Loo-keng Hua

Loo-Keng Hua was one of the top Chinese mathematicians of the twentieth century. He had no formal training, and the only degrees he received were honorary, including a Doctor of Science Degree from the University of Illinois in 1984. Hua was briefly (1948-1950) a mathematics professor at Illinois before returning to China. His work during that time played a major role in the development of Number Theory.

professor Hua in an armchair
Professor Loo-Keng Hua

In 1938 Hua initiated work on what became known as the Goldbach-Waring problem, concerning the writing of integers as sums of powers of primes. Given a natural number k, one asks whether every sufficiently large number can be written as a sum of k-th powers of prime numbers. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Let s(n) denote the minimum possible number of terms in such a sum. Thus s(2) = 4, s(3) = 9, and s(4) = 19. In 2001, it was proved that every sufficiently large odd number can be written as a sum of 21 fifth powers. The general problem remains open today.

While at the Institute for Advanced Study and at the University of Illinois, Hua collaborated with Irving Reiner, publishing three joint papers (1949, 1951, 1952) in the Transactions of the American Math Society. Reiner (at Illinois most of his career) was a superstar in the area of representation theory and these papers are significant. Current Illinois number theorist Kevin Ford suggests that Hua’s best research achievement from his time at Illinois from 1948-50 is his paper providing a great improvement to Vinogradov’s mean value theorem. Further major results on this problem were obtained in a paper in 2016 by Fields medalist Jean Bourgain, Ciprian Demeter (a grad alumnus of Illinois), and Larry Guth.

Hua’s work in Number Theory has inspired much additional research, including considerable work by mathematicians affiliated with the University of Illinois. Loo-Keng Hua died in 1985.

Among the artifacts possessed by art collector Yu Wang are letters between Hua and the Department of Mathematics at Illinois from 1949. Hua requested a small grant to help prepare him for publishing his book Analytic Theory of Numbers. On Oct. 7, 1949, Hua wrote his request letter to Dean Louis N. Rindenour. Three days later Stewart Cairns, then Head of the Department of Mathematics at Illinois, wrote to Dean Rindenour in support of Hua’s request. On Oct. 27, 1949 the Research Board wrote to Hua and appropriated $200 for clerical assistance in the preparation of Analytic Theory of Numbers. ($200 from 1949 is approximately $2400 in 2022.) This grant from Illinois played a crucial role in the dissemination of Hua’s work.

Emeritus Professor John P. D’Angelo met Hua on several occasions. Hua also contributed to the field of Several Complex Variables. Hua and his most important collaborator in that part of mathematics, Liu Qi Keng, organized one of the first international mathematical conferences to be held in China. They were assisted from afar by Joseph Kohn (Princeton) and Yum-Tong Siu (Harvard).

The meeting was held September, 1981 at the beautiful West Lake Hotel in Hangzhou, and it was considered important enough that glimpses of the meeting were shown on Chinese television. As a young mathematician, D’Angelo was blessed to attend this meeting and meet both Hua and Liu. The meeting was scheduled for September, in order that foreign participants could observe the yearly tidal bore, where a big wave crashes the wrong direction on the Qiantang River. The participants also went on a tour of Shanghai, Nanjing, and Beijing after the meeting. The overall experience was fantastic.

Three years later, Heine Halberstam, then Head of the mathematics department at UIUC, spearheaded the awarding to Hua of an honorary degree from Illinois. Hua, who was then confined to a wheel chair, stayed in the Illini Union, with a large entourage. D’Angelo arranged to meet with Hua; the two of them talked about the influence of the Hangzhou conference and complex analysis in general. By then Hua’s interest was in Economics, and he spoke of this work in strong terms, for both its intellectual and practical value. Certainly the Chinese government followed some of his suggestions.

One of Hua’s nineteen books is called Starting with the Unit Circle. From an elementary beginning it brilliantly bridges the fields of Number Theory and Complex Analysis in Several Variables, the subject of the Hangzhou conference.

 

  • Altgeld Hall – Altgeld Hall is the current home of the Department of Mathematics, where Loo-keng Hua worked.

 

MathSciNet lists 166 publications of Hua, which together have been cited more than 2000 times. The reviews of these papers indicate Hua’s great originality and technical power.

To learn more about Hua’s work on the improvement of Vinogradov’s mean value theorem see: See Hua’s paper: An improvement of Vinogradov’s mean-value theorem and several applications, Quart. J. Math. Oxford Ser. 20 (1949), 48–61.

An excellent place to read about Hua’s contributions to Number Theory is his obituary, written by Heine Halberstam, Head of the Dept. of Mathematics at Illinois during the 1980’s. Halberstam, H. Loo-keng Hua: obituaryActa Arith. 51 (1988), no. 2, 99–117.

Perhaps the best source is Hua’s collected works, edited by Heine Halberstam. In the preface Halberstam wrote “The unexpected arrival of Loo-Keng Hua in Europe in the fall of 1978 was for many of us a romantic event, a legend come to life.” See: Loo-Keng Hua Selected Papers, Edited by H. Halberstam, Springer-Verlag, 1982.

See  https://mathshistory.st-andrews.ac.uk/Biographies/Hua/ for additional bibliographic information about Professor Hua.

Photos of Letters courtesy of Ian Wang.

 

Contributors: Dr. John P. D'Angelo