Irving Kaplansky

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Irving Kaplansky
Irving Kaplansky2.jpg
Born March 22, 1917
Died June 25, 2006 (age 89)
Los Angeles
Nationality Canadian, American
Fields Mathematics
Institutions American Mathematical Society
Mathematical Sciences Research Institute
University of Chicago
Columbia University
University of California, Berkley
Alma mater University of Toronto
Harvard University
Doctoral advisor Saunders Mac Lane
Doctoral students Hyman Bass
Susanna S. Epp
Günter Lumer
Eben Matlis
Jacob Matijevic
Donald Ornstein
Ed Posner
Alex F. T. W. Rosenberg
Judith D. Sally
Harold Widom
(entire list)[1]
Known for Kaplansky density theorem
Kaplansky's game
Kaplansky conjecture
Kaplansky's theorem on quadratic forms
Group theory
Hilbert space
Ring theory
Operator algebras
Homological algebra
Topological algebra
Game theory
Field theory
Notable awards William Lowell Putnam Mathematical Competition (1938)
Guggenheim Fellowship (1948)
Jeffery–Williams Prize (1968)
Honorary member of the London Mathematical Society (1987)
Leroy P. Steele Prize (1989)

Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and musician.[2]


Kaplansky or "Kap" as his friends and colleagues called him was born in Toronto, Ontario, Canada, to Polish-Jewish immigrants;[3][4] his father worked as a tailor, and his mother ran a grocery and, eventually, a chain of bakeries.[5][6][7] He attended the University of Toronto as an undergraduate. In his senior year, he competed in the first William Lowell Putnam Mathematical Competition, becoming one of the first five recipients of the Putnam Fellowship, which paid for graduate studies at Harvard University.[5] Administered by the Mathematical Association of America, the competition "is widely considered to be the most prestigious university-level mathematics examination in the world, and its difficulty is such that the median score is often zero or one (out of 120) despite being attempted by students specializing in mathematics." Astoundingly, "through 2015, there have been 144,589 participants...Over the seventy-six competitions between 1938 and 2015 there have been only four perfect scores." Kaplansky only got one question wrong.[8] Below is a table of the first five recipients.

Name (School) Year (s)
George W. Mackey (Rice University) 1938
Irving Kaplansky (University of Toronto) 1938
Michael J. Norris (College of St. Thomas) 1938
Robert W. Gibson (Fort Hays Kansas State) 1938
Bernard Sherman (Brooklyn College) 1938, 1939

After receiving his Ph.D. from Harvard in 1941[9] as Saunders Mac Lane's first student, he remained at Harvard as a Benjamin Pierce Instructor, and in 1944 moved with Mac Lane to Columbia University for one year to collaborate on work surrounding the Manhattan Project with the National Defense Council.[10][11]

He was professor of mathematics at the University of Chicago from 1945 to 1984, and Chair of the department from 1962 to 1967. Kaplansky was the Director of the Mathematical Sciences Research Institute from 1984 to 1992, and the President of the American Mathematical Society from 1985 to 1986.[12]

Kaplansky was also an accomplished amateur musician. He had perfect pitch, studied piano until the age of 15, earned money in high school as a dance band musician, taught Tom Lehrer,[13] and played in Harvard's jazz band in graduate school. He also had a regular program on Harvard's student radio station. After moving to the University of Chicago, he stopped playing for two decades, but then returned to music as an accompanist for student-run Gilbert and Sullivan productions and as a calliope player in football game parades.[5] He often composed music based on mathematical themes. One of those compositions, A Song About Pi, is a melody based on assigning notes to the first 14 decimal places of pi, and has occasionally been performed by his daughter, singer-songwriter Lucy Kaplansky.[14]

Mathematical contributions[edit]

Kaplansky made major contributions to group theory, ring theory, the theory of operator algebras and field theory and created the Kaplansky density theorem, Kaplansky's game and Kaplansky conjecture. He published more than 150 articles and over 20 mathematical books.[2]

Kaplansky was the doctoral supervisor of 55 students including notable mathematicians Hyman Bass, Susanna S. Epp, Günter Lumer, Eben Matlis, Donald Ornstein, Ed Posner, Alex F. T. W. Rosenberg, Judith D. Sally, and Harold Widom. He has over 800 academic descendants, including many through his academic grandchildren David J. Foulis (who studied with Kaplansky at the University of Chicago before completing his doctorate under the supervision of Kaplansky's student Fred Wright, Jr.) and Carl Pearcy (the student of H. Arlen Brown, who had been jointly supervised by Kaplansky and Paul Halmos).[9]

Awards and honors[edit]

Kaplansky was a member of the National Academy of Sciences and the American Academy of Arts and Sciences, Director of the Mathematical Sciences Research Institute, and President of the American Mathematical Society. He was the plenary speaker at the British Mathematical Colloquium in 1966. Won the William Lowell Putnam Mathematical Competition, the Guggenheim Fellowship, the Jeffery-Williams Prize, and the Leroy P. Steele Prize.[15][16]

Selected publications[edit]


  • Kaplansky, Irving (1954). Infinite Abelian groups. [17] revised edn. 1971 with several later reprintings
  • —— (1955). An introduction to differential algebra. University of Chicago Press.  2nd edn. Paris: Hermann. 1957. 
  • —— (1966). Introdução à teoria de Galois, por I. Kaplansky. Pref. de Elon Lages Lima. 
  • —— (1968). Rings of operators. 
  • —— (1969). Fields and rings.  2nd edn. 1972
  • —— (1969). Linear algebra and geometry; a second course.  revised edn. 1974
  • —— (1970). Algebraic and analytic aspects of operator algebras. 
  • —— (1971). Lie Algebras and Locally Compact Groups. University of Chicago Press. ISBN 0-226-42453-7.  several later reprintings
  • —— (1972). Set theory and metric spaces.  2nd edn. 1977
  • —— (September 1974). Commutative Rings. Lectures in Mathematics. University of Chicago Press. ISBN 0-226-42454-5.  1st edn. 1966; revised 1974 with several later reprintings

with I. N. Herstein: —— (1974). Matters mathematical.  2nd edn. 1978



  1. ^ by Tree Mathematics Genealogy Project
  2. ^ a b O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews.
  3. ^ Irving Kaplansky Memoir by Nancy E. Albert
  4. ^ Making Family Stories into Art
  5. ^ a b c Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990), "Irving Kaplansky", More Mathematical People, Harcourt Brace Jovanovich, pp. 118–136 .
  6. ^ Irving Kaplansky — mathematician and author
  7. ^ In memoriam: Irving Kaplansky
  8. ^ The Putnam Competition from 1938-2015 by Joseph A. Gallian
  9. ^ a b Irving Kaplansky at the Mathematics Genealogy Project
  10. ^ Kaplansky, Irving. "Abraham Adrian Albert" 1972.
  11. ^ O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews.
  12. ^ O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews .
  13. ^ Peterson, Ivars. (2013). "A Song about Pi"
  14. ^ Pearce, Jeremy "Irving Kaplansky, 89, a Pioneer in Mathematical Exploration"
  15. ^ Bass, Hyman and Lam, T.Y., (December, 2007) "Irving Kaplansky (1917-2007)." Notices of the American Mathematical Society. (54)11,1477-1493.
  16. ^ O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews.
  17. ^ Baer, Reinhold (1955). "Review: Infinite abelian groups by I. Kaplansky" (PDF). Bull. Amer. Math. Soc. 61 (1): 88–89. doi:10.1090/s0002-9904-1955-09877-x. 


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