Date/Time : 20 Nov 2020 23:30
Venue : via Zoom
Speaker : John Anderson, Princeton University

https://princeton.zoom.us/j/93296784920 (Joint online mathematical general relativity and hyperbolic PDE seminar) Abstract: In this talk, I hope to show a global stability result for a nonlinear anisotropic system of wave equations. This is motivated by studying phenomena involving characteristics with multiple sheets. For the proof, I will describe a strategy for controlling the solution based on bilinear energy estimates. Through a duality argument, this will allow us to prove decay in physical space using decay estimates for the homogeneous wave equation as a black box. The final proof will also require us to exploit a certain null condition that is present when the anisotropic system of wave equations satisfies a structural property involving the light cones of the equations.

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Date/Time : 19 Nov 2020 15:00
Venue : via Zoom
Speaker : Haoyang Chen, Fudan University

Join Zoom Meeting https://nus-sg.zoom.us/j/87823796442?pwd=eHBhYmc4Mi9NNlhTczlCdk5neGROQT09 Meeting ID: 87823796442 Password: 132921 Abstract:  In this talk, we generalize a classic result of Lindblad on the scalar quasilinear wave equation and we show that the Cauchy problem for 3D elastic waves, a physical quasilinear wave system with multiple wave-speeds, is ill-posed in $H^3(R^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties arise from the multiple wave-speeds and its associated non-strict hyperbolicity. We design and combine a geometric approach and an algebraic approach to overcome these difficulties. This is based on joint work with Xinliang An and Silu Yin.

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Date/Time : 15 Jan 2020 15:00
Venue : S17 #06-11
Speaker : Georgios Moschidis, UC Berkeley

Date/Time:15 Jan 2020 15:00 Venue: S17 #06-11 Speaker: Georgios Moschidis, UC Berkeley A proof of the instability of AdS spacetime for the Einstein–scalar field system The AdS instability conjecture is a conjecture about the initial-boundary value problem in general relativity. It states that there exist arbitrarily small perturbations to the initial data of AdS spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes after a sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein–scalar field system. The construction of the unstable family of initial data will require working in a low regularity setting, carefully designing a family of initial configurations of localised matter beams and estimating the exchange of energy taking place between interacting beams over a long period of times, as well as estimating the decoherence rate of the beams. Add to calendar: 

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Date/Time : 09 Dec 2019 15:30
Venue : S17 #05-12 SR4
Speaker : Zhao Lifeng, University of Science and Technology of China

Date/Time:09 Dec 2019 15:30 Venue: S17 #05-12 SR4 Speaker: Zhao Lifeng, University of Science and Technology of China Asymptotic behaviours for energy-subcritical damped Klein-Gordon equations In this talk, we will investigate the long-time behaviour of damped Klein-Gordon equation in the energy subcritical case. In fact, we will establish the soliton resolution for this equation which means that the global bounded solution evolves into the super-position of moving solitons. Moreover, we will give some further information about the resolution. The talk is based on joint works with Ze Li, Raphael Cote, Yvan Martel and Xu Yuan. Add to calendar: 

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Date/Time : 25 Oct 2019 15:00
Venue : S17 #05-11 SR5
Speaker : Xiao Zhang, Guangxi University

Date/Time:25 Oct 2019 15:00 Venue: S17 #05-11 SR5 Speaker: Xiao Zhang, Guangxi University Bondi-Sachs Spacetimes and the Bondi Energy Bondi-Sachs spacetimes were originally introduced by Bondi for axi-symmetric spacetimes, and generalized by Sachs to general asymptotically flat spacetimes for the zero cosmological constant. In this framework, Bondi defined the Bondi energy-momentum at null infinity, which represents the rest energy of spacetimes after the loss due to gravitational radiation. In this talk, we shall discuss, for zero cosmological constant, whether the Bondi energy is nonnegative? How the Bondi energy-momentum relates to the ADM energy-momentum defined at spatial infinity? We shall also discuss the natural boundary condition and the peeling property when the cosmological constant is nonzero. The talk is based on some joint works with W-l. Huang, Yau and F. Xie. Add to calendar: 

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Date/Time : 02 Oct 2019 16:00
Venue : S16 #03-09
Speaker : Yin Silu, Hangzhou Normal University

Date/Time:02 Oct 2019 16:00 Venue: S16 #03-09 Speaker: Yin Silu, Hangzhou Normal University Global well-posedness of elastodynamics In this talk, I will present some classical theory of elastic waves. The motion of elasticity is a wave system with multiple speeds, so Lorentz invariant is absent. For incompressible isotropic elasticity with constant density in 3D, the Cauchy problem with small initial datum were studied by a series work of Sideris and Thomases. We studied the inhomogeneous case in Lagrangian coordinate under the assumption that the density is a small perturbation around a constant state. This is a joint work with Xiufang Cui. Add to calendar: 

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Date/Time : 18 Sep 2019 16:00
Venue : S16 #03-09
Speaker : Chen Haoyang, Fudan University

Date/Time:18 Sep 2019 16:00 Venue: S16 #03-09 Speaker: Chen Haoyang, Fudan University Global regularity for Einstein-Klein-Gordon system with $U(1) \times \mathbb{R}$ isometry group In this paper we prove the global regularity of the 3+1 dimensional Einstein-Klein-Gordon system with a $U(1)\times \mathbb{R}$ isometry group. We reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2+1 dimensional system. We prove that the energy of this system cannot concentrate near the first possible singularity, and hence is small. Then, we show that the global regularity holds for the reduced 2+1 dimensional system with initial data of small energy. Add to calendar: 

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Date/Time : 02 May 2019 16:00
Venue : S17 #06-11
Speaker : Zhang Shuangjian, École Nationale de la Statistique et de l'Administration Économique, Paris, France

Date/Time:02 May 2019 16:00 Venue: S17 #06-11 Speaker: Zhang Shuangjian, École Nationale de la Statistique et de l’Administration Économique, Paris, France On concavity of the principal’s profit maximization facing agents who respond nonliearly to prices A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his own utility, if positive. The principal’s profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal’s price menu. In this talk, we describe a necessary and sufficient condition for the convexity or concavity of the principal’s problem, assuming each agent’s disutility is a strictly increasing but not necessarily affine (i.e. quasilinear) function of the price paid. Concavity when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal’s strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents’ preferences. Moreover, the analytic and geometric interpretations of a specific condition relevant to the concavity of the problem have been explored. This talk represents joint work with Robert McCann. Add to calendar: 

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Date/Time : 18 Apr 2019 16:00
Venue : S17 #05-12 SR4
Speaker : Tian Fangyang, National University of Singapore

Date/Time:18 Apr 2019 16:00 Venue: S17 #05-12 SR4 Speaker: Tian Fangyang, National University of Singapore Laplacian Equations from a Representation-Theoretic Point of View The classical Laplacian equation and wave equation are the basic examples of second order PDE. The fundamental solutions of them, which have been well studied, play an important role in the theory of these equations. A nice feature of these two equations is that the differential operator is invariant under certain classical group, hence one may expect a representation theoretic explanation of these equations. The (indefinite) Laplacian equations serve as a natural generalization of classical Laplacian equation and wave equation. In this talk, I will follow Howe and Tan’s exposition to discuss the (indefinite) Laplacian equation from a representation-theoretic point of view. We will cover some basic representation theory of (sl(2,R), so(2))-module, Weil representation of \tilde{SL(2,R)}, and the fundamental solutions of the indefinite Laplacian Equation. Add to calendar: 

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