帕维尔·亚历山德罗夫Pavel Aleksandrov

像大多数俄罗斯数学家一样，帕维尔·亚历山德罗夫的名字有不同方式转写成罗马字母。除帕维尔·亚历山德罗夫之外，最常见的方式是写作Alexandroff。

帕维尔·亚历山德罗夫的父亲谢尔盖·亚历山德罗维奇帕维尔·亚历山德罗夫毕业于莫斯科大学医学专业，他决定不从事学术职业，而是选择用自己的技能帮助人们，因此在雅罗斯拉夫斯基当全科医生。后来他在博戈罗茨基一家医院担任更高级职位，帕维尔·亚历山德罗夫谢尔盖耶维奇就是在那里出生的。

当帕维尔·亚历山德罗夫·谢尔盖耶维奇一岁时，他的父亲搬到了斯摩棱斯克州立医院，在那里他赢得了非常优秀外科医生的声誉，从此全家就住在斯摩棱斯克。斯摩棱斯克市位于莫斯科以西420公里的第聂伯河畔。帕维尔·亚历山德罗夫·谢尔盖耶维奇的早期教育来自他的母亲Tsezariya Akimovna Aleksandrova，她倾注了自己所有的才华来抚养和教育孩子们。正是从她那里，帕维尔·亚历山德罗夫学会了法语和德语。他的家总是充满音乐，因为他的兄弟姐妹们在这方面都很有天赋。

他母亲给他的良好开端意味着他在斯摩棱斯克就读的文法学校中始终出类拔萃。他的数学老师 Alexsander Romanovich Eiges 很快意识到他的学生在这门学科上有着非凡的天赋，并且（[3] 和 [4]）：-

……在文法学校他学习了天体力学和数学分析。但他的兴趣主要指向数学的基本问题：几何学的基础和non-euclidean geometry。Eiges 对他的学生有恰当的认识，并对他选择数学职业产生了决定性的影响。

1913年，帕维尔·亚历山德罗夫从文法学校毕业，是学校的优等生并获得了金质奖章。当然，此时他已经决定从事数学职业，但他并没有把目标定得像大学教师那么高，而是打算成为一名中学数学教师。Eiges 是他在这个阶段渴望效仿的榜样，因为 Eiges 不仅教了帕维尔·亚历山德罗夫数学，还影响了他对文学和艺术的品味。

帕维尔·亚历山德罗夫于1913年进入莫斯科大学，并立即得到了维亚切斯拉夫·瓦西里耶维奇·斯捷潘诺夫的帮助。维亚切斯拉夫·瓦西里耶维奇·斯捷潘诺夫当时在莫斯科大学工作，比帕维尔·亚历山德罗夫大七岁，但他的家也在斯摩棱斯克，他经常拜访那里的帕维尔·亚历山德罗夫家。维亚切斯拉夫·瓦西里耶维奇·斯捷潘诺夫此时对帕维尔·亚历山德罗夫产生了重要影响，并建议帕维尔·亚历山德罗夫在莫斯科学习的第一年就加入德米特里·叶戈罗夫的讨论班。在帕维尔·亚历山德罗夫学习的第二年，他接触到了刚回到莫斯科的尼古拉·卢津。帕维尔·亚历山德罗夫写道（例如见[3]或[4]）：-

在尼古拉·卢津的讲座之后，我向他请教如何最好地继续我的数学学习，最令我感动的是尼古拉·卢津对他面前这个人的善意——一个18岁的学生……然后我成了尼古拉·卢津的学生，在他最具创造力的时期……在那几年见到尼古拉·卢津，就是见到所谓对科学充满灵感的关系的展现。我从他那里不仅学到了数学，还接受了一课：什么造就了真正的学者，以及大学教授能够而且应该是什么样子。那时，我也看到，追求科学和在科学中培养年轻人是同一个活动的两个方面——学者的活动。

帕维尔·亚历山德罗夫在1915年证明了他的第一个重要结果，即每个不可数的博雷尔集都包含一个完美子集。对集合论而言重要的不仅是这个结果，还有帕维尔·亚历山德罗夫所使用的方法，这些方法后来成为描述集合论中最有用的方法之一。在帕维尔·亚历山德罗夫取得巨大成功之后，尼古拉·卢津做了许多导师可能会做的事，他意识到自己拥有帕维尔·亚历山德罗夫这样一位最伟大的数学天才之一，因此他认为值得请他尝试解决集合论中最大的未解决问题，即连续统假设。

