Hassler Whitney
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Hassler Whitney (March 23, 1907 – May 10, 1989) was an American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and
geometric integration theory In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds. Rather than functions or differential forms, the ...
.


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 The Supreme Court of the State of New York is the trial-level court of general jurisdiction in the New York State Unified Court System. (Its Appellate Division is also the highest intermediate appellate court.) It is vested with unlimited civ ...
judge. His mother, A. Josepha Newcomb Whitney, was an artist and active in politics. He was the paternal nephew of Connecticut Governor and Chief Justice Simeon Eben Baldwin, his paternal grandfather was William Dwight Whitney, professor of Ancient Languages at
Yale University Yale University is a Private university, private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the List of Colonial Colleges, third-oldest institution of higher education in the United Sta ...
, linguist and
Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cul ...
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), a
Steeves Steeves (also Steves) is a surname. Notable people with the name include: *Burpee L. Steeves (1868–1933), American politician from Idaho; lieutenant governor of Idaho 1905–07 * David Steeves (1934–1965), U.S. Air Force officer cleared of gi ...
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. 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. Whitney and his first wife Margaret made an innovative decision in 1939 that influenced the history of modern architecture in New England, when they commissioned the architect Edwin B. Goodell, Jr. to design a new residence for their family in Weston, Massachusetts. They purchased a rocky hillside site on a historic road, next door to another International Style house by Goodell from several years earlier, designed for Richard and Caroline Field. Distinctively featuring flat roofs, flush wood siding, and corner windows—all of which were unusual architectural elements at the time—the Whitney House was also a creative response to its site, in that it placed the main living spaces one floor above ground level, with large banks of windows opening to the south sun and to views of the beautiful property. The Whitney House survives today, along with the Field House, more than 75 years following its original construction; both are contributing structures in the historic Sudbury Road Area. 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 the Yale Mountaineering Society (the precursor to the Yale Outdoors Club) and climbed most of the mountain peaks in Switzerland.


Death

Three years after his third marriage, on 10 May 1989, Whitney died in Princeton, after suffering a stroke. In accordance with his wish, Hassler Whitney's ashes rest atop
mountain A mountain is an elevated portion of the Earth's crust, generally with steep sides that show significant exposed bedrock. Although definitions vary, a mountain may differ from a plateau in having a limited summit area, and is usually higher t ...
Dents Blanches The Dents Blanches (from French, lit. ''White Teeth'') is a mountain in the Chablais Alps on the Swiss Swiss may refer to: * the adjectival form of Switzerland * Swiss people Places * Swiss, Missouri * Swiss, North Carolina *Swiss, West Virg ...
in Switzerland where Oscar Burlet, another mathematician and member of the Swiss Alpine Club, placed them on August 20, 1989.


Academic career

Whitney attended
Yale University Yale University is a Private university, private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the List of Colonial Colleges, third-oldest institution of higher education in the United Sta ...
, where he received baccalaureate degrees in physics and in music, respectively in 1928 and in 1929. Later, in 1932, he earned a
PhD PHD or PhD may refer to: * Doctor of Philosophy (PhD), an academic qualification Entertainment * '' PhD: Phantasy Degree'', a Korean comic series * ''Piled Higher and Deeper'', a web comic * Ph.D. (band), a 1980s British group ** Ph.D. (Ph.D. albu ...
in mathematics at
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
. His doctoral dissertation was ''The Coloring of Graphs'', written under the supervision of George David Birkhoff. At Harvard, Birkhoff also got him a job as Instructor of Mathematics for the years 1930–31, and an Assistant Professorship for the years 1934–35. 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 The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
,
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
, 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 Sciences; Colloquium Lecturer,
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
, 1946; Vice President, 1948–50 and Editor, American Journal of Mathematics, 1944–49; Editor,
Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also ...
, 1949–54; Chairman of the Committee vis. lectureship, 1946–51; Committee Summer Instructor, 1953–54;,
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
; 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 The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communi ...
. 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 1982, 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 In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
. 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 In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in ...
, a fundamental notion in modern
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
and representation theory independently introduced by him and Bartel Leendert van der Waerden in the mid 1930s. In this paper Whitney proved several theorems about the matroid of a graph : one such theorem, now called Whitney's 2-Isomorphism Theorem, states: Given and are graphs with no isolated vertices. Then and are isomorphic if and only if and 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 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 ' ''r'', and proved that, for high enough values of ''r'', a smooth manifold of dimension ''n'' may be embedded in ℝ2''n''+1, and immersed in ℝ2''n''. (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 In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real -dimensional manifold (required also to be Hausdorf ...
".) 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 Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...
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", now referred to as Whitney conditions. 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 Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
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 bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
s) entered the mainstream more quickly.


Teaching

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 '' mathematical 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:Complete bibliography in and . all his published articles, included the ones listed in this section and the preface of the book , are collected in the two volumes and . *. *. *. *. *. *. *. *.


See also

* Loomis–Whitney inequality * Whitney extension theorem * Stiefel–Whitney class * Whitney's conditions A and B * Whitney embedding theorem *
Whitney graph isomorphism theorem In the mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edges of . is constructed in the following way: for each edge in , make a vertex in ; for ever ...
* Whitney immersion theorem *
Whitney inequality In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of smoothness. It was first proved by Hassler Whitney in 1957, and is an important tool in the fiel ...
*
Whitney's planarity criterion In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph ''G'' is planar if and only if its graphic matroid is also cographic (that is, it is the du ...
* Whitney umbrella


Notes


References


Biographical and general references

*. *. *. * *, available from Gallica. *


Scientific references

*. *. *. * ( e-book: ). *. * .


External links

*
Hassler Whitney Page - Whitney Research Group
* ttp://www.icmihistory.unito.it/portrait/whitney.php Hassler Whitney — The First Century of the International Commission on Mathematical Instructionbr>INFORMS
Biography of Hassler Whitney from the Institute for Operations Research and the Management Sciences {{DEFAULTSORT:Whitney, Hassler 1907 births 1989 deaths Scientists from New York City 20th-century 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 College alumni Harvard Graduate School of Arts and Sciences alumni Institute for Advanced Study faculty Mathematicians from New York (state)