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９回

日時：２０１６年６月１０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： A Brief History of Implicit Functions

要約：
An implicit function is a function that is implicitly defined as the solution of an equation with parameters. For example, if F (x, y) is a function of two variables, we may consider the first variable x as a parameter and try to find a function h (x) depending only on this parameter x, such that the equation F (x, h (x))=0 is fulfilled, for all x. In other words, we try to solve the equation F (x, y) = 0 in the form y = h (x). The Implicit Function Theorem gives sufficient conditions for the equation F, which guarantee the existence of a solution h locally. The modern version of this theorem was first proved by Ulisse Dini (1845--1918) in the 1870's. But already Isaac Newton (1642--1727) considered this problem in the 17th century and gave for h an elegant fractional power series solution, the so-called Puiseux Expansion (named after the French mathematician Victor Alexandre Puiseux (1820--1883).)
In this lecture I will explain Newton's solution to the problem, the historic developments with many examples, which led to Dini's theorem and some further developments.
References：
Section 2 of the first reference contains a short history of the Implicit Function Theorem, mostly accessible also for beginners. The second reference, Section (III.8.3), contains a nice account on the Puiseux Expansion.

Steven G. Krantz and Harold R. Parks, The Implicit Function Theorem, History, Theory, and Applications, Birkhäuser, 2002.

Egbert Brieskorn and Horst Körrer, Plane Algebraic Curves, Birkhäuser, 1986.

"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９回
日時：	２０１６年６月１０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：	A Brief History of Implicit Functions
要約：	An implicit function is a function that is implicitly defined as the solution of an equation with parameters. For example, if F (x, y) is a function of two variables, we may consider the first variable x as a parameter and try to find a function h (x) depending only on this parameter x, such that the equation F (x, h (x))=0 is fulfilled, for all x. In other words, we try to solve the equation F (x, y) = 0 in the form y = h (x). The Implicit Function Theorem gives sufficient conditions for the equation F, which guarantee the existence of a solution h locally. The modern version of this theorem was first proved by Ulisse Dini (1845--1918) in the 1870's. But already Isaac Newton (1642--1727) considered this problem in the 17th century and gave for h an elegant fractional power series solution, the so-called Puiseux Expansion (named after the French mathematician Victor Alexandre Puiseux (1820--1883).) In this lecture I will explain Newton's solution to the problem, the historic developments with many examples, which led to Dini's theorem and some further developments. References： Section 2 of the first reference contains a short history of the Implicit Function Theorem, mostly accessible also for beginners. The second reference, Section (III.8.3), contains a nice account on the Puiseux Expansion. Steven G. Krantz and Harold R. Parks, The Implicit Function Theorem, History, Theory, and Applications, Birkhäuser, 2002. Egbert Brieskorn and Horst Körrer, Plane Algebraic Curves, Birkhäuser, 1986.
"http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"