帕维尔·亚历山德罗夫未能解决连续统假设（这并不奇怪，因为正如Cohen在1960年代所表明的，它既不能被证明也不能被否证），他认为自己没有能力从事数学职业。帕维尔·亚历山德罗夫去了诺夫哥罗德-谢韦尔斯基，成为一名戏剧制作人。然后他去了切尔尼科夫，除了戏剧工作外，还讲授俄语和外语，与诗人、艺术家和音乐家成为朋友。在1919年俄国革命时期短暂入狱后，帕维尔·亚历山德罗夫于1920年回到莫斯科。尼古拉·卢津和德米特里·叶戈罗夫在莫斯科大学建立了一个令人印象深刻的研究小组，学生们称之为“Luzitania”，他们与Privalov和维亚切斯拉夫·瓦西里耶维奇·斯捷潘诺夫一起，对帕维尔·亚历山德罗夫的回归非常欢迎。

然而，帕维尔·亚历山德罗夫并没有立即回到莫斯科，因为他在1920-21年回到了斯摩棱斯克的家中，在那里他在大学任教。在此期间，他从事研究工作，大约每月去一次莫斯科，与那里的数学家保持联系，并为考试做准备。大约在这个时候，帕维尔·亚历山德罗夫与“Luzitania”的成员帕维尔·萨穆伊洛维奇·乌雷松变得友好，这种友谊很快发展成重要的数学合作。

1921年通过考试后，帕维尔·亚历山德罗夫被任命为莫斯科大学讲师，讲授实变函数、拓扑学和伽罗瓦理论等多个主题。1922年7月，帕维尔·亚历山德罗夫和帕维尔·萨穆伊洛维奇·乌雷松前往莫斯科附近的博尔舍沃度夏，在那里开始研究拓扑学中的概念。费利克斯·豪斯多夫在其1914年出版的著名著作Grundzüge der MengenlehreⓉ（一般集合论）中，基于莫里斯·弗雷歇等人的工作，创立了拓扑空间和度量空间的理论。帕维尔·亚历山德罗夫和帕维尔·萨穆伊洛维奇·乌雷松此时开始推进这一理论，在可数紧空间方面的工作产生了具有根本重要性的结果。紧空间和局部紧空间的概念正是由他们提出的。

1923年和1924年夏天，帕维尔·亚历山德罗夫和帕维尔·萨穆伊洛维奇·乌雷松访问了哥廷根，他们的成果给埃米·诺特、理查·科朗特和大卫·希尔伯特留下了深刻印象。哥廷根的数学家们对他们关于拓扑空间何时可度量的成果印象尤为深刻。1924年夏天，他们还拜访了波恩的费利克斯·豪斯多夫，他听到两人在拓扑学中开辟的重要新方向后极为着迷。然而，在波恩拜访费利克斯·豪斯多夫时（[3]和[4]）：-

每天帕维尔·亚历山德罗夫和帕维尔·萨穆伊洛维奇·乌雷松都游过莱茵河——这一壮举远非安全，并引起了费利克斯·豪斯多夫的不满。

随后，帕维尔·亚历山德罗夫和帕维尔·萨穆伊洛维奇·乌雷松于1924年8月在荷兰和巴黎拜访了勒伊岑·布劳威尔，之后在布列塔尼的渔村布尔德巴茨度假。当然，数学家在度假时仍继续做数学，两人都在努力工作。8月17日早晨，帕维尔·萨穆伊洛维奇·乌雷松开始撰写一篇新论文，但不幸的是，当天晚些时候他在大西洋游泳时溺水身亡。帕维尔·亚历山德罗夫决心不让他的挚友和合作者的任何想法遗失，他在1925年和1926年的一部分时间里在荷兰与勒伊岑·布劳威尔合作，准备将帕维尔·萨穆伊洛维奇·乌雷松的论文付梓。

