James Alexander (mathematician)

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Alexander at a topology conference in Moscow in 1935
Alexander's horned sphere

James Waddell Alexander II (born September 19, 1888 in Sea Bright , New Jersey , † September 23, 1971 in Princeton (New Jersey) ) was an important topologist , professor at Princeton University and one of the first members of the Institute for Advanced Study .

Life

Alexander came from a long-established and wealthy Princeton family. His paternal ancestors included eminent Presbyterian clergymen, including the first director of Princeton Theological Seminary Archibald Alexander, after whom a street and buildings in Princeton were named. He was the only child of the portrait painter John White Alexander and his wife Elizabeth. His maternal grandfather (also named James Waddell Alexander) was the president of a major life insurance company ( Equitable Life Assurance Society ). In 1917 Alexander married the Russian Natalia Levitzkaja, with whom he had a son and a daughter. His social prestige and circle of acquaintances, which included many respected business people, exceeded that of the other Princeton professors.

Alexander studied mathematics at Princeton and specialized in topology. In 1910 he received his bachelor's degree, in 1911 his master's degree and in 1915 he received his doctorate under Oswald Veblen (Functions which map the interior of the unit circle upon simple regions). Before that he was studying in Europe from 1912 (in Paris and Bologna). In 1915 he became an instructor (as in 1911/12) and a lecturer in Princeton in 1916. During the First World War he served as a lieutenant and most recently as a captain at the US Army Ordnance Office in the Aberdeen Proving Ground (the newly established testing ground for ballistics of the US Army in Maryland). He became Assistant Professor in 1920, Associate Professor in 1926 and Professor in 1928 at Princeton University. Alexander, together with Oswald Veblen, Solomon Lefschetz and others, shaped the development of topology in the USA in the era before the Second World War. In 1933 he was one of the first members of the Institute for Advanced Study at Princeton, where he was a professor until 1947. In 1951 he retired. Since he was a millionaire by inheritance, he waived payment at the institute. He gave a plenary lecture, Some Problems in Topology , at the 1932 International Congress of Mathematicians in Zurich . During the Second World War he was again active (as a civilian) in military research.

Alexander was a passionate mountaineer and Alexander's Chimney , in Rocky Mountain National Park , was named after him. He was even considered one of the outstanding American climbers of his generation. Until 1937 he regularly spent his holidays in Chamonix in the Alps, where he climbed a lot, sometimes with his wife and students (such as Hassler Whitney and Leon Cohen). At Princeton he used to climb into his study at the university building (Fine Hall). Another hobby was dancing (tango) and limericks. After the death of his wife in 1967, his health deteriorated (before that he had already had polio, which left permanent damage). Towards the end of his life he became enthusiastic about amateur radio.

From 1948 he withdrew more and more and gave up his professorship at the IAS, although he kept his office. In 1951 he retired completely and completely disappeared from the mathematician scene. IM James recalled trying to invite Alexander to the festive symposium on the retirement of his longtime colleague at Princeton Solomon Lefschetz in 1953: Alexander initially accepted (he would not go anywhere with a crowd), but then did not show up. Alexander was an avowed socialist and he was viewed with suspicion by Joseph McCarthy's followers. His last public act was the signing of an address of solidarity for Robert Oppenheimer in 1954.

In 1936 he gave the Rouse Ball Lecture at Cambridge University. In 1947 he received an honorary doctorate from Princeton University. In 1928 he became a member of the American Philosophical Society and in 1930 the National Academy of Sciences . In 1933/34 he was Vice President of the American Mathematical Society.

plant

JW Alexander was a pioneer in algebraic topology . He formed the homology theory on the basis of Henri Poincaré and based on it the cohomology theory (around 1936, independent of Andrei Kolmogorow ). In 1928 he received the Bôcher Memorial Prize for this achievement . Alexander-Spanier cohomology was introduced by him in 1935 and expanded by Edwin Spanier in 1948 . The Alexander duality was introduced by him in 1915 in a study of the Jordan-Brouwer decomposition theorem . It made a statement about the relationship between the Betti numbers (dimensions of the homology groups) of a manifold and its complement. It was further developed by Pawel Alexandrow in 1927 and by Lev Pontryagin in 1934 and generalized in the Spaniard-Whitehead duality .

Then he laid the foundations for knot theory . He found the Alexander invariant named after him , which is a module that is given by the homology of the cyclic superposition of the node complement, and finally in 1928 the first polynomial node invariant , which is now called the Alexander polynomial .

Together with Garland Briggs he also found a description of the knot invariants based on translations and manipulations of knot diagrams , later called Reidemeister movements after Kurt Reidemeister , who found them to be independent.

In 1924 he introduced Alexander's Horned Sphere , a pathological topological object. It is created by radially cutting open a torus and attaching a new punctured torus to the two cut surfaces and continuing with it infinitely often. The Alexander sphere itself is topologically a 3-sphere, but its complement is not simply connected, but very complex. It can be seen as a “wild” way of embedding a sphere in three-dimensional Euclidean space, which leads to a sphere minus a Cantor set . The Alexander sphere is a counterexample to the possibility of generalizing Schönflies' theorem to more than two dimensions. Alexander proved, however, that Schoenflies' theorem for smooth or piecewise linear embeddings can be extended to three dimensions (it thus provides an early example of the distinction between the category of topological spaces and piecewise linear or differentiable manifolds).

Before 1920 he also made significant contributions to the theory of algebraic surfaces and to Cremona transformations .

Alexander's lectures were considered outstanding among mathematicians at Princeton.

Namesake

literature

Web links

Individual evidence

  1. ↑ James Alexander in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  2. ↑ IM James, Portrait of Alexander, p. 127
  3. ↑ Alexander, On the Chains of a Complex and Their Duals, Proc. Nat. Acad. USA, Vol. 21, 1935, pp. 509-511
  4. ↑ Alexander, A proof of the invariance of certain constants of analysis situs, Trans. Amer. Math. Soc. , Volume 16, 1915, pp. 148-154
  5. ↑ GS Chogoshvili, Alexander duality , Encyclopedia of Mathematics, Springer
  6. ↑ Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., Vol. 30, 1928, pp. 275-306
  7. ↑ JW Alexander, GB Briggs, On types of knotted curves. Annals of Mathematics, Volume 28, 1926/27, pp. 562-586
  8. ^ Alexander, An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected, Proceedings of the National Academy of Sciences of the USA, Volume 10, 1924, pp. 8-10
  9. ↑ Eric Weisstein, Alexander's horned sphere , Wolfram Mathworld
personal data
SURNAME Alexander, James
ALTERNATIVE NAMES Alexander, James Waddell II (full name); Alexander, James W.
BRIEF DESCRIPTION American topologist
DATE OF BIRTH September 19, 1888
PLACE OF BIRTH Sea Bright
DATE OF DEATH 23rd September 1971
Place of death Princeton, New Jersey