[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む78 (1002ﾚｽ)

現代数学の系譜 工学物理雑談 古典ガロア理論も読む78 http://rio2016.5ch.net/test/read.cgi/math/1571400076/

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614: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/08(金) 16:46:01.10 ID:nN7QsxvT >>613 つづき (1) When was the statement of the monodromy theorem first fully formulated (resp. : proven)? (2) When did the normal form for elliptic curves y^2 = x(x ? 1)(x ? λ), which is by nowadays’ tradition called by many (erroneously?) ‘the Legendre normal form’ first appear? (3) The old ‘Jacobi inversion theorem’ is today geometrically formulated through the geometry of the ‘Jacobian variety J(C)’ of an algebraic curve C of genus g: when did this formulation clearly show up (and so clearly that, ever since, everybody was talking only in terms of the Jacobian variety)? The above questions not only deal with themes of research which were central to Weierstras’ work on complex function theory, but indeed they single out philosophically the importance in mathematics of clean formulations and rigorous arguments. Ath his point it seems appropriate to cite Caratheodory, who wrote so in the preface of his two volumes on ‘Funktionentheorie’ ([Car50]): ‘ The genius of B. Riemann (1826-1865) intervened not only to bring the Cauchy theory to a certain completion, but also to create the foundations for the geometric theory of functions. At almost the same time, K. Weierstras(1815-1897) took up again the above-mentioned idea of Lagrange’s 1 , on the basis of which he was able to arithmetize Function Theory and to develop a system that in point of rigor and beauty cannot be excelled. The Weierstras tradition was carried on in an especially pure form by A. Pringsheim (1850-1941), whose book (1925-1932) is extremely instructive.’ つづく http://rio2016.5ch.net/test/read.cgi/math/1571400076/614

615: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/08(金) 16:46:28.77 ID:nN7QsxvT >>614 つづき Then Caratheodory comments first on the antithesis: ‘During the last third of the 19th Century the followers of Riemann and those of Weierstras formed two sharply separated schools of thought.’2 and then on the sinthesis: ‘ However, in the 1870’s Georg Cantor (1845-1918) created the Theory of Sets. .. With the aid of Set Theory it was possible for the concepts and results of Cauchy’s and Riemann’s theories to be put on just as firm basis as that on which Weierstras ’ theory rests, and this led to the discovery of great new results in the Theory of Functions as well as of many simplifications in the exposition.’ (引用終り) 以上 http://rio2016.5ch.net/test/read.cgi/math/1571400076/615

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