Outline
Mathematics is the foundation of all natural science. By representing the universe in mathematical form, you can resolve the most basic problems. And that leads directly to the "search for truth." The Department of Mathematics does not stop at pure mathematics, but fosters a high level of mathematical perception that contributes to resolving all sorts of natural and social phenomena and putting them into theoretical frameworks.
Professor Hideo Kozono
Tackling fundamental problems to strengthen the intellect by leaps and bounds
Department of Mathematics / Department of Pure and Applied Mathematics (FSE Graduate School)

Q. Professor, I understand your specialty is “non-linear differential equations.” What does that mean?

— First of all, the concept of differential equations has its roots in physics, and Newton’s “equation of motion” which is expressed as F=ma is one such differential equation. When “F” (=force) is taken as the change over time of the motion of an object, it can be represented as a function of place and time. Given the initial conditions shown for this function — place and initial speed — it is possible to predict where an object will be t hours into the future. Newton used this “second-order differential equation of time” to deduce the motions of objects such as the planets.
Today, this approach is used in forecasting the weather. However, a wide variety of factors other than wind force and direction, such as heat and the rotation of the earth, make simple, linear representations of weather impossible; the motion is very complex. The equations used to represent such motion are non-linear differential equations.
Mathematicians like myself do not think of events in physical terms, but understand them in terms of expressions. Mathematicians therefore tackle fundamental problems, starting with whether or not an expression has a solution, and if so, how that solution can be expressed as a function, and if not, whether an approximation can be obtained.
For example, from Galois theory we know that it is not possible to obtain solutions for fifth or higher degree polynomial equations, and sometimes it is also not possible to obtain solutions for non-linear differential equations. Even if a solution is known to exist, it doesn’t follow that a function to express it can be readily found. Between these two there is a gap.

Mathematics is intensive thought training in the “search for truth”

Q. What sort of requirements are there for people undertaking fundamental problems in the Department of Mathematics?

— The Department of Mathematics is a great place for people who like to think, and especially those who like to think about fundamental things.
Studies focused on skills or qualifications tests can only increase knowledge in a specific field. However, those who build their thinking power with mathematics, the foundation of all sorts of scholarship, can contribute to all aspects of human prosperity. This is because greater thinking power makes it possible to study, talk about, and understand even fields that are unfamiliar.
In striving to increase our thinking power, what we are doing is participating in the “search for truth.” There are some who express doubts about the usefulness of such a quest, but I am certain of its value.
For example, in information engineering, the theory of cryptography is a development of algebra. Financial engineering also includes elements of differential equations. In physics, the fundamental question “what is the smallest unit of matter?” has brought us to fundamental particles, and these fundamental particles are making contributions to nuclear energy research.

The mind is trained in a different way, so there is no need to worry about academic paths.

Q. Since mathematics is a fundamental subject, some students may wonder whether it will help them find employment.

— Actually, I was one of those students. Some of my friends said, “if you go into the Department of Mathematics, you’ll just wind up as a teacher.” However, since the mind training in mathematics differs from that in other subjects, we don’t have to worry about academic paths. Mathematicians wind up finding employment not only in the data and communication industries, but in insurance and finance, including many people who find work as actuaries.*
* Actuaries are people who engage in the profession of analyzing the cost of uncertainty and future risk. Also referred to as “statisticians.”

Q. To wind things up, please give us a message for students thinking of entering the Department of Mathematics.

— Mathematics is the best subject possible for training the mind. I recommend it for people who are interested in thinking about fundamental problems. While you’re young, you should not worry too much about whether your studies will prove “useful.” If you are the type of person who likes to work to the benefit of others, no matter field you choose, your studies will bear fruit.
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