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Loo-Keng Hua

Hua might well have remained in England longer, but home was never far from his thoughts and the Japanese invasion of China in 1937 caused him much anxiety. He left Cambridge in 1938 to return to his old university, now as a full professor. However, Quing Hua University was no longer in Beijing; with vast portions of China under Japanese occupation, it had migrated to Kunming, the capital of the southern province of Yunan, where it combined with several other institutions to form the temporary Associated University of the South West. There Hua and his family remained through the World War II years, until 1945, in circumstances of poverty, physical privation, and intellectual isolation. Despite these hardships Hua maintained the level of intensity of his Cambridge period and even exceeded it; by the end of 1945 he had more than 70 publications to his name. During this time he studied Vinogradov's seminal method of estimating trigonometric sums and reformulated it, even in sharper form, in what is now known universally as Vinogradov's mean value theorem. This famous result is central to improved versions of the Hilbert-Waring theorem, and has important applications to the study of the Riemann zeta function. Hua wrote up this work in a booklet that was accepted for publication in Russia as early as 1940, but owing to the war, did not appear (in expanded form) until 1947 as a monograph of the Steklov Institute.

Hua spent three months in Russia in the spring of 1946 at Vinogradov's invitation. Mathematical interaction apart, he was impressed by the organization of scientific activity there, and this experience influenced him when later he reached a position of authority in the new China. In the years ahead, even though Hua's scientific activities branched out in other directions, Hua was always ready to return to Waring's problem, to number theory in general and especially to questions involving exponential sums; thus as late as 1959 he published an important monograph on Exponential Sums and Their Applications in Number Theory for the Enzyklopädie der Matematischen Wissenschaften. His instinct for what was important and his marvellous command of technique make his papers on number theory even now virtually an index to the major activities in that subject during the first half of the twentieth century.

In the closing years of the Kunming period Hua turned his interests to algebra and to analysis, as much as anything for the benefit of his students in the first instance, and soon began to make original contributions in these subjects too. Thus Hua became interested in matrix algebra and wrote several substantial papers on the geometry of matrices. He had been invited to visit the Institute for Advanced Study in Princeton, but because C L Siegel was working there along somewhat similar lines, Hua declined, at first in order to develop his ideas independently. In September 1946, shortly after returning from Russia, Hua did depart for Princeton, bringing with him projects not only in matrix theory but also in functions of several complex variables and in group theory. At this time civil war was raging in China and it was not easy to travel; therefore, the Chinese authorities assigned Hua the rank of general in his passport for the "convenience of travel."

According to his biographer, Hua's "most significant and rewarding research work" during his stay in the United States was on the topic of skew fields, that is, on (non-commutative) division rings, of which the quaternions are a classic example.

There was much else, of course, to distinguish this last major creative period of his life. Hua wrote several papers with H S Vandiver on the solution of equations in finite fields and with I Reiner on automorphisms of classical groups. Much of his algebraic work later provided the basis for the monograph Classical Groups by Wan Zhe Xian and Hua (published by the Shanghai Scientific Press in Chinese in 1963).

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