Bulletin Autumn‧Winter 1980

of which have been used before. It is of course not easy to invent new methods, but neither is the second way easy. To be successful in our attempt, we must have a thorough knowledge of all, not just some, the existing methods which may be useful, and be able to bring them to bear on the specific problems. And this process would require a considerable degree of creativity, other wise the open problem would not have remained open for so long. There are in fact many open problems which are of little significance and we are not in the least enthusiastic about their solu tion. We are only concerned with well-known open problems which are significant. Q. Is there a general criterion fo r judging the value o f a mathematical problem? A. Most branches of mathematics are developed with a specific purpose in view. Plane geometry, for example, was first developed thousands of years ago when the Greeks and Egyptians started to survey their land and fields. Plane geometry has had its effects on the physical world as well as on the study of mathematics. It had been the most important branch of mathematics until the sixteenth and seventeenth centuries when it was found to be inadequate and had ceased to con tribute to mathematical thought, rendering further study on it completely valueless. Another thing is that we w ill not continue with our study of a certain subject if it has already been thoroughly investigated and under stood. Again, we can take as example plane geometry, which we consider as a subject thoroughly understood. Any good mathemati cian, as long as he is willing to spend the time, can solve any problem in the field. Although some people are still carrying on research in the field and have in fact developed some new theorems, we would not consider their efforts worthwhile or the theorems of much value. Q . During your last visit to China, in drawing the attention o f Chinese mathematicians to a num ber o f open problems, were you applying this criterion? A. What I raised are problems in modern differen tial geometry, and these may be divided into (1) problems of special significance and (2) simple and beautiful problems. Differential geometry is a study of lines, circles, curves, two dimensional space, three dimensional space . . . as well as the geometry of curved surfaces. And what I mean by problems of special significance may be construed in two ways. First, they are problems on a certain aspect, on which we have little know ledge, and it is our hope that by raising them we may be able to draw the attention of fellow mathematicians to this aspect. Second, they are the problems whose solution we believe may bring about the settlement of other problems in other fields. Q. You have been to China several times on lecture tours. What do you think o f China's progress in mathematics? A. China's development in mathematics w ill be very promising if it can succeed in training the younger generation properly. It is the younger generation that our hopes lie. As far as I know, Chinese graduates in their twenties never spare themselves in their studies. The four Chinese graduate students who went to study mathematics in America this academic year —one each at the University of California, Berkeley, and Stanford University, and two at California University, Stony Brook 一 are doing very well and have demonstrated their research ability. In fact, the standard of university graduates in China is more or less the same as, or even higher than, that of their American counterparts before they enter graduate school. Therefore, given a favourable mathematical environment, free from external interference, they should be able to achieve some good results in the course of three or four years. But there is one thing which worries me: w ill the outstanding students now studying abroad continue to have a favourable environment for their research when they return to China? If China really wants to avoid wasting talent, the best way is to send more students abroad so that on their return, those whose fields of study are related can be sent to the same institution to teach, thus providing them with a chance to exchange ideas and experiences, which is vital in academic research. Q. I t seems that overseas Chinese mathematicians are very concerned about the development o f mathematics in China. Is this true? A. Yes. Even some of my American friends have a real concern for the development of mathematics in China. Being a Chinese living abroad, I do sincerely hope that China w ill make great pro gress in mathematics, and if I can be of help, I shall do my best. 15