Zheng Fangyang
Original English textA colloquium lectureIn Impressions of Shing-Tung Yau and His Mathematical World (International Press of Boston, 2019). The Chinese translation is published in "Mathematical Humanities" (subscription number:math_hmat), may not be reproduced without permission.
In the fall of 1977, Yau returned to his alma mater, Berkeley, for a one-year visit to consider whether to accept a permanent position. At that time, his paper on the proof of Calabi's conjecture had been widely circulated, and Mr. Chen Shengshen had given a series of lectures on the paper in his seminar class for more than two months. Everyone is looking forward to Qiu Chengtong's arrival. Within weeks of his arrival, he gave a lecture at a seminar, focusing on the application of his proof in algebraic geometry and on the noncompact case (i.e., the existence of a complete Keller-Einstein metric on the divisor coset of an algebraic manifold). He briefly mentioned the results for the non-compact case in [Yau1], but he did not write about most of his known results. His report complemented the discussion in Chen's seminar on the proof of Calabi's conjecture. Later that semester, we learned that he would be giving a lecture at a department-wide academic forum.
At that time, everyone knew that sooner or later he would have to give a lecture aimed at the entire department, because both the seminar reports given by Chen Shengshen and Qiu Chengtong contained too many technical details and were only suitable for geometers. Everyone had long been waiting for a lecture suitable for general audiences. I can no longer remember the title or complete content of his speech. But my ‘amnesia’ regarding content is more subtle, as I’ll explain in detail below.
I don’t think anyone would be surprised if I told you that I didn’t have much mathematical expectations when I went to the lecture. Because I participated in the entire Chen seminar and carefully read and proofread Qiu's paper [Yau1], I already have a sufficient understanding of the details of its proof. So I'm looking forward to a relaxed lecture and enjoyable academic enjoyment. I thought Qiu would tell about the historical background of Calabi’s conjecture and his journey of solving this famous problem. But he did not, and I had to wait until more than forty years later to read his own account of this in [Yau-Nadis]. What I got that day was actually a lecture on how to do mathematics.
Qiu pointed out at the beginning of his lecture that many problems in geometry can be reduced to the solution of partial differential equations (systems). At that time, geometric research on manifolds mainly involved linear equations such as the Laplace operator (harmonic functions, Hodge theory) or the partial operator (complex structures), but Yau believed that we had reached a point where we should focus on nonlinear equations. What are the most important nonlinear equations in geometry? He believed that it should include the harmonic mapping equation, the minimal submanifold equation, and the Monge-Ampere equation. Then for each of the three types of equations, he elaborated on the known results from the perspective of differential geometry in terms of the existence, uniqueness, regularity of the solution, etc., and pointed out what other conclusions should be known but are not yet known. It's so simple, yet so shocking! Against this background, he introduced his own results in these three aspects. He talked about his and Zheng Shaoyuan's discussion of the Monge-Ampere equation [Note: Hans Lewy is a fixture at Berkeley, and he is naturally one of the listeners. When Qiu talked about the Monge-Ampere equation, he turned to Lewy and smiled and said: "I feel very ashamed to discuss the Monge-Ampere equation in front of Professor Lewy." In this way, Yau paid tribute to Lewy, who made foundational contributions to collaborative research on Weyl and Minkowski [Cheng-Yau1], [Cheng-Yau2], and also talked about his own research on the complex Monge-Ampere equation and the proof of Calabi's conjecture [Yau1]. In terms of harmonic mapping, he talks about his collaboration with Richard Schoen [Schoen-Yau1], [Schoen-Yau2], using harmonic mapping to study Mostow rigidity. On minimal submanifolds, I remember him talking about his collaboration with Schoen and Leon Simon [Schoen-Simon-Yau]. He may have also talked about physics, including the positive mass conjecture, etc. I don't remember much about this last point. This amnesia is understandable, after all, so many years have passed, but in fact there are some deeper reasons here.
In countless seminars and academic forums that I have attended, I have heard numerous presentations on harmonic mapping or minimal submanifolds or other geometric studies. I'm used to experts focusing on one narrow aspect and giving updates from the beginning. But now the young man standing on the podium was telling us, in a way that he was unaware of, that he had surveyed the entire field of geometric analysis and decided to attack what he saw as the most fundamental and important problems. Everyone can talk big, but Yau’s ambitions are backed by compelling results. His speech blew me away. Looking back, I'm surprised that I remembered so many details of that talk, because I spent most of the hour asking myself: Why are we failing to do mathematics as he said? My mentor Warren Ambrose once told me that when doing research, most mathematicians will choose those windows that appear to be the easiest to harvest theorems. Choosing those windows that you think contain important secrets and working hard to open them to obtain the truth requires extraordinary courage, good judgment, and excellent abilities. I think that's why so few people make this choice.
With the publication of [Yau2], [Yau3] and numerous subsequent important geometric articles by Yau, his vision has become the main thread of geometric research in the past few decades. But back in 1977, it was a total eye-opener. It left a deep and lasting impression on me and made me realize the importance of looking at the big picture when doing research to avoid getting hung up on minor issues. This self-awareness has helped me a lot, if not to make my mathematics research better, at least to my research in mathematics education. It was my pleasure to express my sincere gratitude to him during my interview in Singapore in 2010 (see [Wu] p. 14). ■
[I would like to thank Larry Francis for his help with this article]
Wu Hongxi
October 5, 2018
(Translation: Zheng Fangyang
November 27, 2021 in Chongqing)
References
On the regularity of the solution of the n-dimensional Minkowski problem
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[Cheng-Yau2] Cheng, S. Y. and Yau, S.-T., On the regularity of the Monge-Ampere equation det(∂2u/∂xi∂xj) = F(x,u). Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68.
[Schoen-Simon-Yau] Schoen, R.; Simon, L.; Yau, S. T., Curvature estimates for minimal hypersurfaces. Acta Math. 134 (1975), no. 3-4, 275–288.
[Schoen-Yau1] Schoen, R. and Yau, S.-T., Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51 (1976), no. 3, 333–341.
[Schoen-Yau2] Schoen, R. and Yau, S.-T., On univalent harmonic maps between surfaces. Invent. Math. 44 (1978), no. 3, 265–278.
[Wu] Wu, H., Mathematics K-12: Crisis in Education (Interview with Wu Hung-Hsi by Leong Yu Kiang). Mathematical Medley Vol. 38, No. 1, June 2012, 2–15.
https://math.berkeley.edu/~wu/Interview-MM.pdf
[Yau1] Yau, S.-T., On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.
[Yau2] Yau, S.-T., Survey on partial differential equations in differential geometry. Seminar on Differential Geometry, pp. 3–71, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982.
[Yau3] Yau, S.-T., Problem section. Seminar on Differential Geometry, pp. 669–706, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982.
[Yau-Nadis] Yau, S.-T., and Nadis, S., The Shape of a Life, Yale University Press, New Haven, CT, 2019. Chinese translation: "The Shape of the Universe", Hunan Science and Technology Press, 2012.
Machine-translated from the Chinese original 伍鸿熙：一次令人难忘的学术讲座 (published 2022-01-14). WeChat original ↗