My research is based on dynamical systems with computer-assisted proof (rigorous numerics).
In particular, I study systematic verification methodologies for “proving” solution structures in concrete dynamical systems with practical data by numerical computations, as well as mathematical theory and numerical computation methods inspired by such computer-assisted studies of dynamical systems.
I aim at obtaining a broad outlook on various theories such as mathematics, materials science and energy science to solve unsolved and challenging issues.
Main topics :
1. Geometric singular perturbation theory and rigorous numerics
2. Finite-time singularities (blow-up solutions, extinction, quenching, canard) from the viewpoint of singularity theory with rigorous numerics
3. Shock waves and rigorous numerics
4. Quantum walks : geometric aspects of dynamics
5. Various problems inspired by combustion and materials science (structural analysis of amorphous-structured materials, optimal arrangement of compounds)