was a Japanese-American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, best known for his eponymous fixed-point theorem.

Biography

Kakutani attended

Tohoku University , or is a Japanese national university located in Sendai, Miyagi in the Tōhoku Region, Japan. It is informally referred to as . Established in 1907, it was the third Imperial University in Japan and among the first three Designated National ...

Sendai is the capital Cities of Japan, city of Miyagi Prefecture, the largest city in the Tōhoku region. , the city had a population of 1,091,407 in 525,828 households, and is one of Japan's 20 Cities designated by government ordinance of Japan, desig ...

, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

Princeton Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...

at the invitation of the mathematician

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

. While there, he also met

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

. Kakutani received his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

in 1941 from

Osaka University , abbreviated as , is a public research university located in Osaka Prefecture, Japan. It is one of Japan's former Imperial Universities and a Designated National University listed as a "Top Type" university in the Top Global University Project. ...

and taught there through

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...

. He returned to the Institute for Advanced Study in 1948, and was given a professorship by

Yale Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the wor ...

in 1949, where he won a students' choice award for excellence in teaching. Kakutani received two awards of the

Japan Academy The Japan Academy ( Japanese: 日本学士院, ''Nihon Gakushiin'') is an honorary organisation and science academy founded in 1879 to bring together leading Japanese scholars with distinguished records of scientific achievements. The Academy is ...

, the Imperial Prize and the Academy Prize in 1982, for his scholarly achievements in general and his work on functional analysis in particular. He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts. Kakutani was married to Keiko ("Kay") Uchida, who was a sister to author

Yoshiko Uchida Yoshiko Uchida (November 24, 1921 – June 21, 1992) was an award-winning Japanese American writer of children's books based on aspects of Japanese and Japanese American history and culture. A series of books, starting with ''Journey to Topaz'' ...

. His daughter,

Michiko Kakutani Michiko Kakutani (born January 9, 1955) is an American writer and retired literary critic, best known for reviewing books for ''The New York Times'' from 1983 to 2017. In that role, she won the Pulitzer Prize for Criticism in 1998. Early life ...

, is a

Pulitzer Prize The Pulitzer Prize () is an award for achievements in newspaper, magazine, online journalism, literature, and musical composition within the United States. It was established in 1917 by provisions in the will of Joseph Pulitzer, who had made h ...

-winning former literary critic for ''

The New York Times ''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...

''.

Work

The

Kakutani fixed-point theorem In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex set, convex, compact set, compact subset of a Euclidean sp ...

is a generalization of

Brouwer's fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simples ...

, holding for generalized

correspondences Correspondence may refer to: *In general usage, non-concurrent, remote communication between people, including letters, email, newsgroups, Internet forums, blogs. Science * Correspondence principle (physics): quantum physics theories must agree ...

instead of

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

s. Its most important uses are in proving the existence of

Nash equilibria In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, and the Arrow–Debreu–McKenzie model of

general equilibrium theory In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...

in microeconomics. Kakutani's other mathematical contributions include

Markov–Kakutani fixed-point theorem In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space ha ...

, another fixed point theorem; the Kakutani skyscraper, a concept in

ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

(a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

that studies

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...

s with an

invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, ...

and related problems); his solution of the

Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...

using the methods of stochastic analysis. The

Collatz conjecture The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...

is also known as the Kakutani conjecture.

Selected articles

*"A generalization of Brouwer's fixed point theorem." Duke Mathematical Journal (1941): 457–459. *"Concrete representation of abstract (L)-spaces and the mean ergodic theorem." Annals of Mathematics (1941): 523–537. *"Concrete representation of abstract (M)-spaces (A characterization of the space of continuous functions)." Annals of Mathematics (1941): 994–1024. *"On equivalence of infinite product measures." Annals of Mathematics (1948): 214–224.

List of books available in English

*''Selected papers'' / Shizuo Kakutani ; Robert R. Kallman, editor (1986)

References

External links

New York Times obituaryBiography, University of St. Andrews/TurnbullThe Lost Theorems of Kakutani, by prof. Stanley Eigen. (PDF)
{{DEFAULTSORT:Kakutani, Shizuo 1911 births 2004 deaths 20th-century Japanese mathematicians 21st-century Japanese mathematicians Probability theorists People from Osaka Japanese emigrants to the United States Tohoku University alumni Laureates of the Imperial Prize Osaka University alumni Osaka University faculty Yale University faculty