Arithmetic Geometry - solving number theoretical problems using geometrical intuition

In the Department of Mathematical Sciences at Keio University, the Bannai Group, led by Professor Kenichi Bannai, is conducting research in number theory. Number theory, which deals with the properties of integers, is known as the "Queen of Mathematics." The Bannai Group is focused especially on arithmetic geometry. Arithmetic geometry utilizes methods and results from algebraic geometry. In this field, number theoretical problems are investigated via the geometric properties of geometric objects defined by algebraic equations. Q. "Humans perceive things in two ways, logically and intuitively. Logic involves calculating things precisely. On the other hand, when using intuition, especially geometric intuition, we look at a problem in a certain geometric way and immediately "know" the answer. If one asks how this can be applied to problems in number theory, for example, consider the problem of finding rational solutions of the equation x2 + y2 = 1. The problem of seeking rational solutions of an algebraic equation is a number theoretical problem. Geometry comes into the picture if one thinks of the equation x2 + y2 = 1 as expressing a unit circle. When I use the word geometric intuition, what I mean is, it is much easier to solve this problem if one thinks that the equation x2 + y2 = 1 is not simply an algebraic equation but also that it represents a circle." Using methods from arithmetic geometry, Andrew Wiles in 1995 solved the Fermat's Last Theorem, which had puzzled mathematicians for 300 years. Number theory can be applied for example to cryptography, which is an important practical application of number theory to society. Q. "Because number theory concerns integers s -- 1, 2, 3, and so on -- one might think it's a very narrow field. At first, I also imagined that number theory was a very narrow topic; I wanted to do mathematics, but number theory did not seem all that interesting. But when I learned that problems in number theory were deeply related to geometry and also to analysis through various interesting analytic functions, I realized that number theory was a very deep field related to a wide range of areas in mathematics. That's what's fascinating about number theory. Rather than being superficial, because number theory deals with integers, which is a very fundamental object of study, it is deeply related to many interesting theories in the forefront of mathematics." Since the dawn of civilization, numbers and equations were used to better understand natural phenomena. Through abstraction, mathematics has greatly expanded its range of application. The Bannai Group will continue to do research, in order to understand through logic and intuition abstract phenomena appearing in number theory.

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