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[小资料] 朗兰兹纲领 Langlands program

2024-4-30 22:43

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[小资料] 朗兰兹纲领 Langlands program

朗兰兹纲领： Langlands program, Langlands programme

朗兰兹： Robert Phelan Langlands

埃尔朗根纲领： Erlangen program, Erlangen Programme

希尔伯特问题： Hilbert problems

布尔巴基： Bourbaki

普朗克： Max Karl Ernst Ludwig Planck

nha toan hoc Robert Langlands.jpg

图1 朗兰兹 Robert Phelan Langlands, 1936-10-06 ~

https://www.hieuhoc.com/media/Photos/nha%20toan%20hoc%20Robert%20Langlands.jpg

图2 朗兰兹 Robert Phelan Langlands, 1936-10-06 ~

https://www.ias.edu/sites/default/files/images/featured-thumbnails/ideas/Langlands_2.png

https://www.ias.edu/ideas/2018/dijkgraaf-rosetta-stone

Robert Langlands drew on the work of Hermann Weyl (above), André Weil HW-SM1.jpg

图3 In his conjectures, now collectively known as the Langlands program, Robert Langlands drew on the work of Hermann Weyl (above), André Weil, and Harish-Chandra, among others with extensive ties to the Institute.

https://www.ias.edu/sites/default/files/styles/one_column_large/public/images/featured-thumbnails/ideas/HW-SM1.jpg?itok=Bh77pEMp

https://www.ias.edu/ideas/modern-mathematics-and-langlands-program

一、朗兰兹纲领, Langlands program / programme

第一次知道“朗兰兹纲领 Langlands programme”，是1998年阅读西蒙·辛格（Simon Singh）的《费马大定理: 一个困惑了世间智者358年的谜 Fermat's last theorem: The story of a riddle that confounded the world's greatest minds for 358 years》，里面有“谷山-志村猜想 Shimura-Taniyama conjecture”与朗兰兹纲领的关系；以及朗兰兹纲领作为数学大统一途径的可能性。

可惜，朗兰兹纲领实在是个太专业的问题。到现在我也弄不太清楚。网上搜集了些资料，慢慢看吧。

“黎曼假设 Riemann Hypothesis”不打算研究了；只思考“素数的分布”吧！

Riemann Hypothesis

https://www.claymath.org/millennium/riemann-hypothesis/

(Enrico Bombieri)

怕将来的研究结果不容易发表，只好“拉大旗作虎皮，包着自己，去吓唬别人。”（鲁迅《且介亭杂文末编·答徐懋庸并关于抗日统一战线问题》）

看上去“朗兰兹纲领 Langlands programme”是面“倍有面子”的大旗，陆续熟悉一下吧！

从历史看，通向重要数学定理或理论的道路，往往不止一条：未来的发展可能是，条条大路通向数学的大统一。

就像“正态分布”就有至少4种不同的推导。

二、一看“数学”的“统一”，我就开心！

因为，俺的偶像爱因斯坦花了不少时间搞“引力、电磁”相互作用的统一！

以至于，看到“数学”的“统一”，也很开心！偶像的统一思想跨界了！

据说，当初布尔巴基 Bourbaki 也有将数学“统一”的意思。虽然至俺今也没看明白。实际上俺也没花精力去看：只是看了介绍布尔巴基的一些科普文献。

现在要思考素数的分布了，没准真的和“朗兰兹纲领, Langlands program, Langlands programme”有关系。所以想真的看看。至少“观看”一下。

Robert Langlands Letter 1967 to André Weil, IAS

Robert Langlands, 1967, Letter to André Weil, Institute for Advanced Study

https://publications.ias.edu/letter-to-weil

A Letter to Weil, part 2 - Institute for Advanced Study

https://publications.ias.edu/book/export/html/2607

真傻年事已高，行将就木。不知来。特此说明。

三、1933年2月17日，普朗克在柏林为德国工程师协会所做演讲中说：

“科学是内在的整体，它被分解为单独的整体不是取决于事物的本身，而是取决于人类认识能力的局限性。实际上存在着从物理到化学，从生物学和人类学到社会学的连续的链条，这是任何一处都不能被打断的链条。”

Denn die Wissenschaft bildet nun einmal sachlish genormmen eine innerkich geschlossene Einheit. Ihre Trennung nach verschiedenen Fächern ist ja nicht der Natur der Sache Begründet, sondern entspingt nur der Begrenztheit des menschlishen Fassungs, vermögens, welche zwangsläufig zu einer Arbeitsheilung führt. In der Tat zieht sich ein kontinuierliches Band von der Physik und Chemie über die Biologie und Anthropologie bis zu den sozialen und Geisteswissenschaften, ein Band, das sich an keiner Stelle ohne Willkür durchschneiden läßt.

