## MATSUE, Kaname MATSUE, Kaname

Associate Professor

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MATSUE, Kaname/ Associate Professor
MATSUE, KanameMATSUE, Kaname（Associate Professor）researcher_infomation research and technical catalogMy 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.

1. geometric singular perturbations (multi-scale dynamics) and numerical computation with guaranteed accuracy

We are developing rigorous methods for the computation of time global orbits of dynamical systems with multiple time scales. In particular, we consider the cases where the time scale parameters are not only "small enough" but also "in the concrete range", and aim to provide a starting point for the application of mathematical theory in various multiscale phenomena.

2. finite-time singularity from the viewpoint of singularity theory (explosive solution, extinction, quench, canard)

We are developing a method to capture various singularities in which the solution of a differential equation becomes undescribable at finite time, using numerical simulations with guaranteed accuracy. Explosions" and "quenches" involving divergence of solutions and their derivatives, "extinctions" where solutions become immobile at finite time, "canard points" where solutions move from stable to unstable manifolds, and many other phenomena that manifest singularities at finite time can be comprehensively captured by using "singularity resolution". By combining numerical calculations with guaranteed accuracy and the basic tools of topology, we aim to rigorously understand the above phenomena using a common method, and to provide a starting point for the analysis of various phenomena.

3. shock wave

We are constructing methods for computing solutions to "conservation law systems" as typified by the equations of motion of gases, especially solutions to Riemannian problems with special initial data. As a starting point, we are studying methods to compute the discontinuous solution (shock wave) and the smooth solution (rarefied wave), which are typical solutions, and the singular shock wave with a delta function at the discontinuity from the viewpoint of the junction of the dynamical system and the explosion solution, using rigorous numerical calculations with guaranteed accuracy. As with the above two projects, we aim to provide a starting point for the analysis of various phenomena described by conservation law systems.

4. quantum walk

The dynamics of "quantum walks" (time evolution of families of unitary operators on graphs), which are considered to be the quantum mechanical counterpart of random walks, and in particular the correspondence with the geometric structure of the set on which the quantum walk is defined. Quantum walks are usually considered on lattices and graphs, but for a comprehensive understanding of their geometric aspects, we construct quantum walks on unitary complexes and study their correspondence with the geometric structure of unitary complexes through analysis of their dynamics, spectra, and so on.

5. issues originating from various scientific fields such as combustion

The dynamics of (premixed) laminar and turbulent flames generated in the combustion process of materials, especially gases, are analyzed from a simplified model derived from first principles of physics. In this project, we focus on planar and spherical propagating flames, and consider how the dynamics varies with the geometrical shape of the flame, and how the stability of the shape and turbulence effects are related. We are also working on mathematical solutions to other problems originating in materials science, such as the "shape" of materials with amorphous structures (e.g., glass), which do not have crystal-like symmetry and periodicity, and the arrangement of media with different properties (e.g., thermal diffusion media) in order to optimize their properties.

Keywords | Dynamical systems, numerical analysis, numerical computation with guaranteed accuracy, singular perturbation theory, differential equations (explosion solutions, shock waves), singularities, topology (including computer-aided), quantum walks, topology optimization, combustion |
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Department | Division of Applied Mathematics |

Link | International Institute for Carbon-Neutral Energy Research (I2CNER) |