Martin David Kruskal (; September 28, 1925 – December 26, 2006)
was an American
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 ...
and
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
. He made fundamental contributions in many areas of mathematics and science, ranging from
plasma physics
Plasma () is a state of matter characterized by the presence of a significant portion of charged particles in any combination of ions or electrons. It is the most abundant form of ordinary matter in the universe, mostly in stars (including th ...
to
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and from
nonlinear analysis to
asymptotic analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing Limit (mathematics), limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very larg ...
. His most celebrated contribution was in the theory of
solitons
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such locali ...
.
He was a student at the
University of Chicago
The University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...
and at
New York University
New York University (NYU) is a private university, private research university in New York City, New York, United States. Chartered in 1831 by the New York State Legislature, NYU was founded in 1832 by Albert Gallatin as a Nondenominational ...
, where he completed his Ph.D. under
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German-American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
in 1952. He spent much of his career at
Princeton University
Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
, as a research scientist at the Plasma Physics Laboratory starting in 1951, and then as a professor of astronomy (1961), founder and chair of the Program in Applied and Computational Mathematics (1968), and professor of mathematics (1979). He retired from
Princeton University
Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
in 1989 and joined the mathematics department of
Rutgers University
Rutgers University ( ), officially Rutgers, The State University of New Jersey, is a Public university, public land-grant research university consisting of three campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's C ...
, holding the David Hilbert Chair of Mathematics.
Apart from serious mathematical work, Kruskal was known for mathematical diversions. For example, he invented the
Kruskal count, a magical effect that has been known to perplex professional magicians because it was based not on sleight of hand but on a mathematical phenomenon.
Personal life
Martin David Kruskal was born to a
Jewish
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
family 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 ...
and grew up in
New Rochelle. He was generally known as Martin to the world and David to his family. His father, Joseph B. Kruskal Sr., was a successful fur wholesaler. His mother,
Lillian Rose Vorhaus Kruskal Oppenheimer, became a noted promoter of the art of
origami
) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a ...
during the early era of television and founded the Origami Center of America in New York City, which later became OrigamiUSA. He was one of five children. His two brothers, both eminent mathematicians, were
Joseph Kruskal
Joseph Bernard Kruskal, Jr. (; January 29, 1928 – September 19, 2010) was an American mathematician, statistician, computer scientist and psychometrician.
Personal life
Kruskal was born to a Jewish family in New York City to a successful fu ...
(1928–2010; discoverer of
multidimensional scaling
Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n objects in a set into a configuration of n points mapped into an ...
, the
Kruskal tree theorem, and
Kruskal's algorithm
Kruskal's algorithm finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that ...
) and
William Kruskal (1919–2005; discoverer of the
Kruskal–Wallis test).
Martin Kruskal's wife, Laura Kruskal, was a lecturer and writer about origami and originator of many new models. They were married for 56 years. Martin Kruskal also invented several origami models including an envelope for sending secret messages. The envelope could be easily unfolded, but it could not then be easily refolded to conceal the deed. Their three children are Karen (an attorney), Kerry (an author of children's books), and
Clyde, a computer scientist.
Research
Martin Kruskal's scientific interests covered a wide range of topics in pure mathematics and applications of mathematics to the sciences. He had lifelong interests in many topics in
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s and
nonlinear analysis and developed fundamental ideas about
asymptotic expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
s,
adiabatic invariant
A property of a physical system, such as the entropy of a gas, that stays approximately constant when changes occur slowly is called an adiabatic invariant. By this it is meant that if a system is varied between two end points, as the time for the ...
s, and numerous related topics.
His Ph.D. dissertation, written under the direction of
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German-American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
and Bernard Friedman at
New York University
New York University (NYU) is a private university, private research university in New York City, New York, United States. Chartered in 1831 by the New York State Legislature, NYU was founded in 1832 by Albert Gallatin as a Nondenominational ...
, was on the topic "The Bridge Theorem For
Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s". He received his Ph.D. in 1952.
In the 1950s and early 1960s, he worked largely on plasma physics, developing many ideas that are now fundamental in the field. His theory of adiabatic invariants was important in fusion research. Important concepts of plasma physics that bear his name include the
Kruskal–Shafranov instability and the
Bernstein–Greene–Kruskal (BGK) modes. With I. B. Bernstein, E. A. Frieman, and R. M. Kulsrud, he developed the MHD (or
magnetohydrodynamic) Energy Principle. His interests extended to plasma astrophysics as well as laboratory plasmas.
