|This article needs additional citations for verification. (April 2013)|
March 23, 1907|
New York City
|Died||May 10, 1989
Institute for Advanced Study
National Science Foundation
National Defense Research Committee
|Alma mater||Yale University|
|Doctoral advisor||George David Birkhoff|
|Doctoral students||James Eells
|Known for||Algebraic topology
Geometric measure 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, as well as in geometric integration theory.
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 was William Dwight Whitney, professor of Ancient Languages at Yale University, 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.
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 died, after suffering a stroke, in Mount Dents Blanches, Switzerland: his ashes were placed on the top of that mountain by Oscar Burlet, another mathematician and member of the Swiss Alpine Club.
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.
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.
In 1969 he was awarded the Lester R. Ford Award for the paper in two parts "The mathematics of Physical quantities" (1968a, 1968b). In 1976 he was awarded the National Medal of Science. In 1980 he was elected honorary member of the London Mathematical Society. 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.
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 independently introduced by him and B. L. van der Waerden in the mid 1930s. In this paper Whitney proved several theorems about the matroid of a graph M(G): one such theorem, now 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.
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 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: later, his work on such topics inspired the researches of 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.
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.
Hassler Whitney published 82 works: 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).
- Whitney, Hassler (January 1932), "Congruent Graphs and the Connectivity of Graphs", American Journal of Mathematics 54 (1): 150–168, doi:10.2307/2371086, JFM 58.0609.01, MR 1506881, Zbl 0003.32804.
- Whitney, Hassler (1933), "2-Isomorphic Graphs", American Journal of Mathematics 55: 245–254, doi:10.2307/2371127, JFM 59.1235.01, MR 1506961, Zbl 0006.37005.
- Whitney, H. (1957), Geometric Integration Theory, Princeton Mathematical Series 21, Princeton, NJ and London: Princeton University Press and Oxford University Press, pp. XV+387, MR 0087148, Zbl 0083.28204.
- Whitney, Hassler (1968a), "The Mathematics of Physical Quantities. Part I", The American Mathematical Monthly 75: 115–138, MR 0228219, Zbl 0186.57901.
- Whitney, Hassler (1968b), "The Mathematics of Physical Quantities. Part II", The American Mathematical Monthly 75: 227–256, MR 0228220, Zbl 0186.57901.
- Whitney, Hassler (1972), Complex analytic varieties, Addison-Wesley Series in Mathematics, Reading-Menlo Park-London-Don Mills: Addison-Wesley, ISBN 0-201-08653-0, MR 0387634, Zbl 0265.32008.
- Whitney, Hassler (1992a), Eells, James; Toledo, Domingo, eds., The collected papers of Hassler Whitney. Volume I., Contemporary Mathematicians, Basel–Boston–Stuttgart: Birkhäuser Verlag, pp. xiv+590, ISBN 0-8176-3558-0, Zbl 0746.01016.
- Whitney, Hassler (1992b), Eells, James; Toledo, Domingo, eds., The collected papers of Hassler Whitney. Volume II., Contemporary Mathematicians, Boston–Basel–Berlin: Birkhäuser Verlag, pp. xiv+596, ISBN 0-8176-3559-9, Zbl 0746.01016.
- 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
- See (Thom 1990, p. 474) and (Chern 1994, p. 465).
- See (Chern 1994, p. 465).
- See (Fowler 1989).
- According to Chern (1994, p. 467).
- O'Connor, JJ and E F Robertson. "Hassler Whitney". Retrieved 4/16/13.
- See (Whitney 1992a, p. xi) or (Whitney 1992b, p. xi), section, "Academic Appointments and Awards".
- See the official list of honorary members redacted by Fisher (2012).
- See (Whitney 1933).
- According to Johnson, Will. "Matroids". Retrieved 5/2/13. .
- See (Oxley 1992, pp. 147–153). Recall that two graphs G and G' are 2-isomorphic if one can be transformed into the other by applying operations of the following types:
- See the review by Federer (1958).
- Harrison, Jenny (1993), "Stokes' theorem for nonsmooth chains", American Mathematical Society, New Series 29 (2): 235–242, arXiv:math/9310231, doi:10.1090/S0273-0979-1993-00429-4, MR 1215309, "Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney."
- Hechinger, Fred. "Learning Math by Thinking".June 10, 1986. http://rationalmathed.blogspot.com/2009/04/learning-math-by-thinking-hassler.html#!/2009/04/learning-math-by-thinking-hassler.html.
- See his complete bibliography in (Whitney 1992a, pp. xii–xiv) or (Whitney 1992b, pp. xii–xiv).
- Chern, Shiing-Shen (September 1994), "Hassler Whitney (23 March 1907-10 May 1989)", Proceedings of the American Philosophical Society 138 (3): 464–467, retrieved 19 April 2013.
- Fisher, Elizabeth (9 November 2012), Full list of Honorary Members, London Mathematical Society, retrieved 14 July 2013.
- Fowler, Glenn (May 12, 1989), "Hassler Whitney, Geometrician; He Eased 'Mathematics Anxiety'", The New York Times, retrieved 9 January 2012.
- Thom, René (1990), "La vie et l'œuvre de Hassler Whitney", Comptes rendus de l'Académie des sciences. Série générale, La Vie des sciences (in French) 7 (6): 473–476, MR 1105198, Zbl 0722.01025, available from Gallica.
- Chirka, Evgeniǐ Mikhaǐlovich (1989), Complex analytic sets, Mathematics and Its Application (Soviet Series) 46, Dordrecht-Boston-London: Kluwer Academic Publishers, ISBN 0-7923-0234-6, MR 1111477, Zbl 0683.32002.
- Federer, Herbert (1958), "Review: Geometric integration theory, by H. Whitney", Bulletin of the American Mathematical Society 64 (1): 38–41, doi:10.1090/s0002-9904-1958-10143-3.
- Oxley, James (1992), Matroid Theory, Oxford Graduate Texts in Mathematics 3, Oxford: The Clarendon Press, Oxford University Press, pp. xii+532, ISBN 0-19-853563-5, MR 1207587, Zbl 0784.05002.
- Shields, Allen (1988), "Differentiable manifolds: Weyl and Whitney", The Mathematical Intelligencer 10 (2): 5–8, MR 0932157, Zbl 0645.01012.
- 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