Departmental Overviews

## Departmental Overviews (Mathematics): The Language of Natural Sciences and the Pursuit of the Essence of Things

*-The Department of Mathematics-*

### Professor Tomohide Terasoma (Head of the Department of Mathematics)

The Department of Mathematics has a special position within the Faculty of Science and the University of Tokyo. Although it is an organization in the Faculty of Science, the Department of Mathematics is connected seamlessly to the Graduate School of Mathematical Sciences, which is an independent graduate school that conducts research on mathematics and mathematical sciences. From ancient times, mathematics has had an inseparable relationship with the development of natural science, including physics; however, modern mathematics is specialized into many fields. In the world of pure mathematics, there were many proofs and theorems that I had thought could never be solved during my lifetime when I was a student. However, Fermat's Last Theorem, which had been unsolved for over 300 years, was proved by Andrew Wiles and the Poincaré conjecture was solved by Perelman. There are also various other kinds of mathematics aside from pure mathematics and a remarkably wide range of topics in mathematics today. Mathematics tackles a vast range of problems ranging from interdisciplinary research in other natural sciences or social sciences to mathematics itself. Common features in mathematics may include approaches to researching the essence of things and common principles as well as research methodologies to determine whether a theorem is logical.

History of the Department of Mathematics

The Graduate School of Mathematical Sciences was established in 1992 when the Department of Mathematics in the Graduate School of Science and the Department of Mathematics in the College of Arts and Sciences became independent. Up until then, the Department of Mathematics in the Faculty of Science had been primarily focused on mathematics but is now strongly involved in a wide range of mathematical sciences. By exchanging their various perspectives with other researchers, researchers in pure mathematics at the Graduate School of Mathematical Sciences are greatly inspired and led to the broad world.

There are a great many mathematicians and researchers who have graduated from the Department of Mathematics in the Faculty of Science, of course, but there are also many graduates who were trained in the Graduate School of Mathematical Sciences and are now applying their various analyzing skills to a variety of occupations in the public sector, government, and education.

Curriculum

I will now introduce the curriculum of the Department of Mathematics in the Faculty of Science. Our curriculum focuses on developing students to master basic skills in mathematics.

** Basic mathematics for second year students**

In the winter semester, second year students learn basic mathematics. There are three main courses in the Department: Algebra & Geometry, Set Theory & Topology, and Complex Analysis. In Algebra & Geometry, students learn about elementary algebra; that is, the abstract algebraic structures of numbers and polynomials. This is considered to be a generalization of linear algebra. Abstraction reveals essential structures and simplifies arguments in proofs. One of the main notions introduced in Set Theory & Topology is topological space, which is the foundation of geometry. Topology is a language that can precisely and unambiguously formulate the abstraction of the concept of continuity. Complex analysis is a developed version of Calculus in the world of complex numbers. A remarkable feature is the fact that many theories gain their richness through development. In this way, as per the figure below, applied mathematics roughly consists of the fields of algebra, geometry and analysis. These fields are closely related and mutually develop one another. It takes a significant amount of time to learn to the level of application, similar to acquiring a second language. There is also a great difference between having a general understanding of these fields and being able to develop logical arguments that others can follow. In order to help students gain a solid understanding of these three fields, they are provided with opportunities to explain solutions to mathematical problems before an audience. As it is difficult for students to notice by themselves whether or not they have grasped the material, faculty members address any uncertainties they may have.

In the beginning of this article, I addressed mathematical research methods, but in terms of research in mathematics, it is essential to prepare for discussions with others or public speaking through practice, and this is likely a new experience for many students. Problem setting in a simple and precise manner can also be quite challenging. For these reasons, there is no doubt that having experience in explaining something accurately is useful even outside of mathematics.

** Seminars for fourth year students **

Image: The Main Conference Room in the Graduate School of Mathematical Sciences Building |

It goes without saying that the most important components in the Department of Mathematics curriculum are its seminars (Mathematics Lecture XA and the Special Lecture on Mathematics) for fourth year students. The field of mathematics can be broadly divided into applied mathematics, which can be further divided into algebra, geometry and analysis, and their related boundary fields. Seminars for fourth year students focus on higher-level specialist texts and students can select which seminars to attend. In general, students have one lecture a week and in the seminar, students also conduct lectures for the instructor in charge. The instructor will point out if they notice even the slightest ambiguities, or long, unnecessary explanations. Furthermore, depending on the instructor, there may be assignments related to upcoming seminars. This volume of work requires students to dedicate an ample amount of time over the course of a week to prepare.

As mentioned earlier, in terms of research in mathematics, it is important to read academic articles and bring awareness to problems that you recognize and also present your theories or methodologies. These are critical elements when it comes to your research. The Department of Mathematics is located in the same building as the Graduate School of Mathematical Sciences and throughout, you will likely see flyers of workshops. Even if you don’t understand the content, the lecture titles alone can elicit excitement.

If you’re someone who simply loves mathematics or wants to use mathematics to discover the cause behind various phenomena, I think it would be worthwhile to consider joining the Department of Mathematics.

▶︎ For more information, please visit the Department of Mathematics homepage:

http://www.ms.u-tokyo.ac.jp/index.html

― *This article is from the 2018 "Departmental Overviews in the School of Science" series in The Rigakubu News* ―

*Translated by Kristina Awatsu, Office of Communication*

― Office of Communication ―