Prof. Yau from the eys of collegues, friends and students

A letter written by Salaff in 1969 for the university administration for Yau’s early graduation:

... A sure mark of mathematical talent in young people is their ability to focus all of their attention on a deductive reasoning problem; in fact it is this preoccupation with the task in symbolic form which has led to the popular stereotype of the mathematician as powerful but absent-minded, a thinker detached from his everyday immediate environment. Because his work involves the drawing of relationships between abstract objects represented by symbols on a piece of paper, the creative mathematician is often seen as a man who scribbles on a tablecloth or on the back of a menu at a restaurant. Yau is probably the sharpest of this kind, from among students or professional mathematicians, that I have ever encountered. When he is faced with a hard problem-it may be from a textbook, one posed during a lecture, or one he may set for himself, he has the ability to concentrate exclusively on that problem, during which process he scribbles almost illegibly on the nearest sheet of paper; he invariably deduces some information after about five minutes. The aspect of this concentrated attack which is most impressive to me is the arbitrariness of the problems which he chooses to work on. For most of us, there is some kind of priority in the way we select our mathematical problems, i.e. we will try to solve problems in our ‘course’, our ‘field’, or on our ‘topic’, and we try to conserve our energy when we are outside our realm of current attention, as much as possible. Yau, along with other preconscious mathematicians, has an intense curiosity about all branches of mathematics, which, coupled with an explosive intellectual energy, permits him to attempt, in good conscience, problems as they arise. This makes him an excellent ‘discursive’ mathematician, who can fit into just about any conversation between mathematicians (about their problem) and contribute something which those mathematicians would consider worthwhile. Jumping around from one problem to another would probably be wasteful for most professionals, for whom hours and days might be required to acclimatize to the new definitions and concepts of a different field, which is why most of us don’t do it. The strain of readjustment is too great and the anti-aesthetic aspect of entering a deductively closed or ‘closable’ subject ‘in the middle’ dissuades us from jumping. For the problem-solver like Yau, however, the aesthetic comes in with the excitement of solution per se rather than in a dispassionate consideration of the whole subject or system.

This innate ability to concentrate exclusively, whether in the face of poor study condition or during a boring and inexpert college lecture, is one of the marks of real scientific poise. Yau has, I found, a surprisingly tolerant attitude to lectures delivered by his teachers, who may well have a weaker grasp of the curricular material than he does. Speaking strictly for myself, I can say that when I was uncertain about subject matter (frequently) Yau would listen with complete attention for 50 minutes. Afterwards, he would come up and say “why don’t you do it this way?”, and then proceed to illuminate the matter at hand. He displays much more patience in the classroom than precocious American students, and is generally more conscientious. He said that he could always learn more during a lecture, even if it was a bad one, than he could by spending the time reading a book about the same material. He likes to hear what different people have to say, and is capable of mentally imposing his own

structure upon a mass of disjointed information if necessary...

D.Stroock, a prominent professor at MIT:

I think that the best way for me to explain my admiration for and wonder at Shing-

Tung Yau’s approach to mathematics is to relate a conversation which I had with him while we were attending a conference in Taiwan. The conference organizers had laid out the red carpet for their guests and one evening arranged for us to see a traditional Chinese puppet show. Much of the show involved tigers threatening, and occasionally eating, local villagers. At the end of the performance I expressed to Yau my surprise that tigers play such a central role in traditional Chinese tales. In response, Yau informed me that tigers were once a serious menace in rural China, but that was before the Chinese had eaten them all.

An eminent geometer at Berkeley Hung-Hsi Wu recalled one of his early experiences with Yau:

Yau was a student in one of my courses, so in this sense, I was his teacher. But the first time I met him, he just popped into my office and asked me a question about the fundamental group of negatively curved manifolds. It was something I was in no position to think about, much less answer. I sensed intuitively right away that here was someone who would go much further in mathematics than I would, and the intuition was confirmed by the time he wrote his Indiana Journal paper on the integrals of harmonic functions in 1974 (appeared in 1976). I am afraid I was his teacher for only one or two years, and became his student the rest of my life.

B.Lawson, a teacher of Yau at Berkeley and currently a distinguished professor at SUNY at Stonybrook, recalls his early days with Yau:

I remember well the day I first met Shing-Tung Yau. There was a reception for faculty and new graduate students at Berkeley. I was entering the second year of my instructorship and I still felt more like an advanced graduate student than a faculty member. Yau had just arrived to begin his graduate studies. We introduced ourselves and began talking. I asked about his plans for the year, and was shocked to learn that he had enrolled in six graduate courses. One of them was advanced Differential Equations, the subject, he said, that most interested him. Another was the course in Riemannian Geometry that I was scheduled to teach that year.

