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MARK KAC
August ~ 6, ~ 914~ctober 25, ~ 984
BY H. P. MCKEAN
POLAND. Mark Kac was born "to the sound of the guns of
August on the 16th clay of that month, 1914," in the town
of Krzemieniec then in Russia, later in PolancI, now in the
Soviet Ukraine (1985,1, p. 61. In this connection Kac liked to
quote Hugo Steinhaus, who, when asked if he had crossed
the border repliecl, "No, but the border crossed me."
In the early days of the century Krzemieniec was a pre-
dominantly Jewish town surrounded by a Polish society gen-
erally hostile to Jews. Kac's mother's family had been mer-
chants in the town for three centuries or more. His father
was a highly educates! person of Galician background, a
teacher by profession, holding degrees in philosophy from
I,eipzig, and in history and philosophy from Moscow.
As a boy Kac was educated at home and at the Lycee of
Krzemieniec, a well-known Polish school of the day. At home
he studied geometry with his father ant! discovered a new
derivation of Cardano's formula for the solution of the cu-
bic a first bite of the mathematical bug that cost Kac pere
five Polish zlotys in prize money. At school, he obtainer] a
splendid general education in science, literature, and history.
He was grateful to his early teachers to the end of his life.
In 1931 when he was seventeen, he entered the John
215

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2~6
BIOGRAPHICAL MEMOIRS
Casimir University of Ew6w, where he obtained the degrees
M. Phil. in 1935 and Ph.D. in 1937.
This was a period of awakening in Polish science. Marian
Smoluchowski had spurrec! a new interest in physics, and
mathematics was developing rapidly: in Warsaw, under Wac-
law Sierpinski, ant! in Ewow, uncler Hugo Steinhaus. In his
autobiography (1985,1, p. 29), Kac called this renaissance
"wonclerful." Most wonderful for him was the chance to study
with Steinhaus, a mathematician of perfect taste, wide cul-
ture, and wit; his adores! teacher who became his true friend
and introduced him to the then undigested subject of prob-
ability. Kac would devote most of his scientific life to this field
and to its cousin, statistical mechanics, beginning with a series
of papers prepared jointly with Steinhaus on statistical in-
depenclence ~1936, I-4; and 1937, I-21.
Kac's student days saw Hitler's rise and consolidation of
power, and he began to think of quitting Poland. In 1938 the
opportunity presented itself in the form of a Polish fellow-
ship to Johns Hopkins in Baltimore. Kac was twenty-four. He
left behind] his whole family, most of whom perished in
Krzemieniec in the mass executions of 1942-43. Years later
he returned, not to Krzemien~ec but to nearby Kiev. ~ re-
member him rapt, sniffing about him ant} saying he had not
smelled such autumn air since he was a boy. On this trip he
met with a surviving female cousin who asked him, at parting,
"Wouic! you like to know how it was in Krzemieniec?" then
a(l(lecl, "No. It is better if you don't know" (1985,1, p. 106~.
These cruel memories and their attendant regrets surely
stood behind Kac's devotion to the plight of Soviet refusniks
and others in like distress. His own life adds poignancy to his
selection of the following quote from his father's hero, Sol-
omon Maimon: "In search of truth ~ left my people, my coun-
try and my family. It is not therefore to be assumed that
shall forsake the truth for any lesser motives" (1985,1, p. 9~.

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MARK KAC
AMERICA
217
Kac came to Baltimore in 1938 and wrote of his reaction
to his new-found land:
"l find it difficult . . . to convey the feeling of decompression, of freedom,
of being caught in the sweep of unimagined and unimaginable grandeur.
It was life on a different scale with more of everything more air to
breathe, more things to see, more people to know. The friendliness and
warmth from all sides, the ease and naturalness of social contacts. The
contrast to Poland . . . defied description." (1985,1, p. 85)
After spending 1938-39 in Baltimore, Kac moved to Ith-
aca, where he would remain until 1961. Cornell was at that
time a fine place for probability: Kai-Lai Chung, Feller, Hunt,
and occasionally the peripatetic Paul Erdos formed, with Kac,
a talented and productive group. His mathematics bloomed
there. He also courter! and married Katherine Mayberry,
shortly finding himself the father of a family. So began, as he
saicl, the healing of the past.
From 1943 to 1947 Kac was associated off end on with the
Racliation Lab at MIT, where he met and began to collaborate
with George UhIenbeck. This was an important event for
him. It reawakened his interest in statistical mechanics and
was a decisive factor in his moving to be with UhIenbeck at
The Rockefeller University in 1962. There DetIev Bronk,
with his inimitable enthusiasm, was trying to build up a small,
top-flight school. While this ideal was not fully realized either
then or afterwards, it afforded Kac the opportunity to im-
merse himself in the statistical mechanics of phase transitions
in the company of Tect Berlin and UhIenbeck, among others.
Retiring in 1981, Kac moved to the University of Southern
California, where he stayer! until his cleath on October 25,
1984, at the age of seventy. He is survives} by his wife Kitty,

