Faculty and Research Themes, Field of Mathematics

We study the structure of manifolds and submanifolds using calculus. For example, we define the direction in which the volume of a manifold decreases more as a geometric quantity by partial differential equations, and identify that geometric quantity for various manifolds.

We mainly research differential geometry. That is, we consider geometric objects primarily using differential methods as the main tool. In particular, we research maps and surfaces related to integrable systems where large symmetry and linearity are hidden behind apparent nonlinearity, such as harmonic maps and constant mean curvature surfaces, which are well-known examples. In addition, we also deal with geometric variational problems and affine differential geometry as related topics.

We study elliptic curves, their higher-dimensional analogs called abelian varieties, and automorphic forms. In recent years, the finite abelian groups formed by rational points of abelian varieties over finite fields handled in this field have been applied in cryptography in connection with the discrete logarithm problem. On the theoretical side, there have been remarkable developments such as the resolution of the Taniyama-Shimura conjecture and the Sato-Tate conjecture. In seminars, we read through literature and papers related to this field, and aim for each student to find a theme and derive original results.

We research commutative algebra and its surroundings. Historically, commutative algebra played a major role in providing the foundations of algebraic geometry, but it is now undergoing new developments and is being very actively researched in connection with singularity theory, computer algebra (Gröbner bases), representation theory, etc. In this course, we mainly adopt combinatorial approaches using the geometry of convex polyhedra and homological algebraic approaches using derived categories and sheaf theory.

We research algebra and topology related to tensor categories. We investigate the structure of tensor categories formed by representations of non-commutative algebras such as group rings, Hopf algebras, subfactor rings, and quantum groups, and use quantum invariants defined using them as tools to examine the geometric properties of knots and 3-dimensional manifolds. Through seminars and other activities, we find research themes ourselves and conduct presentations and discussions to achieve deep understanding.

We research integrable equation systems and their discretization and ultra-discretization. We research the construction of solutions to the equations and the mathematical structures behind the solutions, including symmetries. We also research applications to various fields, such as mathematical models using integrable systems.

We study the global properties of Markov process theory using Dirichlet form theory, which is its analytical counterpart. Also, based on martingale theory, we study optimal stopping theory in optimization theory and derivative pricing theory, which is a field in mathematical finance, through lectures and seminars. In any case, it is not only necessary to have knowledge of probability theory and analysis, but also having broad knowledge of management engineering and economics will lead to a deeper understanding, so please learn about these as needed.

We analyze the probabilistic properties of paths and functionals of Markov processes by examining spectral properties such as compactness of Markov semigroups, bounded continuity of eigenfunctions, and integrability of bases. We define a class of symmetric Markov processes with properties close to one-dimensional diffusion processes, and apply them to the analysis of quasi-stationary distributions by examining their long-time behavior.

Development of computational algebra algorithms represented by Gröbner bases, and their applications to singularity theory, Hamiltonian dynamical systems, chemical reaction dynamics, program verification, etc.

We construct statistical theory for stochastic processes, which are one of the representation models for phenomena with time dependence. In particular, we are interested in estimation methods and simulation methods under non-normality, which is widely observed in time series data.

We research focusing on the theory and numerical analysis of stochastic differential equations. The theory of stochastic analysis and stochastic differential equations plays an important role in various fields such as mathematical finance, physics, and biology. For example, the pricing of financial derivatives is performed using stochastic differential equations (mathematical models). We research the construction of mathematical models based on stochastic analysis and their numerical analysis methods.