On 28 July 1914 Austria attacked Serbia and Russia began to mobilise against Austro-Hungary. Two days later Russia mobilised against Germany who in turn declared war against Russia on 1 August then two days later against France. Austro-Hungary declared war against Russia on 5 August. Many more declarations of war followed. The Austro-Hungarian army was incompetently led and it suffered great losses in the first nine months of the war with over two million soldiers being killed. Badly overstretched with fighting on three fronts, Russia, Romania and Italy, the Austro-Hungarian army recruited new conscripts in 1915. At this stage Radó enlisted as a lieutenant in the Austro-Hungarian army and was sent to the Russian front.
The Russians had been pushed back in 1915 but in 1916, led by General Brusilov, they gained victories over the Austro-Hungarian army. Around 600000 men in the Austro-Hungarian army were killed or captured in the 1916 Russian offensive and Radó was one of those taken prisoner by the Russians. He was sent to a prison camp near Tobolsk in Siberia where he spent most of the following four years. Of course the Revolution in Russia in 1917 changed the nature of the country and its relationship with the Allies for it withdrew from the war with other European countries. When World War I ended towards the end of 1918 Russia was in the midst of a violent internal conflict.
In the camp near Tobolsk Radó had met a fellow prisoner Eduard Helly. After being shot, Helly had been captured by the Russians towards the end of 1915 and after a spell in hospital was by then in the same prison camp Radó. However unlike Radó, who had only just begun his university studies, Helly was already a research mathematician who had made remarkable progress in his work on functional analysis, proving the Hahn-Banach theorem in 1912. In the prison camp Helly acted as mathematics teacher to Radó who was also able to read books on mathematics.
Siberia was a difficult place to be in 1919. It was a stronghold of the White Russian forces which had gathered there. There was also a Czech army consisting of around 50000 escaped prisoners who joined the White Russians. The Japanese saw a chance of expansion and landed a large force in Siberia, while the Americans also landed forces there. None of this had a significant effect on the Russian Revolution but it did mean that Radó remained a prisoner in the midst of bitter conflict. At the end of the War, his own country of Hungary suffered severely for being on the losing side in World War I. It declared its independence from Austria but the Allies severely reduced its size with Romania taking much of its former land and 750000 Magyars found themselves in Czechoslovakia. Eventually in the confusion that was taking place in Siberia Radó was able to escape from the prison camp but returning to his much reduced country forced him to make a most remarkable journey.
Escaping from the prison camp near Tobolsk, Radó made his way north to the Arctic regions of Russia. There Eskimos befriended him and offered him hospitality as he slowly made his way westwards. After a trek of many thousands of miles Radó reached Hungary in 1920. It was five years since he had been studying as a student in Budapest, but now he returned to his studies, this time at University of Szeged. Helly had shown him the fascination of research level mathematics so now it was mathematics rather than civil engineering that he studied.
At University of Szeged his teachers included Alfréd Haar and Frigyes Riesz and, with an interest in analysis coming from his contacts with Helly, he undertook research under Riesz's supervision. His first paper On the roots of algebraic equations was published in 1921 and in the following year he published his first paper on conformal mappings. He wrote his thesis in 1922 and was awarded his doctorate in the following year, remaining at Szeged as an assistant and Privatdozent. Publications came thick and fast; he published three papers in 1923, a further five papers in 1924, four in 1925 and five in 1926. This means that he had written around 20 papers within five years of starting his formal mathematical education, which is a quite remarkable achievement. There was no sign that this outpouring of mathematics was diminishing either as he continued to produce papers at the same rate over the next few years.
Let us mention in particular the paper from this period, namely Über den Begriff der Riemannschen Fläche Ⓣ which Radó published in 1925. It gives necessary and sufficient conditions for the triangulability of topological surfaces and in proving these results he completed work on a problem which had been studied by some of the most famous of mathematicians Riemann, Poincaré, and Weyl. Also during this period Radó married Ida Barabás de Albis on 30 September 1924. They had two children, Judith Viola Radó and Theodore Alexander Radó.
The Rockefeller Foundation awarded Radó a fellowship to enable him to spend 1928 working in Germany; part of the year being spent with Carathéodory in Munich and part with Koebe and Lichtenstein in Leipzig. The following year saw Radó visit the United States where he was a visiting lecturer at Harvard University and Rice University. Then in 1930 he was appointed to the faculty of Ohio State University at Columbus where he established the graduate mathematics programme. He remained on the staff at Ohio State University from this time until his retirement in 1964.
