James Waddell Alexander II

Last updated
James Waddell Alexander II
AlexanderIIJamesWaddel Moscow1935.tif
At a topological conference in Moscow, 1935
Born(1888-09-19)September 19, 1888
DiedSeptember 23, 1971(1971-09-23) (aged 83)
Known for Algebraic topology
Awards Bôcher Memorial Prize (1928)
Scientific career
Fields Topology
Doctoral advisor Oswald Veblen

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).

Contents

Early life, family, and personal life

James was born on September 19, 1888, in Sea Bright, New Jersey. [1] Alexander came from an old, distinguished Princeton family. He was the only child of the American portrait painter John White Alexander 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 on January 11, 1918, a Russian woman. Together, they had two children. [2]

They would frequently spend time, until 1937, in the Chamonix area of France, where he would also climb mountains and hills. Alexander was also a noted mountaineer, having succeeded in many major ascents, e.g. in the Swiss Alps and Colorado Rockies. 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.

Education

He graduated from Princeton University with a Bachelor of Science degree in 1910. He received his Masters of Arts degree in 1911 and his doctoral degree in 1915. [2]

Military career

During World War I, Alexander served with tech staff in the Ordnance Department of the United States Army overseas. He retired as a Captain. [2]

Academic career

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[ citation needed ]) called the Reidemeister moves; and also a means of computing homological invariants from the knot diagram.

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.

Death and legacy

He died on September 23, 1971. [2]

The Alexander's Chimney, in the Rocky Mountain National Park, is named after him.

See also

Related Research Articles

Algebraic topology branch of mathematics

Algebraic topology is a branch of mathematics that 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.

William Thurston mathematician

William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

Vladimir Voevodsky Russian mathematician

Vladimir Alexandrovich Voevodsky was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory.

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

Jean-Pierre Serre French mathematician

Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.

Mikhail Leonidovich Gromov Russian mathematician

Mikhail Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University.

In 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. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.

The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.

In mathematics, Khovanov homology is an oriented link invariant that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial.

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.

Bertram Kostant American mathematician

Bertram Kostant was an American mathematician who worked in representation theory, differential geometry, and mathematical physics.

William Schumacher Massey was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including A Basic Course in Algebraic Topology (ISBN 0-387-97430-X).

Tian Gang Chinese mathematician

Tian Gang is a Chinese mathematician. He is an academician of the Chinese Academy of Sciences and of the American Academy of Arts and Sciences. He is known for his contributions to geometric analysis and quantum cohomology especially Gromov-Witten invariants, among other fields. He has been Vice President of Peking University since February 2017.

Louis Kauffman American mathematician

Louis Hirsch Kauffman is an American mathematician, topologist, and professor of Mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the introduction and development of the bracket polynomial and the Kauffman polynomial.

Robert Mark Goresky is a Canadian mathematician who invented intersection homology with Robert MacPherson.

Kenneth C. Millett is a professor of mathematics at the University of California, Santa Barbara. His research concerns low-dimensional topology, knot theory, and the applications of knot theory to DNA structure; his initial is the "M" in the name of the HOMFLY polynomial.

Andrew Hugh Wallace was a Scottish-American mathematician.

Vladimir Georgievich Turaev is a Russian mathematician, specializing in topology.

Urs Stammbach is a Swiss mathematician, specializing in homological algebra.

References

  1. Staff. A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 1930–1980, p. 43. Institute for Advanced Study, 1980. Accessed November 20, 2015. "Alexander, James Waddell M, Topology Born 1888 Seabright, NJ."
  2. 1 2 3 4 Who Was Who in American History - the Military. Chicago: Marquis Who's Who. 1975. p. 6. ISBN   0837932017.

Sources