哥廷根的氛围对帕维尔·亚历山德罗夫证明非常有益，尤其是在帕维尔·萨穆伊洛维奇·乌雷松去世之后，他从1925年到1932年每年夏天都去那里。他与海因茨·霍普夫成为密友，两人在哥廷根举办了一个拓扑学讨论班。当然，帕维尔·亚历山德罗夫也在莫斯科大学任教，并从1924年起在那里组织了一个拓扑学讨论班。在哥廷根，帕维尔·亚历山德罗夫也讲课并参加埃米·诺特的讨论班。事实上，帕维尔·亚历山德罗夫总是将埃米·诺特和大卫·希尔伯特以及阿姆斯特丹的勒伊岑·布劳威尔和莫斯科的尼古拉·卢津和德米特里·叶戈罗夫列入他的老师之列。

从1926年起，帕维尔·亚历山德罗夫和海因茨·霍普夫成为密切合作的朋友。1926年他们与奥托·纽格包尔在法国南部度过了一段时间。随后帕维尔·亚历山德罗夫和海因茨·霍普夫在1927-28学年于美国普林斯顿度过。这是拓扑学发展的一个重要年份，帕维尔·亚历山德罗夫和海因茨·霍普夫在普林斯顿得以与所罗门·莱夫谢茨、奥斯瓦尔德·维布伦和詹姆斯·韦德尔·亚历山大合作。在普林斯顿的那一年里，帕维尔·亚历山德罗夫和海因茨·霍普夫计划了一部关于Topology的多卷本联合著作，其第一卷直到1935年才问世。这是计划中的三卷中唯一出版的一卷，因为第二次世界大战阻碍了剩余两卷的进一步合作。事实上，在与海因茨·霍普夫的联合著作付印之前，帕维尔·亚历山德罗夫已经开始了他另一段重要的友谊与合作。

1929年，帕维尔·亚历山德罗夫与安德雷·柯尔莫哥洛夫的友谊开始了，他们（[3]和[4]）：-

……沿着伏尔加河、第聂伯河及其他河流，以及在高加索、克里米亚和法国南部旅行了很多地方。

1929年不仅标志着与安德雷·柯尔莫哥洛夫友谊的开始，也标志着帕维尔·亚历山德罗夫被任命为莫斯科大学数学教授。1935年，帕维尔·亚历山德罗夫与安德雷·柯尔莫哥洛夫去了雅尔塔，随后在附近的克里米亚完成了他的Topology书的工作，该书于当年出版。“科马罗夫斯基”时期也在那一年开始（[3]和[4]）：-

在过去的四十年里，莫斯科大学数学史上的许多事件都与科马罗夫卡——莫斯科郊外的一个小村庄——联系在一起。这里有一栋自1935年起归帕维尔·亚历山德罗夫和安德雷·柯尔莫哥洛夫所有的房子。许多著名的外国数学家也访问过科马罗夫卡——雅克·阿达马、莫里斯·弗雷歇、斯特凡·巴拿赫、海因茨·霍普夫、卡齐米日·库拉托夫斯基等人。

1938—1939 年，莫斯科大学的一些顶尖数学家，其中包括 帕维尔·亚历山德罗夫，加入了 苏联科学院 的 弗拉基米尔·安德烈耶维奇·斯捷克洛夫 数学研究所，但同时保留了他们在大学的职位。

帕维尔·亚历山德罗夫在其漫长的职业生涯中撰写了约300部科学著作。早在1924年，他就引入了局部有限覆盖的概念，并将其作为拓扑空间可度量性判据的基础。他在1925年至1929年间的一系列基础性论文中奠定了homology theory的基础。他的方法使组合学和代数拓扑学的论证得以应用于point set topology，并将这些领域结合在一起。帕维尔·亚历山德罗夫在 homology 方面的工作随着他约1928-30年的同调维数理论而向前推进。

帕维尔·亚历山德罗夫是第一个使用“同态的核”这一短语的人，并在1940-41年前后发现了一个正合序列的要素。他研究拓扑空间的连续映射理论。1954年，他针对莫斯科大学一年级学生组织了一个关于这最后一个主题的讨论班，由此展现了他职业生涯中对他至关重要的一个方面，即学生的教育。这在（[3]和[4]）中有描述：-

对于这些学生以及后来者的培养，帕维尔·亚历山德罗夫几乎倾注了全部力量。他对在他指导下学习拓扑学的年轻一代的影响从来都不纯粹是数学上的，无论这种影响多么真实和重要。有拓扑散步的体育日，有持续数天的乘船长途远足，……有横渡伏尔加河或其他宽阔水域的游泳，有在莫斯科郊外山坡上持续数小时的滑雪远足，帕维尔·亚历山德罗夫给这些山坡起了引人注目、奇妙的名字……