In 1930 Radó published the work for which he is most famous, namely his solution to the Plateau Problem. Let us explain a little what the problem is about. Plateau was a physicist who experimented with dipping thin wire frames into a soap solution and examining the soap film which was then stretched across the wire. Because the soap film is extremely light, gravity can be ignored and the soap film forms what is called a minimal surface, that is a surface of minimal area. Any deformation of the surface, such as might be produced by blowing on the film, increases the area of the surface. In fact Plateau did not just examine soap films experimentally, but he also formulated the Plateau problem mathematically. Plateau's mathematical and physical experiments all suggested that for any bounding contour there is always a minimal surface bounded by that contour.
The problem of dealing with a general bounding contour proved extremely difficult. If the bounding contour was planar the problem was easily solved, but the bounding contour might be a complicated curve, even being knotted. Initial progress was made on the problem with proof of the existence of a minimal surface for certain less complicated contours. Schwarz made an important contribution in 1865 as did Riemann at about the same time. Next to contribute were Weierstrass and Darboux but even when he made his contribution in 1914 Darboux wrote about the complexities of the general case:-
Thus far mathematical analysis has not been able to envisage any general method which would permit us to begin the study of this beautiful question.Garnier made a major breakthrough in 1928 followed soon after by independent solutions to the general problem by Douglas and by Radó. Their approaches were very different, Radó's being via conformal mappings. He used conformal mappings of polyhedra, applying a limit theorem to certain approximations to obtain the minimal surface required. His solution appeared 1930 in The problem of least area and the problem of Plateau published in Mathematische Zeitschrift. He published further papers on the Plateau problem in 1930, namely Some remarks on the problem of Plateau and On Plateau's problem. In 1933 he published his impressive text On the Problem of Plateau (reprinted in 1951 and again in 1971) and another major monograph Subharmonic Functions appeared in 1937. Let us remark that the solution to the Plateau problem by both Douglas and by Radó did not exclude the possibility that the minimal surface contained a singularity. In fact it never contains a singularity and this was shown for the first time by Osserman in 1970.
Radó spent 1942 as a visiting professor at the University of Chicago. Of course this was a time when many mathematicians were involved in war related research but he was not involved as this stage. Towards the end of the war, however, he did work for the United States Government as a science consultant to the armed forces. In this capacity he went to Germany at the end of the war to recruit the German scientists which the United States wanted to further its nuclear programme. It was in 1945, the year that the war ended, that Radó was invited to be the American Mathematical Society Colloquium Lecturer. He gave his series of talks on his major contributions on surface area. The lectures he gave at this time formed the basis of his major text Length and Area which was published by the American Mathematical Society in 1948. The book:-
... gives a systematic and detailed exposition of most of the contributions to the theory of the Lebesgue area which have been made by C B Morrey, T Radó and Radó's students.The end of the war also marked a time when Radó took on the role as chairman of the Department of Mathematics of Ohio State University, a position he held from 1946 to 1948. He was invited to speak at the International Congress of Mathematicians in 1950 in Cambridge, Massachusetts and he chose a similar theme to his American Mathematical Society Colloquium Lectures, lecturing on Applications of area theory in analysis.
In 1949 Radó and Reichelderfer announced in their paper On n-dimensional concepts of bounded variation, absolute continuity, and generalized Jacobian their intention to:-
... develop a theory of bounded variation, absolute continuity, and generalized Jacobians which should be comparable in utility and scope with the classical one-dimensional theory.Their theory was fully explained in the important monograph Continuous transformations in analysis : With an introduction to algebraic topology published in 1955.
The Mathematical Association of America invited Radó to give the first Earle Raymond Hedrick Lectures at its meeting in the summer of 1952. In the following year he was elected as vice-president of the American Association for the Advancement of Science. Around this time he lectured on The Mathematical Theory of Rigid Surfaces: An Application of Modern Analysis at a conference at the University of North Carolina sponsored by the National Science Foundation. Among a large number of contributions to the mathematical life of the United States, he served as an editor of the American Journal of Mathematics.
In the last decade of his life, Radó's interests turned to a new area when he found a fascination in the mathematics behind the newly developing ideas in computer science. He worked on logic and theoretical computer science, particularly Turing machines, publishing On non-computable functions in 1962 and Computer studies of Turing machine problems in 1965.
In fact this final publication appeared after he retired in 1964. Sadly it was a short retirement for, after a long illness, he died in New Smyrna Beach, Florida in the following year. He was buried in Bellview Memorial Park, Daytona Beach, Florida.
Article by: J J O'Connor and E F Robertson
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