Alexander studied mathematics and physics at Princeton, where he was a student of Veblen, obtaining a B.S. degree in 1910 and an M.S. degree in 1911. From 1911 to 1912 he served as an instructor in the mathematics department at Princeton. Then in 1912 he went to Europe to further his studies.
During this period abroad, Alexander studied at Paris and Bologna. He returned to Princeton where he submitted his dissertation Functions which map the interior of the unit circle upon simple regions and, in 1915, was awarded his Ph.D. He was appointed as a instructor at Princeton in 1915, being made a lecturer in 1916. Of course this was the time when the United States required the expertise of mathematicians to solve problems which related to the military needs created by World War I. In 1917, after marrying Natalia Levitzkaja on 15 January, later that year Alexander served as a lieutenant in the U.S. Army Ordnance Office at the Aberdeen Proving Ground. This was a military weapons testing site, established in 1917 in Aberdeen, Maryland.
By the end of the war Alexander had reached the rank of captain. Leaving military service, he returned to Princeton where he was an assistant professor from 1920, being promoted to associate professor in 1926 and full professor in 1928. From 1933 until he retired in 1951 he was a member of the Institute for Advanced Study in Princeton. Alexander, however, never drew a salary from the Institute for Advanced Study. He had become a millionaire through inherited wealth and, a rich man, was in no need of a salary. During World War II, Alexander again did war work, but on this occasion he did not rejoin the army. He worked during World War II as a civilian for the U.S. Army Air Force at their Office of Scientific Research and Development.
During the early 1950s senator Joseph R McCarthy whipped up strong feelings against communism in the United States. Alexander, who held left-wing political views, began to come under suspicion from authorities who saw imaginary problems everywhere. Alexander had virtually become a recluse after he retired in 1951 and the McCarthy era resulted in his disappearance from public life. The last time he was seen in public was July 1954 when he signed a statement of support for J Robert Oppenheimer who had lost his security clearance.
In a collaboration with Veblen, he showed that the topology of manifolds could be extended to polyhedra. Before 1920 he had shown that the homology of a simplicial complex is a topological invariant. Alexander's work around this time went a long way to put the intuitive ideas of Poincaré on a more rigorous foundation. Also before 1920 Alexander had made fundamental contributions to the theory of algebraic surfaces and to the study of Cremona transformations.
Soon after arriving in Princeton, Alexander generalised the Jordan curve theorem and continued his work, now exclusively on topology, with an important paper on the Jordan-Brouwer separation theorem. This latter paper contains the Alexander Duality Theorem and Alexander's lemma on the n-sphere. In 1924 he introduced the now famous Alexander horned sphere.
In 1928 he discovered the Alexander polynomial which is much used in knot theory. In the same year the American Mathematical Society awarded Alexander the Bôcher Prize for his memoir, Combinatorial analysis situs published in the Transactions of the American Mathematical Society two years earlier. Knot theory and the combinatorial theory of complexes were the main topics on which he worked over the following few years.
The theory which is now called the Alexander-Spanier cohomology theory, was introduced in 1935 by Alexander but was generalised by Spanier in 1948 to the form seen today. Also around 1935 Alexander discovered cohomology theory, at essentially the same time as Kolmogorov, and the theory was announced in the 1936 Moscow Conference.
Zund, in  writes:-
A mathematician of unusual depth and power, Alexander was a principal figure in the American development of algebraic/combinatorial topology. ... His papers were very carefully written and were very influential in the United States and abroad. Much of his work was of such a basic character that it became common knowledge in topology, with its discoverer being forgotten as a result...Alexander's character is also described in , where he is said to have been:-
... an imposing figure who possessed great charm and a very "youthful" view of mathematics, being one of the first American mathematicians to fully appreciate the use of modern algebraic methods in topology. Colleagues remember his great fondness for limericks and his passion for mountain climbing.Among the many honours bestowed on Alexander was his election to the American Academy of Sciences in 1930.
Article by: J J O'Connor and E F Robertson