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Mark Kac

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Mark Kac
Born(1914-08-03)August 3, 1914
Krzemieniec, Russian Empire (now Ukraine)
DiedOctober 26, 1984(1984-10-26) (aged 70)
California, U.S.
NationalityPolish
CitizenshipPoland, USA
Alma materLwów University
Known forFeynman–Kac formula
Erdős–Kac theorem
Kac–Bernstein theorem
Kac's lemma
Kac process
AwardsChauvenet Prize (1950, 1968)
Birkhoff Prize (1978)
Scientific career
FieldsMathematics
InstitutionsCornell University
Rockefeller University
University of Southern California
Doctoral advisorHugo Steinhaus
Doctoral studentsHarry Kesten
William LeVeque
William Newcomb
Lonnie Cross
Daniel B. Ray
Murray Rosenblatt
Daniel Stroock

Mark Kac (/kɑːts/ KAHTS; Polish: Marek Kac; August 3, 1914 – October 26, 1984) was a Polish American mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. (In the end, the answer was "no", in general.)

Biography

He was born to a Polish-Jewish family; their town, Kremenets (Polish: "Krzemieniec"), changed hands from the Russian Empire (by then the Soviet Union) to Poland after the Peace of Riga, when Kac was a child.[1]

Kac completed his Ph.D. in mathematics at the Polish University of Lwów in 1937 under the direction of Hugo Steinhaus.[2] While there, he was a member of the Lwów School of Mathematics. After receiving his degree he began to look for a position abroad, and in 1938 was granted a scholarship from the Parnas Foundation which enabled him to go work in the United States. He arrived in New York City in November, 1938.[3]

With the onset of World War II, Kac was able to stay in America, while his parents and brother who remained in Kremenets were murdered by the Germans in mass executions in August 1942.[3]

From 1939–61 he was at Cornell University, first as an instructor, then from 1943 as assistant professor and from 1947 as full professor.[4] While there, he became a naturalized US citizen in 1943. In the academic year 1951–1952 Kac was on sabbatical at the Institute for Advanced Study.[5] In 1952, Kac, with Theodore H. Berlin, introduced the spherical model of a ferromagnet (a variant of the Ising model)[6] and, with J. C. Ward, found an exact solution of the Ising model using a combinatorial method.[7] In 1961 he left Cornell and went to The Rockefeller University in New York City. In the early 1960s he worked with George Uhlenbeck and P. C. Hemmer on the mathematics of a van der Waals gas.[8] After twenty years at Rockefeller, he moved to the University of Southern California where he spent the rest of his career.

Work

In his 1966 article with the title "Can one hear the shape of the drum" Kac asked the question whether two resonators ("drums") of different geometrical shapes can have exactly the same set of frequencies ("sound tones"). The answer was negative, meaning that the eigenfrequency set does not uniquely characterize the shape of a resonator.

Reminiscences

  • His definition of a profound truth. "A truth is a statement whose negation is false. A profound truth is a truth whose negation is also a profound truth." (Also attributed to Niels Bohr)
  • He preferred to work on results that were robust, meaning that they were true under many different assumptions and not the accidental consequence of a set of axioms.
  • Often Kac's "proofs" consisted of a series of worked examples that illustrated the important cases.
  • When Kac and Richard Feynman were both Cornell faculty, Kac attended a lecture of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman–Kac formula resulted, which proves rigorously the real case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still unproven. Kac had taught himself about Wiener processes by reading Norbert Wiener's original papers, which were "the most difficult papers I have ever read."[3] Brownian motion is a Wiener process. Feynman's path integrals are another example.
  • Kac's distinction between an "ordinary genius" like Hans Bethe and a "magician" like Richard Feynman has been widely quoted. (Bethe was also at Cornell University.)
  • Kac became interested in the occurrence of statistical independence without randomness. As an example of this, he gave a lecture on the average number of factors that a random integer has. This wasn't really random in the strictest sense of the word, because it refers to the average number of prime divisors of the integers up to N as N goes to infinity, which is predetermined. He could see that the answer was c log log N, if you assumed that the number of prime divisors of two numbers x and y were independent, but he was unable to provide a complete proof of independence. Paul Erdős was in the audience and soon finished the proof using sieve theory, and the result became known as the Erdős–Kac theorem. They continued working together and more or less created the subject of probabilistic number theory.
  • Kac sent Erdős a list of his publications, and one of his papers contained the word "capacitor" in the title. Erdős wrote back to him "I pray for your soul."

