Mark Kac ( ;

Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Polish people, people from Poland or of Polish descent * Polish chicken * Polish brothers (Mark Polish and Michael Polish, born 1970), American twin ...

: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a

Polish-American Polish Americans () are Americans who either have total or partial Polish ancestry, or are citizens of the Republic of Poland. There are an estimated 8.81 million self-identified Polish Americans, representing about 2.67% of the U.S. population, ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. His main interest was

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

. His question, " Can one hear the shape of a drum?" set off research into

spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...

, the idea of understanding the extent to which the

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

allows one to read back the geometry. In the end, the answer was generally "no".

Early life and education

He was born to a

Polish-Jewish The history of the Jews in Poland dates back at least 1,000 years. For centuries, Poland was home to the largest and most significant Jews, Jewish community in the world. Poland was a principal center of Jewish culture, because of the long pe ...

family; their town,

Kremenets Kremenets (, ; ; ) is a city in Ternopil Oblast, western Ukraine. It is the administrative center of the Kremenets Raion, and lies north-east of the Pochaiv Lavra. The city is situated in the historic region of Volhynia and features the 12th-c ...

(

: "Krzemieniec"), changed hands from the

Russian Empire The Russian Empire was an empire that spanned most of northern Eurasia from its establishment in November 1721 until the proclamation of the Russian Republic in September 1917. At its height in the late 19th century, it covered about , roughl ...

(by then

Soviet Ukraine The Ukrainian Soviet Socialist Republic, abbreviated as the Ukrainian SSR, UkrSSR, and also known as Soviet Ukraine or just Ukraine, was one of the constituent republics of the Soviet Union from 1922 until 1991. Under the Soviet one-party m ...

) to

Poland Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...

after the

Peace of Riga The Treaty of Riga was signed in Riga, Latvia, on between Poland on one side and Soviet Russia (acting also on behalf of Soviet Belarus) and Soviet Ukraine on the other, ending the Polish–Soviet War (1919–1921). The chief negotiators o ...

, when Kac was a child.Obituary
in ''Rochester Democrat & Chronicle'', 11 November 1984 Kac completed his Ph.D. in mathematics at the Polish

University of Lwów The Ivan Franko National University of Lviv (named after Ivan Franko, ) is a state-sponsored university in Lviv, Ukraine. Since 1940 the university is named after Ukrainian poet Ivan Franko. The university is the oldest institution of highe ...

in 1937 under the direction of

Hugo Steinhaus Hugo Dyonizy Steinhaus ( , ; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Univers ...

. While there, he was a member of the

Lwów School of Mathematics The Lwów school of mathematics () was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to discuss mathematical problems, ...

. 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 New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive w ...

in November 1938. With the onset of

World War II in Europe The European theatre of World War II was one of the two main theatres of combat during World War II, taking place from September 1939 to May 1945. The Allied powers (including the United Kingdom, the United States, the Soviet Union and Franc ...

, 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 execution Mass killing is a concept which has been proposed by genocide scholars who wish to define incidents of non-combat killing which are perpetrated by a government or a state. A mass killing is commonly defined as the killing of group members witho ...

s in August 1942.

Career

Cornell University

From 1939 to 1961, Kac taught at

Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...

, an

Ivy League The Ivy League is an American collegiate List of NCAA conferences, athletic conference of eight Private university, private Research university, research universities in the Northeastern United States. It participates in the National Collegia ...

university in

Ithaca, New York Ithaca () is a city in and the county seat of Tompkins County, New York, United States. Situated on the southern shore of Cayuga Lake in the Finger Lakes region of New York (state), New York, Ithaca is the largest community in the Ithaca metrop ...

, where he was first an instructor. In 1943, he was appointed an assistant professor, and he became a full professor in 1947. While a professor at Cornell, he became a

naturalized US citizen Citizenship of the United States is a legal status that entails Americans with specific rights, duties, protections, and benefits in the United States. It serves as a foundation of fundamental rights derived from and protected by the Constitu ...

in 1943. From 1943 to 1945, he also worked with

George Uhlenbeck George Eugene Uhlenbeck (December 6, 1900 – October 31, 1988) was a Dutch-American theoretical physicist, known for his significant contributions to quantum mechanics and statistical mechanics. He co-developed the concept of electron spin, alo ...

at the

MIT Radiation Laboratory The Radiation Laboratory, commonly called the Rad Lab, was a microwave and radar research laboratory located at the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts. It was first created in October 1940 and operated until 3 ...

