Mark Kac Autocorrelation Function of some ’Linear’ Stationary Stochastic Processes (med Harald Wergeland) & One Dimensional Gas in Gravity (med Colin J. Thompson)
Abstract
Mark Kac (pronounced Kaats) was born in 1914 in Kremenets (transliterated), a small town in Ukraine. After World War I and the conflicts in its aftermath, the town became part of Poland, with the name Krzemieniec. After World War II it is again Ukrainian. From the age of eleven Kac went to the Polish school in Krzemieniec (Polish was, in fact, his language number four, after Russian, French and Hebrew). At sixteen Kac found a new derivation of the Cardano solution of the cubic equation, and from then on mathematics was his subject of choice.
At the university in Lwów (now Ukrainian Lvov) mathematics was of international standing, due to Hugo Steinhaus and Stefan Banach. Kac gravitated towards Steinhaus, and together they focused on the concept of statistical independence. Their joint efforts considerably strengthened the standing of statistics as a well founded and fruitful part of mathematics. Kac received his doctorate in 1937. Due to Hitler’s rise to power in Germany and increasing anti-Semitism in Poland, Kac had been on the lookout for possibilities to spend time abroad. In 1938 he got a stipend for an academic year in the US. That saved his life. Practically his entire family perished in the war.
After spending his stipend year at Johns Hopkins, Kac got an offer from Cornell University. He happily settled in Ithaca, married, raised two children, and gradually made a reputation for himself in the stimulating mathematics environment of that university. Kac stayed at Cornell until 1961 when he moved to Rockefeller University in New York. There, together with George Uhlenbeck and Ted Berlin, he built a group in mathematics and statistical physics. The first postdoc in the group was Per Christian Hemmer from Trondheim. Kac left Rockefeller for University of Southern California in 1981, where he stayed until his death at the age of 70, in 1984.
From the early collaboration with Steinhaus, Kac had a knack for seeing probabilistic aspects in the most unlikely problems. He was particularly proud of his contribution to number theory with Paul Erdös, where they showed that the number of distinct primes in a large integer is, in a certain sense, distributed according to Gauss’ law. Another famous example of surprising use of stochastics is Kac’s 1966 work (and filmed lecture) called: «Can one hear the shape of a drum?» There he uses the diffusion equation to demonstrate how the spectral density of a membrane, clamped at the boundary, determines the area of the membrane, the length of its boundary and the number of holes in it. This paper stimulated considerable research activity. Finally, in 1992, a concrete example demonstrated that one cannot «hear» the detailed shape.
The contact with Uhlenbeck inspired Kac to think of problems in statistical physics. His ring model beautifully elucidates and lies to rest the reversibility paradoxes associated with the Boltzmann equation. And Kac’s triplet of papers with Uhlenbeck and Hemmer has the status of a classic in modern statistical physics: There they show that the van der Waals equation (the Maxwell construction included) is the exact equation of state for a one-dimensional gas with hard repulsion of short range, plus weak, long range attraction.
As a mathematician Kac was a problem solver with little patience for abstract theory. A characteristic quote is «Axioms come and go, but the striking example is forever!» It is, therefore, not surprising that he was inspired by problems in the periphery of mathematics.
But Mark Kac was not only a first rate mathematician, he was a person full of life, outgoing, generous and witty. Those of us who had the privilege to spend inspiring time close to Mark Kac, are indeed fortunate!