My course was enjoyable to teach, and Yau was constantly in attendance. Then somewhere in the second term he came to see me in office hours. He had been thinking about compact manifolds of non-positive curvature, and thought he could prove the conjecture that when 1 is solvable, such manifolds must be flat. I was amazed. He was still taking my course, but he had obviously moved very far ahead. His proof involved clever use of the Gauss-Bonnet Theorem to establish lots of flat, totally geodesic surfaces. I was taken by the power of the arguments and it occurred to me that these methods might prove that if the fundamental group split as a product, $\pi_1(X) = \Gamma\times\Gamma’$, then the manifold itself must split as a riemannian product $X = Y \times Y’$. We went to work and soon had a proof, at least in the real analytic case. We also showed that if $\pi_1X$ contains a subgroup isomorphic to Zk, then X contains a flat, totally-geodesic torus of dimension k. At the same time Yau wrote down a detailed proof of the conjecture, and thus, by the Spring of his first year, he had already completed a remarkable Ph.D.-thesis.

Yau and I were very taken with some of these theorems – they seemed wild and totally unexpected. Then we learned that at essentially the same time Detlef Gromoll and Joe Wolf had conjectured and proved the same results. In fact their Splitting Theorem and Maximal Torus Theorem were better since they held in the $C^\infty$-case. Thus, by the Spring of his first year, Yau had his first experience with the disappointments of science.

In the end, credit for the theorems was shared and Yau’s thesis appeared in the Annals of Mathematics.

The following year I left Berkeley to visit I.M.P.A. in Brazil, and Yau remained for a second (and last) year of graduate study. Chern became his formal thesis advisor, and Yau started writing papers on a variety of subjects. The range of his interests was great, and whenever he looked into a new subject, papers would appear. I remember that in writing letters for him at the time, I would have to remind people that Yau was only three or four years from having entered graduate school. It was easy for people to forget this and to compare him with much more mature mathematicians.

One very funny aspect of Yau’s early career is that he produced a constant stream of counterexamples to the Calabi Conjecture. Of course they never reached print. He would explain them in private, or give an informal talk at a meeting, and eventually a flaw would be found in his (usually very clever) construction. After a while the counterexamples stopped, and with a few years of intense work Yau come up with a complete proof of the conjecture, thereby establishing one of the greatest results of twentieth century mathematics.

Since that time Yau has transformed the field of differential geometry. He has brought about a sea-change in many areas of the subject. I remember predicting at an early point that Yau’s name would eventually become everywhere dense in the geometry literature. I had no idea that this would become true as well for certain areas of physics.

J.Milgram, a professor at Stanford University:

When Yau first came to Stanford, I wondered at everyone’s enthusiasm. But it didn’t take long to convince me. He was at school and working at all hours. In fact, I can’t remember ever knowing anyone who worked as hard as Yau or as effectively. It seemed like every few weeks he had proved another major conjecture or changed the way major problems were viewed. Before long he had profoundly changed the math department as well. There was a new excitement among the graduate students and even many of the faculty.

J.Coates, an eminent number theorist and a professor at Cambridge University:

I had the good fortune to spend several years with Yau at Stanford in the mid 1970’s, shortly after he had finished his Ph.D at Berkeley. He came to Stanford with the most glowing predictions about his future from Chern and others, and I remember being struck by many things that marked him out from the moment he arrived. Firstly, if one went to the Stanford Mathematics Department late on almost any night, Yau would always be there working, sometimes in the company of Rick Schoen, who was then a graduate student, or Leon Simon, who had recently arrived from Australia. Secondly, Yau regularly attended many courses and seminars outside his own specific area of research, and this kept one on one’s toes in preparing graduate courses for fear of embarrassment at not having the full details of a proof at hand under his watchful eye. Thirdly, Yau awakened my own interest in the history of China through conversations about things that my own education had carefully avoided (for example, after a discussion with Yau, I remember reading with horror the history of the Opium Wars in China, and realizing fully for the first time how excessive power corrupts any country). Altogether, Yau quickly became the focus of the mathematical lives of many of the young people at Stanford in those years, and there was a very happy sense of brotherhood which I will never forget. Since that time, I have never been at the same

institution as Yau, but we still have seen each other on many occasions. I have watched with pleasure to see him gain all the highest international accolades for his own mathematical research, and at the same time to create the world’s leading school of differential geometry.

