Mathematics

# Contemporary Mathematics on Smooth and Singular Figures

Goo Ishikawa , Professor

Faculty of Science/Graduate School of Science (School of Science/Department of Mathematics)

High school : Fukushima Prefectural Fukushima High School

Academic background : Doctorate from Kyoto University

Research areas
geometry
Research keywords
manifolds, singularity, topology
Website
http://www.math.sci.hokudai.ac.jp

## What are you researching? What are your aims?

My speciality is in geometry, where various figures are studied in the framework of contemporary mathematics. By “figures,” you may first think of the triangles and circles you studied in primary and junior high school. The figures studied in contemporary mathematics are in principle the same as this. Since human imagination knows no limits, however, we can conceive preposterous figures and spaces, and contemporary mathematics can be used to seek the substance and applications thereof. Basically, we use generalizations of the equations, vectors, and calculus etc. that you learned in high school to investigate figures, and for that reason this is a subject that anyone is familiar to. One of the main attributes of mathematics is the fact that it is not bound by a single goal. Not having a single specific objective is a particular feature of mathematics. We aim to clarify truths through our research that will be relevant to any age, any subject, and any person.

We say it is about “figures,” but while some figures exist that we can see with our own eyes, higher-dimension shapes also exist that cannot be seen. There are smooth figures and there are singular figures. Smooth figures are called “manifolds.” The sphere and torus (donut-shaped surface) shown in Fig. 1 are two-dimensional manifolds, which can be seen. However, Poincare conjecture on three-dimensional manifolds is a typical geometrical problem on figures that are invisible. Furthermore, a non-smooth point of a figure is called a singular point (Fig. 2: to the left, the shape looks as though the point of a croissant has been attached to it, and to the right, the shape is one in which the hole of the donut has been closed up. The dip in the right hand image is also a singular point). My speciality is to study those figures that have singularities and their applications. In particular, I am engaged in research from topology, in which figures are studied by flexible and continuous deformations (see Citations (1) and (2)). The sphere and torus in Fig. 1 have different connectivity, and are therefore said to be “non-homeomorphic.” The following problem is one of topological problems on figures with singular points. If you spend a little time thinking about it, you will understand how our research is done.

Problem: The surfaces of the two shapes in Fig. 2 will be considered as “homeomorphic” if seen in a certain way. How should we look at them in order to see them as the same? (Hint: For example, gradually expand the “failed donut” in Fig. 2, and imagine yourself going into the interior so that you can see the surface from behind. It will begin to look the same as the shape of the connection on the left.)

## What equipment do you use in your research? What is your style of research?

We mainly use paper, pencils and our own brain. We use the brains of our co-researchers, too, and more recently we have been using computer in our research. The basic work for a mathematician is to look at things creatively from “zero” and discover theorems. At the same time, it is also vital for mathematicians to be able to explain things as simply as possible, in order to share the discoveries they have made and allow them to find applications. I believe that “creativity and sharing” has been behind the development of mathematics so far (see Citation (3) containing a collection of questions and answers regarding mathematics). I continue to have quite an introverted personality since I was a child, and always liked thinking about things. I am still quite introverted (laughs), but that’s probably because I’m still searching to discover big theorems. That’s my style of research. I can become a bit withdrawn when I am busy researching, but I don’t dislike going out for a drink or to karaoke for a change of pace. I like traveling as well. I seem to have gone off topic a bit…

## What do you plan to research in the future?

One of the themes I am interested in at the moment is the topology of null-links in Lagrange-Grassman manifolds (Fig. 3: Two spheres are linked by cords. If you extend the cords and make them go around the spheres, they become entangled or loosened. Fig. 4: Under certain

geometrical conditions, we are investigating the possibility of this type of entanglement). This research is, in fact, incredibly interesting at the moment. I hope to continue finding many more of these kinds of interesting geometric problems, and become a world leader in this type of research. There is no end to the applications of geometry. I hope to be able to come to a deeper understanding of the universe through the means of shapes.

## Do you have a message for readers?

The Department of Mathematics at Hokkaido University is home to some world-famous mathematicians, who are engaged in proactive research. Do you love math? If you love math and are fairly good at it, then come and study contemporary mathematics. Studying contemporary mathematics will open up a wide range of opportunities for you.

## References

(1) Izumiya, S., Ishikawa G., Applied Theory of Singularity, Kyoritsu Shuppan (in Japanese).

(2) Tokunaga, H. et al., Algebraic Curves and Singularity (joint authorship), Kyoritsu Shuppan (in Japanese).

(3) Ishikawa G., Mathematical Problems for All Situations, Nihon Hyoronsha (in Japanese).