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２回 日時： ２０２２年４月１５日（金） 　　　　　　１６：４５−１８：１５ 場所： ４共１１（吉田南４号館） 講師： Helmke, Stefan 助教 題目： Euler and the Basel Problem 要約： In the middle of the 17th century Pietro Mengoli, successor of Bonaventura Cavalieri as professor of mathematics at the University of Bologna, noticed that while the harmonic series diverges, the sum of inverse squares is finite. But in contrast to the other infinite sums he considered, he was unable to compute this one, or even to find satisfying numerical approximations, since it converges so slowly. Mengoli's work was little known at his time, but other mathematicians independently arrived at the same problem, with little more success though. The development of the calculus partially improved the situation so that by the beginning of the 1730's a few methods to compute approximate values for the sum of inverse squares were known, one of them due to Leonhard Euler. And then, in 1735, he suddenly found the exact value! Though his argument was rather controversial and it took him another 7 years before he had a satisfying proof for the formula which led to his result, he did not stop here. In this class we will follow Euler's ideas concerning the sum of inverse squares, now known as the Basel Problem, supplemented with some new insight and later developments. References: An easily readable account on the Basel Problem is contained in Edward Sandifer's book [2]. 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 [3], first published in 1748 in Latin, but with many existing translations, including the English one below. Hiroshi Yuki's book [4] also contains an interesting chapter on the Basel Problem and the Euler Archive [5] has copies of most of Euler's original publications. André Weil, Number Theory; An approach through history; From Hammurapi to Legendre. Birkhäuser, 1983. (Chap. III.17-20.) Charles Edward Sandifer: The Early Mathematics of Leonhard Euler, The Mathematical Association of America, 2007. (Chap. 7, 21, 32 and 44.) Leonhard Euler: Introduction to analysis of the infinite, book I, translated by John D. Blanton, Springer, 1988. (Chap. 9-11.) 結城 浩: 数学ガール､ ソフトバンク クリエイティブ､ ２００７年｡ (第９章) Online Resources: The Euler Archive (http://eulerarchive.maa.org/). "http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"