そういえば、層のHistoryがあったのを思い出したので、貼っておきます ご指摘は、こちらかも ”1951 The Cartan seminar proves theorems A and B, based on Oka's work”が、いま問題の話ですね
(参考) https://en.wikipedia.org/wiki/Sheaf_(mathematics) Sheaf (mathematics) History The first origins of sheaf theory are hard to pin down – they may be co-extensive with the idea of analytic continuation[clarification needed]. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology. ・1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering. ・1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined cochains. ・1943 Norman Steenrod publishes on homology with local coefficients.[18] ・1945 Jean Leray publishes work carried out as a prisoner of war, motivated by proving fixed-point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences.[19] ・1947 Henri Cartan reproves the de Rham theorem by sheaf methods, in correspondence with André Weil (see De Rham–Weil theorem). Leray gives a sheaf definition in his courses via closed sets (the later carapaces). ・1948 The Cartan seminar writes up sheaf theory for the first time. ・1950 The "second edition" sheaf theory from the Cartan seminar: the sheaf space (espace étalé) definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, in several complex variables. ・1951 The Cartan seminar proves theorems A and B, based on Oka's work. ・1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Jean-Pierre Serre,[20] as is Serre duality. 以下略（この倍くらいある） 0082１３２人目の素数さん2024/01/16(火) 13:43:33.30ID:Ai7YhS3I 追加メモ https://en.wikipedia.org/wiki/List_of_important_publications_in_mathematics List of important publications in mathematics
Algebraic geometry Faisceaux Algébriques Cohérents Jean-Pierre Serre Publication data: Annals of Mathematics, 1955 FAC, as it is usually called, was foundational for the use of sheaves in algebraic geometry, extending beyond the case of complex manifolds.
Géométrie Algébrique et Géométrie Analytique Jean-Pierre Serre (1956) In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. (NB While analytic geometry as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.) The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre, now usually referred to as GAGA.
Éléments de géométrie algébrique Alexander Grothendieck (1960–1967) Written with the assistance of Jean Dieudonné, this is Grothendieck's exposition of his reworking of the foundations of algebraic geometry. It has become the most important foundational work in modern algebraic geometry. The approach expounded in EGA, as these books are known, transformed the field and led to monumental advances.
https://en.wikipedia.org/wiki/Shiing-Shen_Chern Shiing-Shen Chern (/tʃɜːrn/; Chinese: 陳省身; pinyin: Chén Xǐngshēn, Mandarin: [tʂʰən.ɕiŋ.ʂən]; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet.
The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds.
History Cobordism had its roots in the (failed) attempt by Henri Poincaré in 1895 to define homology purely in terms of manifolds (Dieudonné 1989, p. 289). Bordism was explicitly introduced by Lev Pontryagin in geometric work on manifolds.
It came to prominence when René Thom showed that cobordism groups could be computed by means of homotopy theory, via the Thom complex construction. Cobordism theory became part of the apparatus of extraordinary cohomology theory, alongside K-theory. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the Hirzebruch–Riemann–Roch theorem, and in the first proofs of the Atiyah–Singer index theorem. 0137１３２人目の素数さん2024/01/23(火) 21:42:15.56ID:UvcFlUMz>>135 >幾何学との関係は、原点を取り除いた複素平面内の単位円板の被覆空間として見なすことができる。複素変数 z と考えると、円板の zn 写像により実現される有限被覆は、穴あき円板の基本群の部分群 n.Z に対応する。 https://ja.wikipedia.org/wiki/%E3%82%AC%E3%83%AD%E3%82%A2%E5%9C%8F#%E3%82%AC%E3%83%AD%E3%82%A2%E5%9C%8F%E6%88%90%E7%AB%8B%E3%81%AE%E7%B5%8C%E7%B7%AF0138１３２人目の素数さん2024/01/23(火) 23:15:44.27ID:VsIuTRfR 知られているすべてのガロア理論が ガロア圏の言葉で表現できるわけではない。 微分体のガロア理論である ピカール・ヴェシオ理論は ガロア圏上では展開できない。 それらのためにグロタンディークによる 淡中圏の理論が構成されている。 0139１３２人目の素数さん2024/01/24(水) 00:26:19.00ID:1i9Un+hN 上空移行の原理について 野口と福田が全然違う例を挙げているのが 興味深い 0140１３２人目の素数さん2024/01/24(水) 07:26:28.28ID:a5OrWVQ3>>139 >上空移行の原理について >野口と福田が全然違う例を挙げているのが >興味深い
(参考) https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture Poincaré conjecture Dimensions Main article: Generalized Poincaré conjecture In 1961, Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem. 0150１３２人目の素数さん2024/01/25(木) 04:45:57.80ID:KZ5ooiqY>>149 ＞コボルディズム理論（次元を一つあげて扱う） これは素人の馬鹿発言 ポイントは、２つの多様体に対して、 「両者を境界とする多様体が存在する」 という性質で類別すること 次元を上げることではない 0151１３２人目の素数さん2024/01/25(木) 05:18:04.32ID:G4VNJ8Al>>150 一次元の円周を境界とする多様体は二次元 0152１３２人目の素数さん2024/01/25(木) 05:54:22.50ID:KZ5ooiqY>>151 だから「ただ１次元上げることに意味がある」というのは馬鹿素人 ２つの多様体を境界にもつ多様体が存在する、というのが重要 0153１３２人目の素数さん2024/01/25(木) 07:23:12.99ID:5C2qXv8g>>152 馬鹿素人（笑 0154１３２人目の素数さん2024/01/25(木) 08:46:15.95ID:G4VNJ8Al>>153 微分トポロジーの有名研究者の言う 上空移行の意味はそういうこと 0155１３２人目の素数さん2024/01/25(木) 08:48:19.54ID:G4VNJ8Al>>152
(参考) https://en.wikipedia.org/wiki/H-cobordism h-cobordism In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps M → W and N → W are homotopy equivalences.
