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９回 日時： ２０１４年６月１３日（金） 　　　　　　１６：３０−１８：００ 場所： 数理解析研究所　４２０号室 講師： HELMKE, Stefan 助教 題目： On Solutions of Polynomial Equations 要約： In the early 16th century, the Italian mathematicians Scipione del Ferro and Niccolò Tartaglia independently found a formula, expressing the roots of a polynomial equation of degree three in terms of radicals, now called Cardano's Formula. Shortly after that, the equation of degree four was solved in a similar way by Lodovico Ferrari. This started a 200 years hunt for a solution of the equation of degree five. In the 18th century, Leonard Euler found a new way to solve the equation of degree four and he thought, he could solve the equation of degree five then too, but failed. This led Joseph Louis Lagrange, in a famous paper, to analyze the reasons, why all of the methods, which were successful in solving polynomial equations of degree four and lower, failed for degrees five and higher. While he himself did not attempt to show that such a solution cannot exist, his work eventually inspired Paolo Ruffini, Niels Henrik Abel and Évariste Galois to prove exactly this. Besides this slightly disappointing result, Lagrange's paper was enormously important for the future developments in algebra. In this lecture, we will learn about Lagrange's beautiful ideas, how to solve polynomial equations. References: L. Euler, Elements of Algebra, Fifth Edition, London, 1840. J. L. Lagrange, Réflexions sur la résolution algebrique des équations, Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin, 1770. P. Pesic, Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability, MIT Press, 2003. "http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"