Masaharu Ishikawa - Appointed in 2018

I study singularity theory and manifold theory. My first encounter with singularities was in an undergraduate lecture on complex analysis. In the two-dimensional wing theory of fluid dynamics, the movement of fluid around a wing is brilliantly described by mathematical formulas. A singularity is a point where information disappears, but conversely, you can observe the strange phenomenon where information about the singularity itself determines the global behavior of the fluid. In graduate school, I studied singularity theory from J.W. Milnor's specialized book on hypersurface singularities and began my research within the framework of using singularities to investigate the behavior of spaces and functions. The seminars in graduate school were styled such that each student worked on a different research topic, which allowed me to learn a great deal of mathematics, including manifold theory, knot theory, dynamical systems, and combinatorics. This diverse experience is still useful to me in various situations today.

A singularity is a point where the derivative of a function (or more generally, a map) vanishes, and it is an object of study that appears not only in mathematics but in almost all academic disciplines. As represented by catastrophe theory, economics is no exception. Furthermore, manifold theory is a theory for understanding space, and this concept of space is also an object of study in all academic disciplines. My research theme is positioned at the intersection of these two theories. The appeal of my research, above all, is its "freedom." Since concepts like singularities and space appear in almost all research fields, I can freely choose the direction and methods of my research. This may be difficult to understand unless you are an expert, but there is a sense of "rigidity" in mathematical tools. Functions (especially complex functions) are rigid, while topology is flexible. There are also many intermediate concepts. By selectively using these tools with varying degrees of rigidity depending on the research subject, I can find clues for my research, which then develop into new theories. Singularity theory and manifold theory are inclusive research fields that allow the free use of both this rigidity and flexibility, offering the exceptional fascination of being able to conduct research with great versatility.

At university, you learn mathematics through lectures and exercises, but a certain amount of self-study is essential for mastering it. While time spent at your desk solving practice problems is necessary (especially to pass exams), it is even more important to spend time reading textbooks and reference books and imagining the meaning of theorems and formulas. Sometimes, take a step back and spend time on vague imaginings, asking yourself, "What is this theorem ultimately trying to say?" The ultimate goal is to develop a proper sense of mathematics. Formulas are written in textbooks, so you don't need to memorize them, and computers can accurately provide answers to complex calculations. The ability to use them correctly is what is required of us. Hints for a proper sense of mathematics are hidden in the words of your professors during university lectures. Listen carefully to the lectures and imagine mathematics in your own mind to cultivate a proper sense and acquire practical mathematical skills.

(Interview conducted in December 2018)