全学共通科目

　　全学共通科目講義（１回生～４回生対象）

現代の数学と数理解析

　　――　基礎概念とその諸科学への広がり

授業のテーマと目的：
数学が発展してきた過程では、自然科学、社会科学などの種々の学問分野で提起される問題を解決するために、既存の数学の枠組みにとらわれない、新しい数理科学的な方法や理論が導入されてきた。また、逆に、そのような新しい流れが、数学の核心的な理論へと発展した例も数知れず存在する。このような数学と数理解析の展開の諸相について、第一線の研究者が、自身の研究を踏まえた入門的・解説的な講義を行う。
数学・数理解析の研究の面白さ・深さを、感性豊かな学生諸君に味わってもらうことを意図して講義し、原則として予備知識は仮定しない。

第６回

日時：２０２１年５月１４日（金）
　　　　　　１６：４５－１８：１５

場所：オンライン
https://panda.ecs.kyoto-u.ac.jp/portal/site/2021-888-N114-001

講師： Helmke, Stefan 助教

題目： Euler and the Basel Problem

要約：
It had been known at least since the 14th century that the harmonic series diverges. But in mid 17th century, it had been observed that, in contrast, the sum of inverse squares converges. However, it converges very slowly and it is therefore rather difficult to compute. So to find the value of this sum was one of the most challenging problems in mathematics of that time. Many famous mathematicians tried to lay their hands on it and failed, including the two Bernoulli brothers from Basel. Then, with the appearance of Leonhard Euler, who also came from Basel and was a student of Johann Bernoulli, the younger of the two brothers, the situation unexpectedly changed and Euler's name is inseparably connected with the solution of this problem, which is thus known as the `Basel Problem'.
In my lecture, I will first briefly review the history of the problem (from Euler's perspective) and then explain his most important contributions to its solution. This will naturally lead to new, even much more interesting problems and I will conclude with a view into the future (again, from Euler's perspective), by shortly reviewing Bernhard Riemann's paper from 1859 on the distribution of prime numbers with its still unsolved `Riemann Hypothesis', one of the most challenging problems of our time!
References: An easily readable account on the Basel Problem is contained in Edward Sandifer's article [2] as well as in Raymond Ayoub's article [3]. A more comprehensive, but also more difficult account can be found in André Weil's book [1]. Also, still readable, is Euler's original book [4], first published in 1748 in Latin, but with many existing translations, including the English one below.

André Weil, Number Theory; An approach through history; From Hammurapi to Legendre. Birkhäuser, 1983. (Chap. III.17-20.)

C. Edward Sandifer, The Early Mathematics of Leonhard Euler, The Mathematical Association of America, 2007. (Chap. 7, 21 and 32.)

Raymond Ayoub, Euler and the Zeta Function, American Mathematical Monthly, Vol. 81 (1974), 1067-1086. This article is reprinted in The Genius of Euler, Reflections on his Life and Work, William Dunham (Editor), The Mathematical Association of America, 2007.

Leonhard Euler, Introduction to analysis of the infinite, book I, translated by John D. Blanton, Springer, 1988. (Chap. 9-11.)

"http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"

授業のテーマと目的：数学が発展してきた過程では、自然科学、社会科学などの種々の学問分野で提起される問題を解決するために、既存の数学の枠組みにとらわれない、新しい数理科学的な方法や理論が導入されてきた。また、逆に、そのような新しい流れが、数学の核心的な理論へと発展した例も数知れず存在する。このような数学と数理解析の展開の諸相について、第一線の研究者が、自身の研究を踏まえた入門的・解説的な講義を行う。数学・数理解析の研究の面白さ・深さを、感性豊かな学生諸君に味わってもらうことを意図して講義し、原則として予備知識は仮定しない。
第６回
日時：	２０２１年５月１４日（金）　　　　　　１６：４５－１８：１５
場所：	オンライン https://panda.ecs.kyoto-u.ac.jp/portal/site/2021-888-N114-001
講師：	Helmke, Stefan 助教
題目：	Euler and the Basel Problem
要約：	It had been known at least since the 14th century that the harmonic series diverges. But in mid 17th century, it had been observed that, in contrast, the sum of inverse squares converges. However, it converges very slowly and it is therefore rather difficult to compute. So to find the value of this sum was one of the most challenging problems in mathematics of that time. Many famous mathematicians tried to lay their hands on it and failed, including the two Bernoulli brothers from Basel. Then, with the appearance of Leonhard Euler, who also came from Basel and was a student of Johann Bernoulli, the younger of the two brothers, the situation unexpectedly changed and Euler's name is inseparably connected with the solution of this problem, which is thus known as the `Basel Problem'. In my lecture, I will first briefly review the history of the problem (from Euler's perspective) and then explain his most important contributions to its solution. This will naturally lead to new, even much more interesting problems and I will conclude with a view into the future (again, from Euler's perspective), by shortly reviewing Bernhard Riemann's paper from 1859 on the distribution of prime numbers with its still unsolved `Riemann Hypothesis', one of the most challenging problems of our time! References: An easily readable account on the Basel Problem is contained in Edward Sandifer's article [2] as well as in Raymond Ayoub's article [3]. A more comprehensive, but also more difficult account can be found in André Weil's book [1]. Also, still readable, is Euler's original book [4], first published in 1748 in Latin, but with many existing translations, including the English one below. André Weil, Number Theory; An approach through history; From Hammurapi to Legendre. Birkhäuser, 1983. (Chap. III.17-20.) C. Edward Sandifer, The Early Mathematics of Leonhard Euler, The Mathematical Association of America, 2007. (Chap. 7, 21 and 32.) Raymond Ayoub, Euler and the Zeta Function, American Mathematical Monthly, Vol. 81 (1974), 1067-1086. This article is reprinted in The Genius of Euler, Reflections on his Life and Work, William Dunham (Editor), The Mathematical Association of America, 2007. Leonhard Euler, Introduction to analysis of the infinite, book I, translated by John D. Blanton, Springer, 1988. (Chap. 9-11.)
"http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"