Human rights concerns

For a number of years Kac was the co-chair of the Committee of Concerned Scientists.[9] He was a co-author of a letter which publicized the case of the scientist Vladimir Samuilovich Kislik[10] and a letter which publicized the case of the applied mathematician Yosif Begun.[11]

Awards and honors

Books

  • Mark Kac and Stanislaw Ulam: Mathematics and Logic: Retrospect and Prospects, Praeger, New York (1968)[17] 1992 Dover paperback reprint. ISBN 0-486-67085-6
  • Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Mathematical Monographs, Mathematical Association of America, 1959.[18]
  • Mark Kac, Probability and related topics in the physical sciences. 1959 (with contributions by Uhlenbeck on the Boltzmann equation, Hibbs on quantum mechanics, and van der Pol on finite difference analogues of the wave and potential equations, Boulder Seminar 1957).[19]
  • Mark Kac, Enigmas of Chance: An Autobiography, Harper and Row, New York, 1985. Sloan Foundation Series. Published posthumously with a memoriam note by Gian-Carlo Rota.[20] ISBN 0-06-015433-0

References

  1. ^ Obituary in Rochester Democrat & Chronicle, 11 November 1984
  2. ^ Mark Kac at the Mathematics Genealogy Project
  3. ^ a b c Kac, Mark (1985). Enigmas of Chance: An Autobiography. New York: Harper and Row. ISBN 0-06-015433-0.
  4. ^ O'Connor, John J.; Robertson, Edmund F., "Mark Kac", MacTutor History of Mathematics Archive, University of St Andrews
  5. ^ Kac, Mark, Community of Scholars Profile, IAS Archived 2013-02-07 at the Wayback Machine
  6. ^ Berlin, T. H.; Kac, M. (1952). "The spherical model of a ferromagnet". Phys. Rev. 86 (6): 821–835. Bibcode:1952PhRv...86..821B. doi:10.1103/PhysRev.86.821.
  7. ^ Kac, M.; Ward, J. C. (1952). "A combinatorial solution of the two-dimensional Ising model". Phys. Rev. 88 (6): 1332–1337. Bibcode:1952PhRv...88.1332K. doi:10.1103/physrev.88.1332.
  8. ^ Cohen, E. G. D. (April 1985). "Obituary: Mark Kac". Physics Today. 38 (4): 99–100. Bibcode:1985PhT....38d..99C. doi:10.1063/1.2814542. Archived from the original on 2013-09-30.
  9. ^ Benguria, Rafael (July 2014). "The centenary of Mark Kac (1914–1984)" (PDF). IAMP News Bulletin: 5–18. (See pages 14–15.)
  10. ^ Gottesman, Max; Kac, Mark Kac; Langer, James (January 1980). "A legacy and a hope". Physics Today. 33 (1): 102. doi:10.1063/1.2913884.
  11. ^ Kac, Mark; Lebowitz, Joel L.; Plotz, Paul H. (12 October 1984). "Yosif Begun". Science. 226 (4671): 114–116. doi:10.1126/science.226.4671.114.c. PMID 17814316. S2CID 239881402.
  12. ^ Kac, Mark (1947). "Random walk and the theory of Brownian motion". Amer. Math. Monthly. 54 (7): 369–391. Bibcode:1947AmMM...54..369K. doi:10.2307/2304386. JSTOR 2304386.
  13. ^ "Mark Kac". American Academy of Arts & Sciences. Retrieved 2022-09-13.
  14. ^ "Mark Kac". www.nasonline.org. Retrieved 2022-09-13.
  15. ^ Kac, Mark (1966). "Can one hear the shape of a drum?". Amer. Math. Monthly. 73, Part II (4): 1–23. doi:10.2307/2313748. JSTOR 2313748.
  16. ^ "APS Member History". search.amphilsoc.org. Retrieved 2022-09-13.
  17. ^ May, Kenneth O. (Spring 1969). "Review of Mathematics and Logic by Mark Kac and Stanislaw Ulam". Isis. 60 (1): 112–113. doi:10.1086/350456.
  18. ^ LeVeque, W. L. (1960). "Review: Statistical independence in probability, analysis and number theory, by Mark Kac. Carus Mathematical Monographs, no. 12". Bull. Amer. Math. Soc. 66 (4): 265–266. doi:10.1090/S0002-9904-1960-10459-4.
  19. ^ Baxter, Glen (1960). "Review: Probability and related topics in the physical sciences, by Mark Kac". Bull. Amer. Math. Soc. 66 (6): 472–475. doi:10.1090/s0002-9904-1960-10500-9.
  20. ^ Birnbaum, Z. W. (1987). "Review: Enigmas of chance; an autobiography, by Mark Kac". Bull. Amer. Math. Soc. (N.S.). 17 (1): 200–202. doi:10.1090/s0273-0979-1987-15563-7.
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