KM's research activities are based on, so-called "Applied Mathematics", in particular
・Dynamical Systems,
・Differential Equations,
・Numerical Analysis, in particular "Rigorous Numerics" or "Computer-Assisted Proofs".
KM's basic mind is to unravel fundamental nature of "strange" or "singular" behavior in differential equations by means of standard machineries in dynamical systems, numerical analysis and application of various mathematics.
"Strange is Interesting". Obviously clear, don't you think ?
KM also engages in applications of these applied mathematics to energy problems such as combustion and related issues.
His particular interests are listed below.
Finite-Time Singularities and Dynamical Systems
Finite-time singularities such as "blow-ups", "finite-time extinction", "quenching" and "collision" in differential equations are KM's primary interests. In particular, fundamental and unified frameworks for characterizing these singularities are his main issues.
Theoretical issues, rather than applications or problems in concrete models, are mainly focused, while applications are also studied as collaborative works.
KM's approach is to study "dynamics at infinity" for ordinary differential equations (ODEs) based on compactifications and time-scale desingularization.
Main achievements by KM and his collaborators consist of
1. development of theory within the class of "asymptotically quasi-homogeneous" vector fields including systematic construction of desingularized vector fields,
https://epubs.siam.org/doi/10.1137/17M1124498
2. characterization of (type-I) blow-ups by means of local (center-)stable manifolds of invariant sets (equilibria, periodic orbits, normally hyperbolic invariant manifolds) at infinity,
https://epubs.siam.org/doi/10.1137/17M1124498
https://arxiv.org/abs/2307.09201
3. case studies for characterizing type-II blow-ups and slower blow-ups by means of nonhyperbolic invariant sets at infinity, as well as unified description of finite-time singularities (extinction, quenching, etc.),
https://www.sciencedirect.com/science/article/pii/S0022039619303274
4. rigorous numerics of sink-type and saddle-type blow-up solutions as well as computer-assisted visualization of blow-up time distributions,
https://link.springer.com/article/10.1007/s00332-023-09900-6
5. machinery to calculate multi-order asymptotic expansions of (type-I) stationary blow-ups with their correspondence to dynamics at infinity,
https://arxiv.org/abs/2211.06865
https://arxiv.org/abs/2211.06868
6. characterization of blow-ups for nonautonomous systems of ODEs.
https://arxiv.org/abs/2307.09201
We can say that blow-up characterization by means of "invariant sets at infinity" provides spatial complexity of blow-up solutions.
Recent studies in this field involve
A. unifying characterization of asymptotic quasi-homogeneity and compactifications,
B. characterizing finite-time singularities beyond normal hyperbolicity,
C. temporal complexity of blow-ups by means of nonautonomous spectrum of linear differential systems,
D. bounded objects by means of dynamics at infinity,
E. characterizing qualitative and quantitative natures of singular shocks in systems of conservation laws (not always strictly hyperbolic), collisions and other possible finite-time singularities.
All above issues are within "finite-dimensional" problems, namely for ODEs.
Natural questions involving "infinite-dimensional" problems such as partially differential equations (PDEs) and delay differential equations (DDEs) are included in our issues, which are still future directions and will be studied once necessary machineries are collected.
Rigorous Numerics in Dynamical Systems
Updated later.
Combustion Science and Engineering
Updated later.
Brain-NeuroScience
Updated later.