James Waddell Alexander II | |
---|---|

At a topological conference in Moscow, 1935 | |

Born | |

Died | September 23, 1971 83) | (aged

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).

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.

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] }

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] }

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.

He died on September 23, 1971.^{ [2] }

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

**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 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 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** 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** 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** 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** 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 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.

- ↑ 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." - 1 2 3 4
*Who Was Who in American History - the Military*. Chicago: Marquis Who's Who. 1975. p. 6. ISBN 0837932017.

- James, I. M., Portrait of Alexander (1888–1971), Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 123–129.
- Cohen, Leon W., James Waddell Alexander (1888–1971), Bull. Amer. Math. Soc. 79 (1973), no. 5, 900—903.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.