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August 20, 2004

Kakutani's Death; Fixed Point Theorems

Shizuo Kakutani, author of the Kakutani Fixed Point Theorem, has died at age 92. I started auditing his Real Analysis class one fall while I was an undergrad. I didn't realize that he was the author of a theorem important for economics, or that real analysis was one of the most useful math courses I could take. Rather, I knew the course was a base one for math majors, and very hard, and I was feeling very self-confident. I didn't last too long. Staying up till the wee hours doing problem sets for a course I was just auditing was too much for me. Still, I remember vividly how Professor Kakutani would clearly exposit series's and sums, filling up blackboard after blackboard in neat handwriting. And I remember his joke about the lady who was surprised at how after so many years in America he still spelled "if" as "iff" (for nonmathematical readers: "iff" means "if and only if" in math).

Alex Tabarrok has a good discussion of fixed point theorems at Marginal Revolution.

One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed.

Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.


Take two pieces of 8*11 paper and lay them on top of one another so that every point on the top paper corresponds with a point on the bottom paper. Now crumple the top piece of paper in anyway that you wish and place it back on top. B's theorem tells us that there must be a point which has not moved, i.e. which lies exactly above the same point that it did initially.


Consider a cupful of coffee. Each point is somewhere in 3-dimensional space. Stir. At least one point ends up in the same place as it began.

Posted by erasmuse at August 20, 2004 12:12 AM

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