Irving Kaplansky: The Architect of Modern Algebra
Irving Kaplansky (1917–2006) was a towering figure in 20th-century mathematics, a scholar whose work bridged the gap between abstract algebraic structures and functional analysis. Known for his
"unfailing sense of the beautiful in mathematics,"Kaplansky was not only a prolific researcher but also a legendary teacher, a talented musician, and a leader who shaped the American mathematical landscape during the Cold War era.
1. Biography: From Toronto to the "Chicago School"
Irving Kaplansky was born on March 22, 1917, in Toronto, Canada, to Polish Jewish immigrants. His father, a tailor, and his mother, who ran a grocery store, encouraged his early intellectual and musical talents.
Education and Early Career
Kaplansky attended the University of Toronto, where he demonstrated early brilliance by becoming a member of the winning team in the first-ever William Lowell Putnam Mathematical Competition in 1938. He earned his B.A. (1938) and M.A. (1939) from Toronto before moving to Harvard University for his doctoral studies. Under the supervision of Saunders Mac Lane, he completed his Ph.D. in 1941 with a dissertation on "Maximal Fields with Valuations."
The Chicago Years
After a brief stint at Harvard and work for the Applied Mathematics Group during World War II, Kaplansky joined the University of Chicago in 1945. This was the beginning of a nearly four-decade tenure. He became a central figure in the "Chicago School" of mathematics, a period of intense productivity that transformed the university into a global hub for algebra and analysis. In 1984, he moved to Berkeley to serve as the Director of the Mathematical Sciences Research Institute (MSRI), a role he held until 1992.
2. Major Contributions: Elegance in Abstraction
Kaplansky’s work is characterized by a drive to find the simplest, most elegant proof for complex problems. He made foundational contributions to several subfields:
- Commutative and Non-commutative Ring Theory: Kaplansky revolutionized the study of rings (sets equipped with addition and multiplication). He developed the theory of primitive rings and made significant strides in understanding the structure of Banach algebras.
- Kaplansky’s Conjectures: He proposed several famous conjectures regarding group rings, some of which remain central puzzles in the field today (notably the "zero-divisor conjecture").
- Functional Analysis: He was a pioneer in the study of C*-algebras and von Neumann algebras. His "Kaplansky Density Theorem" (1951) is a cornerstone of operator theory, providing a vital link between a von Neumann algebra and its dense subalgebras.
- Infinite Abelian Groups: His 1954 monograph provided a systematic treatment of the subject, introducing methods that simplified what was then a fragmented field.
- Differential Algebra: He extended algebraic methods to the study of differential equations, helping to formalize the "Galois theory" of differential fields.
3. Notable Publications
Kaplansky was a master of the mathematical monograph—short, dense, and remarkably clear books that became standard texts for generations of graduate students.
- Infinite Abelian Groups (1954): A classic that brought order to the study of group structures.
- Commutative Rings (1970): Perhaps his most influential book, it reimagined the subject by focusing on the interplay between ring theory and homological algebra.
- Fields and Rings (1969): A concise introduction to Galois theory and the structure of rings.
- Set Theory and Metric Spaces (1972): Noted for its accessibility and pedagogical clarity.
- Lie Algebras and Locally Compact Groups (1971): Bridging the gap between algebra and topology.
4. Awards and Recognition
Kaplansky’s peers recognized him as one of the most versatile mathematicians of his generation.
- The Leroy P. Steele Prize (1983): Awarded by the American Mathematical Society (AMS) for his lifetime of cumulative impact on mathematics.
- President of the American Mathematical Society (1985–1986): A role in which he advocated for increased funding and public appreciation of mathematics.
- National Academy of Sciences: Elected as a member in 1966.
- American Academy of Arts and Sciences: Elected as a fellow in 1962.
- Guggenheim Fellowship (1948): Allowed him to conduct pivotal research in Europe.
5. Impact and Legacy
Kaplansky’s legacy is felt in two primary ways: the mathematical "tools" he left behind and the students he mentored.
The "Kaplansky Style"
He was famous for his "minimalist" writing style. He believed that if a mathematical idea was truly understood, it could be explained simply. This philosophy helped demystify higher algebra for thousands of students.
Academic Genealogy
Kaplansky was an extraordinarily productive mentor, supervising 55 Ph.D. students during his career. Many of his students, such as Hyman Bass and Donald Passman, became leaders in mathematics themselves, ensuring that his rigorous approach to algebra propagated through the academic world.
6. Collaborations and Community
While Kaplansky was a strong individual researcher, his greatest "collaboration" was with the mathematical community at large.
- The Chicago Circle: He worked alongside greats like André Weil, Saunders Mac Lane, and Paul Halmos. This group collectively shifted the focus of American mathematics toward the abstract structuralism pioneered by the Bourbaki group in France.
- MSRI Leadership: As Director of MSRI, he fostered an environment where young researchers could interact with established masters, solidifying the institute's reputation as a premier site for mathematical exchange.
7. Lesser-Known Facts: The Musical Mathematician
Beyond the chalkboard, Irving Kaplansky was a man of diverse passions.
- Professional-Grade Pianist: Kaplansky was a gifted jazz and classical pianist. He often performed at mathematical social gatherings. He famously composed a song titled "A Song About Pi," which used the digits of π to determine the melody (3 = E, 1 = C, 4 = F, etc.).
- A Musical Legacy: His daughter, Lucy Kaplansky, is a well-known American folk singer-songwriter. Irving occasionally performed with her, bridging the worlds of high mathematics and Greenwich Village folk music.
- The "Kaplansky Game": He was known for his sharp wit and love of puzzles. He once remarked that he became a mathematician because it was the only thing he could do that didn't feel like work.
- A "Putnam" Legend: He was the first person to win the Putnam competition and later become a member of the National Academy of Sciences, setting a precedent for the "Putnam-to-Greatness" pipeline in American mathematics.