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→Poincaré and Einstein
他对几何的研究导致了<font color="#ff8000"> 同伦和同调Homotopy and Homology</font>的抽象拓扑定义。他还首先介绍了组合拓扑的基本概念和不变量，如 贝蒂Betti 数和基本群。证明了 n 维多面体的边数、顶点数和面数的一个公式(欧拉-庞加莱定理) ，给出了直观维数概念的第一个精确表达式。
他对几何的研究导致了<font color="#ff8000"> 同伦和同调Homotopy and Homology</font>的抽象拓扑定义。他还首先介绍了组合拓扑的基本概念和不变量，如 贝蒂Betti 数和基本群。证明了 n 维多面体的边数、顶点数和面数的一个公式(欧拉-庞加莱定理) ，给出了直观维数概念的第一个精确表达式。
====Poincaré and Einstein====
====Poincaré and Einstein庞加莱和爱因斯坦====
Einstein's first paper on relativity was published three months after Poincaré's short paper,<ref name="1905 paper" /> but before Poincaré's longer version.<ref name=long /> Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure ([[Einstein synchronisation]]) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on [[special relativity]]. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to [[Hans Vaihinger]] on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's.<ref>{{cite book|series=The Collected Papers of Albert Einstein |url=http://einsteinpapers.press.princeton.edu/vol9-trans/52 |publisher=Princeton U.P. |accessdate=|volume = 9|title = The Berlin Years: Correspondence, January 1919-April 1920 (English translation supplement)|page = 30}} See also this letter, with commentary, in {{cite journal |last=Sass |first=Hans-Martin | authorlink = Hans-Martin Sass|date=1979 |title=Einstein über "wahre Kultur" und die Stellung der Geometrie im Wissenschaftssystem: Ein Brief Albert Einsteins an Hans Vaihinger vom Jahre 1919 |journal=[[Zeitschrift für allgemeine Wissenschaftstheorie]] |volume=10 |issue=2 |pages=316–319 |jstor=25170513 |language=de |doi=10.1007/bf01802352|s2cid=170178963 }}</ref> In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 called ''Geometrie und Erfahrung'' in connection with [[non-Euclidean geometry]], but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."<ref>Darrigol 2004, Secondary sources on relativity</ref>
Einstein's first paper on relativity was published three months after Poincaré's short paper,<ref name="1905 paper" /> but before Poincaré's longer version.<ref name=long /> Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure ([[Einstein synchronisation]]) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on [[special relativity]]. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to [[Hans Vaihinger]] on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's.<ref>{{cite book|series=The Collected Papers of Albert Einstein |url=http://einsteinpapers.press.princeton.edu/vol9-trans/52 |publisher=Princeton U.P. |accessdate=|volume = 9|title = The Berlin Years: Correspondence, January 1919-April 1920 (English translation supplement)|page = 30}} See also this letter, with commentary, in {{cite journal |last=Sass |first=Hans-Martin | authorlink = Hans-Martin Sass|date=1979 |title=Einstein über "wahre Kultur" und die Stellung der Geometrie im Wissenschaftssystem: Ein Brief Albert Einsteins an Hans Vaihinger vom Jahre 1919 |journal=[[Zeitschrift für allgemeine Wissenschaftstheorie]] |volume=10 |issue=2 |pages=316–319 |jstor=25170513 |language=de |doi=10.1007/bf01802352|s2cid=170178963 }}</ref> In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 called ''Geometrie und Erfahrung'' in connection with [[non-Euclidean geometry]], but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."<ref>Darrigol 2004, Secondary sources on relativity</ref>
爱因斯坦关于相对论的第一篇论文发表在庞加莱的短篇论文《1905年论文》发表三个月之后，但在庞加莱的长篇论文发表之前。<ref name=long/>爱因斯坦依靠相对论原理推导出洛伦兹变换，并使用了类似的时钟同步程序（[[爱因斯坦同步]]）庞加莱（1900年）曾描述过，但爱因斯坦的论文很了不起，因为它根本没有参考文献。庞加莱从未承认爱因斯坦在[[狭义相对论]]上的工作。然而，爱因斯坦在1919年5月3日写给[[汉斯-瓦因格]]的信中对庞加莱的观点表示了认同的倾向，当时爱因斯坦认为瓦辛格的总体观点接近于他自己，而庞加莱则接近瓦辛格。<ref>{{cite book|series=The Collected Papers of Albert Einstein |url=http://einsteinpapers.press.princeton.edu/vol9-trans/52 |publisher=Princeton U.P. |accessdate=|volume = 9|title = The Berlin Years: Correspondence, January 1919-April 1920 (English translation supplement)|page = 30}}另见本函及其评注{{cite journal |last=Sass |first=Hans-Martin | authorlink = Hans-Martin Sass|date=1979 |title=Einstein über "wahre Kultur" und die Stellung der Geometrie im Wissenschaftssystem: Ein Brief Albert Einsteins an Hans Vaihinger vom Jahre 1919 |journal=[[Zeitschrift für allgemeine Wissenschaftstheorie]] |volume=10 |issue=2 |pages=316–319 |jstor=25170513 |language=de |doi=10.1007/bf01802352|s2cid=170178963 }}</ref> 在公开场合，爱因斯坦在1921年的一次演讲中追认了庞加莱，他的演讲名为“几何与Erfahrung”，与[[非欧几里德几何]]有关，但与狭义相对论无关。在他去世前几年，爱因斯坦评价庞加莱是相对论的先驱之一，他说：“洛伦兹已经认识到以他命名的变换对于分析麦克斯韦方程组是必不可少的，而庞加莱进一步深化了这一见解……”<ref>Darrigol 2004, Secondary sources on relativity</ref>
<center> Chaotic motion in three-body problem (computer simulation).</center>
<center> Chaotic motion in three-body problem (computer simulation).</center>
< 中心 > 三体问题的混乱运动(计算机模拟) </中心 >
<center> 三体问题的混乱运动(计算机模拟) </center>
====Assessments on Poincaré and relativity====
====Assessments on Poincaré and relativity====