• Physicist Gunnar Nordström, 1881
• Theoretical Physicist Martin David Kruskal, 1925

Nordström graduated high school from Brobergska Skolan in central Helsinki 1899. At first he went on to study mechanical engineering at the Polytechnic institute in Helsinki, later renamed Helsinki University of Technology and today a part of the Aalto University. During his studies he developed an interest for more theoretical subjects, proceeding after graduation to further study for a master's degree in natural science, mathematics and economy at the University of Helsinki (1903 – 1907).

Nordström then moved to Göttingen, Germany, where he had been recommended to go to study physical chemistry. However, he soon lost interest in the intended field and moved to study electrodynamics, a field the University of Göttingen was renowned for at the time. He returned to Finland to complete his doctoral dissertation at the University of Helsinki in 1910, and become a docent at the university. Subsequently he became fascinated with the very novel and soon burgeoning field of gravitation and wanted to move to the Netherlands where scientists with contributions to that field such as Hendrik Lorentz, Paul Ehrenfest and Willem de Sitter were active. Nordström was able to move to Leiden in 1916 to work under Ehrenfest, in the midst of the First World War, due to his Russian passport. Nordström spent considerable time in Leiden where he met a Dutch physics student, Cornelia van Leeuwen, with whom he went on to have several children. After the war he declined a professorship at the University of Berlin, a post awarded instead to Max Born, in order to return to Finland in 1918 and hold at first the professorship of physics and later the professorship of mechanics at the Helsinki University of Technology.

One of the keys to Nordström's success as a scientist was his ability to learn to apply differential geometry to physics, a new approach that also would eventually lead Albert Einstein to the theory of general relativity. Few other scientists of the time in Finland were able to make effective use of this new analytical tool, with the notable exception of Ernst Lindelöf.

During the time in Leiden Nordström solved the field equations for the spherically symmetric charged body, thus extending Hans Reissner's results for a point charge. The metric for a non - rotating charge distribution is nowadays known as the Reissner - Nordström metric. Nordström maintained frequent contact with many of the other great physicists of the era, including Niels Bohr and Albert Einstein. For example, it was Bohr's contributions that helped Nordström to circumvent the Russian censorship of German post to Finland, Finland was at the time a grand duchy of the Russian empire.

The theory for which Nordström was arguably most famous in his own lifetime, his theory of gravitation, was for a long time considered as a competitor to Einstein's theory of general relativity which was published in 1915, after Nordström's theory. In 1914 Nordström introduced an additional space dimension to his theory, which provided coupling to electromagnetism. This was the first of the extra dimensional theories, which later came to be known as Kaluza - Klein theory. Kaluza and Klein, whose names are commonly used today for the theory, did not publish their work until the 1920s. Some speculations as to why Nordström's contribution fell into oblivion are that his theory was partly published in Swedish and that Einstein in a later publication referenced to Kaluza alone. Today extra dimensions and theories thereof are widely researched, debated and even looked for experimentally.

Nordström's theory of gravitation was subsequently experimentally found to be inferior to Einstein's as it did not predict the bending of light which was observed during the solar eclipse in 1919. However, Nordström and Einstein were in friendly competition or by some measure even cooperating scientists, not rivals. This can be seen from Nordström's public admiration of Einstein's work, as demonstrated by the two occasions on which Nordström nominated Einstein for the Nobel prize in physics for his theory of relativity. Einstein never received the Nobel prize for the theory, as the first experimental evidence presented in 1919 could at the time still be disputed and there was not yet a consensus or even general understanding in the scientific community of the complex mathematical models that Einstein, Nordström and others had developed. Nordström's scalar theory is today mainly used as a pedagogical tool when learning general relativity.

Today there is a limited public knowledge of Nordström's contributions to science even in Finland. However, after his death a number of Finnish physicists and mathematicians devoted their time to the theory of relativity and differential geometry presumably due to the legacy he left. On the other hand, the most notable opponent of general relativity in the Finnish scientific world was Hjalmar Mellin, the previous rector of the Helsinki University of Technology where Nordström held a professorship.

Nordström died in December 1923, at the age of 42, from pernicious anemia. The illness was perhaps caused by exposure to radioactive substances. Nordström was known to be interested in radioactivity, and a firm adherent to the then widespread misconception that radioactivity was beneficial for one's health. Among his publications there is one from 1913 regarding the measurement of the radioactive emancipation power of different springs and ground waters in Finland. Nordström was known for experimenting with radioactive substances and for enjoying the Finnish sauna tradition using water from a spring rich in radium.

He was a student at the University of Chicago and at New York University, where he completed his Ph.D. under Richard Courant in 1952. He spent much of his career at Princeton University, 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 in 1989 and joined the mathematics department of Rutgers University, holding the David Hilbert Chair of Mathematics.

Apart from his research, Kruskal was known as a mentor of younger scientists. He worked tirelessly and always aimed not just to prove a result but to understand it thoroughly. And he was notable for his playfulness. He invented the Kruskal Count, a magical effect that has been known to perplex professional magicians because – as he liked to say – it was based not on sleight of hand but on a mathematical phenomenon.

Martin David Kruskal was born in New York City 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 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 (born 1928; discoverer of multidimensional scaling, the Kruskal tree theorem, and Kruskal's algorithm) and William Kruskal (1919 – 2005; discoverer of the Kruskal – Wallis test).

Martin Kruskal was married to Laura Kruskal, his wife of 56 years. Laura was well known as a lecturer and writer about origami and originator of many new models. Martin, who had a great love of games, puzzles and word play of all kinds, also invented several quite unusual origami models including an envelope for sending secret messages (anyone who unfolded the envelope to read the message would have great difficulty refolding it to conceal the deed).

Martin and Laura traveled extensively to scientific meetings and to visit Martin’s many scientific collaborators. Laura used to call Martin “my ticket to the world.” Wherever they went, Martin would be hard at work and Laura would often keep busy teaching origami workshops in schools and institutions for elderly people and people with disabilities. Martin and Laura had a great love of traveling and hiking.

Their three children are Karen, Kerry, and Clyde, who are known respectively as an attorney, an author of children’s books, and a mathematician.

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 equations and nonlinear analysis and developed fundamental ideas about asymptotic expansions, adiabatic invariants and numerous related topics.

His Ph.D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic “The Bridge Theorem For Minimal Surfaces.” 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. Martin Kruskal's work in plasma physics is considered by some to be his most outstanding.

In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of black hole in General Relativity. A spherically symmetric black hole 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 horizon 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” 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.

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 Gary Deem) 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 nondispersively 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,” 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 nonlinear lattice by Enrico Fermi, John Pasta and Stanislaw Ulam, 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 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 and George Stokes because their linear water wave theories were unable to explain them. Joseph Boussinesq (1871) and Lord Rayleigh (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, 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.

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. 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. 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.

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 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. The solutions are also intimately related to representation theory, 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 numbers. 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 is whether sufficiently well behaved surreal functions possess definite integrals. A positive answer would have deep implications in analysis. Over the last years of his life, Kruskal worked with much dedication on defining surreal integration. Much was learned but important parts of this project remain unfinished. At the time of his death, he 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 Behavior; 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. 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.