Mark Kac
Mark Kac  

Born 
Krzemieniec 
August 3, 1914
Died  October 26, 1984 California 
(aged 70)
Residence  USA 
Nationality  Polish 
Citizenship  Poland, USA 
Alma mater  Lwów University 
Known for 
Feynman–Kac formula Erdős–Kac theorem 
Awards 
Chauvenet Prize (1950, 1968) Birkhoff Prize (1978) 
Scientific career  
Fields  Mathematics 
Institutions 
Cornell University Rockefeller University University of Southern California 
Doctoral advisor  Hugo Steinhaus 
Doctoral students 
Harry 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. He was born to a PolishJewish family; their town, Kremenets (Polish: "Krzemieniec"), changed hands from the Russian Empire to Poland when Kac was a child.^{[1]} 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.)
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 Western Ukraine were murdered by the Germans in the mass executions in Krzemieniec in August 1942.^{[4]}
From 193961 he was at Cornell University, first as an instructor, then from 1943 as assistant professor and from 1947 as full professor.^{[5]} 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.^{[6]} In 1952, Kac, with Theodore H. Berlin, introduced the spherical model of a ferromagnet (a variant of the Ising model)^{[7]} and, with J. C. Ward, found an exact solution of the Ising model using a combinatorial method.^{[8]} 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.^{[9]} After twenty years at Rockefeller, he moved to the University of Southern California where he spent the rest of his career.
Contents
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 positive, meaning that the eigenfrequency set does not uniquely characterize the shape of a resonator.
Reminiscences
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 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 on the Cornell faculty he went to a lecture of Feynman's and saw that the two of them were working on the same thing from different directions. The FeynmanKac 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 learned 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."
Awards and honors
 1950 — Chauvenet Prize for 1947 expository article^{[10]}
 1959 — member of the American Academy of Arts and Sciences
 1965 — member of the National Academy of Sciences
 1968 — Chauvenet Prize (and 1967 Lester R. Ford Award) for 1966 expository article^{[11]}
 1971 — Solvay Lecturer at Brussels
 1980 — Fermi Lecturer at the Scuola Normale, Pisa
Books
 Mark Kac and Stanislaw Ulam: Mathematics and Logic: Retrospect and Prospects, Praeger, New York (1968) Dover paperback reprint.
 Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Mathematical Monographs, Mathematical Association of America, 1959.^{[12]}
 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).^{[13]}
 Mark Kac, Enigmas of Chance: An Autobiography, Harper and Row, New York, 1985. Sloan Foundation Series. Published posthumously with a memoriam note by GianCarlo Rota.^{[14]}
See also
References
 ^ Obituary in Rochester Democrat & Chronicle, 11 November 1984
 ^ Mark Kac at the Mathematics Genealogy Project
 ^ ^{a} ^{b} Mark Kac, Enigmas of Chance: An Autobiography, Harper and Row, New York, 1985. ISBN 0060154330
 ^ M Kac, Enigmas of chance: an autobiography (California, 1987)
 ^ O'Connor, John J.; Robertson, Edmund F., "Mark Kac", MacTutor History of Mathematics archive, University of St Andrews.
 ^ Kac, Mark, Community of Scholars Profile, IAS Archived 20130207 at the Wayback Machine.
 ^ Berlin, T. H.; Kac, M. (1952). "The spherical model of a ferromagnet". Phys. Rev. 86: 821–835. doi:10.1103/PhysRev.86.821.
 ^ Kac, M.; Ward, J. C. (1952). "A combinatorial solution of the twodimensional Ising model". Phys. Rev. 88: 1332–1337. Bibcode:1952PhRv...88.1332K. doi:10.1103/physrev.88.1332.
 ^ Cohen, E. G. D. (April 1985). "Obituary: Mark Kac". Physics Today. 38 (4): 99–100. doi:10.1063/1.2814542. Archived from the original on 20130930.
 ^ Kac, Mark (1947). "Random walk and the theory of Brownian motion". Amer. Math. Monthly. 54: 369–391. doi:10.2307/2304386.
 ^ Kac, Mark (1966). "Can one hear the shape of a drum?". Amer. Math. Monthly. 73, Part II: 1–23. doi:10.2307/2313748.
 ^ 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/S000299041960104594.
 ^ 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/s000299041960105009.
 ^ 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/s027309791987155637.
External links
Wikiquote has quotations related to: Mark Kac 
 O'Connor, John J.; Robertson, Edmund F., "Mark Kac", MacTutor History of Mathematics archive, University of St Andrews.
 Mark Kac at the Mathematics Genealogy Project
 National Academy of Sciences Biographical Memoir
 1914 births
 1984 deaths
 20thcentury American mathematicians
 Cornell University faculty
 Guggenheim Fellows
 Institute for Advanced Study visiting scholars
 Polish emigrants to the United States
 American people of PolishJewish descent
 Presidents of the Institute of Mathematical Statistics
 Probability theorists
 Polish mathematicians
 Polish Jews
 University of Southern California faculty
 People with acquired American citizenship