Mathematics is the basis for policy judgment and decision-making

My high school was an industrial high school, divided into four fields: design, crafts, systems, and business. I was in the business field, but I couldn't decide what I wanted to study at university. It was during that time that I learned SFC has a flexible curriculum where you can do whatever you want, so I decided to find my own path of learning at SFC.

When I first enrolled, I was interested in mathematical sociology and started studying mathematics as its foundation. Being in an environment with so many people who have diverse backgrounds and skills was very stimulating. To be honest, I was initially overwhelmed by how amazing everyone around me was, but I think that's precisely why I was motivated to tackle the basics one step at a time. As I studied diligently day by day, I became engrossed in the charm of mathematics, which is full of novel ideas and mysterious phenomena.

The first time I was surprised was when I was studying set theory, a fundamental field of mathematics. I believe one of the purposes of set theory is to deal with infinity. For example, let's consider which set has more numbers: the set of integers, like "...0, 1, 2, 3...", or the set of rational numbers that can be expressed as fractions, like "...1/2, 1/3, 1/4...".

Intuitively, you might think there are more rational numbers, right? After all, between the integers 0 and 1, there are countless rational numbers like 1/2, 1/3, 1/4, and so on. However, it has been proven that the infinite sets of integers and rational numbers contain the same number of elements. The proof involves demonstrating that each number in the set of integers can be put into a one-to-one correspondence with each number in the set of rational numbers, and that concept was shocking to me.

Currently, I belong to the Atsushi Kanazawa Seminar and am focusing my research on geometry. Geometry deals with space, and it is divided into various fields depending on the approach to that space. For example, differential geometry analyzes the curvature of space and flows on space, whereas topology analyzes the general shape and connectivity of space.

Let me explain using the steel desktop in front of me, the cylindrical case of sanitizing wipes on it, and a bundle of coiled cords. The desktop is flat with right-angled edges, and it has no smoothly curved parts like the side of the case. In differential geometry, which studies the curvature of space, these two are considered different spaces. On the other hand, topology focuses on general connectivity and shape, so it considers the desk and the case to be the same space. Can you imagine that if you were to squeeze the desktop from its left and right edges toward the center and smooth out the corners, you could make it the same shape as the case?

However, if we consider the bundle of cords to be a donut, topology would see the desktop and the donut as different spaces. This is because the donut has a hole, but the desktop does not. No matter how much you bend the desktop, you cannot create a hole. One of the objects of analysis in topology is the number of holes in a space.

A geometry that studies the curvature of space and a geometry that studies the number of holes in a space. In differential geometry and topology, what is analyzed about space is fundamentally different. However, there is an amazing theorem that shows that although they seem completely different, they are actually closely related. It's called the Gauss-Bonnet theorem.

The curvature of space is called "curvature." A simple explanation of this theorem is that for a special curved surface like the surface of a sphere or a donut, if you calculate the sum of the curvature over the entire space, the result is 2π times the Euler characteristic of that surface. The Euler characteristic is a quantity determined by the number of holes in a space, and it is an important quantity in topology. This is a mysterious phenomenon in mathematics where two fields with fundamentally different ways of thinking are actually connected in the depths.

The Faculty of Policy Management is often thought of as a humanities faculty. However, since everything must be considered based on data, science-related fields such as statistics and data analysis are also very important. Since mathematics is applied in these fields, I believe that mathematics is a discipline that forms the basis for various important issues, including policy judgment and decision-making.

All such fields of mathematics have originated from phenomena around us. For example, graph theory. This is a theory about graphs, which are composed of a set of points and a set of lines, and its origin lies in a puzzle-like problem from around the 18th century. It was born from and evolved out of the "one-stroke drawing" problem: "What route should one take to cross every bridge over a river exactly once and tour the entire town?" Mathematics is, by its nature, a discipline connected to everyday life.

I was the same way, but I think it's natural not to know what you want to do during your high school years, so there's no need to panic. On the other hand, I don't think doing nothing is the right approach. You won't figure out what you want to do unless you go to a stimulating environment, keep your antennae up, and expand your knowledge on your own, so I hope you will make an effort to take the initiative and change your environment.

In that sense, coming to SFC will completely change your environment. I think it's a very stimulating place. If you approach your studies with a proactive stance, I believe it will respond to you fully.