Time: 14:00-15:00, Wednesday, December 4 2024
Venue: E4-233, Yungu Campus
Host: Thierry De Pauw, ITS
Speaker: Siran Li, Shanghai Jiao Tong University
Biography：Siran Li obtained his B.A. from Columbia University (2013) and D.Phil. from the University of Oxford (2017, under the supervision of Gui-Qiang Chen). After undertaking a postdoctoral instructorship at Rice University with Robert Hardt and a visiting assistant professorship at New York University-Shanghai, he joined Shanghai Jiao Tong University in September 2021 as a tenure-track associate professor. The main research interests of Siran include the analysis of PDE arising from isometric immersions, fluid dynamics, and rough path theory.
Title: Mixed-type Partial Differential Equations and the Isometric Immersions Problem
Abstract: This talk is about a classical problem in differential geometry and global analysis: the isometric immersions of Riemannian manifolds into Euclidean spaces. We focus on the PDE approach to isometric immersions, i.e., the analysis of Gauss--Codazzi--Ricci equations, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges. Connections with nonlinear elasticity and fluid mechanics will also be discussed.
