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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Date/Time : 17 Dec 2020 09:00
Venue : via Zoom
Speaker : Yau Wing Li, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/87836014196?pwd=MGdrTmVWN1NuWHFCbFg3YTBjMEZsUT09 Meeting ID: 878 3601 4196 Password: 339627 Abstract: Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H. It is consistent with the Langlands functoriality conjecture.

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Date/Time : 17 Dec 2020 15:00
Venue : via Zoom
Speaker : Jue Liu , Tsinghua University

Join Zoom Meeting https://nus-sg.zoom.us/j/83066226180?pwd=NllOVE1TNXZxTXhqVGFyYVRxMlQ1UT09 Meeting ID: 830 6622 6180 Password: 886072 One of the fundamental questions in mathematical relativity is the weak cosmic censorship conjecture proposed by Penrose, which states that for generic asymptotically flat spacetime the singularities are invisible to the infinity observers. This is a global problem involved the whole future evolution of the asymptotically flat initial data. A suitable local version is the trapped surfaces conjecture proposed by Christodoulou, which states that given a first singularity, for generic initial data defined on an arbitrary domain containing the intersection of the past of the singularity and the initial hypersurface, the maximal future development must contain a trapped surface. In this talk, I will review the previous works on the formation of trapped surfaces and introduce the recent work by Junbin Li and me about the trapped surfaces conjecture when the first singularity is a spherical naked singularity constructed by Christodoulou.

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Date/Time : 16 Dec 2020 22:00
Venue : via Zoom
Speaker : Pengyu Le, University of Michigan

Join Zoom Meeting https://nus-sg.zoom.us/j/87148806876?pwd=VXlaNjlIaGNpK0c1SnY4YTQ1WUs1QT09 Meeting ID: 871 4880 6876 Password: 380377 Abstract: The Penrose inequality is a renowned inequality in GR proposed by Penrose in 1973. It conjectures a simple relation between the total mass of the spacetime and the mass of a black hole inside it. The original physical reasoning supporting this inequality by Penrose relates it to some most important and difficult problems in GR, i.e. the final state of the evolution of black holes and naked singularities (the weak cosmic censorship conjecture). A major breakthrough was achieved independently by Huisken, Ilmanen and by Bray, proving the Riemannian Penrose inequality. The Riemannian Penrose inequality is of course not the end of the story. There is another important case of the inequality on null hypersurfaces which is still open. The speaker will present his work confirming the null Penrose inequality on null hypersurfaces in a perturbed Schwarzschild spacetime. A major ingredient of the work is the study of perturbations of null hypersurfaces.

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Date/Time : 11 Dec 2020 14:00
Venue : via Zoom
Speaker : Zhang Lei, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82676230389?pwd=UmJtUTN2dzhycTJ2UTFrWHpZVXpTZz09 Meeting ID: 826 7623 0389 Password: 654751 Abstract: Fourier coefficients of automorphic representations play a fundamental role in number theory and representation theory, the theory of automorphic forms in particular. For instance, they are used to count the number of representations of a natural number as the sum of four squares and now they play a crucial role in our approach to the celebrated Gan-Gross-Prasad Conjecture. One of my main goals is to justify the Local-Global principle of those Fourier coefficients. Motivated by this, I investigated various cases and connections with other topics in the past several years. I will report my results and my future research plan in this talk.

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Date/Time : 09 Dec 2020 15:00
Venue : via Zoom
Speaker : Tsviqa Lakrec , The Hebrew University of Jerusalem, Israel

Join Zoom Meeting https://nus-sg.zoom.us/j/81609880468?pwd=Wk5Dak1QK0ZpL1lkWVo5S0VOVVB5dz09 Meeting ID: 816 0988 0468 Password: 692705 Abstract: I will discuss a quantitative equidistribution result for the random walk on a torus arising from the action of the group of affine transformations. This is a joint work with Weikun He and Elon Lindenstrauss. Under the assumption that the Zariski closure of the group generated by the linear part acts strongly irreducibly on R^d, we give quantitative estimates (depending only on the linear part of the random walk) for how fast this random walk equidistributes, unless the initial point and the translation part of the affine transformations can be perturbed so that the random walk is trapped in a finite orbit of small cardinality. In particular, we prove that the random walk equidistributes in law to the Haar measure if and only if the random walk is not trapped in a finite orbit. This extends the results of Bourgain-Furman-Lindenstrauss-Mozes and He-de Saxcé on quantitative equidistribution of the random walk arising from the action of the automorphism group of the torus. I will also mention a new joint work with He and Lindenstrauss that extends these results to nilmanifolds.

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