根据《王媛. 为一段流浪的名人名言找到故乡[J]. 图书馆建设, 2019, (2): 158-162. 发布日期: 2019-03-26.》第 162 页图片“这一页就有这句的德文原文！”录入的“科学是内在的整体。它被分解为单独的部门不是取决于事物的本质，而是取决于人类认识能力的局限性。实际上存在着由物理学到化学，通过生物学和人类学到社会科学的连续的链条。”

道德楷模希尔伯特也说过：“数学是一个有机体，其生命力的必要条件是各部分不可分割的结合。”

As Hilbert put it, "Mathematics is an organism for whose vital strength the indissoluble union of the parts is a necessary condition."

参考资料：

[1] (英)西蒙·辛格著. 薛密翻译. 费马大定理: 一个困惑了世间智者358年的谜[M]. 上海：上海译文出版社，1998.

费马大定理 Simon Singh 1998 裁剪.png

fermat-s-last-theorem-the-story-of-a-riddle-that-confounded-the-world-s-greatest.jpg

https://librariaarcana.ro/8373-thickbox_default/fermat-s-last-theorem-the-story-of-a-riddle-that-confounded-the-world-s-greatest-minds-for-358-years-simon-singh.jpg

[2] 2022-12-23，朗兰兹，R. /Langlands,Robert/胡作玄，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=419008&Type=bkzyb&SubID=61734

朗兰兹最重要贡献是制定朗兰兹纲领，该纲领是把数论、群表示论、非交换调和分析与自守形式论结合在一起的理论体系，它推广了阿贝尔类域论、海克理论、自守函数论以及可约群的表示论等，其中包括大量的猜想，但只有一些特殊的情形获证。

[3] Robert Phelan Langlands, MacTutor History of Mathematics

https://mathshistory.st-andrews.ac.uk/Biographies/Langlands/

[4] 中国科学院，2016-03-21，【紫光阁】朗兰兹纲领：一项伟大的数学工程

https://www.cas.cn/cm/201603/t20160321_4550124.shtml

[5] 韩扬眉. 朗兰兹纲领：一项伟大的数学工程[N]. 中国科学报, 2020-08-21 第1版要闻.

https://news.sciencenet.cn/dz/dzzz_1.aspx?dzsbqkid=34916

https://bbs.sciencenet.cn/plus.php?mod=iframelogin

https://news.sciencenet.cn/dz/dzzz_1.aspx?dzsbqkid=34916

[6] 2023-08-18，埃尔朗根纲领/Erlangen program/几何学分支编写组，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=318393&Type=bkzyb&SubID=149704

[7] Erlangen program. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Erlangen_program

[8] 2022-12-23，布尔巴基/Bourbaki/胡作玄，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=418792&Type=bkzyb&SubID=61734

布尔巴基学派的主要思想是主张数学的统一性，认为数学结构是数学统一性的基础。数学的基本结构可分为代数结构（群、环、域）、序结构（格）、拓扑结构三大类型，两种或多种结构可以复合成更为复杂的结构，如拓扑群、拓扑向量空间乃至微分流形和李群等。通过结构的分析，数学的各个分支也就在统一数学的框架之内，形成一个严整的体系。

[9] Nicolas Bourbaki, MacTutor History of Mathematics

https://mathshistory.st-andrews.ac.uk/

Bourbaki: the pre-war years

https://mathshistory.st-andrews.ac.uk/HistTopics/Bourbaki_1/

Bourbaki's decision to use the axiomatic method throughout brought with it the necessity of a new arrangement of mathematics' various branches. It proved impossible to retain the classical division into analysis, differential calculus, geometry, algebra, number theory, etc. Its place was taken by the concept of structure, which allowed definition of the concept of isomorphism and with it the classification of the fundamental disciplines within mathematics.

For example there are algebraic structures, order structures, and topological structures. All three of these structures are present in the concept of the real numbers, for example, and certainly not in an independent way but interlinked in a complex fashion. Also it was decided that Bourbaki would never generalise from special cases but would always deduce special cases from the most general. The consequence of this approach was a strong logical ordering on the way that the mathematical building was constructed.