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218
BIOGRAPHICAL MEMOIRS
his son Michael, his daughter Deborah, and his grand-
chilciren.
MATHEMATICAL WORK
Independence and the Normal Law
In the beginning was the notion of statistical indepen-
clence to which Steinhaus introduced Kac. The basic idea is
that the probability of the joint occurrence of indepenclent
events should be the product of their incliviclual probabilities,
as in I/2 x I/2 = I/4 for a run of two heads in the tossing
of an honest coin. The most famous consequence of this type
of inclependence is the fact that, if #(n) is the number of
heads in n tosses of such an honest coin, then the normal law
of errors hoIcis:
Pea (~21~ c be ~ ~ (27r)-~2e-X22d.x
in which P signifies the probability of the event indicated
between the brackets, the subtracted n/2 is the mean of #(n),
and the approximation to the right-handecl integral im-
proves inclefinitely as n gets large. The fact goes back to A.
cle Moivre (1667-1754) and was extended to a vague but
much more inclusive statement by Gauss and Laplace. It was
put on a better technical footing by P. L. Cebysev (~821-~890)
and A. A. Markov (~856-1922), but as Poincare complained,
"Tout le monde y croft (la loi des erreurs) parce que les mathe'mati-
ciens s'imaganent que c'est un fait d'observation et les observateurs
que c'est un the'oreme de mathe'matiques." The missing ingredient,
suppliecl by Steinhaus, was an unambiguous concept of in-
depenclence. But that was only the start. All his life Kac de-
lighted in extending the sway of the normal law over new
and unforseen clomains. ~ mention two instances:

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MARK KAC
219
Let W~, ,Mn be n ~ 2) independent frequencies meaning
that no integral combination of them vanishes. Then
T-l measure [O s t C T a c sin ()It I/ N~ sin (I)nt C bl
~
Jb
(fir)- 1/2 e-x2/2 dx
Ja
for large T and n in which measure signifies the sum of the
lengths of the several subintervals on which the indicated
inequality takes place. In short sinusoids of independent fre-
quencies behave as if they were statistically independent
though strictly speaking they are not (1937 2; 1943 2).
On another occasion Kac looker! to a vastly different do-
main: Let d(n) be the number of ctistinct prime divisors of
the whole number n = ~ 2 3 .... Then for large N
Non N an (~)~g2 cub i;b(21r)-~2e-x22`lx
in which # denotes the number of integers having the prop-
erty indicated in the brackets and [g2n is the iterated loga-
rithm [g(Ign). In short there is some kind of statistical inde-
pendence in number theory too. Kac maple this beautiful
discovery jointly with P. Erdos (1940 4).
These and other examples of statistical independence are
explained in Kac s delightful Carus Monograph Independence
in Probability, Analysis and Number Theory (I 95 ~ I).
Brownian Motion and Integration in Function Space
The Brownian motion typified by the incessant move-
ment of dust motes in a beam of sunlight was first discussed
from a physical standpoint by M. Smoluchowski and A.
Einstein (1905). N. Wiener later put the discussion on a solic!

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220
BIOGRAPHICAL MEMOIRS
mathematical footing. Kac was introduced to both develop-
ments during his association with MIT from 1943 to 1947.
Now the statistical law of the Brownian motion is normal:
if Aft) is the displacement of the Brownian traveler in some
fixed direction, then
PLa c x~t) ' b1
fib
= J (2Ut)-l/2e-X2/2tdX
a
The fact is that Brownian motion is nothing but an ap-
· . . · .
proxlmatlon to honest coln-tosslng:
x~t) ~ ~ x t+ (the number of heacis in T tosses)
(the number of taits)],
in which T is the whole number nearest to tN and N is large.
The normal law for coin-tossing cited before is the simplest
version of this approximation. Kac, with the help of UhIen-
beck, perceived the general principal at work, of which the
following is a pretty instance: Let ptn) be the number of times
that heads outnumber tails in n tosses of an honest coin.
Then the arcsine law holds:
PEn-~p~n) c c]
IrC dx
~_ ~ =
~ Jo W] - x)
arcsine By,
the right-hand side being precisely the probability that the
Brownian path x~t): O c t c I, starting at x(O) = 0, spends a
total time, T ' c, to the right of the origin (1947,21.
Kac's next application of Brownian motion was suggested
In a quantum-mechanical form by R. Feynman. It has to do
with the so-called elementary solution eft,x,y) of the
Schrodinger equation:
.

