James Waddell Alexander II
Encyclopedia
James Waddell Alexander II (September 19, 1888 – September 23, 1971) was a mathematician
and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen
, Solomon Lefschetz
, and others. He was one of the first members of the Institute for Advanced Study
(1933–1951), and also a professor at Princeton University
(1920–1951).
and Elizabeth Alexander. His maternal grandfather, James Waddell Alexander, was the president of the Equitable Life Assurance Society
. Alexander's affluence and upbringing allowed him to interact with high society in America and elsewhere. He married Natalia Levitzkaja in 1917, a Russia
n woman, and they would frequently spend time, until 1937, in the Chamonix
area of France
, where he would also climb mountains and hills.
He was a pioneer in algebraic topology
, setting the foundations for Henri Poincaré
's ideas on homology theory
and furthering it by founding cohomology theory, which developed gradually in the decade after he gave a definition of cochain. For this, in 1928 he was awarded the Bôcher Memorial Prize
. He also contributed to the beginnings of knot theory
by inventing the Alexander invariant of a knot, which in modern terms is a graded module obtained from the homology of a "cyclic covering" of the knot complement
. From this invariant, he defined the first of the polynomial knot invariants.
With Garland Briggs, he also gave a combinatorial description of knot invariance based on certain moves, now (against the history) called the Reidemeister moves; and also a means of computing homological invariants from the knot diagram.
Alexander was also a noted mountaineer
, having succeeded in many major ascents, e.g. in the Swiss Alps
and Colorado Rockies
. The Alexander's Chimney, in the Rocky Mountain National Park
, is named after him. When in Princeton, he liked to climb the university buildings, and always left his office window on the top floor of Fine Hall open so that he could enter by climbing the building.
Towards the end of his life, Alexander became a recluse. He was known as a socialist and his prominence brought him to the attention of McCarthyists
. The atmosphere of the McCarthy era pushed him into a greater seclusion. He was not seen in public after 1954, when he appeared to sign a letter supporting J. Robert Oppenheimer.
He died on September 23, 1971.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
, Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
, and others. He was one of the first members of the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
(1933–1951), and also a professor at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
(1920–1951).
Biography
He was born on September 19, 1888. Alexander came from an old, distinguished Princeton family. He was the only child of the American portrait painter John White AlexanderJohn White Alexander
John White Alexander was an American portrait, figure, and decorative painter and illustrator.-Biography:thumb|“Isabella and the Pot of Basil”, oil on canvas, 1897, [[Museum of Fine Arts, Boston]]...
and Elizabeth Alexander. His maternal grandfather, James Waddell Alexander, was the president of the Equitable Life Assurance Society
AXA Equitable Life Insurance Company
AXA Equitable Life Insurance Company, formerly The Equitable Life Assurance Society of the United States, also known as The Equitable, was founded by Henry Baldwin Hyde in 1859. In 1991, AXA, a French insurance company, acquired majority control of The Equitable...
. Alexander's affluence and upbringing allowed him to interact with high society in America and elsewhere. He married Natalia Levitzkaja in 1917, a Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n woman, and they would frequently spend time, until 1937, in the Chamonix
Chamonix
Chamonix-Mont-Blanc or, more commonly, Chamonix is a commune in the Haute-Savoie département in the Rhône-Alpes region in south-eastern France. It was the site of the 1924 Winter Olympics, the first Winter Olympics...
area of France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
, where he would also climb mountains and hills.
He was a pioneer in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, setting the foundations for Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
's ideas on homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
and furthering it by founding cohomology theory, which developed gradually in the decade after he gave a definition of cochain. For this, in 1928 he was awarded the Bôcher Memorial Prize
Bôcher Memorial Prize
The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 . It is awarded every five years for a notable research memoir in analysis that has appeared during the past six years in a recognized North American...
. He also contributed to the beginnings of knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
by inventing the Alexander invariant of a knot, which in modern terms is a graded module obtained from the homology of a "cyclic covering" of the knot complement
Knot complement
In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...
. From this invariant, he defined the first of the polynomial knot invariants.
With Garland Briggs, he also gave a combinatorial description of knot invariance based on certain moves, now (against the history) called the Reidemeister moves; and also a means of computing homological invariants from the knot diagram.
Alexander was also a noted mountaineer
Mountaineer
-Sports:*Mountaineering, the sport, hobby or profession of walking, hiking, trekking and climbing up mountains, also known as alpinism-University athletic teams and mascots:*Appalachian State Mountaineers, the athletic teams of Appalachian State University...
, having succeeded in many major ascents, e.g. in the Swiss Alps
Swiss Alps
The Swiss Alps are the portion of the Alps mountain range that lies within Switzerland. Because of their central position within the entire Alpine range, they are also known as the Central Alps....
and Colorado Rockies
Rocky Mountains
The Rocky Mountains are a major mountain range in western North America. The Rocky Mountains stretch more than from the northernmost part of British Columbia, in western Canada, to New Mexico, in the southwestern United States...
. The Alexander's Chimney, in the Rocky Mountain National Park
Rocky Mountain National Park
Rocky Mountain National Park is a national park located in the north-central region of the U.S. state of Colorado.It features majestic mountain views, a variety of wildlife, varied climates and environments—from wooded forests to mountain tundra—and easy access to back-country trails...
, is named after him. When in Princeton, he liked to climb the university buildings, and always left his office window on the top floor of Fine Hall open so that he could enter by climbing the building.
Towards the end of his life, Alexander became a recluse. He was known as a socialist and his prominence brought him to the attention of McCarthyists
McCarthyism
McCarthyism is the practice of making accusations of disloyalty, subversion, or treason without proper regard for evidence. The term has its origins in the period in the United States known as the Second Red Scare, lasting roughly from the late 1940s to the late 1950s and characterized by...
. The atmosphere of the McCarthy era pushed him into a greater seclusion. He was not seen in public after 1954, when he appeared to sign a letter supporting J. Robert Oppenheimer.
He died on September 23, 1971.
See also
- Alexander horned sphereAlexander horned sphereThe Alexander horned sphere is a wild embedding of a sphere into space, discovered by . It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:...
- Alexander polynomialAlexander polynomialIn mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
- Alexander cochain
- Alexander-Spanier cohomologyAlexander-Spanier cohomologyIn mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomologytheory for topological spaces, introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of A. D...
- Alexander dualityAlexander dualityIn mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin...
- Alexander's trick