Over the last decade, I have seen at first hand his titanic efforts to build up mathematics in China in a way that will be worthy of the greatness and longevity of Chinese civilization. I think it is fair to say that no other mathematician of our times has come close to his remarkable successes in raising money to establish and run Research Institutes in Hong Kong, Beijing, and Hangzhou, to fund three major International Congresses of Chinese Mathematicians, and to fund many other high level research congresses in China. I have been very conscious that number theory has been one of the big beneficiaries of Yau’s strong desire to see all major fields of mathematical research flourish in China in the future. I hope these few words convey something of what I feel are Yau’s great qualities and achievements, and his passion for the finest things in mathematics.

F.Catanese, a prominent algebraic geometer recalls his visits to Yau at IAS and UCSD:

I have first met Shing Tung Yau 25 years ago, when he was already famous for having solved the Calabi conjecture and several other difficult and important problems. I visited the IAS in Princeton, where he was a permanent Member, for the whole academic year 1981-82, on occasion of a special program in algebraic geometry. I remember vividly the early fall times when we were playing volleyball in front of Fuld Hall and discussing mathematical plans. His quickness to discuss things and his openness had a big impact on me. I remember that when he moved to San Diego and tried to build a very strong mathematical center there, including algebraic geometry, he invited me (as part of his plans) to visit for one year. The enthusiasm of my response was immense: I was fascinated at the idea of the fantastic possibilities of interaction between algebraic surfaces, complex differential geometry, and differential topology (in 1986 it was clear that Mike Freedman was going to be the second Fields medaillist in UCSD). I also wanted to learn a lot of new things! Some health and family problems on my side, some University conflicts on the other, made it so that I could only visit for the summer term, enjoying the privilege of Yau’s friendship, and participating actively into the intense work of Yau’s Department (I mean, an incredibly effficient organization, parallel to the ’other’ Department): Yau’s colloquium, Yau’s seminars, which he was attending with his 15 or so graduate students from morning to evening, snoring from time to time but always ready to wake up at the crucial point ... At the parties and at the seminars I always felt at ease with a person like him who was always going exactly to the point, in and outside mathematics, without any sort of hypocrisy.

If one looks at his achievements, it is simply amazing that a person who gave so incredibly many and so significant contributions in so many diverse fields, found also the way to care so much for so many students, to very actively engage in editorial activities (trying always to recruit the best papers) and in so many other projects. Last time I visited him in Harvard, he was more busy than usual, working hard on the weekends with Hamilton, but we always found the way to talk mathematics at lunch or dinner, and I benefited from some useful suggestions which later motivated my starting a new topic of research. As I told him on occasion of his election as a foreign Member of the Accademia Nazionale dei Lincei, he is not only regarded as the successor of Chern in the field of differential geometry, but many hope that he may continue to play a similar role for chinese mathematics in the future.

Harvard colleague Clifford Taubes says:

Yau is a terrific inspiration for me; he is a tremendously creative individual and a fantastic source for mathematics. Meanwhile, he is a very kind and loyal soul and so a role model in the ways of life.

Wilfred Schmid confirmed Taubes’ opinion:

Yau is a great friend and a model colleague. I admire him not only for his deep, wide-ranging mathematical work, but also for his seemingly boundless energy. On top of his research activities, full teaching load, and numerous Ph.D. students, he finds time to guide the activities of research institutes in Hong Kong and mainland China. He is knowledgeable about the state of K-12 mathematics education in China, and uses his considerable influence to improve the situation. No other mathematician I know does so many different things, so well, all at the same time.

Y.Wang, an eminent number theorist in China, expresses similar opinions:

A very valuable feat of Yau is that he dares to tell the truth. His criticism of the Chinese education system has been widely publicized and appreciated by millions of Chinese people. In many newspaper interviews, he has voiced his opinions about corruption in the academic world in China and the need to improve the quality of mathematical research and education. It takes great courage to raise such questions against a system.

Though I work in an area far from Yau’s, I can feel his broad and deep knowledge of mathematics. He has great insights into both mathematics and human nature. He always replies to my inquiries. When he comes to Beijing, his schedule is often very tight, and I always try to find chances during car rides to learn from him.

D. Gross, the Dean of Harvard College, emphasized:

Everyone knows about Yau’s deep contributions to mathematics and physics, as well as his devotion to improving mathematical education in China. But few realize what a profound effect he has had on the Harvard mathematics department, both at the level of graduate and post-doctoral training and in the recruitment of senior faculty.

