Hassler Whitney

Hassler Whitney
Born	(1907-03-23)March 23, 1907 New York City
Died	May 10, 1989(1989-05-10) (aged 82) Dents Blanches
Fields	Mathematics
Institutions	Harvard University Institute for Advanced Study Princeton University National Science Foundation National Defense Research Committee
Alma mater	Yale University
Doctoral advisor	George David Birkhoff
Doctoral students	James Eells Wilfred Kaplan Paul Olum Herbert Robbins
Known for	Contributions in algebraic topology, differential topology, singularity theory
Notable awards	National Medal of Science (1976) Wolf Prize (1983) Steele Prize (1985)

Hassler Whitney (23 March 1907 – 10 May 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.

Biography

Life

Hassler Whitney was born on March 23, 1907, in New York City, where his father Edward Baldwin Whitney was the First District New York Supreme Court judge. His mother, Josepha (Newcomb) Whitney, was an artist and active in politics. His paternal grandfather, William Dwight Whitney, was a Yale University Professor of Ancient Languages, linguist, and Sanskrit scholar. Whitney was the great grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-grandson of American founding father Roger Sherman. His maternal grandparents were astronomer and mathematician Simon Newcomb (1835-1909) and Mary Hassler Newcomb (the granddaughter of the first superintendent of the Coast Survey Ferdinand Hassler). His great uncle was the first to survey Mount Whitney.

Throughout his life he pursued two particular hobbies with excitement: music and mountain-climbing. An accomplished player of the violin and the viola, Whitney played with the Princeton Musical Amateurs. He would run outside, 6 to 12 miles every other day. As an undergraduate, with his cousin Bradley Gilman, Whitney made the first ascent of the Whitney Gilman ridge on Cannon Mountain, New Hampshire in 1929. It was the hardest and most famous rock climb in the East. He was a member of the Swiss Alpine Society and climbed most of the mountain peaks in Switzerland.

He married Margaret R. Howell, May 30, 1930; children: James Newcomb, Carol, Marian; married Mary Barnett Garfield, January 16, 1955; children: Sarah Newcomb, Emily Baldwin; and married Barbara Floyd Osterman, February 8, 1986.

Academic career

Whitney attended Yale University where he received a baccalaureate degree in physics in 1928 and in music in 1929. He earned a Ph.D. in mathematics at Harvard University in 1932. His doctorate was awarded for a dissertation The Coloring of Graphs written under the supervision of George David Birkhoff. He was Instructor of Mathematics at Harvard University, 1930–31, 1933–35; NRC Fellow, Mathematics, 1931–33; Assistant Professor, 1935–40; Associate Professor, 1940–46, Professor, 1946–52; Professor Instructor, Institute for Advanced Study, Princeton University, 1952–77; Professor Emeritus, 1977–89; Chairman of the Mathematics Panel, National Science Foundation, 1953–56; Exchange Professor, Collège de France, 1957; Memorial Committee, Support of Research in Mathematical Sciences, National Research Council, 1966–67; President, International Commission of Mathematical Instruction, 1979–82; Research Mathematicians, National Defense Research Committee, 1943–45; Construction of the School of Mathematics. Recipient, National Medal of Science, 1976, Wolf Prize, Wolf Foundation, 1983; and a Steele Prize in 1985.

He was a member of the National Academy of Science; Colloquium Lecturer, American Mathematical Society, 1946; Vice President, 1948–50 and Editor, American Journal of Mathematics, 1944–49; Editor, Mathematical Reviews, 1949–54; Chairman of the Committee vis. lectureship, 1946–51; Committee Summer Instructor, 1953–54; Steele Prize, 1985, American Mathematical Society; American National Council Teachers of Mathematics, London Mathematical Society (Honorary), Swiss Mathematics Society (Honorary), Académie des Sciences de Paris (Foreign Associate); New York Academy of Sciences.

