Hassler Whitney (March 23, 1907 – May 10, 1989) was an American

^{''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 ^{''r''}, and proved that, for high enough values of ''r'', a smooth manifold of dimension ''n'' may be embedded in ℝ^{2''n''+1}, and immersed in ℝ^{2''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".) 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

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)

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, structure, space, models, and change.
History
On ...

. He was one of the founders of singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

, and did foundational work in manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s, embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is gi ...

s, immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...

s, characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...

es, and geometric integration theory.
Biography

Life

Hassler Whitney was born on March 23, 1907, in New York City, where his fatherEdward Baldwin Whitney
Edward Baldwin Whitney (August 16, 1857 – January 5, 1911) was an American lawyer and judge.
Life
Edward Baldwin Whitney was born August 16, 1857. His father was linguist William Dwight Whitney (1827–1894) of the New England Dwight family. Hi ...

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
Simeon Eben Baldwin (February 5, 1840 – January 30, 1927) was an American jurist, law professor, and politician who served as the 65th governor of Connecticut.
Education
The son of jurist, Connecticut governor and U.S. Senator Roger Sherman ...

, his paternal grandfather was William Dwight Whitney
William Dwight Whitney (February 9, 1827June 7, 1894) was an American linguist, philologist, and lexicographer known for his work on Sanskrit grammar and Vedic philology as well as his influential view of language as a social institution. He was ...

, professor of Ancient Languages at Yale University
Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the wo ...

, linguist and Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...

scholar. Whitney was the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin
Roger Sherman Baldwin (January 4, 1793 – February 19, 1863) was an American politician who served as the 32nd Governor of Connecticut from 1844 to 1846 and a United States senator from 1847 to 1851. As a lawyer, his career was most notable ...

, and the great-great-grandson of American founding father Roger Sherman
Roger Sherman (April 19, 1721 – July 23, 1793) was an American statesman, lawyer, and a Founding Father of the United States. He is the only person to sign four of the great state papers of the United States related to the founding: the Cont ...

. His maternal grandparents were astronomer and mathematician Simon Newcomb
Simon Newcomb (March 12, 1835 – July 11, 1909) was a Canadian–American astronomer, applied mathematician, and autodidactic polymath. He served as Professor of Mathematics in the United States Navy and at Johns Hopkins University. Born in Nov ...

(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 g ...

descendant, and Mary Hassler Newcomb, granddaughter of the first superintendent of the Coast Survey Ferdinand Rudolph Hassler
Ferdinand Rudolph Hassler (October 6, 1770 – November 20, 1843) was a Swiss-American surveyor who is considered the forefather of both the National Oceanic and Atmospheric Administration (NOAA) and the National Institute of Standards and Techn ...

. His great uncle Josiah Whitney
Josiah Dwight Whitney (November 23, 1819 – August 18, 1896) was an American geologist, professor of geology at Harvard University (from 1865), and chief of the California Geological Survey (1860–1874). Through his travels and studies in the ...

was the first to survey Mount Whitney
Mount Whitney (Paiute: Tumanguya; ''Too-man-i-goo-yah'') is the highest mountain in the contiguous United States and the Sierra Nevada, with an elevation of . It is in East–Central California, on the boundary between California's Inyo and Tu ...

.
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 atopmountain
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 (topography), summit area, and ...

Dents Blanches in Switzerland where Oscar Burlet, another mathematician and member of the Swiss Alpine Club, placed them on August 20, 1989.
Academic career

Whitney attendedYale University
Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the wo ...

, where he received baccalaureate degrees in physics and in music, respectively in 1928 and in 1929. Later, in 1932, he earned a PhD 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 higher le ...

. His doctoral dissertation was ''The Coloring of Graphs'', written under the supervision of George David Birkhoff
George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during ...

.
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 university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

, 1952–77; Professor Emeritus, 1977–89; Chairman of the Mathematics Panel, National Science Foundation
The National Science Foundation (NSF) is an independent agency of the United States government that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National I ...

, 1953–56; Exchange Professor, Collège de France
The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris n ...

, 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
The National Defense Research Committee (NDRC) was an organization created "to coordinate, supervise, and conduct scientific research on the problems underlying the development, production, and use of mechanisms and devices of warfare" in the Un ...

, 1943–45; Construction of the School of Mathematics.
He was a member of the National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...

; 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 pu ...

, 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
The New York Academy of Sciences (originally the Lyceum of Natural History) was founded in January 1817 as the Lyceum of Natural History. It is the fourth oldest scientific society in the United States. An independent, nonprofit organization wi ...

.
Honors

In 1947 he was elected member of theAmerican 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 communit ...

.
In 1969 he was awarded the Lester R. Ford Award
Lester is an ancient Anglo-Saxon surname and given name. Notable people and characters with the name include:
People
Given name
* Lester Bangs (1948–1982), American music critic
* Lester W. Bentley (1908–1972), American artist from Wisc ...

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 ongraph 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 conne ...

. Many of his contributions were to the graph-coloring, and the ultimate computer-assisted solution to the four-color problem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sha ...

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
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 appl ...

and representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

independently introduced by him and Bartel Leendert van der Waerden
Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics.
Biography
Education and early career
Van der Waerden learned advanced mathematics at the University of Amster ...

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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

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 ℝCharles Fefferman
Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr. '43 University Professor of Mathematics. He was awarded the Fields Medal in 1978 for his contri ...

.
In a 1936 paper, Whitney gave a definition of a smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

of class ' Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was lon ...

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 structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Definition
A smooth structure on a manifold M is ...

s on a given topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...

.
He was one of the major developers of cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

, and characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...

es, 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 In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965.
A stratification of a topological space is a finite filtration by closed subsets ...

. 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
Jenny Harrison is a professor of mathematics at the University of California, Berkeley.
Education and career
Harrison grew up in Tuscaloosa, Alabama. On graduating from the University of Alabama, she won a Marshall Scholarship which she used to ...

.
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
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...

, 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 every po ...

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
Mathematical anxiety, also known as math phobia, is anxiety about one's ability to do mathematics.
Math Anxiety
Mark H. Ashcraft defines math anxiety as "a feeling of tension, apprehension, or fear that interferes with math performance" (2002, p. ...

,'' 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
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if ''A'' is a closed subset of a Euclidean space, then it is possible to e ...

*Stiefel–Whitney class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...

* Whitney's conditions A and B
*Whitney embedding theorem
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 Hausdorff ...

*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 every ...

*Whitney immersion theorem
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, ...

* Whitney inequality
* Whitney's planarity criterion
*Whitney umbrella
frame, Section of the surface
In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dime ...

Notes

References

Biographical and general references

*. *. *. * *, available from Gallica. *Scientific references

*. *. *. * (e-book
An ebook (short for electronic book), also known as an e-book or eBook, is a book publication made available in digital form, consisting of text, images, or both, readable on the flat-panel display of computers or other electronic devices. Alt ...

: ).
*.
* .
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)