Mark Kac ( ;
Polish
Polish may refer to:
* Anything from or related to Poland, a country in Europe
* Polish language
* Poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...
: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a
Polish American
Polish Americans ( pl, Polonia amerykańska) are Americans who either have total or partial Polish ancestry, or are citizens of the Republic of Poland. There are an estimated 9.15 million self-identified Polish Americans, representing about 2.83 ...
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, structure, space, models, and change.
History
On ...
. His main interest was
probability theory
Probability theory 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 expressing it through a set ...
. 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 operators in a variety of mathematical spaces. It is a result ...
, 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
Kremenets ( uk, Крем'янець, Кременець, translit. ''Kremianets'', ''Kremenets''; pl, Krzemieniec; yi, קרעמעניץ, Kremenits) is a city in Ternopil Oblast (province) of western Ukraine. It is the administrative center o ...
(
Polish
Polish may refer to:
* Anything from or related to Poland, a country in Europe
* Polish language
* Poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...
: "Krzemieniec"), changed hands from the
Russian Empire
The Russian Empire was an empire and the final period of the Russian monarchy from 1721 to 1917, ruling across large parts of Eurasia. It succeeded the Tsardom of Russia following the Treaty of Nystad, which ended the Great Northern War. ...
(by then
Soviet Ukraine
The Ukrainian Soviet Socialist Republic ( uk, Украї́нська Радя́нська Соціалісти́чна Респу́бліка, ; russian: Украи́нская Сове́тская Социалисти́ческая Респ ...
) to
Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populou ...
after the
Peace of Riga
The Peace of Riga, also known as the Treaty of Riga ( pl, Traktat Ryski), was signed in Riga on 18 March 1921, among Poland, Soviet Russia (acting also on behalf of Soviet Belarus) and Soviet Ukraine. The treaty ended the Polish–Soviet Wa ...
, when Kac was a child.
[Obituary](_blank)
in ''Rochester Democrat & Chronicle'', 11 November 1984
Kac completed his Ph.D. in mathematics at the Polish
University of Lwów
The University of Lviv ( uk, Львівський університет, Lvivskyi universytet; pl, Uniwersytet Lwowski; german: Universität Lemberg, briefly known as the ''Theresianum'' in the early 19th century), presently the Ivan Franko Na ...
in 1937 under the direction of
Hugo Steinhaus
Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 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 Un ...
.
While there, he was a member of the
Lwów School of Mathematics
The Lwów school of mathematics ( pl, lwowska szkoła matematyczna) 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 ...
. 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 or NYC, is the most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the most densely populated major city in the Un ...
in November 1938.
With the onset of
World War II in Europe, Kac was able to stay in America, while his parents and brother, who had remained in Kremenets, were
murdered by the Germans in
mass execution
Capital punishment, also known as the death penalty, is the state-sanctioned practice of deliberately killing a person as a punishment for an actual or supposed crime, usually following an authorized, rule-governed process to conclude that t ...
s in August 1942.
From 1939 to 1961, Kac taught at
Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
, first as an instructor, then from 1943 as an assistant professor and from 1947 as a full professor.
While there, 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 Constituti ...
in 1943. During the 1951–1952 academic year, Kac was on
sabbatical at the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
. In 1952, Kac, with
Theodore H. Berlin, introduced the spherical model of a
ferromagnet
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
(a variant of the
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
) and, with
J. C. Ward, found an exact solution of the Ising model using a combinatorial method. 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 classified ...
in New York City. In the early 1960s, he worked with
George Uhlenbeck
George Eugene Uhlenbeck (December 6, 1900 – October 31, 1988) was a Dutch-American theoretical physicist.
Background and education
George Uhlenbeck was the son of Eugenius and Anne Beeger Uhlenbeck. He attended the Hogere Burgerschool (High S ...
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
, mottoeng = "Let whoever earns the palm bear it"
, religious_affiliation = Nonsectarian—historically Methodist
, established =
, accreditation = WSCUC
, type = Private research university
, academic_affiliations =
, endowment = $8.1 ...
where he spent the rest of his career.
Work
In his 1966 article titled "
Can one hear the shape of the drum", Kac asked the question whether two
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 resonator ...
s ("drums") of different
geometrical shapes can have exactly the same set of frequencies ("sound tones"). The answer was negative, meaning that the
eigenfrequency
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
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
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
)
*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
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
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 The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman were both Cornell faculty, Kac attended a present ...
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 process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
es by reading
Norbert Wiener's original papers, which were "the most difficult papers I have ever read."
[ ]Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
is a Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
. Feynman's path integrals are another example.
*Kac's distinction between an "ordinary genius" like Hans Bethe
Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won the 1967 Nobel ...
and a "magician" like Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
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
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed lim ...
, and the result became known as the Erdős–Kac theorem In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
. They continued working together and more or less created the subject of probabilistic number theory
In mathematics, Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underlying it is that different prime numbers are, in ...
.
*Kac sent Erdős a list of his publications, and one of his papers contained the word "capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
" 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. He was a co-author of 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.
Awards and honors
* 1950 — Chauvenet Prize
The Chauvenet Prize is the highest award for mathematical expository writing. It consists of a prize of $1,000 and a certificate, and is awarded yearly by the Mathematical Association of America in recognition of an outstanding expository article ...
for 1947 expository article
* 1959 – member of the American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) 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, a ...
* 1965 – member of the National Academy of Sciences
* 1968 – Chauvenet Prize
The Chauvenet Prize is the highest award for mathematical expository writing. It consists of a prize of $1,000 and a certificate, and is awarded yearly by the Mathematical Association of America in recognition of an outstanding expository article ...
(and 1967 Lester R. Ford Award
Lester is an ancient Anglo-Saxon surname and given name. Notable people and characters with the name include:
People
Given name
* Lester Bangs (1948–1982), American music critic
* Lester W. Bentley (1908–1972), American artist from Wisc ...
) for 1966 expository article
* 1969 – member of the American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
* 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)
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 to ...
, 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 on finite difference analogues of the wave and potential equations, Boulder Seminar 1957).
* 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
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
Cornell University faculty
Institute for Advanced Study visiting scholars
Polish emigrants to the United States
Presidents of the Institute of Mathematical Statistics
Probability theorists
Lwów School of Mathematics
American people of Polish-Jewish descent
University of Southern California faculty
People with acquired American citizenship
People from Kremenets
Members of the American Philosophical Society