Of course, Yau’s reputation can be larger than life. When Harvard’s President, Larry Summers, toured China with some of our faculty, Yau received more attention from the Chinese media than the President did.

Sir Michael Atiyah shares similar feelings:

I have known Yau for many years and have always admired his energy and dynamism as well as his technical mathematical prowess. He has played a big part in training excellent Ph.D. students, many from China, and in assisting the development of mathematics both in Hong Kong and the mainland China.

In recent years Yau has been very active in the exciting interface between Geometry and Theoretical Physics and his example has been followed by many of his students and colleagues.

His friend and the donor of the Morningside Center of Math, Ronnie Chan said:

Shing-Tungs scientific accomplishments are well-known to all. Few of his academic stature, however, have devoted as much time to educating the young. For decades he spends several months each year training students in mainland China, Taiwan, and Hong Kong, including those at his alma mater the Chinese University of Hong Kong. He is above politics and narrow allegiance to any particular institution.

The wealth of a nation, to a good degree, is a function of how advanced are its sciences, which in turn depends on the rigor of research. It is thus not surprising that countries with the highest standards in this regard are among the richest....

China is an ancient civilization with a superb history of literary studies and the arts, a source of immense joy and pride to its people. However, although the country had enjoyed periods when sciences flourished, it had fallen behind the West in the past few centuries. To catch up, a tradition of rigor in research and a healthy respect for related institutions must be established.

Shing-Tung is not only interested in college students; he also wants to inspire and nurture younger talents. It is no surprise that he regularly brings the world’s renowned mathematicians to the Morningside Center of Mathematics at the Chinese Academy of Sciences in Beijing, a feat that anyone with less stature would not be able to do. He also brings equally prominent mathematicians to adjudicate the Hang Lung Mathematics Awards competition for high school students in Hong Kong.

Shing-Tung’s knowledge and love for Chinese literature is legendary among friends. It is a legacy of his late father. He once gave a lecture to my students using 22 ancient Chinese literary texts to illustrate the pursuit of mathematics. Another incident demonstrating this passion of his occurred at an event hosted by the former Chinese President Jiang Zemin. Desiring to remind his guests who were all Nobel and Field’s Medal laureates of Chinese descent, of the richness and depth of Chinese culture, President Jiang gave the first line of a poem. No one was able to continue except Shing-Tung; he shocked his host by reciting the long piece in its entirety.

Everyone knows that Shing-Tung has an amazing mind–perhaps only a few of this caliber each century–but unknown to many is his extraordinarily kind heart. He feels for the less fortunate in society. This quality, although innate, should have been strengthened by his upbringing–he had an exceptionally kind-hearted and learned father who unfortunately died when Shing-Tung was still young. It must have also been enforced by his wonderful wife.

His collaborator J.Smoller, a math professor at the University of Michigan, said:

A thing that I particularly liked very much about Yau was his unflinching support of young mathematicians, and his setting them off in good directions; his many fine students can attest to that. In 1998, he introduced me to Felix Finster, a young German mathematician who was one of his postdocs. Finster was an expert on the Dirac equation, and the three of us began to work together on particle-like, and black-hole solutions of the Dirac equation coupled to Einsteins equations, and other fields (Maxwell, and Yang/Mills). This was an exciting time for all of us, as we made several surprising discoveries, and led to an on-going collaboration which we still have to this day.

I first met Yau at a lecture that he gave in 1978 in Bonn, Germany. His reputation was considerable even at his relatively young age, and I was very interested in hearing this rising star. I well remember how his talk excited me, and I went up to him afterwards to clarify certain aspects. What I found particularly impressive was his easy-going and friendly manner.

F.Bogomolov, an eminent algebraic geometer at Courant institute:

Professor Shing-Tung Yau has unquestionably been one of the world’s leading mathematicians.

I was fortunate enough to meet and talk with Prof. Yau at many different stages of my life. I met him first at the ICM in Helsinki in 1978, and I vividly remember our discussions for many hours on the benches of the park and during the walks on the streets of Helsinki till late at night. These were extremely interesting and enlightening conversations. His recent solution of the Calabi conjecture uncovered a whole world of new results and directions, and at that time they just started to roll on. We discussed all types of problems in geometry and many of his ideas were completely new to me.

