# John R. Isbell

**John Rolfe Isbell** (October 27, 1930 – August 6, 2005)[1] was an American mathematician, for many years a professor of mathematics at the University at Buffalo (SUNY).

## Biography

Isbell was born in Portland, Oregon, the son of an army officer from Isbell, a town in Franklin County, Alabama.[2][3][4] He attended several undergraduate institutions, including the University of Chicago, where professor Saunders Mac Lane was a source of inspiration.[3][4] He began his graduate studies in mathematics at Chicago, briefly studied at Oklahoma A&M University and the University of Kansas,[5] and eventually completed a Ph.D. in game theory at Princeton University in 1954 under the supervision of Albert W. Tucker.[3][4][6] After graduation, Isbell was drafted into the U.S. Army, and stationed at the Aberdeen Proving Ground.[3] 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.[7]

## Research

Isbell published over 140 papers under his own name, and several others under pseudonyms. 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.[8] Isbell published other articles using two additional pseudonyms, **M. G. Stanley** and **H. C. Enos**, publishing two under each.[4][8]

Many of his works involved topology and category theory:

- He was "the leading contributor to the theory of uniform spaces".[7]
- Isbell duality is a form of duality arising when a mathematical object can be interpreted as a member of two different categories; a standard example is the Stone duality between sober spaces and complete Heyting algebras with sufficiently many points.[9]
- Isbell was the first to study the category of metric spaces defined by metric spaces and the metric maps between them, and did early work on injective metric spaces and the tight span construction.[10]

In abstract algebra, Isbell found a rigorous formulation for the Pierce–Birkhoff conjecture on piecewise-polynomial functions.[11] He also made important contributions to the theory of median algebras.[12]

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.[13]

## See also

## References

- Birth date from an excerpt of "The Harloe-Kelso Genealogy" by C. B. Harloe (1943), accessed 2011-03-23; death date from death announcement in the Buffalo News, August 28, 2005, reproduced by usgwarchives.net, accessed 2011-03-23. Magill (1996) also states his birth date as 1930, but Henriksen (2006) states it as 1931.
- Harloe (1943).
- Magill, K. D., Jr. (1996), "An interview with John Isbell",
*Topology Communications*,**1**(2). - Henriksen, Melvin (2006), "John Isbell 1931–2005",
*Topology Communications*,**11**(1). - The University of Kansas had professors Ainsley Diamond and Nachman Aronszajn, who had previously been professors at Oklahoma A&M. The two moved to Kansas after Oklahoma A&M had instituted a requirement that instructors sign a strict loyalty oath. Ainsley Diamond, as a quaker, had refused to sign the loyalty oath.
- John Rolfe Isbell at the Mathematics Genealogy Project.
- Announcement of Isbell's death in
*Topology News*, October 2005. - The seminar on functional analysis at the University of Washington has been called the "Rainwater seminar".
Phelps, Robert R. (2002). Melvin Henriksen (ed.). "Biography of John Rainwater".

*Topological Commentary*.**7**(2). - Barr, Michael; Kennison, John F.; Raphael, R. (2008), "Isbell duality" (PDF),
*Theory and Applications of Categories*,**20**(15): 504–542. - Isbell, J. R. (1964), "Six theorems about injective metric spaces",
*Commentarii Mathematici Helvetici*,**39**(1): 65–76, doi:10.1007/BF02566944. - Madden, James J. (1999), "The Pierce–Birkhoff Conjecture",
*International Conference and Workshop on Valuation Theory*, archived from the original on 2011-06-08. - Isbell, John R. (August 1980), "Median algebra",
*Transactions of the American Mathematical Society*,**260**(2): 319–362, doi:10.2307/1998007, JSTOR 1998007. - Soifer, Alexander (2008),
*The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators*, New York: Springer, p. 29, ISBN 978-0-387-74640-1.

## External resources

### Mathematical Reviews

- Mathematical Reviews. "
**John R. Isbell**". Retrieved 2011-04-02. - Pseudonyms used by Isbell (and other mathematicians):
- Mathematical Reviews. "John Rainwater". Retrieved 2011-04-02.
- Mathematical Reviews. "M. G. Stanley". Retrieved 2011-04-02.
- Mathematical Reviews. "H. C. Enos". Retrieved 2011-04-02.