Whitney, Hassler

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WHITNEY, HASSLER

(b. New York, New York, 23 March 1907;

d. Princeton, New Jersey, 10 May 1989), mathematician, topology, geometry.

One of the most creative mathematicians of the twentieth century, Whitney contributed fundamental notions and results to several areas of mathematics. His impact was most forcefully felt in topology, where he founded the field of differential topology.

Hassler Whitney was born to an accomplished family; his father, Edward Baldwin Whitney, was a state supreme court judge and his mother, Josepha Newcomb, an artist, was active in politics. His grandfathers were William D. Whitney, a noted Sanskrit scholar, and Simon Newcomb, a renowned astronomer and political economist. While young, Whitney spent two years in Switzerland, where he took up mountaineering, a lifelong passion. Among climbers, he is known for the first ascent of the Whitney-Gilman ridge on Cannon cliff in New Hampshire, which he made with his cousin in 1929. He is considered a pioneer in the fast and light style of climbing. His passion of mountaineering was shared by some of his colleagues in topology with whom he climbed in Switzerland.

Whitney went to Yale University, where he graduated in 1928 with a degree in physics. He stayed on a year longer to earn a degree in music in 1929; he was an accomplished player of the violin and the viola, and a member of local orchestras during his life. He earned a PhD in mathematics at Harvard University in 1932, under the direction of George Birkhoff, on the subject of graph coloring. By 1930 he was already an instructor at Harvard, and he remained there until 1952, a full professor from 1946. He became a permanent member of the faculty at the Institute for Advanced Study in Princeton in 1952, where he stayed until retirement in 1977.

Whitney’s thesis on graph theory was motivated by the four-color conjecture, that every planar map of connected countries may be colored using four colors; he became interested in this famous problem as an undergraduate and remained so throughout his career. The problem of coloring a map may be studied by assigning to a planar map its dual graph constructed as follows: place a point, a vertex, in each country; then join two points together with a line, an edge, if there is a border shared by two countries. This set of vertices, edges, and adjacency data form the dual graph. If you can color the vertices of this graph in such a way that no edge joins vertices of the same color, then you have colored the original map. Whitney studied the number of colorings of a graph as a function of the number of colors used. This function is a polynomial, now called the chromatic polynomial of the graph. He derived formulas for the coefficients of this polynomial using what he called a “logical expansion,” which is a version of the principle of inclusion-exclusion for which he gave several other applications. Borrowing from the apparatus of combinatorial topology, he introduced in 1932 a combinatorial generalization of the dual of a planar graph, now called the Whitney dual. He proved that the existence of this dual is equivalent to a graph being planar.

During the years 1931–1933, Whitney was named a National Research Council Fellow, and he spent the academic year 1931–1932 in Princeton. He finished his thesis while there, and began to shift his research to topology, a subject being developed actively at Princeton. Whitney was attracted to problems that presented an immediate and usually elementary challenge. For example, he noticed common features in the descriptions of the edge-sets of graphs and of the linear independence of columns of a matrix. His abstraction of these ideas, which he named matroids, developed into a central idea in combinatorics.

While at Yale he got interested in the Whyburn problem on the existence of nonconstant differentiable functions with a prescribed set of critical points. In a series of papers in the early 1930s, he developed the analysis of differentiable functions on Euclidean space, proving results about when a given function could be extended from a given domain to all of space with a given degree of differentiability. Around this time the notion of a manifold was given a working, abstract definition in the 1932 work of Oswald Veblen and J. H. C. Whitehead. Their abstract definition provided a foundation to differential geometry by unifying the many competing definitions of manifold. Equipped with his results on extensions of functions, Whitney proved that the abstract manifolds of Veblen and Whitehead can always be identified with a subset of Euclidean space. This theorem provided many concrete properties that could be used to study manifolds; for example, distances make sense inside a Euclidean space, and so a manifold inherits a notion of distance from its embedding. This shows that abstract manifolds always have a Riemannian structure. Any particular embedding of a manifold determines a set of directions at each point of the manifold that are either tangent to it, or perpendicular to the tangent directions. This decomposition into tangent directions and normal directions played a key role in Whitney’s later research.

