Print  Font Size: 

Loo-Keng Hua

At this time Quing Hua University was the leading Chinese institution of higher education, and its faculty was in the forefront of the endeavour to bring the country's mathematics and science abreast of knowledge in the West, a formidable task after several hundred years of stagnation. During 1935-36 Hadamard and Norbert Wiener visited the university; Hua eagerly attended the lectures of both and created a good impression. Wiener visited England soon afterward and spoke of Hua to G H Hardy. In this way Hua received an invitation to come to Cambridge, England, and he arrived in 1936 to spend two fruitful years there. By now he had published widely on questions within the orbit of Waring's problem (also on other topics in diophantine analysis and function theory) and he was well prepared to take advantage of the stimulating environment of the Hardy-Littlewood school, then at the zenith of its fame. Hua lived on a $1,250 per annum scholarship awarded by the Culture and Education Foundation of China; it is interesting to recall that this foundation derived its funds from reparations paid by China to the United States following wars waged in China by the United States and several other nations in the previous century. The amount of the grant imposed on him a Spartan regime. Hardy assured Hua that he could gain a PhD in two years with ease, but Hua could not afford the registration fee and declined; of course, he gave quite different reasons for his decision.

During the Cambridge period Hua became friendly with Harold Davenport and Hans Heilbronn, then two young research fellows of Trinity College - one a former student of Littlewood and the other Landau's last assistant in Göttingen - with whom he shared a deep interest in the Hardy-Littlewood approach to additive problems akin to Waring's. They helped to polish the English in several of Hua's papers, which now flowed from his pen at a remarkable rate; more than 10 of his papers date from this time, and many of these appeared in due course in the publications of the London Mathematical Society.

About the only easy thing about Waring's problem is its statement: In 1770 Waring asserted without proof (and not in these words) that for each integer k 2 there exists an integer s = s(k) depending only on k such that every positive integer N can be expressed in the form

N = x1k + x2k + ... +xsk

where the xj(j = 1, 2, ... , s) are non-negative integers. In that same year Lagrange had settled the case k = 2 by showing that s(2) = 4, a best possible result; after that, progress was painfully slow, and it was not until 1909 that Hilbert solved Waring's problem in its full generality. His argument rested on the deployment of intricate algebraic identities and yielded rather poor admissible values of s(k). In 1918 Hardy and Ramanujan returned to the case k = 2 in order to determine the number of representations of an integer as the sum of s squares by means of Fourier analysis, an approach inspired by their famous work on partitions, and they succeeded. This encouraged Hardy and Littlewood in 1920 to apply a similar method for general k, and they devised the so-called circle method to tackle the general Hilbert-Waring theorem and a host of other additive problems, Goldbach's problem among them. During the next 20 years the machinery of the circle method came to be regarded about as difficult as anything in the whole of mathematics; even today, after numerous refinements and much progress, the intricacies of the method remain formidable.

This is the background against which Hua set to work as a young man, and it is probably fair to say that it is for his contributions in this area that Hua's name will remain best remembered: notably for his seminal work on the estimation of trigonometric sums, singly or on average.

To do a quick search, highlighting any word(s) then click Help!