Mark Kac

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Mark Kac
Born(1914-08-03)August 3, 1914
DiedOctober 26, 1984(1984-10-26) (aged 70)
CitizenshipPoland, USA
Alma materLwów University
Known forFeynman–Kac formula
Erdős–Kac theorem
Kac–Bernstein theorem
Kac's lemma
Kac process
Kac ring
AwardsChauvenet Prize (1950, 1968)
Birkhoff Prize (1978)
Scientific career
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, the idea of understanding the extent to which the spectrum allows one to read back the geometry. In the end, the answer was generally "no".

Early life and education[edit]

He was born to a Polish-Jewish family; their town, Kremenets (Polish: "Krzemieniec"), changed hands from the Russian Empire (by then Soviet Ukraine) 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 in Europe, Kac was able to stay in the United States, while his parents and brother, who had remained in Kremenets, were murdered by the Nazis in mass executions in August 1942.[3]


Cornell University[edit]

From 1939 to 1961, Kac taught at Cornell University, an Ivy League university in Ithaca, New York, where he was first an instructor. In 1943, he was appointed an assistant professor, and he became a full professor in 1947.[4]

While a professor at Cornell, he became a naturalized US citizen in 1943. From 1943 to 1945, he also worked with George Uhlenbeck at the MIT Radiation Laboratory.[3] During the 1951–1952 academic year, 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 1956, he introduced a simplified mathematical model known as the Kac ring, which features the emergence of macroscopic irreversibility from completely time-symmetric microscopic laws. Using the model as an analogy to molecular motion, he provided an explanation for Loschmidt's paradox.[8]

Rockefeller University[edit]

In 1961, Kac left Cornell and went to The Rockefeller University in New York City.

He worked with George Uhlenbeck and P. C. Hemmer on the mathematics of a van der Waals gas.[9] After twenty years at Rockefeller, he moved to the University of Southern California where he spent the rest of his career.

In his 1966 article, "Can one hear the shape of the drum", Kac asked whether the geometric shape of a drum is uniquely defined by its sound. The answer was negative, meaning two different resonators can have identical set of eigenfrequencies.

Human rights[edit]

Kac was the co-chair of the Committee of Concerned Scientists.[10] He co-authored a letter, which publicized the case of the scientist Vladimir Samuilovich Kislik[11] and a letter which publicized the case of the applied mathematician Yosif Begun.[12]

Awards and honors[edit]


  • Mark Kac and Stanislaw Ulam: Mathematics and Logic: Retrospect and Prospects, Praeger, New York (1968)[18] 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.[19]
  • 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).[20]
  • 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.[21] ISBN 0-06-015433-0


  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): 821–35. 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. ^ Thompson, Colin J (1986). "The contributions of Mark Kac to mathematical physics". The Annals of Probability. 14: 1129–1138.
  9. ^ 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.
  10. ^ Benguria, Rafael (July 2014). "The centenary of Mark Kac (1914–1984)" (PDF). IAMP News Bulletin: 5–18. (See pages 14–15.)
  11. ^ Gottesman, Max; Kac, Mark Kac; Langer, James (January 1980). "A legacy and a hope". Physics Today. 33 (1): 102. Bibcode:1980PhT....33a.102G. doi:10.1063/1.2913884.
  12. ^ 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.
  13. ^ 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.
  14. ^ "Mark Kac". American Academy of Arts & Sciences. Retrieved 2022-09-13.
  15. ^ "Mark Kac". Retrieved 2022-09-13.
  16. ^ 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.
  17. ^ "APS Member History". Retrieved 2022-09-13.
  18. ^ 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.
  19. ^ 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.
  20. ^ 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.
  21. ^ 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.

External links[edit]