Mark Kac

Short description: Polish-American Mathematician

Mark Kac

Born|

Krzemieniec, Russian Empire (now Ukraine )

Died| October 26, 1984(1984-10-26) (aged 70)

California , U.S.

Nationality| Polish
Citizenship| Poland, USA
Alma mater| Lwów University
Known for| Feynman–Kac formula
Erdős–Kac theorem
Kac–Bernstein theorem
Kac's lemma
Kac process
Kac ring
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. 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".

## Contents

* 1 Early life and education
* 2 Career
* 2.1 Cornell University
* 2.2 Rockefeller University
* 2.3 Human rights
* 3 Awards and honors
* 4 Books
* 5 References
* 6 External links

## Early life and education

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 Germans in mass executions in August 1942.[3]

## Career

### Cornell University

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

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

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

* 1950 — Chauvenet Prize for 1947 expository article[13]
* 1959 – member of the American Academy of Arts and Sciences[14]
* 1965 – member of the National Academy of Sciences[15]
* 1968 – Chauvenet Prize (and 1967 Lester R. Ford Award) for 1966 expository article[16]
* 1969 – member of the American Philosophical Society[17]
* 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)[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

## References

1. ↑ Obituary in Rochester Democrat & Chronicle, 11 November 1984
2. ↑ Mark Kac at the Mathematics Genealogy Project
3. ↑ 3.0 3.1 3.2 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, http://www-history.mcs.st-andrews.ac.uk/Biographies/Kac.html .
5. ↑ Kac, Mark, Community of Scholars Profile, IAS
6. ↑ Berlin, T. H.; Kac, M. (1952). "The spherical model of a ferromagnet". Phys. Rev. 86 (6): 821–35. doi:10.1103/PhysRev.86.821. Bibcode: 1952PhRv...86..821B. http://prola.aps.org/abstract/PR/v86/i6/p821_1.
7. ↑ Kac, M.; Ward, J. C. (1952). "A combinatorial solution of the two-dimensional Ising model". Phys. Rev. 88 (6): 1332–1337. doi:10.1103/physrev.88.1332. Bibcode: 1952PhRv...88.1332K.
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. doi:10.1063/1.2814542. Bibcode: 1985PhT....38d..99C. http://www.physicstoday.org/resource/1/phtoad/v38/i4/p99_s2?bypassSSO=1.
10. ↑ Benguria, Rafael (July 2014). "The centenary of Mark Kac (1914–1984)". IAMP News Bulletin: 5–18. http://www.iamp.org/bulletins/old-bullletins/201407.pdf. (See pages 14–15.)
11. ↑ Gottesman, Max; Kac, Mark Kac; Langer, James (January 1980). "A legacy and a hope". Physics Today 33 (1): 102. doi:10.1063/1.2913884. Bibcode: 1980PhT....33a.102G.
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.
13. ↑ Kac, Mark (1947). "Random walk and the theory of Brownian motion". Amer. Math. Monthly 54 (7): 369–391. doi:10.2307/2304386. Bibcode: 1947AmMM...54..369K. http://www.maa.org/programs/maa-awards/writing-awards/random-walk-and-the-theory-of-brownian-motion.
14. ↑ "Mark Kac" (in en). https://www.amacad.org/person/mark-kac.
15. ↑ "Mark Kac". http://www.nasonline.org/member-directory/deceased-members/54016.html.
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. http://www.maa.org/programs/maa-awards/writing-awards/can-one-hear-the-shape-of-a-drum-0.
17. ↑ "APS Member History". https://search.amphilsoc.org/memhist/search?creator=Mark+Kac&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced.
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. https://www.ams.org/journals/bull/1960-66-04/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. https://www.ams.org/journals/bull/1960-66-06/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. https://www.ams.org/journals/bull/1987-17-01/S0273-0979-1987-15563-7/.

