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Kylin lecture series
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Time: Mar. 25 - May 12 2022
Venue: Tencent Video \Taizhou Hall \Zoom Video

Generic dynamics of mean curvature flows

Jinxin Xue(Tsinghua University)

Mar. 25 2022, 19:30-23:30(HK Time) -(Tencent Video :839 393 053)

Abstract: Mean curvature flow (MCF) is a way of evolving a hypersurface in Euclidean space according to a velocity field that is the negative mean curvature at each point of the hypersurface. Singularities always develop under MCF, so it is crucial to analyze singularities. We study mean curvature flow from a perspective of dynamical systems. We show how generic MCF avoids some unstable singularities and how dynamics is related to geometric information of the flow. This talk is based a series of joint works with Ao Sun.


Collective oscillations in adaptive cell populations (1)(2)

Lei-Han Tang(Hong Kong Baptist University)

Apr. 13 2022, 16:00-17:00( HK Time) - (Taizhou  HALL)

Apr. 14 2022, 16:00-17:00( HK Time) - (Taizhou HALL)

Abstract: Cell-density-dependent rhythmic behavior has been suggested to coordinate opulation level activities such as cell migration and embryonic development. Quantitative description of the oscillatory phenomenon is hitherto hampered by incomplete knowledge of the underlying intracellular processes, especially when isolated cells appear to be quiescent. Here we report a nonequilibrium hermodynamic scenario where adaptive sensing drives the oscillation of a dissipative signaling field through stimulated energy release. We prove, on eneral grounds, that daptation by individual cells leads to phase reversal of the linear response function in a certain frequency domain, in violation of the fluctuation-dissipation theorem (FDT). As the cell density increases beyond a threshold, an oscillating signal in a suitable frequency range becomes self-sustained. We find this overarching principle to be at work in several natural and synthetic oscillatory systems where cells communicate through a chemical signal. Applying the theoretical cheme to 2D bacterial suspensions, we found that swimming cells of sufficiently high density pontaneously develop a weak ircular motion with a laminar flow profile in the thin fluid layer. The theoretical results are compared with weak collective oscillations discovered earlier in Yilin Wu's lab, which can be considered as a vector version of our basic theory.


Singularities and complicated orbits in N-body problem

Jinxin Xue(Tsinghua University)

Apr. 15 2022, 19:00-20:00(HK Time)- (Tencent Video : 176 479 748)

Abstract: Singularities are crucial for the study of dynamics of evolutionary differential equations. In this talk we give an overview of the singularities in N-body problem as well as various orbits with complicated dynamics. We shall compare singularities in N-body problem with that in other differential equations such as mean curvature flows.


Numerical Study on Nonlinear-Expectation

Xingye Yue(Soochow University)

Apr. 18 2022, 19:00-20:00 (HK Time) - (Tencent Video: 324 985 588)

Abstract:We will present some numerical methods for a fully nonlinear PDE which is related to the G-Expectation or nonlinear expectation introduced by Shige Peng. Numerical experiments will be carried out to show the efficiency, accuracy and stability of the proposed methods. The effect of the artificial boundary conditions is also numerically investigated. Some numerical analysis is given to show the convergence of the numerical solutions to the viscous solutions of the original G-equation.


Uniqueness of BV solution for compressible Euler equations

Geng Chen(University of Kansas)

Apr. 23 2022, 9:30-12:00 HK Time - (Zoom video :937 1356 1299)


Abstract: Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data. In this talk, we will present our recent proof on uniqueness of BV solution. As a major breakthrough for system of hyperbolic conservation laws in 1990's, solutions have been proved to be unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In the paper of this talk, we show that these solutions are stable in a larger class of weak (and possibly not even BV) solutions of the system. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the BV theory, in the case of systems with two unknowns. Hence, the uniqueness of BV solution is proved. This is a joint work with Sam Krupa and Alexis Vasseur.