Since then I have met with him many times, and each conversation with him contained an element of discovery, bringing some completely new and formidable idea, or a remarkable question. Professor Yau has a striking ability to summarize the essence of a problem or an idea in very short and well focused form. His sharp questions are always directed straight to the point of the problem and help to uncover its hidden sense, wether it concerns mathematics or some other matter. He is a brave thinker who never stops his quest and is never afraid to challenge a difficult problem.

The impact of Yau on mathematics is described by S. Donaldson, a Fields medalist:

Yau is a towering figure in modern differential geometry. His characteristic approach, blending geometry with PDE theory, has set the tone for vast parts of the subject over the past quarter century.

Apart from his own renowned research results, he has also done a huge amount to foster the development of the subject in a whole variety of ways. On a personal level,

I recall as a graduate student gaining a large part of my orientation in the field from his Princeton Seminar volume, and particularly from his masterful concluding survey. At the same time, I remember studying in awe his proof of the Calabi conjecture and trying to understand it. Now, 23 years later, I am still trying to understand it!

This opinion is also expressed by B.Wong, a prominent professor at UC Riverside:

There is no doubt in anybody’s mind that Shing Tung Yau will be regarded as one of the several greatest mathematicians in the twentieth/twenty first centuries. In fact, many mathematicians have commented that Yau really deserves another Fields Medal for his contributions after 1983; many important disciplines of Mathematics and Mathematical Physics have been dramatically reshaped upon the impact of his work and energy.

Shing Tung is a loyal soul with an extremely kind heart. I personally witnessed in many circumstances that he gave an unflinching support to numerous young mathematicians, and friends when their careers were in hardship. The extent of encouragement and support that Yau had offered to many mathematicians and friends (I myself was among one of them) were so critical and unforgettable. I also admire him very much for his unselfish devotion to nurturing the younger generations of mathematicians, both in the United States and China; he is never interested in obtaining monetary return for his endless effort of doing this. For me, Shing Tung is a great mathematician, a brave man with lofty ideals, and a good friend, all in one.

Fan Chung Graham, an eminent mathematician on discrete mathematics comments:

Yau sees clearly the connections between the discrete and the continuous in ways that only a true master of the first rank can.

C.-N.Yang, a Nobel laureate in physics, said:

Prof. S.-T. Yau is one of today’s leading mathematicians in the world. Unique among all distinguished mathematicians, he has made first-rate contributions to, and lasting impacts on, both physics and mathematics.

C. Vafa, a physicist at Harvard, commented:

The work of Yau and his collaborators have had great impact for theoretical physics and in particular string theory. Calabi-Yau manifolds are among the ‘standard toolkit’ for string theorists today.

Yau’s interests in and impact on physics are summarized by E.Witten, an eminent physicist and a Fields medalist at IAS, Princeton:

Yau has been tirelessly interested for all these years in theoretical physics and especially in areas of theoretical physics that are related to differential geometry and can be understood by differential equations. I am always amazed at the breadth of his interests, and all the areas of which he is in the forefront. A lot of things in string theory nowadays depend on Yau’s work, whether through applications to physics that Yau made himself, or ways that other people found to apply Yau’s mathematical ideas

and results to physics. And Yau’s enthusiasm is so infectious; he always makes other people excited that there are so many wonderful new things to be found.

Y.-T. Siu, an eminent mathematician at Harvard University and a friend of Yau, commented on the significance of Calabi-Yau manifolds:

After the solution of the Calabi conjecture, the so-called String Theory was proposed. The Calabi conjecture provides an essential piece to the model of String Theory, and Yau’s contribution in this direction is enormous.

Yau’s contribution to physics is explained by Andrew Strominger, a physicist at Harvard University:

The work of Prof. S.-T. Yau transcends pure mathematics and has had a deep and lasting impact on physics. To cite two important examples, his proof of the positive energy theorem in general relativity finally demonstrated - sixty years after its discovery - that Einstein’s theory is consistent and stable. His proof of the Calabi conjecture allowed physicists - using Calabi-Yau compactification - to show that string theory is a viable candidate for a unified theory of nature. More generally, he plays a key role in the fertile dialogue between modern string theory and mathematics, and is uniquely able to communicate with both physicists and mathematicians from a wide variety of backgrounds.

