目：How round is a Jordan curve?

间：2021.09.03（星期五），09:00-10:00

点：数学院南楼N204    腾讯会议：909 401 102

要：A Jordan curve is a simple loop on the sphere. We recently introduced the conformally invariant Loewner energy to measure the roundness of a Jordan curve. Initially, the definition is motivated by describing asymptotic behaviors of Schramm-Loewner evolution (SLE), a probabilistic model of random curves of importance in statistical mechanics. Intriguingly, this energy is shown to be finite if and only if the curve is a Weil-Petersson quasicircle, an interesting class of Jordan curves that has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, and string theory and is studied since the eighties. The myriad of perspectives on this class of curves is both luxurious and mysterious. In my talk, I will overview the basics of Loewner energy, SLE, and Weil-Petersson quasicircles and show you how ideas from probability theory inspire many new results Weil-Petersson quasicircles.
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目：Observation, Stabilization and Resolvent estimate: Beyond the geometric control condition

间：2021.09.10（星期五），16:00-17:00

点：数学院南楼N204    腾讯会议：206 257 109

要：I will present some recent results on control and stabilization for the wave and Schrödinger equations, using semiclassical analysis. I will focus on two simple operatros: the flat Laplacian on torus and the Bouendi-Grushin operator. The talk is a general introduction. I will first discuss the links of concepts in the control theory for linear evolution equations (such as the exact controllability, observability, stabilization) and resolvent estimates (for stationary problems). Next, a quick review for some classical results will be given, in order to motivate the study of geometric effects on control problems.
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目：Long time behavior of 2d water waves

间：2021.09.17（星期五），09:00-10:00

点：数学院南楼N219    腾讯会议：874 326 514

要：In this talk, we will talk about the long time behavior of the two dimensional water waves. In the first part we discuss the long time existence of water waves with concentrated vorticity. We show that the water waves with small initial slope and velocity, coupled with a pair of counter-rotating point vortices which travels downward, remains small and smooth for a long time. In the second part we discuss the NLS approximation of the water waves. In particular, I will discuss the rigorous justification of the Peregrine soliton and the nonlinear instability of the Stokes waves.