Basic Philosophy: an Old Chinese Poem Says
نویسندگان
چکیده
This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S. S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I [724] worked on fundamental groups of manifolds with non-positive curvature. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much interested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi's works on affine geometry in 1972 at Berkeley, S. Y. Cheng told me about these inspiring lectures. By 1973, Cheng and I started to work on some problems mentioned in Chern's lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [556] published his results right before us by different arguments. Nevertheless our ideas are useful in handling other problems in affine geometry, and my knowledge about Monge-Ampère equations started to broaden in these years. Chern was very pleased by my work, especially after I [729] solved the problem of Calabi on Kähler Einstein metric in 1976. I had been at Stanford, and Chern proposed to the Berkeley Math Department that they hire me. I visited Berkeley in 1977 for a year and gave a course on geometric analysis with emphasis on isometric embedding. Chern nominated me to give a plenary talk at the International Congress in Helsinki. The talk [730] went well, but my decision not to stay at Berkeley did not quite please him. Nevertheless he recommended me for a position on the faculty at the Institute for Advanced Study. Before I accepted a faculty position at the Institute, I organized a special year on geometry in 1979 at the Institute at the invitation of Borel. That was an …
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