[笔记，科普，资料] The Millennium Prize Problems 千禧年大奖难题

千禧年大奖难题： The Millennium Prize Problems

克雷数学研究所： Clay Mathematics Institute

未解问题： unsolved problems

https://www.claymath.org/millennium-problems/

一、The Millennium Prize Problems 千禧年大奖难题

https://www.claymath.org/millennium-problems/

   In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.

   The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France. Three lectures were presented: Timothy Gowers spoke on The Importance of Mathematics; Michael Atiyah and John Tate spoke on the problems themselves.

【机器翻译】，个别之处进行了人工修改。

   为了庆祝新千年的数学，马萨诸塞州剑桥克雷数学研究所（CMI）设立了七个奖题。这些奖项旨在记录数学家在第二个千年之交所面临的一些最困难的问题；提高公众的认识，在数学中，边界仍然是开放的，并且存在大量未解决的重要问题；强调努力解决最深刻、最困难问题的重要性；表彰具有历史意义的数学成就。

   2000年5月24日，在巴黎法兰西学院举行的一次会议上宣布了这些奖项。举办了三场讲座：Timothy Gowers谈到了数学的重要性；Michael Atiyah和John Tate亲自谈到了这些问题。

   《数学文化》里的介绍：千禧年大奖难题之始与未终

https://www.global-sci.org/intro/article_detail/mc/18380.html

   七个难题如下，顺序依照难题英文名称的字母序排列，括号中所列的是写下难题之具体陈述的作者姓名：

   (1) Birch-Swinnerton-Dyer 猜想 (Andrew Wiles)

   (2) Hodge 猜想 (Pierre Deligne)

   (3) Navier-Stokes 方程解的存在性与光滑性 (Charles Fefferman)

   (4) P/NP 问题 (Stephen Cook)

   (5) 庞加莱猜想 (John Milnor)

   (6) 黎曼假设 (Enrico Bombieri)

   (7) Yang-Mills 规范场存在性与质量间隙 (Arthur Jaffe 和Edward Witten)

   这七个难题中，黎曼假设是唯一一个在“希尔伯特问题”里就出现过的，已历经一百多年却仍然巍然屹立。它在山顶的风景如此令人迷醉，山间的雾霭却乱人眼眸，人们手执大斧披荆斩棘，却始终无法找到一条通往山巅的清晰途径。

二、Unsolved problems 未解问题

2.1  Birch and Swinnerton-Dyer Conjecture, Birch-Swinnerton-Dyer 猜想

https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/

2.2  Hodge Conjecture, Hodge 猜想

https://www.claymath.org/millennium/hodge-conjecture/

2.3  Navier-Stokes Equation, Navier-Stokes 方程解的存在性与光滑性

https://www.claymath.org/millennium/navier-stokes-equation/

2.4  P vs NP, P/NP 问题

https://www.claymath.org/millennium/p-vs-np/

2.5  Riemann Hypothesis, 黎曼假设

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

2.6  Yang-Mills & The Mass Gap, Yang-Mills 规范场存在性与质量间隙

https://www.claymath.org/millennium/yang-mills-the-maths-gap/

三、Solved problems 已解决的问题

3.1  Poincaré Conjecture 庞加莱猜想

https://www.claymath.org/millennium/poincare-conjecture/

   已经被佩雷尔曼( Григорий Яковлевич Перельман, 格里戈里·雅科夫列维奇·佩雷尔曼, Grigori Yakovlevich Perelman, 1966-06-13 ~ ) 于 2002~2003 年之间解决。

   2020-01-18，克雷数学研究所宣布佩雷尔曼获奖：

https://www.claymath.org/wp-content/uploads/2022/06/Poincare-press-release.pdf

https://www.claymath.org/millennium/poincare-conjecture/

   The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture.  The citation for the award reads:

   The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.

   克莱数学研究所（CMI）今天宣布，俄罗斯圣彼得堡的Grigoriy Perelman博士因解决庞加莱猜想而获得千年奖。颁奖词如下：

   克莱数学研究所特此将解决庞加莱猜想的千年奖授予格里戈里·佩雷尔曼。

图1  格里戈里·雅科夫列维奇·佩雷尔曼 Григорий Яковлевич Перельман, Grigori Yakovlevich Perelman, 1966-06-13 ~

https://fr.rbth.com/tech/91348-grigori-perelman-mathematicien-russe-refus-prix

   佩雷尔曼的3篇论文贴在 arxiv :

[P1] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org, November 11, 2002, 长度39页

doi:  10.48550/arXiv.math/0211159

https://arxiv.org/abs/math/0211159

[P2] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv.org, March 10, 2003, 长度22页

doi:  10.48550/arXiv.math/0303109

https://arxiv.org/abs/math/0303109

[P3] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv.org, July 17, 2003, 长度7页

doi:  10.48550/arXiv.math/0307245

https://arxiv.org/abs/math/0307245

参考资料：

[1] The Millennium Prize Problems, Clay Mathematics Institute

https://www.claymath.org/millennium-problems/

[1-2] Arthur Jaffe, 薛博卿译. 千禧年大奖难题之始与未终[J]. 数学文化, 2020, 11(4): 65-74.