Work

Research

Whitney's earliest work, from 1930 to 1933, was on graph theory. Many of his contributions were to the graph-coloring, and the ultimate computer-assisted solution to the four-color problem relied on some of his results. His work in graph theory culminated in a 1933 paper, where he laid the foundations for matroids, a fundamental notion in modern combinatorics and representation theory. For example, Whitney proved theorems about the matroid of a graph M(G). Matroids were discovered independently by Hassler Whitney and B. L. van der Waerden in the mid 1930s. One such theorem, called Whitney's 2-Isomorphism Theorem, states: Given G and H are graphs with no isolated vertices. Then M(G) and M(H) are isomorphic if and only if G and H are 2-isomorphic. Two graphs G and G' are 2-isomorphic if one can be transformed into the other by operations of the following: 1)Vertex identification 2)Vertex cleaving 3)Twisting.

Whitney's lifelong interest in geometric properties of functions also began around this time. His earliest work in this subject was on the possibility of extending a function defined on a closed subset of ℝⁿ to a function on all of ℝⁿ with certain smoothness properties. A complete solution to this problem was found only in 2005 by Charles Fefferman.

In a 1936 paper, Whitney gave a definition of a smooth manifold of class C ^r, and proved that, for high enough values of r, a smooth manifold of dimension n may be embedded in ℝ²ⁿ⁺¹, and immersed in ℝ²ⁿ. (In 1944 he managed to reduce the dimension of the ambient space by 1, provided that n > 2, by a technique that has come to be known as the "Whitney trick".) This basic result shows that manifolds may be treated intrinsically or extrinsically, as we wish. The intrinsic definition had been published only a few years earlier in the work of Oswald Veblen and J.H.C. Whitehead. These theorems opened the way for much more refined studies: of embedding, immersion and also of smoothing: that is, the possibility of having various smooth structures on a given topological manifold.

He was one of the major developers of cohomology theory, and characteristic classes, as these concepts emerged in the late 1930s, and his work on algebaic topology continued into the 40s. He also returned to the study of functions in the 1940s, continuing his work on the extension problems formulated a decade earlier, and answering a question of Schwartz in a 1948 paper On Ideals of Differentiable Functions.

Whitney had, throughout the 1950s, an almost unique interest in the topology of singular spaces and in singularities of smooth maps. An old idea, implicit even in the notion of a simplicial complex, was to study a singular space by decomposing it into smooth pieces (nowadays called "strata"). Whitney was the first to see any subtlety in this definition, and pointed out that a good "stratification" should satisfy conditions he termed "A" and "B". The work of René Thom and John Mather in the 1960s showed that these conditions give a very robust definition of stratified space. The singularities in low dimension of smooth mappings, later to come to prominence in the work of René Thom, were also first studied by Whitney.

His book Geometric Integration Theory gives a theoretical basis for Stokes' theorem applied with singularities on the boundary and later inspired the generalization found by Jenny Harrison.

These aspects of Whitney's work have looked more unified, in retrospect and with the general development of singularity theory. Whitney's purely topological work (Stiefel–Whitney class, basic results on vector bundles) entered the mainstream more quickly.

Outside of Advanced Mathematics

Teaching the Youth

In 1967, he became involved full-time in educational problems, especially at the elementary school level. He spent many years in classrooms, both teaching mathematics and observing how it is taught. He spent four months teaching pre-algebra mathematics to a classroom of seventh graders and conducted summer courses for teachers. He traveled widely to lecture on the subject in the United States and abroad.He worked toward removing the mathematics anxiety, which he felt leads young pupils to avoid mathematics. Whitney spread the ideas of teaching mathematics to students in ways that relate the content to their own lives as opposed to teaching them rote memorization.

Death

Whitney divorced 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. Three years later on May 10, 1989, Whitney passed away, after suffering a stroke, in Mount Bents Blanches, Switzerland, where they left his body to rest in peace.

See also

• Loomis–Whitney inequality
• McShane–Whitney extension theorem
• Stiefel–Whitney class
• Whitney's conditions A and B
• Whitney embedding theorem
• Whitney graph isomorphism theorem
• Whitney immersion theorem
• Whitney trick
• Whitney umbrella

Biographical references

• Fowler, Glenn (May 12, 1989), "Hassler Whitney, Geometrician; He Eased 'Mathematics Anxiety'", The New York Times, retrieved 9 January 2012 

External links

• Hassler Whitney at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Hassler Whitney", MacTutor History of Mathematics archive, University of St Andrews .
• Hassler Whitney Page - Whitney Research Group
• Interview with Hassler Whitney about his experiences at Princeton
• Hassler Whitney — The First Century of the International Commission on Mathematical Instruction