In the seminal paper of 1966, V. I. Arnold discovered that geodesics on the group of volume preserving diffeomorphisms are flows of the ideal fluid whose velocity field satisfies the incompressible Euler equation in hydrodynamics. Many other interesting nonlinear PDEs in mathematical physics turned out to fit in this framework, such as the Korteweg-de Vries equation and the quasigeostrophic equation (or Hasegawa-Mima equation). Geometry helps us to study the qualitative behavior of the solution of PDEs. For example, Arnold gave the first mathematical proof of the impossibility of weather prediction by computing that sectional curvatures of the atmospheric configuration space are negative. In this talk, I will survey these geometric approaches to PDEs. In particular, I will share recent progress on the geometry of quantomorphism group and quasigeostrophic motion.
Jae Min Lee is a postdoctoral research associate in the School of Mathematics and Statistics at the University of Sydney. Jae Min obtained his Ph.D. in Mathematics in 2018 from the CUNY Graduate Center, NY USA, under the supervision of Stephen Preston. Previously, Jae Min held a postdoctoral position at the KTH Royal Institute of Technology, Sweden. He has been in Sydney since 2022. Jae Min’s main research interests are global behaviors of the solution of nonlinear PDEs and their stability around equilibria.