https://www.global-sci.org/intro/article_detail/mc/18380.html

https://www.global-sci.org/intro/articles_list/mc/2043.html

[2] 2023-04-19，佩雷尔曼，G.Y. /Perelman,Grigori Yakovlevich/李心灿，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=198957&Type=bkzyb&SubID=61734

[3] Grigori Yakovlevich Perelman, MacTutor History of Mathematics Archive

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

推荐阅读：

[1] The Nobel Prize, The Nobel Foundation

https://www.nobelprize.org/prizes/

[2] A.M. Turing Award, Association for Computing Machinery

https://amturing.acm.org/

   The A.M. Turing Award, sometimes referred to as the "Nobel Prize of Computing," was named in honor of Alan Mathison Turing (1912–1954), a British mathematician and computer scientist. He made fundamental advances in computer architecture, algorithms, formalization of computing, and artificial intelligence. Turing was also instrumental in British code-breaking work during World War II.

   A.M.图灵奖，有时被称为“诺贝尔计算奖”，是为了纪念英国数学家和计算机科学家Alan Mathison Turing（1912-1954）而设立的。他在计算机体系结构、算法、计算形式化和人工智能方面取得了根本性进展。图灵在第二次世界大战期间也在英国的密码破译工作中发挥了重要作用。

[3] The Abel Prize, The Norwegian Academy of Science and Letters

https://abelprize.no/

https://abelprize.no/page/about-abel-prize

   The Abel Prize – International Prize of Mathematics – Awarded yearly

   The Abel Prize is named after Niels Henrik Abel, Norway’s greatest mathematician throughout the times. Abel left lasting marks on the mathematical world. His mathematics have served as a base for a number of major technological breakthroughs, there amongst the development of the internet.  The Abel Prize was established by the Norwegian Parliament (The Storting) in 2002, on the occasion the 200-year anniversary of his birth.

   阿贝尔奖-国际数学奖-每年颁发

   阿贝尔奖以挪威历史上最伟大的数学家尼尔斯·亨里克·阿贝尔的名字命名。阿贝尔在数学界留下了不可磨灭的印记。他的数学为互联网的发展提供了许多重大技术突破的基础。阿贝尔奖由挪威议会（挪威议会）于2002年在他诞辰200周年之际设立。

[4] SCIENCE, 2005-01-07, Special Issue 125th Anniversary, 01 JULY 2005, VOL 309, ISSUE 5731

https://science.sciencemag.org/content/309/5731

[4-2] In Praise of Hard Questions

https://www.science.org/doi/10.1126/science.309.5731.76

https://www.science.org/toc/science/309/5731

[4-3] 125

https://www.science.org/doi/10.1126/science.1115951

   Although tempted to review the 25 years of progress since 1980, my colleagues and I went with Fred instead and decided to contemplate the future, this time by posing 25 “Big Questions” along with 100 smaller ones. The choice reflects our belief that questions are more important than answers in shaping the future of science.

   尽管我很想回顾自1980年以来的25年进展，但我和我的同事们还是和弗雷德一起决定思考未来，这次提出了25个“大问题”和100个小问题。这一选择反映了我们的信念，即在塑造科学的未来方面，问题比答案更重要。

[4-4] 2019-05-17，《Science 125个前沿问题解读》一书即将出版 |《科学通报》

https://blog.sciencenet.cn/blog-528739-1179598.html

[4-5] 2017-01-22，Science125个科学前沿问题系列解读2016年度汇总|《科学通报》

https://blog.sciencenet.cn/blog-528739-1029192.html

[4-6] 2016-03-18，Science 125个科学前沿问题系列解读 |《科学通报》

https://blog.sciencenet.cn/blog-528739-963412.html

[5] 等你求解！上海交大携手《科学》杂志向全球发布125个科学问题, 2021-04-11, 上海交通大学 

https://news.sjtu.edu.cn/mtjj/20210412/145693.html

[5-2] SCIENCE, 2021-04-11, 125 questions: Exploration and discovery, In honor of Shanghai Jiao Tong University’s 125th Anniversary

https://www.sciencemag.org/collections/125-questions-exploration-and-discovery

相关链接：

[1] 2024-04-01，[笔记，数学文化] “千禧年大奖难题”，“发现全新的研究方向或领域”，后者更难能可贵

https://blog.sciencenet.cn/blog-107667-1427807.html

[2] 2024-07-23，[笔记，科普，资料] 诺贝尔奖相关知识与资料

https://blog.sciencenet.cn/blog-107667-1443474.html

[3] 2022-07-07，[小资料] 真数学原创需要多长时间（怀尔斯、佩雷尔曼）

https://blog.sciencenet.cn/blog-107667-1346288.html

[4] 2017-04-07，[求证真伪] 一无所有的数学国王（Grisha Perelman, 佩雷尔曼）

https://blog.sciencenet.cn/blog-107667-1047347.html

[5] 2010-03-21，Grigori Perelman: Millennium Prize of Clay Mathematics Institute 与天才的心灵

https://blog.sciencenet.cn/blog-107667-304994.html

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