My research interests are in "Noncommutative Differential Geometry". They are also included in "Poisson geometry" and "Mathematical Physics" in a broad sense.
In particular, I am mainly studying the following topics:
The explicit construction of noncommutative differentiable manifolds using "deformation quantization",
Noncommutative deformations of mathematical objects using ``star product",
Generalizations of mathematical concepts in ordinary differential geometry to those in noncommutative differential geometry,
Applications to theoretical physics, such as (classical and quantum) field theory, string theory and quantum gravity.
In recent years, I have focused on the deformation quantization for some Kähler manifolds (e.g. locally (Hermitian) symmetric ones, complex Grassmannians, etc.) and the mathematical formulation for quantum field theories from the perspective of deformation quantization as an application.
In the future, I would also like to extend my research interests to not necessarily Kähler manifolds, e.g. "generalized Kähler manifolds", "Hessian manifolds", ``symplectic statistical manifolds", and so on. As another outlook, I am also interested in applications of deformation quantization to twistor theory, integrable systems such as KdV equations, mirror symmetry, and quantum field theories such as matrix models.
-Keywords-
(Formal or non-formal) Deformation quantization
Star product
Noncommutative differentiable manifolds
Kähler manifolds
Locally Hermitian symmetric spaces
Hyper-Kähler manifolds
Complex Grassmannians (including complex projective space)
Fock representations
Twistor theory
Noncommutative integrable systems
Noncommutative field theories
Noncommutative instantons
Quantum field theories
Matrix models