Research

My work lies in differential geometry in the tradition of Élie Cartan: moving frames, structure equations, and geometric structures modeled on Lie groups and homogeneous spaces. A central theme is that, in many special geometric problems, Cartan's structure equations admit a deformation by a spectral parameter and become integrable systems.

The research program can be organized around the following themes.

Research themes

1. Integrable Systems and Surface Geometry
   Constant mean curvature surfaces, constant Gaussian curvature surfaces, loop group methods, DPW construction, and harmonic maps.

2. Teichmüller Theory and Harmonic Maps
   Complex landslide flow, harmonic maps into hyperbolic space, and links with geometric structures.

3. Minimal Lagrangian and Product Geometries
   Minimal Lagrangian surfaces in complex space forms, product spaces, and quadrics.

4. Statistical, Hessian, and Affine Geometry
   Statistical manifolds, alpha-connections, affine differential geometry, Hessian geometry, and convex optimization.

5. Discrete Differential Geometry
   Discrete constant curvature surfaces, discrete affine spheres, and integrable discretizations.

Short description for students

微分幾何学では、曲線、曲面、多様体などの「曲がった空間」を、曲率、接続、微分形式、Lie 群、微分方程式などを用いて調べます。私の研究室では、平均曲率一定曲面、ガウス曲率一定曲面、曲線の時間発展、戸田格子、sine-Gordon 方程式、sinh-Gordon 方程式、調和写像、統計多様体、凸最適化など、幾何と微分方程式が交わるテーマを扱っています。