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MARK KAC
~ 64f/13t = 32/2 - V(X)4f.
Y21
The formula states that, with the left-hand imaginary unit
removed:
eft,X,y) = foxy I X (47rt)-~/2e_(X_y)2'4t
in which the final factor is the free elementary solution (for
V = 0) and Exy is the Brownian mean taken over the class
of paths starting at x(O) = x ant! ending at x(~) = y. This is
not really as explicit as it looks, as the mean is not readily
expressible in closed form for any but the simplest cases, but
it floes exhibit just how e depends upon V in a transparent
way. It can be used very electively, as Kac illustrated by a
beautiful derivation of the WKB approximation of classical
quantum mechanics (1946,31.
~ will describe one more application of Brownian motion
contained in Kac's Chauvenet Prize paper, Can One Hear the
Shape of a Drum? (1976,1~. The story goes back to H. Wey1's
proof of a conjecture of H. A. Lorentz. Let D be a plane
region bounded by one or more nice curves, holes being per-
mitted, anc! let oh, W2, etc., be its fundamental tones, i.e., let
_~2~, ~22, etc., be the eigenvalues of Laplace's operator ~ =
d2/8x2~ + 82/3x2 acting upon smooth functions that vanish at
the boundary of D. Then Lorentz conjectured and Wey}
proved:
#in: O)n C (do ~ U-~2 X the area of D
for large m. Kac found a remarkably simple proof of this fact
based upon the self-evident principle that the Brownian trav-
eller, starting inside D, does not fee] the boundary of D until
it gets there. He also speculated as to whether you could de-
duce the shape of D (up to rigid motions) if you could "hear"
all of its fundamental tones and showed that, indeed, you can

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222
BIOGRAPHICAL MEMOIRS
hear the length of the boundary, and the number of holes if
any. The full question is still open.
Statistical Mechanics
As noted before, Kac's interest in this subject had been
reawakened by UhIenbeck at MIT. A famous conundrum of
the fielcl was the superficial incompatibility of the (obvious)
irreversibility of natural processes and the reversibility of the
underlying molecular mechanics. Boltzmann struggled con-
tinually with the problem, best epitomized by UhIenbeck's
teachers, P. and T. Ehrenfest, in what they called the "dog-
flea" model. Kac's debut in statistical mechanics was to pro-
vide its complete solution, put forth in his second Chauvenet
Prize paper (1947,4~.
Next, Kac took up Boltzmann's equation describing the
development, in time, of the distribution of velocities in a
dilute gas of like molecules subject to streaming and to col-
lisions (in pairs). ~ think this work was not wholly successful,
but it did prompt Kac to produce a stimulating stucly of
Boltzmann's idea of"molecular chaos" (Stosszahiansatz) and
a typically elegant, Kac-type "caricature" of the Boltzmann
equation itself.
~ pass on to the eminently successful papers on phase
transitions. The basic question which Maxwell and Gibbs an-
swerec! in principle is this: How does steam know it should
be water if the pressure is high or the temperature is low, and
how does that come out of the molecular model? There are
as many variants of the question as there are substances. A
famous one is the Ising mode! of a ferromagnet, brilliantly
solved by L. Onsager in the two-climensional case. Kac and I.
C. Warc! found a different ant! much simpler derivation
(1952,21. The related "spherical model" invented by Kac was
solved by T. Berlin ~ ~ 952, ~ ).
But to my mind, Kac's most inspiring work in this line is

