Date/Time : 20 Oct 2021 09:00
Venue : via Zoom
Speaker : Siqi He, Stony Brook University

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https://nus-sg.zoom.us/j/88336974987?pwd=SVVRVStaVTJ6STVEL1JGRSsxQjVFdz09

Meeting number：883 3697 4987
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Abstract: Special Lagrangian submanifolds are a distinguished class of real calibrated submanifolds defined in a Calabi-Yau manifold, which are calibrated by the real part of the holomorphic volume form. Given a compact, smooth special Lagrangian submanifold, Mclean proved that the space of nearby special Lagrangian submanifolds of it could be parametrized by the harmonic 1-forms. In this talk, we will discuss some recent progress on generalizing Mclean’s result to the branched deformations. We will describe how to use multi-valued harmonic functions to construct branched nearby deformations.

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Date/Time : 26 May 2021 15:00
Venue : via Zoom
Speaker : Wen Deng, Chinese Academy of Sciences

Join Zoom Meeting https://nus-sg.zoom.us/j/83402872988?pwd=UWxhOEFKam9oYnk1N1FqT2JCdnFzdz09 Meeting ID: 834 0287 2988 Password: 471879 We consider the stability problem of the Couette flow for the 2D Boussinesq system. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non self-adjoint operator in the perturbation equation that makes the stability possible. In this work, we prove the nonlinear stability of the Couette flow for 2D Boussinesq system with full dissipation and with only vertical dissipation in Sobolev spaces. This is a joint work with J. Wu and P. Zhang.

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Date/Time : 19 May 2021 15:00
Venue : via Zoom
Speaker : Yanlin Liu, Beijing Normal University

Join Zoom Meeting https://nus-sg.zoom.us/j/86563869492?pwd=MlRhL0k5VDhRbVFXTmFUVCtWZ0hsZz09 Meeting ID: 865 6386 9492 Passcode: 488917 We show that any unique global solution (here we do not require any smallness condition beforehand) to 3-D axisymmetric Navier-Stokes equations in some scaling invariant spaces must eventually become a small solution. In particular, we show that the limits of $\|\omega^\theta(t)/r\|_{L^1}$ and $\|u^\theta(t)/\sqrt r\|_{L^2}$ are all $0$ as $t$ tends to infinity. And by using this, we can refine some decay estimates for the axisymmetric solutions.

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Date/Time : 28 Apr 2021 15:00
Venue : via Zoom
Speaker : Chao Liu, Huazhong University of Science and Technology

Join Zoom Meeting https://nus-sg.zoom.us/j/81653239587?pwd=U2lCTS9qV01IejN3VDJ2eDB4dml2UT09 Meeting ID: 816 5323 9587 Password: 918779 Motivated by the astrophysical problems of star formations from molecular clouds, we make the first step on the possible behaviors of certain molecular clouds. This talk (1) establishes the diffuse boundary problem of Euler-Poisson system for describing the evolution of molecular clouds; (2) proves the local existence, uniqueness and continuation principle of the classical solution to the diffuse boundary problem; (3) proves the classical solution (without any symmetry condition) to the diffuse problem blows up at finite time if there is no the first class of global solution and the data is admissible (large scale, irregularly-shaped, expanding and rotational molecular clouds); (4) proves certain singularities can be removed from the boundary if the data is strongly admissible. This result partially answers Makino’s conjecture (1992) on the finite blowup of any tame solution without symmetries and gives the possibilities of star formations, fragmentation and possibilities of formations of shocks and physical vacuum boundary in perfect fluids with Newtonian self-gravity. This talk is based on the article arXiv:2102.11550.

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Date/Time : 27 Apr 2021 15:00
Venue : via Zoom
Speaker : Jinxin Xue, Tsinghua University

Join Zoom Meeting https://nus-sg.zoom.us/j/86384771548?pwd=RnMzcXJrcUtERmFrbHVXMUFrejVsZz09 Meeting ID: 863 8477 1548 Password: 244323 In this talk, we study the dynamics of geodesic flows in perturbed blackhole backgrounds using the techniques of Hamiltonian dynamics. The central object in our work is the blackhole shadow. It gives a normally hyperbolic invariant submanifold in the phase space and is robust under perturbations. We find a gap structure within the blackhole shadow and also use it to construct long chaotic orbit with large oscillation of angular momentum. Moreover, similar methodology applies to the event horizon of Kerr to give chaotic orbit skirting around the event horizon.

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Date/Time : 18 Mar 2021 14:30
Venue : via Zoom
Speaker : Hao Liu, University of Hong Kong

Join Zoom Meeting https://nus-sg.zoom.us/j/81674259454?pwd=MmtaT2ZWUWlCeDQ1UE5oWXByeUpxZz09&from=addon Meeting ID: 816 7425 9454 Password: 778322 Abstract:  In this talk, we are going to talk about the global weak solutions to the gmCH equations which are constructed through peakons. We use a double mollification to the ODE system for trajectories of N-peakon weak solutions which has the crucial weak consistency property. With the global N-peakon weak solutions, we can use the mean-field approximation method to obtain global weak solutions for general initial data.

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Date/Time : 17 Mar 2021 16:30
Venue : via Zoom
Speaker : Te Li, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/83795761997?pwd=ZFdUTXBIVGlHMHZhZlpjRmlFYnRDQT09 Meeting ID: 837 9576 1997 Password: 483694 Abstract:  We study the incompressible Navier-Stokes equations in dimensions d = 2 or 3 and address the hydrodynamic stability problem at high Reynolds. Our talk will mainly focus on the stability of laminar equilibria, around which the fluid moves in well-ordered layers. In joint works with Dongyi Wei and Zhifei Zhang, we established the (nonlinear) stability of Oseen vortices, Kolmogorov flow and Couette flow. In particular, our work on Oseen vortices confirmed Gallay’s conjecture. Pseudospectral bound and wave-operator method play crucial roles in our proofs.

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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 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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