D.Gieseker, an eminent algebraic geometer at UCLA, explains:

The impact of Yau’s work on algebraic geometry is not solely from the interaction with physics. Important conjectures of Severi and Bombieri were also established as corollaries of the Calabi conjecture, which by the way was proved at UCLA during a visit by Yau. Yau also made the most important contribution in the case that c1 > 0 and conjectured its relation to the stability in the sense of geometric invariant theory in algebraic geometry. This has motivated the important work of Donaldson on scalar curvature and stability. Another important result of Donaldson-Uhlenbeck-Yau is that a holomorphic vector bundle is stable in the sense of Mumford if and only if there exists an Hermitian-Yang-Mills metric on it. This has many important consequences in algebraic geometry, for example, the characterization of certain symmetric spaces, Chern number inequalities for stable bundles, and the restriction of the fundamental groups of a Kahler manifold.

Yau’s work in algebraic geometry alone would make him one of the world’s preeminent mathematicians, not to mention all his work in geometry.

The past few decades have seen the reemergence of a remarkably fruitful interaction between mathematics and physics, in particular between string theory and algebraic geometry. This impact by physics arose from physical arguments suggesting that certain unexpected and mysterious mathematical relationships exist that are not evident from a standard mathematical view point, the mirror conjecture being a prime example of such a relation. Yau has been the epicenter of this interaction. It is impossible to overstate the importance of Yau’s proof of the Calabi conjecture, as it is the absolutely essential bridge between string theory and algebraic geometry. This development has flowered in many directions, in particular the proof of the mirror conjecture, the SYZ program and the Yau-Zaslow conjecture.

His collaborator Peter Li explains:

It was during the special year on Geometry at the Institute for Advanced Study (1979-80) when Borel asked Yau if he knew a good upper estimate for the heat kernel for the Laplace operator. I believe Borel wanted to prove that the heat operator on certain locally symmetric spaces is of trace class. In any case, since in my thesis I worked on estimating the heat kernel, Yau, Cheng and I started working on this problem. At that time, little was known about the behavior of the heat kernel with respect to the geometry of the underlying manifold. Cheng, Yau and I wrote a paper proving an upper bound for the heat kernel using geometric quantities. Cheeger, Gromov, and Taylor later gave a generalization of our work. After a year or two, Yau suggested that we looked into the gradient estimate method he used in 1974 for harmonic functions (I also used it for eigenfunctions in my thesis) to get bounds for the heat kernel. It turned out that the estimate one can get is sharp, not only for the gradient bound, but also for the bounds (both upper and lower) for the heat kernel on manifolds with nonnegative Ricci curvature. After refinement, this method also applied to manifolds with Ricci curvature bounded from below as well as more general parabolic Schrodinger operators. As it turned out, the maximum principal argument to yield the parabolic gradient estimate became extremely useful in dealing with geometric flow type problems. This philosophy and the argument were substantially amplified by Hamilton, which he systematically and successfully applied to many nonlinear parabolic equations. Of course, the most famous one was the so-called “Li-Yau-Hamilton” inequality for the Ricci flow equations. Moreover, a portion of the recent work of Perelman was also motivated and related to our estimate on the parabolic Schr¨odinger equation. In particular, Perelman’s reduced distance is parallel to the parabolic distance in my joint paper with Yau. As to our estimate on the heat kernel, it also became very useful in geometric analysis. In many situations the heat kernel estimate can be used to gain control over various quantities that are important to the theory of partial differential equations on manifolds, such as the Sobolev constant, the Greens function, and the Poincar´e inequality.

Other than the obvious fact that Yau is a top-notch mathematician, his impact in mathematics has gone beyond his theorems. He has been very generous in supervising graduate students and mentoring postdoctoral fellows, in terms of his time, his energy, and more importantly his insights in the subject. One example is his famous set of open problems in geometry. Many mathematicians are unwilling to share their insights with the public because they would rather save it for their own use and for the use of their students and proteges. Yau’s problem set is a counter-example to this phenomenon. As far as I know, his problem set is still extremely influential and continues to generate a lot of interesting mathematics.

Dennis Sullivan, a versatile topologist shows Yau’s strong geometric intuition and conviction:

Once when Yau was a professor at the IAS (Princeton) and I was visiting there I mentioned to him how awkward it was to write papers using Brownian motion and Riemannian geometry because there was not a convenient reference connecting the two cultures. I was remembering Yau’s result that a p-summable harmonic function on a large Riemannian manifold has to be zero for p larger than one, and the fact due to Lucy Garnett that the same result follows for p equal to one in an intuitively obvious way using Brownian motion in a dynamical argument. Yau answered that he had tried to use random path arguments to construct bounded holomorphic functions on large negatively curved complex manifolds and the method did not seem to lead anywhere, and so the lack of references might be justified. He particularly mentioned the problem of finding bounded harmonic functions on large real manifolds of negative curvature.

