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
  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 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.

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