**Hassler Whitney**'s father was Edward Baldwin Whitney, a judge, and his mother was A Josepha Newcomb. Edward's father was William Dwight Whitney who was a linguist and one of the foremost Sanskrit scholars of his time, noted especially for his classic work, *Sanskrit Grammar* (1879). Josepha's father was Simon Newcomb who has a biography in this archive. Certainly Hassler had two very famous grandfathers.

Whitney attended Yale University where he received his first degree in 1928, then continued to undertake mathematical research at the University of Harvard from where his doctorate was awarded in 1932. His doctorate was awarded for a dissertation *The Coloring of Graphs* written under Birkhoff's supervision. Whitney was a keen mountaineer all his life and he made a particularly famous climb while an undergraduate. His grandson James writes:-

[

Whitney]made the first ascent of the Whitney Gilman ridge on Cannon cliff, New Hampshire in1929with his cousin Bradley Gilman. This knife edge ridge is700feet high and is one of the most beautiful climbs on the east coast. ... Whitney subscribed to the "fast and light" school of mountaineering(and rock climbing). ... Hassler moved quickly with less protection than is considered usual today. On the Whitney Gilman ridge, he climbed without pitons(or anything else)for protection. He did not even protect the belays, instead using a comfortable and secure stance. On some climbs, especially on climbs in the Swiss Alps, he did use pitons for protection and/or belays.

Whitney married Margaret R Howell on 30 May 1930; they had three children, James Newcomb Whitney, Carol Whitney, and Marian Whitney. He continued to work at Harvard, being appointed an instructor in mathematics from 1930 until 1935, although the years 1931-33 were spent as a National Research Council Research Fellow at Harvard and Princeton. From 1935 he was promoted to assistant professor, then from 1940 associate professor. From 1943 to 1945 he was a member of the Applied Mathematics Panel of the National Defense Research Committee. John McCleary writes in his article *Hassler Whitney, the Applied Mathematics Panel, and Airborne Weapons Accuracy*:-

The Applied Mathematics Panel was established in late1942to provide mathematical expertise to the divisions of the National Defense Research Committee. It employed academic mathematicians at several sites. Hassler Whitney, then at Harvard, joined the group at Columbia working on questions concerning fire control systems, that is, systems which control the aiming of weaponry, especially on aircraft and rockets. His involvement was typical of the problems handled by the Applied Mathematics Panel.

Harvard made Whitney a full professor in 1946 and he held this professorship until he accepted an offer from the Institute for Advanced Study at Princeton of a professorship in 1952. After Whitney and his wife were divorced he married Mary Barnett Garfield on 16 January 1955; they had two children, Sarah Newcomb Whitney and Emily Baldwin Whitney.

Banchoff, reviewing [2], writes:-

In the collected papers of Hassler Whitney, two things are apparent - the wide range of his interests and innovations and the solitary nature of his research. He worked almost entirely alone, although he kept up with the developments in the fields in which he initiated new ideas. His work contains carefully done and complete details, including descriptions of the key examples that motivated the research.

He then goes on to give details of the topics on which Whitney worked:-

As impressive as each individual paper is, it is even more impressive to see them grouped together in two volumes, the first including papers in graphs and combinatorics, differentiable functions and singularities, and analytic spaces, and the second containing contributions to manifolds, bundles and characteristic classes, topology and algebraic topology, and geometric integration theory. A final section on other topics includes nine papers on logic, geometry, and the mathematics of physical quantities, for the last of which he received a Lester Ford Award.

Whitney's doctoral thesis was on graph theory, in particular making a major contribution to the four colour problem. Following this he published a number of papers on graph theory such as *A theorem on graphs* (1931), *Non-separable and planar graphs* (1932), *Congruent graphs and the connectivity of graphs* (1932), *The coloring of graphs* (1932), *A numerical equivalent of the four color map problem* (1937).

His main work, however, was in topology, particularly in the theory of manifolds. Continuing work started by Veblen and Henry Whitehead, Whitney produced fundamental work in differential topology in 1935. In particular he proved theorems about the embedding of an n-dimensional differentiable manifold in Euclidean space and he discovered characteristic classes at the same time as Stiefel. The term Stiefel-Whitney characteristic classes is often used today. He wrote a survey paper *Topological properties of differentiable manifolds* in 1937 which includes the many of the recent contributions he had made. In 1939 he gave his famous duality and product theorems: the term Whitney duality is now used.

Other work on algebraic varieties and integration theory was important. He published the book *Geometric integration theory* In 1957 which describes his work on the interactions between algebraic topology and the theory of integration. After an introduction, the chapters of the book are:-

Grassmann algebra; Differential forms; Riemann integration theory; Smooth manifolds; Abstract integration theory; Some relations between chains and functions; General properties of chains and cochains; Chains and cochains in open sets; Flat cochains and differential forms; Lipschitz mappings; Chains and additive set functions.

This topic had been the subject of the lecture which Whitney gave to the International Congress of Mathematicians, held in Cambridge, Massachusetts in 1950. His second book *Complex analytic varieties* was published in 1972.

In addition to research at the frontiers of mathematical research, Whitney was also interested in mathematics teachings in schools. Zund writes [8]:-

Whitney became actively involved in mathematical education at the elementary school level. He gave a number of lectures on this topic, conducted summer courses for teachers, and on one occasion spent four months teaching pre-algebra mathematics to a seventh grade class of students.

Outside mathematical research and teaching mathematics Whitney contributed in many ways to his subject. He was chairman of the National Science Foundation mathematics panel from 1953 until 1956. He was editor of the *American Journal of Mathematics* from 1944 to 1949, then editor of *Mathematical Reviews* from 1949 until 1954. He was honoured by being elected to the National Academy of Sciences (United States) in 1945, and he was also elected to the Academy of Sciences (Paris) and the Swiss Mathematical Society. He was American Mathematical Society Colloquium Lecturer in 1946 and he was vice-president of the American Mathematical Society from 1948 to 1950.

Ulam writing about Whitney said:-

He was friendly, but rather taciturn - psychologically of a type one encounters in this country more frequently than in central Europe - with wry humour, shyness but self-assurance, a probity which shines through, and a certain genius for persistent and deep follow-through in mathematics.

Princeton was to remain Whitney's base from 1952 until he retired in 1977. The year before he retired he was awarded the National Medal of Science. Then in 1983 he received the Wolf Prize:-

... for his fundamental work in algebraic topology, differential geometry and differential topology.

Two years later he was awarded the Steele Prize.

After Whitney was divorced from his second wife he married Barbara Floyd Osterman on 8 February 1986. He was nearly 79 years old at the time of his third marriage.

**Article by:** *J J O'Connor* and *E F Robertson*

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