由于对数学的杰出贡献，帕维尔·亚历山德罗夫获得了许多荣誉。他于1932年至64年担任莫斯科数学会主席，1958年至62年担任国际数学家大会副主席，自1929年起为苏联科学院通讯院士，1953年起为正式院士。许多其他学会也选举帕维尔·亚历山德罗夫为成员，包括哥廷根科学院、Austrian Academy of Sciences、哈雷的利奥波尔迪纳科学院、Polish Academy of Sciences、National Academy of Sciences of the United States、London Mathematical Society、美国哲学会和荷兰数学会。

他编辑了几种数学期刊，特别是著名的苏联期刊Uspekhi Matematicheskikh Nauk，并且他获得了许多苏联奖项，包括1943年的斯大林奖和五枚列宁勋章。

今天，莫斯科国立大学一般拓扑学与几何学系是俄罗斯集合论拓扑学研究的领先中心。在帕维尔·亚历山德罗夫 于1982年11月去世后，他曾经担任讲席的高等几何与拓扑学系的同事们致信莫斯科大学校长A A Logunov，提议由帕维尔·亚历山德罗夫 以前的一名学生担任系主任，以保持帕维尔·亚历山德罗夫 的科学学派。1982年12月28日，校长发布通告，成立一般拓扑学与几何学系。Vitaly Vitalievich Fedorchuk 被选为系主任。

同样为了纪念帕维尔·亚历山德罗夫 在莫斯科大学对拓扑学的贡献以及他与莫斯科数学会 的合作，每年五月举行一次年度拓扑学研讨会Aleksandrov Proceedings。

Like most Russian mathematicians there are different ways to transliterate Aleksandrov's name into the Roman alphabet. The most common way, other than Aleksandrov, is to write it as Alexandroff.

Pavel Sergeevich Aleksandrov's father Sergej Aleksandrovich Aleksandrov was a medical graduate from Moscow University who had decided not to follow an academic career but instead had chosen to use his skills in helping people and so he worked as a general practitioner in Yaroslavskii. Later he worked in more senior positions in a hospital in Bogorodskii, which is where he was when Pavel Sergeevich was born.

When Pavel Sergeevich was one year old his father moved to Smolensk State hospital, where he was to earn the reputation of being a very fine surgeon, and the family lived from this time in Smolensk. The city of Smolensk is on the Dnieper River 420 km west of Moscow. Pavel Sergeevich's early education was from his mother, Tsezariya Akimovna Aleksandrova, who applied all her considerable talents to bringing up and educating her children. It was from her that Aleksandrov learnt French and also German. His home was one that was always filled with music as his brothers and sisters all had great talent in that area.

The fine start which his mother gave him meant that he always excelled at the grammar school in Smolensk which he attended. His mathematics teacher Alexsander Romanovich Eiges soon realised that his pupil had a remarkable talent for the subject and ([3] and [4]):-

... at grammar school he studied celestial mechanics and mathematical analysis. But his interest was mainly directed towards fundamental problems of mathematics: the foundations of geometry and non-euclidean geometry. Eiges had a proper appreciation of his pupil and exerted a decisive influence on his choice of a career in mathematics.

In 1913 Aleksandrov graduated from the grammar school being dux of the school and winning the gold medal. Certainly at this time he had already decided on a career in mathematics, but he had not set his sights as high as a university teacher, rather he was aiming to become a secondary school teacher of mathematics. Eiges was the role model whom he was aspiring to match at this stage, for Eiges had done more than teach Aleksandrov mathematics, he had also influenced his tastes in literature and the arts.

