John R. Isbell
Encyclopedia
John Rolfe Isbell was an American mathematician, for many years a professor of mathematics at the University of Buffalo (SUNY).
, the son of an army officer. He attended several undergraduate institutions, including the University of Chicago
, where professor Saunders Mac Lane
was a source of inspiration. He began his graduate studies in mathematics at Chicago, briefly studied at Oklahoma A&M
and the University of Kansas
, and eventually completed a Ph.D. in game theory
at Princeton University
in 1954 under the supervision of Albert W. Tucker
. After graduation, Isbell was drafted into the U.S. Army, and stationed at the Aberdeen Proving Ground
. In the late 1950s he worked at the Institute for Advanced Study
in Princeton, New Jersey
, from which he then moved to the University of Washington
and Case Western Reserve University
. He joined the University at Buffalo in 1969, and remained there until his retirement in 2002.
s. Isbell published the first paper by John Rainwater
, a fictitious mathematician, who had been invented by graduate students at the University of Washington in 1952. After Isbell's paper, other mathematicians have published papers using the name "Rainwater" and have acknowledged "Rainwater's assistance" in articles. Isbell published other articles using two additional pseudonyms, M. G. Stanley and H. C. Enos, publishing two under each.
Many of his works involved topology
and category theory
:
In abstract algebra
, Isbell found a rigorous formulation for the Pierce–Birkhoff conjecture
on piecewise-polynomial functions. He also made important contributions to the theory of median algebra
s.
In geometric graph theory
, Isbell was the first to prove the bound χ ≤ 7 on the Hadwiger–Nelson problem
, the question of how many colors are needed to color
the points of the plane in such a way that no two points at unit distance from each other have the same color.
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Biography
Isbell was born in Portland, OregonPortland, Oregon
Portland is a city located in the Pacific Northwest, near the confluence of the Willamette and Columbia rivers in the U.S. state of Oregon. As of the 2010 Census, it had a population of 583,776, making it the 29th most populous city in the United States...
, the son of an army officer. He attended several undergraduate institutions, including the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
, where professor Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
was a source of inspiration. He began his graduate studies in mathematics at Chicago, briefly studied at Oklahoma A&M
Oklahoma State University–Stillwater
Oklahoma State University–Stillwater is a land-grant, sun-grant, coeducational public research university located in Stillwater, Oklahoma, USA. OSU was founded in 1890 under the Morrill Act...
and the University of Kansas
University of Kansas
The University of Kansas is a public research university and the largest university in the state of Kansas. KU campuses are located in Lawrence, Wichita, Overland Park, and Kansas City, Kansas with the main campus being located in Lawrence on Mount Oread, the highest point in Lawrence. The...
, and eventually completed a Ph.D. in game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
at 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....
in 1954 under the supervision of Albert W. Tucker
Albert W. Tucker
Albert William Tucker was a Canadian-born American mathematician who made important contributions in topology, game theory, and non-linear programming....
. After graduation, Isbell was drafted into the U.S. Army, and stationed at the Aberdeen Proving Ground
Aberdeen Proving Ground
Aberdeen Proving Ground is a United States Army facility located near Aberdeen, Maryland, . Part of the facility is a census-designated place , which had a population of 3,116 at the 2000 census.- History :...
. In the late 1950s he worked at the 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...
in Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, from which he then moved to the University of Washington
University of Washington
University of Washington is a public research university, founded in 1861 in Seattle, Washington, United States. The UW is the largest university in the Northwest and the oldest public university on the West Coast. The university has three campuses, with its largest campus in the University...
and Case Western Reserve University
Case Western Reserve University
Case Western Reserve University is a private research university located in Cleveland, Ohio, USA...
. He joined the University at Buffalo in 1969, and remained there until his retirement in 2002.
Research
Isbell published over 140 papers under his own name, and several others under pseudonymPseudonym
A pseudonym is a name that a person assumes for a particular purpose and that differs from his or her original orthonym...
s. Isbell published the first paper by John Rainwater
John Rainwater
John Rainwater is the pseudonym of a fictitious mathematician, in whose name mathematicians publish papers. Rainwater has worked mainly in functional analysis, particularly in the geometric theory of Banach spaces and in convex functions...
, a fictitious mathematician, who had been invented by graduate students at the University of Washington in 1952. After Isbell's paper, other mathematicians have published papers using the name "Rainwater" and have acknowledged "Rainwater's assistance" in articles. Isbell published other articles using two additional pseudonyms, M. G. Stanley and H. C. Enos, publishing two under each.
Many of his works involved topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
:
- He was "the leading contributor to the theory of uniform spaceUniform spaceIn the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
s". - Isbell duality is a form of dualityDuality (mathematics)In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
arising when a mathematical object can be interpreted as a member of two different categoriesCategory (mathematics)In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
; a standard example is the Stone dualityStone dualityIn mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation...
between sober spaceSober spaceIn mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...
s and complete Heyting algebraComplete Heyting algebraIn mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
s with sufficiently many points. - Isbell was the first to study the category of metric spacesCategory of metric spacesIn category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
defined by metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s and the metric maps between them, and did early work on injective metric spaceInjective metric spaceIn metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces...
s and the tight spanTight spanIn metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes...
construction.
In abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, Isbell found a rigorous formulation for the Pierce–Birkhoff conjecture
Pierce–Birkhoff conjecture
In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated, albeit in non-rigorous and vague wording, in the 1956 paper of Garrett Birkhoff and Richard S....
on piecewise-polynomial functions. He also made important contributions to the theory of median algebra
Median algebra
In mathematics, a median algebra is a set with a ternary operation < x,y,z > satisfying a set of axioms which generalise the notion of median or majority function, as a Boolean function.The axioms are# < x,y,y > = y...
s.
In geometric graph theory
Geometric graph theory
In mathematics, a geometric graph is a graph in which the vertices or edges are associated with geometric objects or configurations. Geometric graph theory is a specialization of graph theory that studies geometric graphs...
, Isbell was the first to prove the bound χ ≤ 7 on the Hadwiger–Nelson problem
Hadwiger–Nelson problem
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance one from each other have the same color. The answer is unknown, but has been narrowed down to...
, the question of how many colors are needed to color
Graph coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the...
the points of the plane in such a way that no two points at unit distance from each other have the same color.
Mathematical Reviews
- Pseudonyms used by Isbell (and other mathematicians):
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