Bourbaki: the post-war years

https://mathshistory.st-andrews.ac.uk/HistTopics/Bourbaki_2/

Bourbaki struggled in the seventies and the eighties to formulate new directions. [There was] a failed project about several complex variables. There were attempts at homotopy theory, at spectral theory of operators, at the index theorem, at symplectic geometry. But none of these projects went beyond a preliminary stage. Bourbaki could not find a new outlet, because they had a dogmatic view of mathematics: everything should be set inside a secure framework. That was quite reasonable for general topology and general algebra, which were already solidified around 1950. Most people agree now that you do need general foundations for mathematics, at least if you believe in the unity of mathematics. But I believe now that this unity should be organic, while Bourbaki advocated a structural point of view.

[10] 朗兰兹纲领，数学术语，百度百科

https://baike.baidu.com/item/%E6%9C%97%E5%85%B0%E5%85%B9%E7%BA%B2%E9%A2%86?fromModule=lemma_search-box

[11] 数学中「神奇」的大统一理论——朗兰兹纲领 - 知乎，2018

https://zhuanlan.zhihu.com/p/100696067

[12] 2022-12-23，希尔伯特问题/Hilbert’s problems/胡作玄，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=418952&Type=bkzyb&SubID=61734

[13] Hilbert problems. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Hilbert_problems

Hilbert's ninth problem.

Proof of the most general law of reciprocity in any number field

Solved by E. Artin (1927; see Reciprocity laws). See also Class field theory, which also is relevant for the 12th problem. The analogous question for function fields was settled by I.R. Shafarevich (the Shafarevich reciprocity law, 1948); see [a46]. All this concerns Abelian field extensions. The matter of reciprocity laws and symbols for non-Abelian field extensions more properly fits into non-Abelian class field theory and the Langlands program, see also below.

Hilbert's twelfth problem.

Extension of the Kronecker theorem on Abelian fields to any algebraic realm of rationality.

For Abelian extensions of number fields (more generally, global fields and also local fields) this is (more or less) the issue of class field theory. For non-Abelian extensions, i.e. non-Abelian class field theory and the much therewith intertwined Langlands program (Langlands correspondence, Langlands–Weil conjectures, Deligne–Langlands conjecture), see e.g. [a25], [a27]. See also [a21] for two complex variable functions for the explicit generation of class fields.

[14] Shimura-Taniyama conjecture. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Shimura-Taniyama_conjecture

Let E be an elliptic curve over the rational numbers, and let L(E,s) denote its Hasse–Weil L-series. The curve E is said to be modular if there exists a cusp form f of weight 2 on Γ0(N), for some N, such that L(E,s)=L(f,s). The Shimura–Taniyama conjecture asserts that every elliptic curve over Q is modular. Thus, it gives a framework for proving the analytic continuation and functional equation for L(E,s). It is prototypical of a general relationship between the L-functions attached to arithmetic objects and those attached to automorphic forms (cf. also Automorphic form), as described in the far-reaching Langlands program.

[15] A.W. Knapp, "Introduction to the Langlands program" T.N. Bailey (ed.) et al. (ed.) , Representation theory and automorphic forms , Amer. Math. Soc. (1997) pp. 245–302

A. W. Knapp. Introduction to the Langlands Program [J]. Proceedings of Symposia in Pure Mathematics, 1997, 61: 245–302.

http://virtualmath1.stanford.edu/~conrad/JLseminar/refs/Knappintro.pdf

[16] BOOK REVIEWS. Introduction to the Langlands program, by J. Bernstein and S. Gelbart (Editors), with contributions by D. Bump, J. W. Cogdell, D. Gaitsgory, E. de Shalit, E. Kowalski, S. S. Kudla, Birkh¨auser, Boston, 2003, x + 281 pp., $39.95, ISBN 0-8176-3211-5. BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY, 2004, Volume 41, Number 2, Pages 257–266.

https://www.ams.org/journals/bull/2004-41-02/S0273-0979-04-01007-9/S0273-0979-04-01007-9.pdf

[17] Weisstein, Eric W. "Langlands Program." From MathWorld--A Wolfram Web Resource.

https://mathworld.wolfram.com/LanglandsProgram.html

[18] Kelly Devine Thomas, Modern Mathematics and the Langlands Program, 2010, Institute for Advanced Study

https://www.ias.edu/ideas/modern-mathematics-and-langlands-program

[19] Alex Kontorovich, What Is the Langlands Program?, 2022-06-01, Institute for Advanced Study

https://www.ias.edu/news/2022/what-is-the-langlands-program

相关链接：

[1] 2024-04-28，[资源，统一场，P vs NP] 何为相等？

https://blog.sciencenet.cn/blog-107667-1431879.html

[2] 2021-08-06，谷山丰（谷山豊，Yutaka Taniyama），不应该忘却的纪念

https://blog.sciencenet.cn/blog-107667-1298658.html