Mark Kac

Mark Kac ( born August 3, 1914 in Kremenez, † October 26, 1984 in California; Marek Kac ), pronounced [ kaʦ ], was a Polish- American mathematician.

Kac was born as the son of Jewish parents in the Government General of Warsaw, a part of the Russian Empire. 1915, the family was evacuated to Russia, and he received home schooling from his father who learned French from his governess, Polish but only after his return to Poland in 1921., Where he studied at the Lviv University in Lwów in Hugo Steinhaus, with whom he also friends was mathematics and in 1937 received his doctorate. He tried for a scholarship in the USA and finally came in 1939 at Cornell University in Ithaca, New York, where he became professor in 1947. In 1961 he joined the Rockefeller University in New York City and in 1981 at the University of Southern California.

His main area of ​​work was the probability theory, especially its application in statistical mechanics. With Richard Feynman gave a path integral solution of the Fokker -Planck equation ( describing the time evolution of a probability distribution under drift and diffusion, for example, brown shear movement ), the Feynman - Kac formula, for example, with Monte Carlo method can be approximated. He was with Paul Erdős, with whom he collaborated on several occasions, a pioneer in the application of probability theory in number theory, where George Pólya already had given a probabilistic interpretation of the prime number theorem. Kac and Erdős be such a "discharge" in American Journal of Mathematics Vol 62, 1940, p 738 Precisely, they proved that the number of prime factors of a number is normally distributed ( set of Erdős - Kac ). Furthermore, Kac has worked on the theory of phase transitions and over exact models in statistical mechanics (spherical model of Kac / Berlin, Physical Review, 1952, etc.).

Also famous is his essay Can one hear the shape of a drum? , American Mathematical Monthly in 1966, which won the 1968 Chauvenet Prize of the American Mathematical Society for outstanding mathematical expositions. Therein it comes from the spectrum ( eigenvalues ​​) of the Laplace operator (which also has connections to the probability and the stationary heat conduction equation is the Laplace equation) to close to its geometric shape in an area. Hermann Weyl had in 1915 found that the asymptotic spectrum of eigenvalues ​​is determined by the volume of the area. In general the answer is negative ( counter-examples provide so-called isospektrale areas, namely, different shape, identical spectrum). Stark gained importance to the field after the Atiyah-Singer index theorem correlations between the topology of a manifold and the spectrum of the Laplace or Dirac operator ( as it were the " square root " of the Laplacian ) pointed to this manifold and 1970 in the years proofs of this theorem have been found by means of the heat equation (diffusion).

In 1978 he was awarded the George David Birkhoff - Prize for Applied Mathematics.

Among his doctoral students were Harry Kesten and Murray Rosenblatt.

Works

  • Stanislaw Ulam with: Mathematics and Logic: Retrospect and Prospects. Praeger, New York, 1968, Dover paperback reprint.
  • Enigmas of Chance: An Autobiography. Harper and Row, New York, 1985. Sloan Foundation Series. Posthumously published with afterword by Gian- Carlo Rota.
  • Statistical independence in probability, analysis and number theory. (Carus Mathematical Monographs No.. 12), MAA and John Wiley, 1959.
  • Probability, number theory and statistical physics. (Reprint of Kac's work with his commentary and autobiographical note), 1979.
  • Probability and related topics in the physical sciences. 1959 ( with contribution of Uhlenbeck for the Boltzmann equation and Hibbs quantum mechanics, Boulder Seminar 1957).
  • Random walk and the theory of Brownian motion. American Mathematical Monthly, 1947, p 369 ( also received the Chauvenet Prize ).
  • On applying mathematics - reflections and examples. Quarterly Journal of Applied Mathematics, Vol 30, 1972, p 17
  • With Ward: A combinatorial solution of the two dimensional Ising model. Physical Review Vol 88, 1952, 1332.
  • Can one hear the shape of a drum? , American Mathematical Monthly, Volume 73, 1966, pp. 1-23
Kremenets Fokker–Planck equation Feynman–Kac formula Laplace-Operator Laplace's equation Asymptote Atiyah–Singer index theorem Topological space Dirac-Operator Gian-Carlo Rota Carolyn S. Gordon Integrated Authority File
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