Research Topics

① Geometric analysis on manifolds
② Geometric invariants on submanifolds
③ Moduli Theory of periodic minimal surfaces

Key Words

Spectral Geometry, Minimal surfaces, Morse index, Nullity, Signature

Research Topics

① Differential geometry related to integrable systems
② Geometric variational problems
③ Affine differential geometry

Key Words

Harmonic maps, Minimal surfaces, Surfaces with constant mean curvature, Integrable systems, Variational problems, Affine differential geometry

Research Topics

① The arithmetic of abelian varieties with complex or quaternionic multiplication
② The relationship between abelian varieties and automorphic forms

Key Words

Abelian Variety, Complex Multiplication, Quaternionic Multiplication, Field of Moduli, Field of Definition, Defining Equation, Rational Point

Applications

Crytography Theory

Research Topics

① Combinatorial commutative algebra
② Application of the derived category and sheaf theory to the above area
③ Oriented Matroid

Key Words

Stanley-Reisner Ring, Derived Category, Constructible Sheaf, Local Duality, Dualizing Complex, Poicare- Verdier Duality, (Affine)Oriented Matroid

Applications

Computational Algebra

Research Topics

① Quantum invariants of knots and 3-manifolds
② Representation categories of Hopf algebras

Key Words

Topology, Tensor Category, Representation, Topological Field Theory, Hopf Algebra, Quantum Group, Knot, 3-Manifold, Subfactor, Triangulation

Applications

Natural Science

Research Topics

① Discrete dynamical systems and their integrability criteria
② Cellular automata and ultra-discrete equations

Key Words

Algebraic entropy, Ultra-discrete equations, Arithmetic dynamical systems, Singularity confinement

Research Topics

① Global path properties for Markov Processes
② Regularity problem for Dirichlet forms
③ Construction of Feller semi-groups in terms of integro-differential operators

Key Words

Symmetric Stable-like Processes, Jump-diffusions, Dirichlet Forms, Martingale Additive Functionals, Recurrence, Transience, Regularities, Ergodicity

Applications

Estimates for Stationarity of Markov Chains, Determination of Option Prices in the Discrete Model Case, Foundations of Probabilistic Risk Analysis

Research Topics

① Stochastic calculus of symmetric Markov processes
② Spectral properties of symmetric Markov operators
③ Large deviations for additive functionals
④ Quasi-stationary distribution, penalization problem
⑤ Criticality theory for Schrödinger operators

Key Words

Symmetric Markov process, Additive functional, Large deviation, Dirichlet form

Research Topics

① Computational Algebra
② Applied Singularity Theory
③ Hamiltonian Dynamical Systems

Key Words

mixed-module, comprehensive Groebner basis, singularity theory, dynamical systems

Applications

multi-objective optimization, program verification, chemical reaction dynamics

Research Topics

① Estimation theory
② Model　Selection
③ Simulation　method

Key Words

Lévy process,information criterion,stochastic differential equation,misspecified model

Applications

High frequency data analysis

Research Topics

① Stochastic calculus

Key Words

Euler-Maruyama scheme, Monte Carlo method, stochastic differential equation, Dyson Brownian motions

Applications

Mathematical finance