Zu Chongzhi Research Seminar

Date and Time (China standard time): Friday, December 12, 9:00 am – 10:00 am

Location: WDR 1007

Zoom: 974 8661 5780, Passcode: dkumath

Title: Mixed-type Partial Differential Equations and the Isometric Immersion Problem

Speaker: Siran Li, Shanghai Jiao Tong University

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 (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 Uhlenbeck gauges. Connections with nonlinear elasticity and fluid mechanics will also be discussed. 

Joint work with Reza Pakzad (Toulon), Armin Schikorra (Pittsburgh), and Xiangxiang Su (Shanghai Jiao Tong).

Bio: Siran Li works in the analysis of partial differential equations (PDEs), with a focus on the well-posedness theory of mixed-type PDEs arising from fluid dynamics and differential geometry. Siran received his D.Phil. degree in mathematics from the University of Oxford in 2017 under the supervision of Gui-Qiang Chen. He was a G. C. Evans instructor at Rice University from 2017 to 2020, mentored by Robert Hardt, and a visiting assistant professor at NYU-Shanghai in 2020-2021. Siran joined Shanghai Jiao Tong University as a tenure-track associate professor in September 2021.