**Research interests**

My research interests are in Optimal Transport and Partial Differential Equations. In particular, I’m interested in Monge-Ampere type equations that comes from optimal transport. Also, I’m interested in dynamic setting of optimal transport problem and its gradient flow structure.

- Holder regularity of solutions to generated Jacobian equations
- ArXiv : https://arxiv.org/abs/2004.12004
- In this paper, I generalized local Holder regularity result of G.Loeper(https://arxiv.org/abs/math/0504137) to generated Jacobian equation case.

- Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function
- ArXiv : https://arxiv.org/abs/2010.14471
- In this paper, I show Loeper’s condition and quantitatively quasi-convexity are equivalent when the cost function is C^2 with Lipschitz mixed hessian.

**Study**

For PDEs, I took PDE courses, such as ‘*Hyperbolic & Dispersive Equations’*, ‘*Linear and Nonlinear Parabolic Equations*‘, and ‘*Regularity for Second Order Elliptic Equations*‘. The PDE that I mainly study is the Monge-Ampere equation. It was one of topics for my comprehensive test, and I studied the equation with the book ‘*The Monge-Ampere equations and its applications*‘ by *A.Figalli*, and with many papers of many authors.

For Optimal Transport, I study with books such as ‘*Topics in optimal transportation’ and ‘Optimal transport Old and new*‘ by *C.Villani* and read many papers of many authors. One topic of optimal transport that I’m interested in is geometric view of optimal transport, which can be found in the paper ‘*Continuity, curvature, and the general covariance of optimal transport*‘ by *Y-H.Kim* and *R.Mccann*. Since it seems that there are some relation semi-Riemannian geometry, I studied the book ‘*Semi-Riemannian geometry with applications to relativity*‘ by *B.O’neill* in the reading course with *Willie Wong*. Another topic that I’m interested in is R.Mccann’s displacement interpolation and gradient flow structure of optimal map. I study ‘*Gradient Flows*‘ by* L.Ambrosio, N.Gigli*, and *G.Savare* for this topic.