## External links

| Wikimedia Commons has media related to Mark Kac.

|

* O'Connor, John J.; Robertson, Edmund F., "Mark Kac", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Kac.html .
* Mark Kac at the Mathematics Genealogy Project
* National Academy of Sciences Biographical Memoir

* v
* t
* e

Chauvenet Prize recipients

* 1925 G. A. Bliss
* 1929 T. H. Hildebrandt
* 1932 G. H. Hardy
* 1935 Dunham Jackson
* 1938 G. T. Whyburn
* 1941 Saunders Mac Lane
* 1944 R. H. Cameron
* 1947 Paul Halmos
* 1950 Mark Kac
* 1953 E. J. McShane
* 1956 Richard H. Bruck
* 1960 Cornelius Lanczos
* 1963 Philip J. Davis
* 1964 Leon Henkin
* 1965 Jack K. Hale and Joseph P. LaSalle
* 1967 Guido Weiss
* 1968 Mark Kac
* 1970 Shiing-Shen Chern
* 1971 Norman Levinson
* 1972 François Trèves
* 1973 Carl D. Olds
* 1974 Peter D. Lax
* 1975 Martin Davis and Reuben Hersh
* 1976 Lawrence Zalcman
* 1977 W. Gilbert Strang
* 1978 Shreeram S. Abhyankar
* 1979 Neil J. A. Sloane
* 1980 Heinz Bauer
* 1981 Kenneth I. Gross
* 1982 No award given.
* 1983 No award given.
* 1984 R. Arthur Knoebel
* 1985 Carl Pomerance
* 1986 George Miel
* 1987 James H. Wilkinson
* 1988 Stephen Smale
* 1989 Jacob Korevaar
* 1990 David Allen Hoffman
* 1991 W. B. Raymond Lickorish and Kenneth C. Millett
* 1992 Steven G. Krantz
* 1993 David H. Bailey, Jonathan M. Borwein and Peter B. Borwein
* 1994 Barry Mazur
* 1995 Donald G. Saari
* 1996 Joan Birman
* 1997 Tom Hawkins
* 1998 Alan Edelman and Eric Kostlan
* 1999 Michael I. Rosen
* 2000 Don Zagier
* 2001 Carolyn S. Gordon and David L. Webb
* 2002 Ellen Gethner, Stan Wagon, and Brian Wick
* 2003  Thomas C. Hales
* 2004 Edward B. Burger
* 2005 John Stillwell
* 2006 Florian Pfender & Günter M. Ziegler
* 2007 Andrew J. Simoson
* 2008 Andrew Granville
* 2009 Harold P. Boas
* 2010 Brian J. McCartin
* 2011 Bjorn Poonen
* 2012 Dennis DeTurck, Herman Gluck, Daniel Pomerleano & David Shea Vela-Vick
* 2013 Robert Ghrist
* 2014 Ravi Vakil
* 2015 Dana Mackenzie
* 2016 Susan Marshall & Donald R. Smith
* 2017 Mark Schilling
* 2018 Daniel J. Velleman
* 2019 Tom Leinster

* v
* t
* e

John von Neumann Lecturers

* Lars Ahlfors (1960)
* Mark Kac (1961)
* Jean Leray (1962)
* Stanislaw Ulam (1963)
* Solomon Lefschetz (1964)
* Freeman Dyson (1965)
* Eugene Wigner (1966)
* Chia-Chiao Lin (1967)
* Peter Lax (1968)
* George F. Carrier (1969)
* James H. Wilkinson (1970)
* Paul Samuelson (1971)
* Jule Charney (1974)
* James Lighthill (1975)
* René Thom (1976)
* Kenneth Arrow (1977)
* Peter Henrici (1978)
* Kurt O. Friedrichs (1979)
* Keith Stewartson (1980)
* Garrett Birkhoff (1981)
* David Slepian (1982)
* Joseph B. Keller (1983)
* Jürgen Moser (1984)
* John W. Tukey (1985)
* Jacques-Louis Lions (1986)
* Richard M. Karp (1987)
* Germund Dahlquist (1988)
* Stephen Smale (1989)
* Andrew Majda (1990)
* R. Tyrrell Rockafellar (1992)
* Martin D. Kruskal (1994)
* Carl de Boor (1996)
* William Kahan (1997)
* Olga Ladyzhenskaya (1998)
* Charles S. Peskin (1999)
* Persi Diaconis (2000)
* David Donoho (2001)
* Eric Lander (2002)
* Heinz-Otto Kreiss (2003)
* Alan C. Newell (2004)
* Jerrold E. Marsden (2005)
* George C. Papanicolaou (2006)
* Nancy Kopell (2007)
* David Gottlieb (2008)
* Franco Brezzi (2009)
* Bernd Sturmfels (2010)
* Ingrid Daubechies (2011)
* John M. Ball (2012)
* Stanley Osher (2013)
* Leslie Greengard (2014)
* Jennifer Tour Chayes (2015)
* Donald Knuth (2016)
* Bernard J. Matkowsky (2017)
* Charles F. Van Loan (2018)

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