Pavel S. Aleksandrov Biography

Pavel S. Aleksandrov laid the foundation for the field of mathematics known as topology. In addition to writing the first comprehensive textbook on the subject, Aleksandrov introduced several basic concepts of topology and its offshoots, homology and cohomology, which blend topology and algebra. His important work in defining and exploring bicompact (compact or locally compact) spaces laid the groundwork for research done by other mathematicians in these fields.

The youngest of the six children of Sergei Aleksandrovich Aleksandrov and Tsezariia Akimovna Zdanovskaia, Pavel Sergeevich Aleksandrov was born in Bogorodsk, Russia, on May 7, 1896. A year later the family moved to Smolensk, where Aleksandrov's father became head doctor in the state hospital. Although educated mainly in public schools, Aleksandrov learned German and French from his mother, who was skilled in languages.

In grammar school Aleksandrov developed an interest in mathematics under the guidance of Aleksandr Eiges, his arithmetic teacher. Aleksandrov entered the University of Moscow in 1913 as a mathematics student, and achieved early success when he proved the importance of Borel sets after hearing a lecture by Nikolai Nikolaevich Luzin in 1914. Aleksandrov graduated in 1917 and planned to continue his studies. However, after failing to reach similar results on his next project-- Georg Cantor's continuum hypothesis (since acknowledged unsolvable; that is, it can be neither proved nor disproved)--Aleksandrov dropped out of the mathematical community and formed a theater group in Chernigov, a city situated seventy-seven miles north of Kiev, in the Ukraine. Besides participating in the theater group, he lectured publicly on various topics in literature and mathematics. He also was involved in political support of the new Soviet government, for which he was jailed briefly in 1919 by counterrevolutionaries.

Later that same year, Aleksandrov suffered a lengthy illness, during which he decided to return to Moscow and mathematics. To help himself catch up, he enlisted the help of another young graduate student, Pavel Samuilovich Uryson. The two immediately became close friends and colleagues. After a brief, unsuccessful marriage in 1921 to his former teacher's sister, Ekaterina Romanovna Eiges, Aleksandrov joined some fellow graduate students in renting a summer cottage. There, he and Uryson began their study of the new field of topology, the branch of mathematics that deals with properties of figures related directly to their shape and invariant under continuous transformation (that is, without cutting or tearing). In topology, often called rubber-sheet geometry, a cylinder and a sphere are equivalent, because one can be shaped (or transformed) into the other. A doughnut, however, is not equivalent to a sphere, because it cannot be shaped or stretched into a sphere. No textbooks were available on the subject, only articles by Maurice Fréchet, Felix Hausdorff, and a few others. Nonetheless, from these articles, Uryson and Aleksandrov came up with their first major topological discovery: the theorem of metrization. Metrization is the process of deriving a specific measurement for the abstract concept of a topological space. In order to do this, Aleksandrov and Uryson first had to develop definitions of topological spaces. They initially defined a bicompact space (now known as compact and locally compact spaces), whose property is that for any collection of open sets (or groups of elements) that contains it (the interior of a sphere is an example of an open set). There is a subset of the collection with a finite number of elements that also contains it. Prior to their work, the concept of space was too abstract to be applicable to other mathematical fields; Aleksandrov and Uryson's research led to the acceptance of topology as a valid field of mathematical study.

With this result, the pair rose to fame within the mathematical community, gaining the approval of such notable scholars as Emmy Noether, Richard Courant, and David Hilbert. In 1924 Uryson and Aleksandrov went to Holland and visited with Luitzen Egbertus Jan Brouwer, who suggested that they publish their studies on topology. Aleksandrov and Uryson went on to the seaside in France for a spell of work and relaxation that ended tragically when Uryson drowned while swimming. In the aftermath of his friend's death, Aleksandrov lost himself in his work, conducting a seminar on topology that he and Uryson had begun organizing in 1924, and spending 1925 to 1926 working with Brouwer in an attempt to get his research into a form suitable for publication. During this time he further developed his theories of topology and compact space, with an eye to applying topology to the investigation of complex problems.

In 1927 Aleksandrov left Europe for a year to continue his work with a new friend and colleague, Heinz Hopf, at Princeton. Aleksandrov had met Hopf during the summer of 1926 in Göttingen, which along with Paris was considered to be the mathematical hub of Europe. It was in Göttingen in 1923 that Aleksandrov and Uryson first presented their results outside the U.S.S.R., and it was Aleksandrov's preferred summer residence until 1932. There he worked with others, including Noether, who gave the topological work of Aleksandrov and Hopf its algebraic bent. This may have led to Aleksandrov's growing interest in homology, the offshoot of topology incorporating algebra. Homology had first been developed by the French mathematician Jules Henri Poincaré, but only for certain types of topological spaces. In 1928 Aleksandrov made a major step in expanding the field when he was able to generalize homology to other topological spaces.

In 1934 Aleksandrov at last received his doctorate from the University of Moscow. The next year, he would issue his most famous work. After much difficult research, the first volume of Aleksandrov and Hopf's still-classic work Topologie was published (the remaining two volumes would not be published until after World War II, though they were completed sooner). In the tome they outlined, often for the first time, many basic concepts of this branch of mathematics. They also introduced the definition of cohomology, which is the "dual" theory, or mirror image, of homology. Cohomologists consider the same topics as homologists, but from a different vantage point, providing different results. The publication achieved, Aleksandrov settled in a small town outside of Moscow with his friend and colleague Andrey Nikolayevich Kolmogorov. They stayed together here, teaching at the University of Moscow, for the rest of their lives.

Always concerned with the younger generation of mathematicians, Aleksandrov in later years crafted ground-breaking textbooks in the fields of topology, homology, and group theory, which studies the properties of certain kinds of sets. He guided his students--noted mathematicians such as A. Kuros, L. Pontriagin, and A. Tikhonov--to great heights. He also led the mathematical community in Moscow, presiding over that city's mathematical society for more than thirty years. In 1979 Aleksandrov wrote his autobiography. He died three years later in Moscow on November 16, 1982.