Research & Study

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.

1. Holder regularity of solutions to generated Jacobian equations
   • Pure and Applied Analysis 3-1 (2021), 163–188.
     • DOI 10.2140/paa.2021.3.163
   • ArXiv : https://arxiv.org/abs/2004.12004
   • In this paper, I generalized local Holder regularity result of G.Loeper(https://doi.org/10.1007/s11511-009-0037-8) to generated Jacobian equation case.
2. 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.
3. Conditions for existence of single valued optimal transport maps on conex boundaries with nontwisted cost
   • ArXiv : https://arxiv.org/abs/2308.06826
   • Calculus of Variations and Partial Differential Equations, Vol. 64 (2025)
     • DOI : 10.1007/s00526-025-02974-y
   • Joint work with Jun Kitagawa
   • We study regularity of optimal transportation problem on a C^1 convex body with Euclidean distance square cost function, which does not satisfy so-called twisted condition
4. Convergence analysis of t-SNE as a gradient flow for point cloud on a manifold
   • ArXiv : https://arxiv.org/abs/2401.17675
   • Joint work with Hau-Tieng Wu
   • We establish theoretical background of boundedness result of the dimension reduction algorithm called t-SNE.
5. Quadratic optimal transportation problem with a positive semi definite structure on the cost function
   • ArXiv : https://arxiv.org/abs/2408.05161
   • We introduce the quadratic optimal transportation problem which is motivated from some problems in the dimension reduction algorithms and data science.
6. An alternative definition for c-convex functions and another synthetic statement of MTW condition
   • ArXiv : https://arxiv.org/abs/2505.12063
   • We define the alternatice c-convex functions and show that the MTW condition holds if and only if the c-convexity and the alternative c-convexity are equivalent.