# 大学院教育・講義

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## 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４回 日時： ２０１８年５月１１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 the Newton-Puiseux Series 要約： A plane algebraic curve is the set of pairs (x,y) satisfying a relation f(x,y)=0, where f is a polynomial in two variables. In 1669, Isaac Newton (1642--1727) published an interesting method to describe such a curve locally around one of its points, in which y is expressed as a fractional power series in x, (if f contains some power of y.) This series is now called the Newton-Puiseux series, named after Newton himself of course, and the French mathematician Victor Alexandre Puiseux (1820--1883), who in 1850 wrote the first more formal treatise on this subject. For practical purposes -- certainly Newton's original motivation -- the series had never been of great importance, but theoretically it played an important role in the foundations of algebraic geometry at least until the 1960's. In this lecture, I will explain Newton's original idea with some examples and how his method developed over the centuries. References： The following two books include an elementary introduction to plane algebraic curves, in particular the Newton-Puiseux series. Egbert Brieskorn and Horst Knörrer, Plane Algebraic Curves, Birkhäuser, 1986. In particular Section (III.8.3). Robert J. Walker, Algebraic curves Springer-Verlag, 1978, (originally published by Princeton University Press in 1950). Newton's original papers can be found in his collected works together with English translations. The second is a refinement of the first, but was only published posthumous. Unfortunately, there seems to be no English translation of Puiseux's paper. Isaac Newton, De analysi per æquationes numero terminorum infinitas, 1669. Isaac Newton, De methodis serierum et fluxionum, 1671. Victor A. Puiseux, Recherches sur les fonctions algébriques, J. Math. Pures Appl. 15, 1850, 365--480. "http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"

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