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MARK KAC
223
contained in the three papers written jointly with P. C.
Hemmer and UhIenbeck (1963,1-3), in which they relater!
the phase transition of a one-dimensional mode} of a gas to
the splitting of the lowest eigenvalue of an allied integral
equation and derived, for that model, the (previously ad hoc)
van der Waal's equation of state, Maxwell's rule of equal areas
included a real tour de force.
PERSONAL APPRECIATION
~ am sure ~ speak for all of Kac's friends when ~ remember
him for his wit, his personal kinkiness, and his scientific style.
One summer when ~ was quite young and at loose ends, ~
went to MIT to stucly mathematics, not really knowing what
that was. ~ hac! the luck to have as my instructor one M. Kac
ant! was enchanted not only by the content of the lectures
but by the person of the lecturer. ~ had never seen mathe-
matics like that nor anybody who could impart such (to me)
difficult material with so much charm.
As ~ understood! more fully later, his attitude toward the
subject was in itself special. Kac was fond of Poincare's dis-
tinction between GocI-given and man-made problems. He
was particularly skillful at pruning away superfluous details
from problems he considered to be of the first kind, leaving
the question in its simplest interesting form. He mistrusted
as insufficiently digestecl anything that required fancy tech-
nical machinery to the extent that he would sometimes in-
sist on clumsy but elementary methods. ~ user! to kic! him
that he tract made a career of noting with mock surprise that
ex = ~ + x + x2/2 + etc. when the whole thing could have
been done without expanding anything. But he clid wonders
with these sometimes awkward tools. Indeed, he loved com-
putation (Desperazionsmatematik includecI) and was a prodi-
gious, if secret, calculator all his life.
~ cannot close this section without a Kac story to illustrate

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224
BIOGRAPHICAL MEMOIRS
his wit and kindliness. Such stories are innumerable, but ~
reproduce here a favorite Kac himself recorded in his auto-
biography:
"The candidate [at an oral examination] was not terribly god in math-
ematics at least. After he had failed a couple of questions, I asked him a
really simple one . . . to describe the behavior of the function 1/z in the
complex plane. 'The function is analytic, sir, except at z = 0, where it has
a singularity,' he answered, and it was perfectly correct. 'What is the sin-
gularity called?' I continued. The student stopped in his tracks. 'Look at
me,' I said. 'What am I?' His face lit up. 'A simple Pole, sir,' which was the
correct answer." ( 1985,1, p. 126)

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MARK KAC
HONORS, PRIZES, AND SERVICE
1950 Chauvenet Prize, Mathematical Association of
America
1959 American Academy of Sciences
1963 Lorentz Visiting Professor, Leiden
1965 National Academy of Sciences
1965-1966 Vice President, American Mathematical Society
1966-1967 Chairman, Division of Mathematical Sciences,
National Research Council
Chauvenet Prize, Mathematical Association of
America
Nordita Visitor, Trondheim
Visiting Fellow, Brasenose College, Oxford
American Philosophical Society
Royal Norwegian Academy of Sciences
Solvay Lecturer, Brussels
Alfred Jurzykowski Award
G. D. Birkhoff Prize, American Mathematical Society
Kramers Professor, Utrecht
Fermi Lecturer, Scuola Normale, Pisa
968
968
969
969
969
971
976
978
980
980
225

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226
BIOGRAPHICAL MEMOIRS
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With H. Steinhaus. Sur les fonctions independantes II. Stud.
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1937
With H. Steinhaus. Sur les fonctions independantes IV. Stud.
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Quelques remarques sur les zeros des integrales de Fourier. J. Lon-
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1939
Note on power series with big gaps. Am. J. Math., 61:473-76.
On a characterization of the normal distribution. Am. J. Math.,
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MARK KAC
227
With E. R. van Kampen. Circular equidistribution and statistical
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With E. R. van Kampen and A. Wintner. On Buffon's needle prob-
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On a problem concerning probability and its connection with the
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1941
With R. P. Agnew. Translated functions and statistical indepen-
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Note on the distribution of values of the arithmetic function dime.
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Two number theoretic remarks. Rev. Ciencias, 43: 177-82.
1942
Note on the partial sums of the exponential series. Rev. Univ. Nac.
Tucuman Ser. A., 3:151-53.
1943
On the average number of real roots of a random algebraic equa-
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On the distribution of values of trigonometric sums with linearly
independent frequencies. Am. J. Math., 65:609-15.
Convergence of certain gap series. Ann. Math., 44~2~:411-15.

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1944
With Henry Hurwitz,3r. Statistical analysis of certain types of ran-
dom functions. Ann. Math. Stat., 15: 173 -81.
1945
Random walk in the presence of absorbing barriers. Ann. Math.
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With R. P. Boas, Jr. Inequalities for Fourier transforms of positive
functions. Duke Math. I., 12: 189-206.
A remark on independence of linear and quadratic forms in-
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16:400-1.
1946
On the distribution of values of sums of the type It. Ann.
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With A. J. F. Siegert. On the theory of random noise in radio re-
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With T. H. Berlin. The spherical model of a ferromagnet. Phys.
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