Thinking about this later that night it seemed intuitively obvious using random paths that such bounded harmonic functions should exist abundantly in Yau’s problem. However it was not easy the next day to describe the argument because of the kind of difficulty already mentioned about references and different math cultures. Milnor was listening and he suggested I try to write a formal argument. This was possible and about a year later I was happy to present the rigorous argument to Yau. Yau’s response was unyielding: “If one could do it that way one could also do it by geometry!” he insisted. In fact, Yau was right because Michael Anderson about the same time independently found the same result about bounded harmonic functions on simply connected negatively curved manifolds using a geometrically natural convexity construction.

The moral of the story seems to be that having a definite perspective and sticking to it is an effective asset in doing and stimulating research whether or not in a given instance one is right or one is wrong or one is both right and wrong.

Yau’s contributions are summarized nicely by Phong, a colleague at Columbia University:

The range of fields - differential geometry, non-linear partial differential equations, general relativity, algebraic geometry and string theory - which have been fundamentally reshaped by Professor Shing-Tung Yau’s works is nothing short of astonishing. There is hardly a research direction in modern geometric analysis which does not make essential use of his works, or is not deeply influenced by a research program which he has laid out. He will surely be counted among the greatest figures in the long history of mathematics.

Yau’s wife Yu-Yun explained:

One of his goals in life is to help the Math society in China be first rate in the world. He has spent almost all his spare time in the last 15 years or so to reach this goal. That

involves majority if not all his winter and summer vacation months. It was a sacrifice for the family but we went along with him.

To accomplish this objective, he basically has a two prong approach: one is to train many students, the other to raise funding for establishing the Math Institutes in China. He spent a lot of time both in school and at home to educate the students. Fund raising was not his call, however he went ahead for the necessity. He holds this basic principle of not using a penny from the raised fund for personal expenses, including travel and lodging.

The efforts he made at our home to groom some students individually probably are more than what he did for our children. And he tries to protect his students, particularly the newer PhDs, on their research work and if possible their jobs, sometimes at his expense of friendship with other professional contacts.

S.-S. Lin, an eminent mathematician in Taiwan, recalls:

Yau first came to Taiwan to attend a joint conference of the United States and Taiwan on nonlinear PDE in 1985. The group was led by T.-P. Liu, and the members included J. Glimm, L. Nirenberg, P. Rabinowitz. D. Stroock and Yau, all VIPs in the field of mathematics in the USA. This event represented the first opportunity for many local mathematicians to meet the best mathematicians in their own hometown. They had prepared the meeting for almost two years by improving their own mathematical results and polishing their English. Before the meeting, stories were published in newspapers about Yau, his great achievements in mathematics, and his winning of Fields Medal, and he was recognized as a hero. The meeting was very successful. Thereafter, Yau visited Taiwan frequently to meet people and promote mathematics. In 1990, he was invited by Dr. C.-S. Liu, then the President of National Tsinghua University, Hsinchu, to visit the university for one year as a chair professor. During that year, many mathematicians visited especially to seek his advice. A few years later, he convinced Dr. C.-S. Liu, now the chairman of National Science Council, to create the National Center of Theoretical Sciences(NCTS), which was established at Hsinchu in 1998. He was the chairman of the Advisory Board of the NCTS until 2005 and was followed by H. T. Yau of Harvard University. He has significantly promoted the development of mathematics in Taiwan by making friends, encouraging people, teaching students and helping organizing many mathematical activities over the last two decades.

Similar opinions on Yau’s influence on mathematics in Taiwan are shared by C.-S. Lin, a prominent mathematician in Taiwan:

He visited Taiwan for his sabbatical leave, at Hsing-Hwa university. He gave a course on isometric embedding, and also conducted many seminars. His style of treating mathematics is so grandaus that it almost was a shock to many young students. Even after he left Taiwan, his affect was still very big. Some of most talented students have been attracted to mathematics, for example, Mu-Toa Wang (Columbia) and Ching-Lung Wang (Central Univ.) went to study under the supervision of Yau. Now they are already becoming leading figures of mathematics in Taiwan. Another thing I want to mention is the National center for theoretical sciences in Tsing-Hwa University. This center is playing the most important role of mathematics activities in Taiwan. The idea of establishing this center came also from Yau. He suggested it to the head of the National Science Council of Taiwan, and it was immediately accepted. Yau has made a great effort to help the birth of the center. He was the chief adviser of the center from the begining. He helped build up the whole strcture of center. Yau constantly visits Taiwan and gives a lot of inspirations to young people.

