Hassler Whitney
Hassler Whitney  

Whitney in April 1973


Born  New York City 
March 23, 1907
Died  May 10, 1989 Princeton, New Jersey 
(aged 82)
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  Algebraic topology Differential topology Graph theory Geometric measure theory Singularity theory 
Notable awards  National Medal of Science (1976) Wolf Prize (1983) Leroy P. Steele Prize (1985) 
Hassler Whitney (March 23, 1907 – May, 10 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration theory.
Contents
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.^{[1]} His mother, Josepha (Newcomb) Whitney, was an artist and active in politics.^{[2]} His paternal grandfather was William Dwight Whitney, professor of Ancient Languages at Yale University, linguist, and Sanskrit scholar.^{[2]} Whitney was the great grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the greatgreatgrandson of American founding father Roger Sherman. His maternal grandparents were astronomer and mathematician Simon Newcomb (18351909), a Steeves descendant, and Mary Hassler Newcomb, granddaughter of the first superintendent of the Coast Survey Ferdinand Rudolph Hassler. His great uncle Josiah Whitney was the first to survey Mount Whitney.^{[3]}
He married three times: his first wife was Margaret R. Howell, married on the 30 May 1930. They had three children, James Newcomb, Carol and Marian. After his first divorce, on January 16, 1955 he married Mary Barnett Garfield. He and Mary had two daughters, Sarah Newcomb and Emily Baldwin. Finally, Whitney divorced his second wife and married Barbara Floyd Osterman on 8 February 1986.
Throughout his life he pursued two particular hobbies with excitement: music and mountainclimbing. 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 WhitneyGilman 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.^{[4]}
Death
Three years after his third marriage, on 10 May 1989, Whitney died in Princeton,^{[5]} after suffering a stroke.^{[6]} In accordance with his wish, Hassler Whitney ashes rest atop mountain Dents Blanches in Switzerland where Oscar Burlet, another mathematician and member of the Swiss Alpine Club, placed them on August 20, 1989.^{[7]}
Academic career
Whitney attended Yale University where he received baccalaureate degrees in physics and in music, respectively in 1928 and in 1929.^{[2]} Later, in 1932, He earned a PhD in mathematics at Harvard University.^{[2]} His doctoral dissertation was The Coloring of Graphs, written under the supervision of George David Birkhoff.^{[8]}^{[9]} At Harvard, Birkhoff also got him a job as Instructor of Mathematics for the years 1930–31,^{[10]} and an Assistant Professorship for the years 1934–35.^{[11]} Later on he held the following working positions: 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 Mathematician, National Defense Research Committee, 1943–45; Construction of the School of Mathematics.
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;, 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.
Honors
In 1947 he was elected member of the American Philosophical Society.^{[12]} In 1969 he was awarded the Lester R. Ford Award for the paper in two parts "The mathematics of Physical quantities" (1968a, 1968b).^{[13]} In 1976 he was awarded the National Medal of Science. In 1980 he was elected honorary member of the London Mathematical Society.^{[14]} In 1983 he received the Wolf Prize from the Wolf Foundation, and finally, in 1985, he was awarded the Steele Prize from the American Mathematical Society.
Work
Research
Whitney's earliest work, from 1930 to 1933, was on graph theory. Many of his contributions were to the graphcoloring, and the ultimate computerassisted solution to the fourcolor problem relied on some of his results. His work in graph theory culminated in a 1933 paper,^{[15]} where he laid the foundations for matroids, a fundamental notion in modern combinatorics and representation theory independently introduced by him and Bartel Leendert van der Waerden in the mid 1930s.^{[16]} In this paper Whitney proved several theorems about the matroid of a graph M(G): one such theorem, now called Whitney's 2Isomorphism 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 2isomorphic.^{[17]}
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 ℝ^{n} to a function on all of ℝ^{n} 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 ℝ^{2n+1}, and immersed in ℝ^{2n}. (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 algebraic 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 Laurent 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.
In his book Geometric Integration Theory he gives a theoretical basis for Stokes' theorem applied with singularities on the boundary:^{[18]} later, his work on such topics inspired the researches of Jenny Harrison.^{[19]}
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.
