Date/Time : 01 Dec 2020 09:00
Venue : via Zoom
Speaker : Sun Zheng, Ohio State University

Join Zoom Meeting https://nus-sg.zoom.us/j/87275619715?pwd=WVJrYXVyVVFZbk5meXhna3kycTJEdz09 Meeting ID: 872 7561 9715 Password: 617596 In scientific and engineering computing, we usually encounter numerical discretizations of time-dependent equation systems associated with conserved or decreasing energy. The strong stability of a time stepping method refers to its ability to preserve the monotone energy-decay law at the discrete level, which may provide improved robustness for simulations. In this talk, we present a unified study on strong stability of explicit Runge-Kutta methods for linear semi-negative problems. In particular, we prove a both necessary and sufficient condition for strong stability of arbitrary odd-order methods.

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Date/Time : 01 Dec 2020 16:00
Venue : via Zoom
Speaker : Filippo Bracci, Rome "Tor Vergata", Italy

Join Zoom Meeting https://nus-sg.zoom.us/j/84662919254?pwd=bXk4WWU2NTFPTEptNkZ3cjBLbGcrZz09 Meeting ID: 846 6291 9254 Password: 206241 Gromov hyperbolic spaces are well studied since their introduction. In 2000 Balogh and Bonk proved that bounded strongly pseudoconvex domains are Gromov hyperbolic with respect to the Kobayashi distance and the natural boundary is homeomorphic to the Gromov boundary. It was however only in the last five years that Gromov hyperbolicity was started to be used in depth in complex analysis. The most useful tool in Gromov hyperbolicity is the so-called “shadowing lemma”. In this talk I will present a general introduction to Gromov hyperbolicity in complex analysis and review some recent results and applications to one and several complex variables (such as the study of trajectory of semi-complete holomorphic vector fields in the unit disc and boundary extension results for biholomorphisms in scv without assuming boundary regularity of the domains).

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Date/Time : 25 Nov 2020 09:00
Venue : via Zoom
Speaker : Tong Jiajun, University of California, Los Angeles

Join Zoom Meeting https://nus-sg.zoom.us/j/87962970385?pwd=QnRENCtzSW55VXArejBpWUtXcmlyUT09 Meeting ID: 879 6297 0385 Password: 808752 Abstract:  We study a two-phase tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. Instead of considering evolution of cell densities, our focus will be on the dynamics of two moving interfaces in the model that separate the tumor, the normal tissue, and the exterior vacuum. We will discuss well-posedness results on the interface evolution starting from a nearly radial initial configuration, under the assumption that the tumor has lower mobility than the normal tissue, which is in line with the well-known Saffman-Taylor condition in viscous fingering. We will briefly discuss possible ill-posedness scenarios when these conditions are violated, as well as implications of our results on further analysis of the tumor growth models. This is joint work with Inwon Kim.

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Date/Time : 23 Nov 2020 09:00
Venue : via Zoom
Speaker : Yao Yao, Georgia Institute of Technology

Join Zoom Meeting https://nus-sg.zoom.us/j/81762064402?pwd=Z250ZmtKTFVhUDNqTzFlN0pvdldoQT09 Meeting ID: 817 6206 4402 Password: 845588 Abstract:  For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] <= E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will talk about a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. I will also discuss its connection with some nonlocal PDEs such as the aggregation-diffusion equation.

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Date/Time : 20 Nov 2020 22:30
Venue : via Zoom
Speaker : Elena Giorgi, Princeton University

https://princeton.zoom.us/j/93296784920 (Joint online mathematical general relativity and hyperbolic PDE seminar) Abstract: The Kerr-Newman spacetime is the most general explicit black hole solution, and represents a stationary rotating charged black hole. Its stability to gravitational and electromagnetic perturbations has eluded a proof since the 80s in the black hole perturbation community, because of “the apparent indissolubility of the coupling between the spin-1 and spin-2 fields in the perturbed spacetime”, as put by Chandrasekhar. We will present a derivation of the Teukolsky and Regge-Wheeler equations in Kerr-Newman in physical space and use it to obtain a quantitative proof of stability

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Date/Time : 20 Nov 2020 09:00
Venue : via Zoom
Speaker : Xiaochuan Tian, University of California, San Diego

Join Zoom Meeting Meeting ID: 879 3214 5555 Password: 996723 Abstract: Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations.  While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved.  We present in this talk numerical methods for nonlocal models characterized by a length parameter which measures the range of nonlocal interactions.  Several numerical methods will be discussed with a focus on asymptotic compatibility of the schemes that are robust under the change of the nonlocal length parameter.

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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 : 19 Nov 2020 09:00
Venue : via Zoom
Speaker : Shen Junliang, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/88455891763?pwd=Sm5ISjNHSnpDVDMzYkFVd2ZBTit2QT09 Meeting ID: 884 5589 1763 Password: 869110 Abstract:  The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In this talk, we will focus on interactions between topology of Lagrangian fibrations and Hodge theory in general hyper-Kaehler geometries. Such connections shed new light on both the P=W conjecture for Hitchin systems and the Lagrangian base conjecture for compact hyper-Kähler manifolds.

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