Yau’s care for his children is recalled by D. Christodoulou, a prominent geometric analyst at ETH, Zurich:

During the spring of 1984 , Yau’s family had already moved to California, but Yau himself had stayed behind. So I remember that we were working in his office at the IAS far into the night. At about 1 AM, which is 10PM pacific time, he would pick up the phone and start singing beautiful Chinese songs! Yau is full of surprises, I thought to myself, now he wants to become a great opera singer. As I later found out, these songs were lullabies for his children who were just going to bed at that time.

I had obtained a Ph.D. in physics from Princeton University in 1971, so at the time I first met Yau I already had a 10 year career as a physicist, however my career as a mathematician was just beginning. I was very fond of his openness and I interacted a lot with him for about 5 years. These were formative years for me, during which I learned from Yau the method and approach of geometric analysis. It is this approach which I have followed since, and I see my own work as being an extension of this geometric analysis approach to problems involing evolution in time, were the basic equations are hyperbolic (such as the Einstein equations of general relativity or the equations of continuum mechanics).

F. Hirzebruch, an eminent mathematician, commented:

I admire Shing-Tung Yau as a great mathematician and teacher, as a supporter of international cooperation, as a founder of institutes, as an editor of journals and as organizer of many important meetings. He seems to have infinite energy. Let me mention two occasions where I personally enjoyed Yau’s activities: I am grateful that he attended the conference “Hirzebruch 65” (Emmy Noether Research Institute, Bar-Ilan University, Israel, 1993) and lectured there on his famous work about Khler-Einstein manifolds. The fundamental Miyaoka-Yau inequality $c_1^2\leq 3c_2$ holds for the Chern numbers of algebraic surfaces of general type. The equality $c_1^2 = 3c_2$  characterizes the ball quotients among the surfaces of general type. All this was basic for the book jointly with Gottfried Barthel and Thomas Hfer (“Gradenkonfigurationen und Algebraische Flachen”, Vieweg 1987) and for my cooperation with Paula Cohen (compare our review on the book “Commensurabilities among lattices in PU(1,n)” by Pierre Deligne and G. Daniel Mostow, in Bulletin AMS 32 (1995)). Yau mentions in his Bar-Ilan talk that we discussed such matters in Berkeley in 1981. Secondly, I am grateful to Yau because he organized a splendid conference sponsored by the Journal of Differential Geometry in Harvard in 1999 “to honour the four mathematicians who founded Index Theory” and edited two books, on the conference and on reminiscences of Atiyah, Bott, Hirzebruch and Singer.

J.Jost, the Director of Max Planck Institute for Mathematics in the Sciences, Leipzig, describes his experience with Yau:

I learned from Yau that in order to solve a problem you should try out as many ideas as possible. Even if none of them works, understanding why they don’t work gives you so much insight into the problem that eventually its essence becomes completely clear, and then you can solve it very easily.

Winnie Li, a prominent number theorist at Penn State University, commented:

I never had any close contact with Yau, and I have always admired him from distance. I appreciate his support of activities in number theory. His leadership has a profound influence to generations of mathematicians.

Yau’s impact goes beyond research and education in mathematics, as E. Lieb, a professor at Princeton University, commented:

One important aspect of Yau's work, in addition to his mathematical contributions, is his social service to the community. In particular, he is one of the very few people who had the will, the energy, and the ability to do something about the high cost of mathematical publication by starting a publishing house, International Press, that produces books and journals at modest prices, and with a far sighted policy about copyright. It has been my great pleasure to work with him on several projects involving publishing. Our community is in his debt for this aspect of his work alone, even without counting his manifold other contributions.

I.M.Singer, an institute professor at MIT and an Abel Prize winner, commented:

Yau is a mathematics department all to himself. He holds seminars all the time, his students are very active all the time. They become part of his family.

M.Stern, a professor at Duke University, recalls the way Yau advises his students:

Yau’s interest in and support of his students continued after they graduated. In my case, he offered career advice and collaborations, and put me in contact with future collaborators. In one instance, his introduction to a string theorist needing my expertise led to a fruitful ten year collaboration.

Christine Taylor recalls affectionately Yau as an advisor:

Having Yau as an advisor is like having another parent, not just for the few years of graduate school, but also for life. We can always count on his support. