The VI AMMCS International Conference

Waterloo, Ontario, Canada | August 14-18, 2023

AMMCS 2023 Plenary Talk

The isometric immersions problem: from perspectives of PDE, geometry, and physics

Siran Li (Shanghai Jiao Tong & New York University)

We report our recent work on a classical problem in differential geometry: isometric immersions and/or embeddings of Riemannian and semi-Riemannian manifolds. The underlying PDE is the system of Gauss-Codazzi-Ricci equations. Existence of isometric immersions is studied under various curvature conditions, via elliptic and hyperbolic PDE techniques. Weak continuity of isometric immersions is investigated with the help of the theory of compensated compactness. Connections to other problems in mathematical physics, including fluid dynamics, harmonic maps, and Coloumb gauges, will be discussed.
Our talk contains joint work with Gui-Qiang Chen, Reza Pakzad, Armin Schikorra, and Marshall Slemrod.

Siran Li grew up in Beijing, studied mathematics from 2009 to 2013 at Columbia University, New York, and obtained the D.Phil. degree in 2017 from the University of Oxford, under the supervision of Gui-Qiang Chen. He did postdoctoral researches with Bob Hardt from 2017 to 2020 at Rice University, Houston, and spent the year 2018 - 2019 at McGill and Concordia Universities in Montreal as a CRM-ISM postdoctoral fellow. Siran moved back to China in 2020, first working at New York University-Shanghai. He joined Shanghai Jiao Tong University as an associate professor in September 2021.
Siran's main research interests include analysis of partial differential equations, geometric measure theory, and applied mathematics. One of the main themes of his mathematical work is the analysis of singularities, or structures of weak regularities, which arise naturally from problems in differential geometry and fluid dynamics. His recent research projects concern with isometric immersions of manifolds, the theory of compensated compactness, and the Navier-Stokes equations.