In another work of his time in Princeton, Whitney studied the curves that solve a differential equation as a topological and geometric object, introducing functions that allow cross sections and tubes to be defined. These ideas were current in the study of certain manifolds of three dimensions introduced by Karl Johannes Herbert Seifert. Whitney greatly generalized this class of manifolds with the introduction of his sphere-spaces in 1935. A sphere-space consists of a base space and a total space, where the base space is a parameter space for a family of spheres whose union makes up the total space. In this context, the embedding of a manifold gave rise to two sphere-spaces with base space the manifold, one from the tangent directions and the other from the normal directions. He also introduced invariants of a sphere-space, defined by mapping simplices of the base space to collections of unit orthogonal directions in a sphere identifiable with the sphere at a given point. The set of all such collections of unit orthogonal directions in Euclidean space was being studied independently in Switzerland by Heinz Hopf and Eduard Stiefel; the manifold of such collections became known as a Stiefel manifold. The invariants were identified with classes in the homology of the base space with coefficients in the homology of the Stiefel manifolds, and these elements are called the characteristic classes of the sphere-space. Whitney’s work on the embeddings of manifolds implied that the characteristic classes of the tangent sphere-space and normal sphere-space were related by what is now called Whitney duality.

Whitney traveled to Moscow in 1935 to participate in the International Conference in Topology organized by P. S. Alexandroff. The conference brought together an international community of researchers for whom common problems and newly found methods could be shared. Here Whitney presented his work on embeddings and on sphere-spaces, and he learned of Stiefel’s parallel work from Hopf. He also learned of the independent discoveries by James W. Alexander and A. N. Kolmogoroff of a product structure in a simplicial complex. However, their definitions were flawed. Eduard Cech and Whitney in the coming months independently figured out how to rectify the definition. Whitney’s paper established the terminology of cohomology, coboundary, cocycle, cup product, and cap product now used in topology. He was also able to apply the new structures to give proofs of Hopf’s foundational results on the classification of continuous functions from an n-dimensional complex to an n-dimensional sphere.

In the years leading up to World War II, Whitney extended his results on embeddings, proving that an n-dimensional manifold can be embedded in 2n- dimensional Euclidean space. To reduce this dimension further is not possible, but it is possible to immerse the manifold in (2n-1)-dimensional Euclidean space, that is, to find a mapping that is locally nice, but might cross itself in places where the crossing is well-behaved. In proving the result on immersions, Whitney began forging the foundations for the study of singularities of differentiable mappings, a subject he developed further after the war.

During the war, Whitney became involved in the war effort through the Applied Mathematics Panel, for which he served as a consultant. He worked with the group at Columbia University on problems of fire control, that is, weapons guidance on airplanes, boats, and on the ground. A coworker, Mina Rees, commented that Whitney “turned out to have an absolute genius for airplane problems from guidance studies.” His involvement was deep and, though he was naturally shy, he brought his findings before military officials in an effort to change policy, and he was successful.

After the war, Whitney turned his research from topology to focus on singularities of differentiable mappings, integration theory, and complex algebraic geometry. After joining the permanent faculty at the Institute for Advanced Study in Princeton, he wrote two books, on geometric integration theory and on complex analytic varieties, which are unique in their geometric approach to these subjects. Late in his career, he returned to the four-color conjecture when one of the first attempts at a computer proof suggested a simpler argument. With William T. Tutte, Whitney analyzed the work of Y. Shimamoto and disagreed with the outcome of the computer results. Tutte and Whitney were correct, and this approach to a proof of the four-color theorem was abandoned.

By the time of his retirement from the institute, Whitney had turned his energies to mathematics education. His writings on education emphasized the need to let young learners explore “naturally, … finding one’s way through problems of new sorts, and taking responsibility for the results.” He involved himself thoroughly in this research, visiting schools, and talking with teachers. He also served as president of the International Commission on Mathematical Instruction, 1979–1982. For his pioneering work in mathematics, Whitney was honored with the National Medal of Science in 1976, the Wolf Foundation Prize in 1982, and the Steele Prize from the American Mathematical Society for his seminal work in 1985.

BIBLIOGRAPHY

The mathematical works of Whitney can be found in Collected Papers, Vols. I and II. Edited and with a preface by James Eells and Domingo Toledo. Boston: Birkhäuser, 1992.