Shizuo Kakutani

Background

Shizuo Kakutani was born on August 28, 1911, in Osaka, Japan. His father, Kakujiro Kakutani, was a lawyer, and his older brother Seiichi studied physics at Kyoto University. Seiichi introduced Shizuo to mathematical ideas during visits home, helping encourage an interest that eventually led him away from the legal career his father had initially expected one of his sons to pursue.

Kakutani grew up during a period when Japan was developing modern research universities and expanding its scientific institutions. After his brother's early death from typhoid fever, his father allowed him to pursue mathematics. Kakutani later established his professional life in the United States, where he became a central figure in twentieth-century mathematical analysis while retaining strong connections with Japanese mathematical institutions.

Education

Kakutani studied at Tohoku University in Sendai from 1930 to 1934. Tohoku, founded as Japan's third imperial university in 1907, had developed a strong tradition in scientific and mathematical research. At Tohoku, Kakutani studied analysis under Tatsujirō Shimizu and built the foundation for his later work in complex analysis and functional analysis.

He pursued graduate study at Osaka University and received his Doctor of Philosophy in 1941. Osaka University, whose modern predecessor was founded in the nineteenth century and which became an imperial university in 1931, was developing into an important center of advanced scientific research. Kakutani's doctoral work grew out of his research on complex analysis and Riemann surfaces, which had already attracted international attention before the completion of his degree.

Career

Kakutani began his academic career at Osaka University in 1934 as a teaching assistant in its newly formed mathematics department. He published early work in complex analysis, functional analysis, and ergodic theory, collaborating with Kiyosi Yosida and other Japanese mathematicians. He later became an assistant professor at Osaka University and continued teaching and research through the Second World War.

Hermann Weyl invited Kakutani to the Institute for Advanced Study in Princeton, where he was a Member in 1940–1941 and again in 1941–1942. The Institute's School of Mathematics brought together leading researchers in analysis, topology, mathematical physics, and related fields. Seminars associated with Weyl and John von Neumann provided the setting in which Kakutani developed the result later known as the Kakutani fixed-point theorem, a generalization of Brouwer's theorem from single-valued functions to set-valued correspondences.

Kakutani returned to Japan during the war and resumed his work at Osaka University. He came back to the Institute for Advanced Study in 1948–1949 and joined Yale University in 1949 as an assistant professor of mathematics. Yale, founded in 1701 and organized around a long-standing liberal-arts and research tradition, became the principal institutional setting for the remainder of his career. He advanced to a professorship and held the Eugene Higgins Professorship of Mathematics before retiring in 1982 as Professor Emeritus.

At Yale, Kakutani worked across functional analysis, probability theory, Brownian motion, Markov processes, complex analysis, and ergodic theory. His Kakutani fixed-point theorem became a key existence tool in John Nash's work on noncooperative-game equilibria and in the general-equilibrium theory associated with Kenneth Arrow and Gérard Debreu. His Kakutani skyscraper became an important construction in ergodic theory for organizing and analyzing measure-preserving transformations and stochastic processes. He was also an influential teacher of undergraduate and graduate mathematics and delivered a plenary lecture at the International Congress of Mathematicians in 1950.

Achievements

• Kakutani developed the fixed-point theorem published in his 1941 paper “A Generalization of Brouwer's Fixed Point Theorem.” The theorem extended Brouwer's result to upper-semicontinuous set-valued mappings and became an important mathematical foundation for equilibrium proofs in game theory and general-equilibrium economics.

• Kakutani created the Kakutani skyscraper, a construction in ergodic theory that organizes the behavior of measure-preserving transformations into levels and provides a useful way to analyze dynamical and stochastic processes.

• His research helped advance functional analysis, probability theory, Brownian motion, Markov processes, complex analysis, and ergodic theory. His work connected abstract analysis with applications in economics and the mathematical study of random systems.

• Kakutani served as a plenary speaker at the International Congress of Mathematicians held in Cambridge, Massachusetts, in 1950.

Works

• journal article

  • A Generalization of Brouwer's Fixed Point Theorem

    (Kakutani's foundational paper generalized Brouwer's fixed...)

    1941

  • Concrete Representation of Abstract (M)-Spaces

    (A research paper in functional analysis concerning the re...)

    1941

  • Representation of (M)-Spaces

    (A co-authored contribution with H. F. Bohnenblust on the ...)

    1941

• selected papers

  • Shizuo Kakutani: Selected Papers

    (A two-volume collection of Kakutani's mathematical papers...)

    1986

Membership

Kakutani was a member of the American Mathematical Society, the Mathematical Society of Japan, and the Connecticut Academy of Arts and Sciences. He was also elected to the American Academy of Arts and Sciences in 1959, joining a learned society founded in 1780 to recognize achievement in the sciences, arts, and public life.

• American Mathematical Society , United States

• Mathematical Society of Japan , Japan

• Connecticut Academy of Arts and Sciences , United States

• Elected a Fellow.

  American Academy of Arts and Sciences , United States

  1959

Personality

Colleagues and former students described Kakutani as mild, gracious, courteous, warm, and deeply dedicated to mathematics. Yale accounts emphasized his demanding but encouraging teaching, his personal interest in students, and his ability to stimulate sustained effort without intimidating those who found the subject difficult.

Quotes from others about the person

• Yale students regarded Kakutani's analysis courses as among the most formative experiences of their education. Colleagues remembered his warmth, encouragement, and humane interest in students and their families, while mathematical contemporaries described him as a gentleman and scholar whose dedication left a lasting impression.

Connections

Kakutani married Keiko Kay Uchida in 1952. They had one daughter, Michiko Kakutani, who became a prominent literary critic and received the Pulitzer Prize for Criticism in 1998.

His professional relationships included extended collaborations and intellectual exchanges with mathematicians such as Kiyosi Yosida, Hermann Weyl, John von Neumann, Warren Ambrose, Paul R. Halmos, and Arshag Hajian. His doctoral students and younger colleagues later carried his approaches in analysis, probability, and ergodic theory into further research.

Father:

Kakujiro Kakutani

A lawyer in Osaka who initially expected one of his sons to enter the family legal practice.

Brother:

Seiichi Kakutani

Studied physics at Kyoto University and introduced Shizuo to mathematical ideas.

Wife:

Keiko Kay Uchida

Kakutani married Uchida in 1952.

Daughter:

Michiko Kakutani

A literary critic who served as chief book critic of The New York Times and won the Pulitzer Prize for Criticism in 1998.

References

• Institute for Advanced Study profile of Shizuo Kakutani Documents Kakutani's memberships at the Institute for Advanced Study in 1940–1942 and 1948–1949, his 1941 Osaka University doctorate, and the development and significance of his fixed-point theorem.
• In Memoriam: Yale Mathematician Shizuo Kakutani Known for His Work in Functional Analysis and Probability Yale's memorial account documents Kakutani's birth in Osaka, doctoral degree, Yale career, research in functional analysis and probability, teaching, family, and the Academy and Imperial Prizes.
  2004
• Shizuo Kakutani biography A biographical account of Kakutani's family background, education at Tohoku University and Osaka University, work at Osaka and Yale, mathematical contributions, personality, and honors.
  2001
• Shizuo Kakutani The American Academy of Arts and Sciences profile records Kakutani as a Yale mathematician and educator elected to the Academy in 1959.
• Shizuo Kakutani: Selected Papers Publisher information for the collection of Kakutani's selected mathematical papers, edited by Robert R. Kallman and issued by Birkhäuser.
  1986