In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
in
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. A spherically symmetric spacetime can be described by the
Schwarzschild solution, which was discovered in the early days of general relativity. However, in its original form, this solution only describes the region exterior to the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive c ...
of the black hole. Kruskal (in parallel with
George Szekeres) discovered the maximal analytic continuation of the Schwarzschild solution, which he exhibited elegantly using what are now called
Kruskal–Szekeres coordinates.
This led Kruskal to the astonishing discovery that the interior of the black hole looks like a "
wormhole
A wormhole is a hypothetical structure that connects disparate points in spacetime. It can be visualized as a tunnel with two ends at separate points in spacetime (i.e., different locations, different points in time, or both). Wormholes are base ...
" connecting two identical,
asymptotically flat universes. This was the first real example of a wormhole solution in general relativity. The wormhole collapses to a singularity before any observer or signal can travel from one universe to the other. This is now believed to be the general fate of wormholes in general relativity. In the 1970s, when the
thermal nature of black hole physics was discovered, the wormhole property of the Schwarzschild solution turned out to be an important ingredient. Nowadays, it is considered a fundamental clue in attempts to understand
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
.
Kruskal's most widely known work was the discovery in the 1960s of the
integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time. These developments began with a pioneering computer simulation by Kruskal and
Norman Zabusky (with some assistance from
Harry Dym) of a nonlinear equation known as the
Korteweg–de Vries equation (KdV). The KdV equation is an asymptotic model of the propagation of nonlinear
dispersive waves. But Kruskal and Zabusky made the startling discovery of a "solitary wave" solution of the KdV equation that propagates non-dispersively and even regains its shape after a collision with other such waves. Because of the particle-like properties of such a wave, they named it a "
soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
", a term that caught on almost immediately.
This work was partly motivated by the near-
recurrence paradox that had been observed in a very early computer simulation of a
certain nonlinear lattice by
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian and naturalized American physicist, renowned for being the creator of the world's first artificial nuclear reactor, the Chicago Pile-1, and a member of the Manhattan Project ...
,
John Pasta,
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, ...
and
Mary Tsingou at
Los Alamos in 1955. Those authors had observed long-time nearly recurrent behavior of a one-dimensional chain of anharmonic oscillators, in contrast to the rapid thermalization that had been expected. Kruskal and Zabusky simulated the KdV equation, which Kruskal had obtained as a
continuum limit of that one-dimensional chain, and found solitonic behavior, which is the opposite of thermalization. That turned out to be the heart of the phenomenon.
Solitary wave phenomena had been a 19th-century mystery dating back to work by
John Scott Russell
John Scott Russell (9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventnor, Isle of Wight) was a Scottish civil engineer, naval architecture, naval architect and shipbuilder who built ''SS Great Eastern, Great Eastern'' in collaboration with Is ...
who, in 1834, observed what we now call a soliton, propagating in a canal, and chased it on horseback. In spite of his observations of solitons in wave tank experiments, Scott Russell never recognized them as such, because of his focus on the "great wave of translation," the largest amplitude solitary wave. His experimental observations, presented in his Report on Waves to the British Association for the Advancement of Science in 1844, were viewed with skepticism by
George Airy
Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, as well as the Lucasian Professor of Mathematics from 1826 to 1828 and the seventh Astronomer Royal from 1835 to 1881. His many achievements inc ...
and
George Stokes because their linear water wave theories were unable to explain them.
Joseph Boussinesq (1871) and
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh ( ; 12 November 1842 – 30 June 1919), was an English physicist who received the Nobel Prize in Physics in 1904 "for his investigations of the densities of the most important gases and for his discovery ...
(1876) published mathematical theories justifying Scott Russell's observations. In 1895,
Diederik Korteweg and
Gustav de Vries formulated the KdV equation to describe shallow water waves (such as the waves in the canal observed by Russell), but the essential properties of this equation were not understood until the work of Kruskal and his collaborators in the 1960s.