The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds.
The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture.
Background Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The h-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "Whitney trick" of Hassler Whitney, which geometrically untangles homologically-tangled spheres of complementary dimension in a manifold of dimension >4. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that the trick fails to work in lower dimensions, which have no room for entanglement. (引用終り)
寄り道ですが １）グロタンディクの分解原理は、別にリンクあり（下記） これは、1957 "American Journal of Mathematics"で、彼がアメリカ滞在時の仕事なのだろう ２）かれは、1955〜1957年にアメリカにいて、"Tôhoku paper"を書いた。フランス国籍がなく仏ではアカデミックポストは困難だった ”In 1957 he was invited to visit Harvard by Oscar Zariski”とあるが、he refused to sign a pledge promising not to work to overthrow the United States government（機械訳：アメリカ合衆国政府を転覆させるために働かないと約束する誓約書への署名を彼が拒否した） のでダメになったという ３）1958 IHÉSへ。IHÉSは、無国籍のグロタンディクのために作られたという
(参考) https://en.wikipedia.org/wiki/Birkhoff%E2%80%93Grothendieck_theorem (Redirected from Grothendieck splitting principle) Birkhoff–Grothendieck theorem In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over CP^1 is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck (1957, Theorem 2.1),[1] and is more or less equivalent to Birkhoff factorization introduced by George David Birkhoff (1909).[2] References 1. Grothendieck, Alexander (1957). "Sur la classification des fibrés holomorphes sur la sphère de Riemann". American Journal of Mathematics. 79 (1): 121–138.
https://en.wikipedia.org/wiki/Alexander_Grothendieck Alexander Grothendieck Studies and contact with research mathematics In Nancy, he wrote his dissertation under those two professors on functional analysis, from 1950 to 1953.[29] At this time he was a leading expert in the theory of topological vector spaces.[30] In 1953 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he had refused to take French nationality (as that would have entailed military service against his convictions). He stayed in São Paulo (apart from a lengthy visit in France from October 1953 - March 1954) until the end of 1954. His published work from the time spent in Brazil is still in the theory of topological vector spaces; it is there that he completed his last major work on that topic (on "metric" theory of Banach spaces).
Grothendieck moved to Lawrence, Kansas at the beginning of 1955, and there he set his old subject aside in order to work in algebraic topology and homological algebra, and increasingly in algebraic geometry.[31][32] It was in Lawrence that Grothendieck developed his theory of Abelian categories and the reformulation of sheaf cohomology based on them, leading to the very influential "Tôhoku paper".[33]
In 1957 he was invited to visit Harvard by Oscar Zariski, but the offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government—a refusal which, he was warned, threatened to land him in prison. The prospect of prison did not worry him, so long as he could have access to books.[34]
IHÉS years In 1958, Grothendieck was installed at the Institut des hautes études scientifiques (IHÉS), a new privately funded research institute that, in effect, had been created for Jean Dieudonné and Grothendieck. (引用終り) 以上 0160１３２人目の素数さん2024/01/25(木) 11:57:53.00ID:glB93F6O>>157 馬鹿素人でなければそのように理解するでしょう 0161１３２人目の素数さん2024/01/25(木) 12:01:28.47ID:zxKJrX2I 補足 グロタンディークと圏論 これがピッタリの組み合わせだったのかも （下記”数学史 グロタンディーク”など ） (参考) https://twilog.togetter.com/Auf_Jugendtraum/month-1905/2 数学の歩みbot@Auf_Jugendtraum 2019年05月28日(火)24 tweetssource 5月28日@Auf_Jugendtraum 数学の歩みbot@Auf_Jugendtraum グロタンディークは，まるで川のない所に洪水を起こすような，バキュームクリナーに大きな機関車をつけて数学の世界を走る回るような人物だった．（広中平祐）
(参考) https://en.wikipedia.org/wiki/Surgery_theory Surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). A relatively easy argument using Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to the same cobordism class.[1] Attaching handles and cobordisms A surgery on M not only produces a new manifold M′, but also a cobordism W between M and M′. The trace of the surgery is the cobordism (W; M, M′), with 略
https://en.wikipedia.org/wiki/John_Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal, the Wolf Prize, the Abel Prize and all three Steele prizes.
