Hisao Kato (Geometry)

Where You Will Come Across the Most Extraordinary Spaces:
Geometric Topology and Chaotic Dynamical Systems

Hisao Kato

Please tell us about your research.

I study geometric topology and topological dynamical systems. I research geometrical and dynamical properties of compact metric spaces and separable metric spaces. I also study properties of continuous mappings utilizing the theory of topological spaces, geometric and algebraic topology, topological dynamical systems, and ergodic theory. In recent years, my primary focus has been on solving the complicated invariant sets of geometrical and dynamical structures appearing in dynamical systems. Generally, dynamical systems of continuous mappings are known to produce complicated topologies. Moreover, invariant sets led by attractors often display very complicated geometrical structures, and these complicated geometrical and dynamical structures in compact metric spaces are research topics of considerable interest.

Urysohn Universal Space:
Space Containing All Geometrical Targets (Separable Metric Spaces) Isometrically

Please tell us about the feature or the main attraction of your research.

In geometric topology, all separable metric spaces included in the Euclidean spaces, Hilbert space, Urysohn universal space, and functional spaces are topics for research. Urysohn universal space is an extremely large space in which all geometrical targets are isometrically included. In my research, spaces are not limited to locally favorable and familiar spaces (such as manifolds and polyhedra) because in a world that may seem strange and unnatural, many mysterious and wild spaces can appear. Examples include spaces that have recently become familiar as strange attractors in chaotic dynamical systems. Geometrical figures with fractal structures are also a target of my research, but the study of this space is still in its infancy. There are many wondrous spaces defying imagination. The interesting aspect of geometric topology is that we encounter these mysterious spaces, and we seek theories for describing and understanding these spaces.

Strange Attractors (Solenoids)

What kind of research is being conducted by the students in the laboratory?

Students are working on geometric topology and topologicaldynamical systems; in particular, they are studying the theory of topological spaces, continuum theory, dimension theory, ANR theory, ergodic theory, and entropy.

Homogeneous Continuum Not Dividing the Plane (Pseudo-Arc)

Please convey a message to the students.

Please use your own head to think about crucial things (of course, mathematics is one of the crucial things). The key is to think as much as possible for yourself. Please avoid the habit of simply and unquestioningly accepting things written down or said by other people. Instead, take time to understand them. By thinking and using your training, you yourself can significantly improve your abilities. Moreover, life is a long road. Your health is of utmost importance for realizing your dreams. Please train yourself physically and mentally. At present, the world is full of many technological conveniences, but your growth will probably be enhanced by limiting your reliance on technology and making more use of your own abilities.