Solitonic behavior suggested that the KdV equation must have conservation laws beyond the obvious conservation laws of mass, energy, and momentum. A fourth conservation law was discovered by
Gerald Whitham and a fifth one by Kruskal and Zabusky. Several new conservation laws were discovered by hand by
Robert Miura, who also showed that many conservation laws existed for a related equation known as the Modified Korteweg–de Vries (MKdV) equation. With these conservation laws, Miura showed a connection (called the Miura transformation) between solutions of the KdV and MKdV equations. This was a clue that enabled Kruskal, with
Clifford S. Gardner,
John M. Greene, and Miura (GGKM), to discover a general technique for exact solution of the KdV equation and understanding of its conservation laws. This was the
inverse scattering method, a surprising and elegant method that demonstrates that the KdV equation admits an infinite number of
Poisson-commuting conserved quantities and is completely integrable. This discovery gave the modern basis for understanding of the soliton phenomenon: the solitary wave is recreated in the outgoing state because this is the only way to satisfy all of the conservation laws. Soon after GGKM,
Peter Lax famously interpreted the inverse scattering method in terms of
isospectral deformations and
Lax pairs.
The inverse scattering method has had an astonishing variety of generalizations and applications in different areas of mathematics and physics. Kruskal himself pioneered some of the generalizations, such as the existence of infinitely many conserved quantities for the
sine-Gordon equation
The sine-Gordon equation is a second-order nonlinear partial differential equation for a function \varphi dependent on two variables typically denoted x and t, involving the wave operator and the sine of \varphi.
It was originally introduced by ...
. This led to the discovery of an inverse scattering method for that equation by M. J.
Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur (AKNS). The sine-Gordon equation is a relativistic wave equation in 1+1 dimensions that also exhibits the soliton phenomenon and which became an important model of solvable
relativistic field theory. In seminal work preceding AKNS,
Zakharov and Shabat discovered an inverse scattering method for the nonlinear Schrödinger equation.
Solitons are now known to be ubiquitous in nature, from physics to biology. In 1986, Kruskal and Zabusky shared the
Howard N. Potts Gold Medal from the Franklin Institute "for contributions to mathematical physics and early creative combinations of analysis and computation, but most especially for seminal work in the properties of solitons". In awarding the 2006 Steele Prize to Gardner, Greene, Kruskal, and Miura, the American Mathematical Society stated that before their work "there was no general theory for the exact solution of any important class of nonlinear differential equations". The AMS added, "In applications of mathematics, solitons and their descendants (kinks, anti-kinks, instantons, and breathers) have entered and changed such diverse fields as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences. Nonlinearity has undergone a revolution: from a nuisance to be eliminated, to a new tool to be exploited."
Kruskal received the
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral science, behavior ...
in 1993 "for his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution".
In an article surveying the state of mathematics at the turn of the millennium, the eminent mathematician Philip A. Griffiths wrote that the discovery of integrability of the KdV equation "exhibited in the most beautiful way the unity of mathematics. It involved developments in computation, and in mathematical analysis, which is the traditional way to study differential equations. It turns out that one can understand the solutions to these differential equations through certain very elegant constructions in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. The solutions are also intimately related to
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, in that these equations turn out to have an infinite number of hidden symmetries. Finally, they relate back to problems in elementary geometry."
In the 1980s, Kruskal developed an acute interest in the
Painlevé equations. They frequently arise as symmetry reductions of soliton equations, and Kruskal was intrigued by the intimate relationship that appeared to exist between the properties characterizing these equations and completely integrable systems. Much of his subsequent research was driven by a desire to understand this relationship and to develop new direct and simple methods for studying the Painlevé equations. Kruskal was rarely satisfied with the standard approaches to differential equations.
The six
Painlevé equations have a characteristic property called the Painlevé property: their solutions are single-valued around all singularities whose locations depend on the initial conditions. In Kruskal's opinion, since this property defines the Painlevé equations, one should be able to start with this, without any additional unnecessary structures, to work out all the required information about their solutions. The first result was an asymptotic study of the Painlevé equations with
Nalini Joshi, unusual at the time in that it did not require the use of associated linear problems. His persistent questioning of classical results led to a direct and simple method, also developed with Joshi, to prove the Painlevé property of the Painlevé equations.