Aleksandrov entered Moscow University in 1913 and immediately he was helped by Stepanov. Stepanov, who was working at Moscow University, was seven years older than Aleksandrov but his home was also in Smolensk and he often visited the Aleksandrov home there. Stepanov was an important influence on Aleksandrov at this time and suggested that Aleksandrov join Egorov's seminar even in the first year of his studies in Moscow. In Aleksandrov's second year of study he came in contact with Luzin who had just returned to Moscow. Aleksandrov wrote (see for example [3] or [4]):-

After Luzin's lecture I turned to him for advice on how best to continue my mathematical studies and was struck most of all by Luzin's kindness to the man addressing him - an 18-year old student ... I then became a student of Luzin, during his most creative period ... To see Luzin in those years was to see a display of what is called an inspired relationship to science. I learnt not only mathematics from him, I received also a lesson in what makes a true scholar and what a university professor can and should be. Then, too, I saw that the pursuit of science and the raining of young people in it are two facets of one and the same activity - that of a scholar.

Aleksandrov proved his first important result in 1915, namely that every non-denumerable Borel set contains a perfect subset. It was not only the result which was important for set theory, but also the methods which Aleksandrov used which turned out to be one of the most useful methods in descriptive set theory. After Aleksandrov's great successes Luzin did what many a supervisor might do, he realised that he had one of the greatest mathematical talents in Aleksandrov so he thought that it was worth asking him to try to solve the biggest open problem in set theory, namely the continuum hypothesis.

After Aleksandrov failed to solve the continuum hypothesis (which is not surprising since it can neither be proved or disproved as was shown by Cohen in the 1960s) he thought he was not capable of a mathematical career. Aleksandrov went to Novgorod-Severskii and became a theatre producer. He then went to Chernikov where, in addition to theatrical work, he lectured on Russian and foreign languages, becoming friends with poets, artists and musicians. After a short term in jail in 1919 at the time of the Russian revolution, Aleksandrov returned to Moscow in 1920. Luzin and Egorov had built up an impressive research group at the University of Moscow which the students called 'Luzitania' and they, together with Privalov and Stepanov, were very welcoming to Aleksandrov on his return.

It was not an immediate return to Moscow for Aleksandrov, however, for he spent 1920-21 back home in Smolensk where he taught at the University. During this time he worked on his research, going to Moscow about once every month to keep in touch with the mathematicians there and to prepare himself for his examinations. At around this time Aleksandrov became friendly with Urysohn, who was a member of 'Luzitania', and the friendship would soon develop into a major mathematical collaboration.

After taking his examinations in 1921, Aleksandrov was appointed as a lecturer at Moscow university and lectured on a variety of topics including functions of a real variable, topology and Galois theory. In July 1922 Aleksandrov and Urysohn went to spend the summer at Bolshev, near to Moscow, where they began to study concepts in topology. Hausdorff, building on work by Fréchet and others, had created a theory of topological and metric spaces in his famous book Grundzüge der Mengenlehre Ⓣ published in 1914. Aleksandrov and Urysohn now began to push the theory forward with work on countably compact spaces producing results of fundamental importance. The notion of a compact space and a locally compact space is due to them.

In the summers of 1923 and 1924 Aleksandrov and Urysohn visited Göttingen and impressed Emmy Noether, Courant and Hilbert with their results. The mathematicians in Göttingen were particularly impressed with their results on when a topological space is metrisable. In the summer of 1924 they also visited Hausdorff in Bonn and he was fascinated to hear the major new directions that the two were taking in topology. However while visiting Hausdorff in Bonn ([3] and [4]):-

Every day Aleksandrov and Urysohn swam across the Rhine - a feat that was far from being safe and provoked Hausdorff's displeasure.

Aleksandrov and Urysohn then visited Brouwer in Holland and Paris in August 1924 before having a holiday in the fishing village of Bourg de Batz in Brittany. Of course mathematicians continue to do mathematics while on holiday and they were both working hard. On the morning of 17 August Urysohn began to write a new paper but tragically he drowned while swimming in the Atlantic later that day. Aleksandrov determined that no ideas of his great friend and collaborator should be lost and he spent part of 1925 and 1926 in Holland working with Brouwer on preparing Urysohn's paper for publication.

The atmosphere in Göttingen had proved very helpful to Aleksandrov, particularly after the death of Urysohn, and he went there every summer from 1925 until 1932. He became close friends with Hopf and the two held a topological seminar in Göttingen. Of course Aleksandrov also taught in Moscow University and from 1924 he organised a topology seminar there. At Göttingen, Aleksandrov also lectured and participated in Emmy Noether's seminar. In fact Aleksandrov always included Emmy Noether and Hilbert among his teachers, as well as Brouwer in Amsterdam and Luzin and Egorov in Moscow.

