Bulletin Autumn‧Winter 1980

proved a theorem', meaning that the methods employed by them are not rigorous enough. But, to be fair to them, we should take into consider ation the fact that they are often faced with some very difficult problems which are too urgent to wait for a rigorous solution. They are thus forced to resort to less rigorous theories and methods. As for scientists in other fields, they, too, use less rigorous methods. Physicists or engineers rely a lot on computers for solution of problems they are unable to work out. They are willing to accept the solution of the computer even though they are aware that the proof of its correctness requires more advanced calculation processes. This practice, unacceptable to most pure mathe maticians, may nevertheless be justifiable from the practical point of view, as it may be the only way to expedite matters. Q. What i f the solutions are found to be erroneous afterwards? A. They may still be able to proceed with their research despite minor errors in the imperfect solutions they have accepted, but as the errors accumulate there may eventually come a point when it is impossible to go on. But, they do not have to wait very long before mathematicians become interested in these problems and attempt their own solution. Generally speaking, we are usually able to rectify these errors in time, before anything serious happens. It can be seen that the work of mathemati cians and that of other scientists are complemen tary to each other. The weakness of scientists lies in not coming up to the standard of rigour required. The weakness of pure mathematicians, on the other hand, lies in the fact that being used to thinking in abstract terms, their minds are sometimes less adapted to practical problems, and that in their relentless desire to satisfy the requirement of rigour, they may end up with proofs which are of no practical value at all. Q. Your solution o f Calabi's conjecture is an impor tant breakthrough in mathematics. What is the significance o f your solution? A. Whether it is a breakthrough depends on how you look at it. From the standpoint of geometry, the solution may be looked upon as a theorem which can be stated simply, a theorem, compre hensible even to students equipped with only a general knowledge of the concept of curvature. The solution of Calabi's conjecture has some practical value as well. Differential geometry is the study of geometric forms and the behaviour of the change of curvature. When I attempted a solution of Calabi's conjecture, I was fully aware that its solution would contribute to a deeper understanding of curvature. It turned out that the solution of this conjecture has a great signi ficance not only for differential geometry itself, but also for other fields such as algebraic geometry and even the theory of relativity. It has also led to the solution of many problems which have been hitherto insoluble. Q. Your other outstanding achievement is the proof o f the positive mass conjecture, which is said to have something to do with the black hole phenomenon. A. Yes, it helps to show that the black hole pheno menon, though extraordinary, is based on many of the basic physical phenomena. This concept should be viewed in the light of the general theory of relativity. The development of the general theory of relativity and differential geometry cannot be separated. The general theory of relativity was first developed by Einstein on the basis of mathematical concepts. What we call time and space is viewed from the standpoint of geometry. Curvature, then, represents gravitation. After 1915, the general theory of relativity, in turn, affected greatly differential geometry. The study of curved space in differential geometry is in timately bound up with the general theory of relativity. About twenty years ago, several British scientists put forth the hypothesis of the black hole, deduced from the concepts of differential geometry. This hypothesis renewed the interest of physicists in the study of the general theory of relativity, using the method of differential geometry. I started my study in positive mass conjecture with a student because in recent years it was discovered that the answers to many of the unsolved problems of general relativity could in fact be found by using the old methods in differential geometry. Q. Some o f the problems you have tried to solve must be open problems ， and being open pro blems, there might be others working on them as well What is the key to success then? A. There are two ways of approaching awell-known open problem. One is by using new methods completely unknown to our predecessors, the other is by employing a barrage of methods all 14