. During the 1951–1952 academic year, Kac was on

sabbatical A sabbatical (from the Hebrew: (i.e., Sabbath); in Latin ; Greek: ) is a rest or break from work; "an extended period of time intentionally spent on something that’s not your routine job." The concept of the sabbatical is based on the Bi ...

at the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

. In 1952, Kac, with Theodore H. Berlin, introduced the spherical model of a

ferromagnet Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromag ...

, a variant of the

Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...

, and, with J. C. Ward, found an exact solution of the Ising model using a combinatorial method. In 1956, he introduced a simplified mathematical model known as the

Kac ring In statistical mechanics, the Kac ring is a toy model introduced by Mark Kac in 1956 to explain how the second law of thermodynamics emerges from T-symmetry, time-symmetric interactions between molecules (see reversibility paradox). Although artif ...

, which features the emergence of

macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenome ...

irreversibility from completely time-symmetric

microscopic The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as the scale betwe ...

laws. Using the model as an analogy to molecular motion, he provided an explanation for

Loschmidt's paradox In physics, Loschmidt's paradox (named for Josef Loschmidt), also known as the reversibility paradox, irreversibility paradox, or ' (), is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics ...

Rockefeller University

In 1961, Kac left Cornell and went to

The Rockefeller University The Rockefeller University is a private biomedical research and graduate-only university in New York City, New York. It focuses primarily on the biological and medical sciences and provides doctoral and postdoctoral education. It is classifi ...

. He worked with

and P. C. Hemmer on the mathematics of a van der Waals gas. After twenty years at Rockefeller, he moved to the

University of Southern California The University of Southern California (USC, SC, or Southern Cal) is a Private university, private research university in Los Angeles, California, United States. Founded in 1880 by Robert M. Widney, it is the oldest private research university in ...

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 A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, ''shape'' excludes informat ...

of a drum is uniquely defined by its sound. The answer was negative, meaning two different

resonator A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a reso ...

s can have identical set of eigenfrequencies.

Human rights

Kac was the co-chair of the

Committee of Concerned Scientists The Committee of Concerned Scientists (CCS) is an independent international organization devoted to the protection and advancement of human rights and scientific freedom of scientists, physicians, engineers, and scholars. History The Committee wa ...

. He co-authored a letter which publicized the case of the scientist Vladimir Samuilovich Kislik and a letter which publicized the case of the applied mathematician

Yosif Begun Iosif Ziselovich Begun, sometimes spelled Yosef (born 9 July 1932, , ) is a former Soviet refusenik, prisoner of conscience, human rights activist, author and translator. Over the course of 17 years, Begun was imprisoned three times and spent ove ...

Awards and honors

* 1950 —

Chauvenet Prize The Chauvenet Prize is an annual award given by the Mathematical Association of America in recognition of an outstanding expository article on a mathematical topic. It consists of a prize of $1,000 and a certificate. The Chauvenet Prize was the ...

for 1947 expository article * 1959 – member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

* 1965 – member of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

* 1968 –

(and 1967

Lester R. Ford Award ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expositor ...

) for 1966 expository article * 1969 – member of the

American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...

* 1971 – Solvay Lecturer at Brussels * 1980 – Fermi Lecturer at the Scuola Normale, Pisa

Books

* Mark Kac and

Stanislaw Ulam Stanislav and variants may refer to: People *Stanislav (given name), a Slavic given name with many spelling variations (Stanislaus, Stanislas, Stanisław, etc.) Places * Stanislav, Kherson Oblast, a coastal village in Ukraine * Stanislaus County, ...

: ''Mathematics and Logic: Retrospect and Prospects'', Praeger, New York (1968
1992 Dover paperback reprint.
* Mark Kac, ''Statistical Independence in Probability, Analysis and Number Theory'',

Carus Mathematical Monographs The ''Carus Mathematical Monographs'' is a monograph series published by the Mathematical Association of America.Drake, Miriam A. (2003). ''Encyclopedia of Library and Information Science: Lib-Pub.'' CRC Press, Books in this series are intended t ...

, Mathematical Association of America, 1959. * 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 Van der Pol (also "Van de Pol", "Van de Poll", "Van den Pol" or "Van Pol") is a Dutch, toponymic surname A toponymic surname or habitational surname or byname is a surname or byname derived from a place name,Gian-Carlo Rota Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, proba ...

References

External links

* *
National Academy of Sciences Biographical Memoir
{{DEFAULTSORT:Kac, Mark 1914 births 1984 deaths 20th-century American mathematicians American people of Polish-Jewish descent Cornell University faculty Institute for Advanced Study visiting scholars Lwów School of Mathematics Members of the American Philosophical Society People from Kremenets Naturalized citizens of the United States Polish emigrants to the United States Presidents of the Institute of Mathematical Statistics Probability theorists Mathematical physicists University of Southern California faculty