In the later part of his career, one of Kruskal's chief interests was the theory of
surreal number
In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...
s. Surreal numbers, which are defined constructively, have all the basic properties and operations of the real numbers. They include the real numbers alongside many types of infinities and infinitesimals. Kruskal contributed to the foundation of the theory, to defining surreal functions, and to analyzing their structure. He discovered a remarkable link between surreal numbers, asymptotics, and exponential asymptotics. A major open question, raised by Conway, Kruskal and Norton in the late 1970s, and investigated by Kruskal with great tenacity, is whether sufficiently well behaved surreal functions possess definite integrals. This question was answered negatively in the full generality, for which Conway et al. had hoped, by Costin, Friedman and Ehrlich in 2015. However, the analysis of Costin et al. shows that definite integrals do exist for a sufficiently broad class of surreal functions for which Kruskal's vision of asymptotic analysis, broadly conceived, goes through. At the time of his death, Kruskal was in the process of writing a book on surreal analysis with O. Costin.
Kruskal coined the term
asymptotology to describe the "art of dealing with applied mathematical systems in limiting cases". He formulated seven Principles of Asymptotology: 1. The Principle of Simplification; 2. The Principle of Recursion; 3. The Principle of Interpretation; 4. The Principle of Wild Behaviour; 5. The Principle of Annihilation; 6. The Principle of Maximal Balance; 7. The Principle of Mathematical Nonsense.
The term asymptotology is not so widely used as the term
soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
. Asymptotic methods of various types have been successfully used since almost the birth of science itself. Nevertheless, Kruskal tried to show that asymptotology is a special branch of knowledge, intermediate, in some sense, between science and art. His proposal has been found to be very fruitful.
[Dewar R. L. Asymptotology – a cautionary tale. ANZIAM J., 2002, 44, 33–40.]
Awards and honors
Kruskal's honors and awards included:
* Gibbs Lecturer,
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
(1979);
*
Dannie Heineman Prize,
American Physical Society
The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of ...
(1983);
*
Howard N. Potts Gold Medal, Franklin Institute (1986);
* Award in Applied Mathematics and Numerical Analysis, National Academy of Sciences (1989);
* National Medal Of Science (1993);
* John von Neumann Lectureship, SIAM (1994);
* Honorary DSc,
Heriot–Watt University
Heriot-Watt University () is a Public university, public research university based in Edinburgh, Scotland. It was established in 1821 as the School of Arts of Edinburgh, the world's first mechanics' institute, and was subsequently granted univers ...
(2000);
* Maxwell Prize, Council For Industrial And Applied Mathematics (2003);
*
Steele Prize, American Mathematical Society (2006)
* Member of the National Academy of Sciences (1980) and the American Academy of Arts and Sciences (1983)
* Elected a
Foreign Member of the Royal Society (ForMemRS) in 1997
* Elected Foreign Member of the
Russian Academy of Sciences
The Russian Academy of Sciences (RAS; ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across the Russian Federation; and additional scientific and social units such ...
(2000)
* Elected a
Fellow of the Royal Society of Edinburgh
Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently distinguished in their subject". ...
(2001)
See also
*
Kruskal count in recreational mathematics
References
External links
In Memoriam: Martin David Kruskal*
Solitons, Singularities, Surreals and Such: A Conference in Honor of Martin Kruskal's Eightieth Birthday*
ttps://www.princeton.edu/pr/pwb/07/0205/2a.shtml Princeton University Weekly Bulletin Obituary, 2007-02-05Chris Eilbeck/Heriot–Watt University, Edinburgh, UKObituaries – Martin David KruskalSociety for Industrial and Applied Mathematics, 2007-04-11
{{DEFAULTSORT:Kruskal, Martin David
1925 births
2006 deaths
20th-century American mathematicians
21st-century American mathematicians
20th-century American Jews
Foreign members of the Royal Society
Members of the United States National Academy of Sciences
Fellows of the American Physical Society
National Medal of Science laureates
Courant Institute of Mathematical Sciences alumni
Princeton University faculty
American relativity theorists
Rutgers University faculty
University of Chicago alumni
Scientists from New Rochelle, New York
20th-century American physicists
Howard N. Potts Medal recipients
Jewish American physicists
Mathematicians from New York (state)
Fellows of the Royal Society of Edinburgh
21st-century American Jews
Recreational mathematicians