Teaching activity
Teaching the youth
In 1967, he became involved fulltime in educational problems, especially at the elementary school level. He spent many years in classrooms, both teaching mathematics and observing how it is taught.^{[20]} He spent four months teaching prealgebra 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.
Selected publications
Hassler Whitney published 82 works:^{[21]} all his published articles, included the ones listed in this section and the preface of the book Whitney (1957), are collected in the two volumes Whitney (1992a, pp. xii–xiv) and Whitney (1992b, pp. xii–xiv).
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See also
 Loomis–Whitney inequality
 Whitney extension theorem
 Stiefel–Whitney class
 Whitney's conditions A and B
 Whitney embedding theorem
 Whitney graph isomorphism theorem
 Whitney immersion theorem
 Whitney inequality
 Whitney embedding theorem
 Whitney umbrella
Notes
 ↑ Thom (1990, p. 474) and Chern (1994, p. 465).
 ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} Chern (1994, p. 465)
 ↑ According to Chern (1994, p. 465) and Thom (1990, p. 474): Thom cites Josiah Whitney explicitly while Chern simply states that:"... a great uncle was the first to survey Mount Whitney".
 ↑ Fowler (1989).
 ↑ Kendig (2013, p. 18) clarifies Princeton, NJ as his correct death place.
 ↑ According to Kendig (2013, p. 18). Kendig writes also that being him apparently in good health, the physicians attributed the cause of the stroke to the treatments for prostate cancer he was undergoing.
 ↑ The story on his resting place story is reported by Chern (1994, p. 467): see also Kendig (2013, p. 18).
 ↑ O'Connor, JJ and E F Robertson. "Hassler Whitney". Retrieved 20130416.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ See Kendig (2013, pp. 8–10).
 ↑ See (Kendig 2013, p. 9).
 ↑ See (Kendig 2013, pp. 9–10).
 ↑ See (Chern 1994, p. 465).
 ↑ Whitney (1992a, p. xi) and Whitney (1992b, p. xi), section, "Academic Appointments and Awards".
 ↑ See the official list of honorary members redacted by Fisher (2012).
 ↑ Whitney (1933).
 ↑ According to Johnson, Will. "Matroids" (PDF). Retrieved 5 February 2013.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ According to Oxley (1992, pp. 147–153). Recall that two graphs G and G' are 2isomorphic if one can be transformed into the other by applying operations of the following types:
 Vertex identification
 Vertex cleaving
 Twisting.
 ↑ See Federer's review (1958).
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Hechinger, Fred. "Learning Math by Thinking".June 10, 1986. http://rationalmathed.blogspot.com/2009/04/learningmathbythinkinghassler.html#!/2009/04/learningmathbythinkinghassler.html.
 ↑ Complete bibliography in Whitney (1992a, pp. xii–xiv) and Whitney (1992b, pp. xii–xiv).
References
Biographical and general references
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 Fisher, Elizabeth (9 November 2012), Full list of Honorary Members (PDF), London Mathematical Society, retrieved 14 July 2013<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 Fowler, Glenn (May 12, 1989), "Hassler Whitney, Geometrician; He Eased 'Mathematics Anxiety'", The New York Times, retrieved 9 January 2012<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 Kendig, Keith (August 2013), "Hassler Whitney", Celebratio Mathematica, 2013 (1), retrieved November 27, 2014<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
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Scientific references
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 Epstein, Marcelo (2004), "Whitney's Geometric Integration and Its Use in Continuum Mechanics", in Capriz, Gianfranco; Grioli, Giuseppe; Magenes, Enrico; Pitteri, Mario; PodioGuidugli, Paolo, Whence the Boundary Conditions in Modern Continuum Physics? (Roma 14–16 ottobre 2002), Atti dei Convegni Lincei, 210, Roma: Accademia Nazionale dei Lincei, pp. 127–137<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
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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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 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
 1907 births
 1989 deaths
 People from New York City
 20thcentury American mathematicians
 Geometers
 Graph theorists
 Topologists
 Members of the United States National Academy of Sciences
 Members of the American Philosophical Society
 National Medal of Science laureates
 Wolf Prize in Mathematics laureates
 Harvard University faculty
 Princeton University faculty
 Yale University alumni
 Harvard University alumni
 Institute for Advanced Study faculty
 Sherman family