My mathematical research lies in the general area of nonlinear Partial Differential Equations(PDEs) and infinite dimensional Riemannian Geometry. In particular, I study PDEs that are related with incompressible Euler’s equation for ideal fluid; Camassa-Holm, Quasi-Geostrophic equations, etc. These equations arise as geodesics on the group of diffeomorphisms endowed with an invariant metric, and are called Euler-Arnold equations. Diffeomorphism groups can have rich mathematical structures such as Lie group and Banach manifold. One can solve problems about the PDEs using this infinite dimensional Riemannian geometric interpretations, or study the geometry of the diffeomorphism group on its own right.
 Global Lagrangian Solutions of the Camassa-Holm equation, submitted, [arXiv]