Hassler Whitney

# Hassler Whitney

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Hassler Whitney was an American mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He was one of the founders of singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...

, and did foundational work in manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s, embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

s, immersions, and characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...

es.

## Work

Whitney's earliest work, from 1930 to 1933, was on graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. 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 1935 paper, where he laid the foundations for matroids, a fundamental notion in modern combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

and representation theory
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 studiesmodules over these abstract algebraic structures...

.

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
Charles Fefferman
Charles Louis Fefferman is an American mathematician at Princeton University. His primary field of research is mathematical analysis....

.

In a 1936 paper, Whitney gave a definition of a smooth manifold of class Cr, and proved that, for high enough values of r, a smooth manifold of dimension n may be embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

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
Oswald Veblen
Oswald Veblen 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.-Life:...

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
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....

s on a given topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

.

He was one of the major developers of cohomology theory, and characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...

es, 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 Schwarz 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
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...

and John Mather
John Mather
John Norman Mather is a mathematician at Princeton University known for his work on singularity theory and Hamiltonian dynamics...

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
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...

, were also first studied by Whitney.

His book Geometric Integration Theory gives a theoretical basis for Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...

applied with singularities on the boundary and later inspired the generalization found by Jenny Harrison
Jenny Harrison
Jenny Harrison is a professor of mathematics at UC Berkeley. She specializes in geometric analysis and areas in the intersection of algebra, geometry, and geometric measure theory...

.

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
Stiefel–Whitney class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney class is an example of a \mathbb Z_2characteristic class associated to real vector bundles.-General presentation:...

, basic results on vector bundle
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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

s) entered the mainstream more quickly.

## Career

He received his Ph.D. from Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...

in 1928; his Mus.B., 1929; Sc.D. (Honorary), 1947; and Ph.D. from Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...

, under George David Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....

, in 1932.

He was Instructor of Mathematics at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...

, 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
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

, Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

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

, 1953–56; Exchange Professor, Collège de France
Collège de France
The Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue Saint-Jacques and Rue des Écoles...

, 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
National Defense Research Committee
The National Defense Research Committee 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 United States from June 27, 1940 until June 28, 1941...

, 1943–45; Construction of the School of Mathematics. Recipient, National Medal of Science
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and...

, 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
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...

, 1946; Vice President, 1948–50 and Editor, American Journal of Mathematics, 1944–49; Editor, Mathematical Reviews
Mathematical Reviews
Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses of many articles in mathematics, statistics and theoretical computer science.- Reviews :...

, 1949–54; Chairman of the Committee vis. lectureship, 1946–51; Committee Summer Instructor, 1953–54; Steele Prize, 1985, American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...

; American National Council Teachers of Mathematics, London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...

(Honorary), Swiss Mathematics Society (Honorary), Académie des Sciences de Paris (Foreign Associate); New York Academy of Sciences
New York Academy of Sciences
The New York Academy of Sciences is the third oldest scientific society in the United States. An independent, non-profit organization with more than members in 140 countries, the Academy’s mission is to advance understanding of science and technology...

.

## Family

Hassler Whitney was the son of First District New York Supreme Court
New York Supreme Court
The Supreme Court of the State of New York is the trial-level court of general jurisdiction in thestate court system of New York, United States. There is a supreme court in each of New York State's 62 counties, although some smaller counties share judges with neighboring counties...

judge Edward Baldwin Whitney
Edward Baldwin Whitney
-Life:Edward Baldwin Whitney was born August 16, 1857. His father was linguist William Dwight Whitney of the new England Dwight family. His mother was Elizabeth Wooster Baldwin, daughter of US Senator and Governor of Connecticut Roger Sherman Baldwin....

and Josepha (Newcomb) Whitney, the grandson of Yale University Professor of Ancient Languages William Dwight Whitney
William Dwight Whitney
William Dwight Whitney was an American linguist, philologist, and lexicographer who edited The Century Dictionary.-Life:William Dwight Whitney was born in Northampton, Massachusetts on February 9, 1827. His father was Josiah Dwight Whitney of the New England Dwight family...

, the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin
Roger Sherman Baldwin
Roger Sherman Baldwin was an American lawyer involved in the Amistad case, who later became the 17th Governor of Connecticut and a United States Senator.-Early life:...

, and the great-great-great-grandson of American founding father Roger Sherman
Roger Sherman
Roger Sherman was an early American lawyer and politician, as well as a founding father. He served as the first mayor of New Haven, Connecticut, and served on the Committee of Five that drafted the Declaration of Independence, and was also a representative and senator in the new republic...

.

Hassler Whitney's maternal grandparents were astronomer and celestial mechanician Simon Newcomb
Simon Newcomb
Simon Newcomb was a Canadian-American astronomer and mathematician. Though he had little conventional schooling, he made important contributions to timekeeping as well as writing on economics and statistics and authoring a science fiction novel.-Early life:Simon Newcomb was born in the town of...

and Mary Hassler Newcomb (the granddaughter of the first superintendent of the Coast Survey - Ferdinand Hassler).

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.

### Named after Whitney

• Loomis–Whitney inequality
• McShane–Whitney extension theorem
• Stiefel–Whitney class
Stiefel–Whitney class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney class is an example of a \mathbb Z_2characteristic class associated to real vector bundles.-General presentation:...

• Whitney's conditions A and B
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...

• Whitney embedding theorem
• Whitney graph isomorphism theorem
• Whitney immersion theorem
• Whitney trick
• Whitney umbrella
Whitney umbrella
right|frame|240px|Section of the surfaceIn mathematics, the Whitney umbrella is a self-intersecting surface placed in three dimensions...