From 1926 Aleksandrov and Hopf were close friends working together. They spent some time in 1926 in the south of France with Neugebauer. Then Aleksandrov and Hopf spent the academic year 1927-28 at Princeton in the United States. This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander. During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935. This was the only one of the three intended volumes to appear since World War II prevented further collaboration on the remaining two volumes. In fact before the joint work with Hopf appeared in print, Aleksandrov had begun yet another important friendship and collaboration.

In 1929 Aleksandrov's friendship with Kolmogorov began and they ([3] and [4]):-

... journeyed a lot along the Volga, the Dnieper, and other rivers, and in the Caucuses, the Crimea, and the south of France.

The year 1929 marks not only the beginning of the friendship with Kolmogorov but also the appointment of Aleksandrov as Professor of Mathematics at Moscow University. In 1935 Aleksandrov went to Yalta with Kolmogorov, then finished the work on his Topology book in the nearby Crimea and the book was published in that year. The 'Komarovski' period also began in that year ([3] and [4]):-

Over the last forty years, many of the events in the history of mathematics in the University of Moscow have been linked with Komarovka, a small village outside Moscow. Here is the house owned since 1935 by Aleksandrov and Kolmogorov. Many famous foreign mathematicians also visited Komarovka - Hadamard, Fréchet, Banach, Hopf, Kuratowski, and others.

In 1938-1939 a number of leading mathematicians from the Moscow University, among them Aleksandrov, joined the Steklov Mathematical Institute of the USSR Academy of Sciences but at the same time they kept their positions at the University.

Aleksandrov wrote about 300 scientific works in his long career. As early as 1924 he introduced the concept of a locally finite covering which he used as a basis for his criteria for the metrisability of topological spaces. He laid the foundations of homology theory in a series of fundamental papers between 1925 and 1929. His methods allowed arguments of combinatorial and algebraic topology to be applied to point set topology and brought together these areas. Aleksandrov's work on homology moved forward with his homological theory of dimension around 1928-30

Aleksandrov was the first to use the phrase 'kernel of a homomorphism' and around 1940-41 he discovered the ingredients of an exact sequence. He worked on the theory of continuous mappings of topological spaces. In 1954 he organised a seminar on this last topic aimed at first year students at Moscow University and in this he showed one of the aspects of his career which was of major importance to him, namely the education of students. This is described in ([3] and [4]):-

To the training of these students and those who came after them, Aleksandrov literally devoted all his strength. His influence on the class of young men studying topology under him was never purely mathematical, however real and significant that was. There were physical days exercise on topological walks, in long outings lasting several days by boat, ... in swimming across the Volga or other broad stretches of water, in skiing excursions lasting for hours on the slopes outside Moscow, slopes to which Aleksandrov gave striking, fantastic names...

Many honours were given to Aleksandrov for his outstanding contribution to mathematics. He was president of the Moscow Mathematical Society from 1932 to 64, vice president of the International Congress of Mathematicians from 1958 to 62, a corresponding member of the USSR Academy of Sciences from 1929 and a full member from 1953. Many other societies elected Aleksandrov to membership including the Göttingen Academy of Sciences, the Austrian Academy of Sciences, the Leopoldina Academy in Halle, the Polish Academy of Sciences, the National Academy of Sciences of the United States, the London Mathematical Society, the American Philosophical Society, and the Dutch Mathematical Society.

He edited several mathematical journals, in particular the famous Soviet Journal Uspekhi Matematicheskikh Nauk, and he received many Soviet awards, including the Stalin Prize in 1943 and five Orders of Lenin.

Today the Department of General Topology and Geometry of Moscow State University is Russia's leading centre of research in set-theoretic topology. After Aleksandrov's death in November 1982, his colleagues from the Department of Higher Geometry and Topology, in which he had held the chair, sent a letter to Moscow University's rector A A Logunov proposing that one of Aleksandrov's former students should become Head of the Department, to preserve Aleksandrov's scientific school. On 28 December 1982 the rector issued a circular creating the Department of general topology and Geometry. Vitaly Vitalievich Fedorchuk was elected Head of the Department.

Also in memory of Aleksandrov's contributions to topology at Moscow University and his work with the Moscow Mathematical Society, there is an annual topological symposium Aleksandrov Proceedings held every May.