# Professor Fumihiro Ushitaki

Area and Subject Taught Topology (1) Theory of Topological Transformation Groups and Related Topics (2) Development of Teachers' Training Programs in Arithmetic and Mathematics Doctor of Science, Osaka University Transformation Groups, Borsuk-Ulam Theorem, Isovariant Maps,Development of Teachers' Training Programs in Arithmetic and Mathematics Not Public

## Research Overview

The symmetry of figures is an important concept in geometry. When a figure has symmetry, a symmetric transformation is obtained by moving the points on the figure to their corresponding symmetric points. Furthermore, if two symmetric transformations are applied successively, the result becomes another symmetric transformation. The reverse of a symmetric transformation is also a symmetric transformation. This shows that we can go beyond looking at each symmetric transformation individually, and consider all possibilities of symmetric transformation by introducing the algebraic concept of a group (constituted by combining transformations). Investigating the subject in such a framework clarifies the properties of figures containing symmetry. The purpose of the theory of transformation groups, the field of my research, is to use this approach to study problems relating to the symmetry of figures from the viewpoint of topology ("soft geometry"). However, since the figures are considered in terms of soft geometry, the term "symmetry" takes on a meaning quite different from that we learn in high school. Yet just as we learn in high school that there is a fixed point for a rotation, and there is a fixed line for a reflection, so the theory of transformation groups addresses problems such as fixed points, and the movement of points by transformations. At present, I am studying symmetry-preserving correspondences between figures with symmetry, and their properties.

## Notable Publications and Works in the Last Three Years

1. F. Ushitaki: Transformation Groups and Finite Topological Spaces, RIMS kôkyûroku, 2016 (2016) 132-141
2. 西川信廣，牛瀧文宏：「学校と教師を変える小中一貫教育〜教育政策と授業論の観点から〜」ナカニシヤ出版　2015年
3. I. Nagasaki and F. Ushitaki: On G-bi-isovariant equivalence between G-representation spaces, RIMS kôkyûroku, 1922 (2014) 60-64
4. 牛瀧文宏監修，子ども学力向上研究会著：「「算数の教え方」がわかる本」メイツ出版　2014年
5. 長崎生光監修，牛瀧文宏編集：「初歩からの線形代数」講談社（2013）