Lu Qi.

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Date/Time : 23 Dec 2020 22:00
Venue : via Zoom
Speaker : Martin Lesourd, Harvard University

Join Zoom Meeting https://nus-sg.zoom.us/j/82136576826?pwd=TFVpYjhOMGZRMVErdzNSZVI1aVVZUT09 Meeting ID: 821 3657 6826 Password: 706367 Abstract: Christodoulou’s 2008 breakthrough on the formation of trapped surfaces in general relativity has engendered a great deal of activity and has brought the fundamental conjecture of Weak Cosmic Censorship into closer view. This conjecture amounts to a global existence problem for the hyperbolic PDEs of the Einstein vacuum system, and there are currently no non-trivial examples of it. I’ll describe some of the landscape that has arisen since Christodoulou’s seminal result, as well as recent work with Nikos Athanasiou in which we construct examples of the conjecture that are conditional on the validity of another problem in general relativity: the stability of the Kerr black hole solution.

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