# In Memoriam: Yale Mathematician Shizuo Kakutani Known for His Work in Functional Analysis and Probability

Yale mathematician Shizuo Kakutani, who invented a tool known as the Kakutani skyscraper that was used to organize random processes such as coin flipping, died this week in New Haven. He was 92.

Kakutani, the Eugene Higgins Professor of Mathematics at Yale, joined the Yale faculty in 1949 as an assistant professor and retired in 1982.

He was known to mathematicians for his influential work in the fields of ergodic theory, functional analysis, and Brownian motion, as it relates to probability theory. One tool he developed, the Kakutani fixed-point theorem, was a key step in the original proof of the existence of Nash equilibria, the theorem for which John Forbes Nash received his Nobel Prize.

A native of Osaka, Japan, Kakutani received his Ph.D. from Osaka University in 1941 and taught at Osaka until 1949. During his time at the university he spent two years as a Research Member at the Institute for Advanced Study at Princeton.

To economists, Kakutani was known for a mathematical tool used to prove theorems about social systems. The Kakutani fixed-point theorem also was used to prove a famous 1954 theorem by the economists Kenneth Arrow and Gerard Debreu which says that there are prices for goods that balance supply and demand in a complex economy. Both economists also won Nobel Prizes, partly for this work.

The Kakutani skyscraper, a fundamental tool he devised for use in ergodic theory, is used to describe a random process, such as coin flipping. The skyscraper is a way of organizing the process into a picture with levels that look like the floors of an office tower. This picture makes it easier to understand the properties of the process. In the case of coin flipping, tossing the coin corresponds to ascending one floor in the skyscraper.

Kakutani in 1982 received two major awards of the Japan Academy, the Imperial Prize and the Academy Prize, for his scholarly achievements in general and his work on functional analysis in particular. He was a member of the American Mathematical Society, the Mathematical Society of Japan, and the Connecticut Academy of Arts and Sciences.

Although known in the international community of mathematicians and economists for his researches, he was extraordinarily successful as a teacher of advanced undergraduate courses. Students of his courses in analysis regarded their experience with Professor Kakutani as the high water mark of their education at Yale, despite the challenging nature of the problems that he assigned. His warmth and encouragement coaxed many a student to a level of effort never experienced previously.

At post-colloquium parties at the home of colleagues, he took a genuine interest in conversation with colleagues’ children. Passing him in department corridors frequently consisted of more than a cursory hello.

He is survived by his wife of 52 years, the former Keiko Kay Uchida, and his daughter Michiko, the chief book critic of The New York Times.

A memorial service is planned for later this fall.