 # Introduction

## What is mathematics?

Mysterious relations among various numbers and the beautiful nature of space figures and shapes offer us perpetual fascination. For example, Euler's theorem asserts that, for any convex polyhedron, (the number of sides) - (the number of edges) + (the number of vertices) = 2. As a matter of fact, a similar theorem is known to hold also for polyhedra with holes, dents and bulges. Mathematics is a discipline used in the quest for such mysterious but beautiful relations thoroughly and deeply.

You must have studied differentiation and integration in high school. As you might remember, Isaac Newton discovered calculus. Given the travel distance of a body as a function of time, its differentiation produces velocity; further differentiation yields acceleration. In this manner, one aspect of mathematics is to describe various phenomena in nature.

On the other hand, it is often the case that a theory invented by a mathematician conversely contributes to explanation of an actual physical phenomenon. For example, Bernhard Riemann, mathematician in the 19th century, invented geometry in a curved space, i.e. Riemannian geometry. Albert Einstein applied it successfully to describe the general theory of relativity in the 20th century.

Such a relation with mathematics is found not only in physics but in chemistry, biology, information science, engineering, social science, and other fields. For example, the binary number system notation that is so commonly used for computer science was discovered in research of pure mathematics; cryptography is based on number theory. The importance of mathematics in various academic fields is likely to continue to increase.

Many outstanding research investigations have been conducted at the Mathematical Institute since its foundation in 1911. Cutting-edge research results have been produced continually, e.g., by Professor Tadao Tannaka, who is famous for the Tannaka duality theorem, and Professor Shigeo Sasaki, known for the theory of Sasaki manifolds. Moreover, active study is in progress today in various fields, such as analysis, algebra, and geometry, in rivalry with researchers all over the world.

The outstanding characteristics of the Mathematical Institute include a collection of mathematical literature with value and scale that lead the country. Over 60,000 foreign and Japanese books and journals are kept in the stacks of the Mathematical Archives on the third floor of the mathematics building. It is a truly comfortable environment with no inconvenience for studying mathematics. The Mathematical Institute publishes the Tohoku Mathematical Journal, an academic journal specialized in mathematics. This prestigious journal is on shelves in libraries throughout the world. It was published as the first European-language magazine of mathematics in Japan at the same time as the foundation of the Mathematical Institute.

### Curriculum

Aside from mathematics courses in interdisciplinary education, Introductory Mathematics A [basic mathematics] is held during Semester 1 as a special course for first year students of the Mathematical Institute. This course is designed so that students can advance smoothly from high school mathematics to college mathematics, partly by making students solve problems in class. Introductory Mathematics B [set theory], held during Semester 2, explains the notion of the infinite set, the most fundamental subject for studying modern mathematics.

Fundamental knowledge is studied in courses by the third year to explain modern mathematics comprehensively. See the curriculum in the attached table for detailed course guides. Courses for senior students range over various fields that are more specialized than the courses taken during the third year. In addition to these usual courses, some intensive courses are held by instructors from other universities, through which students can get a more varied taste of modern mathematics.

A seminar is scheduled in the fourth year. Using textbooks of their specialty written in English and under instruction of faculty, students conduct studies for a year in small groups of about five. This seminar is a requisite course is the most important subject at the Mathematical Institute.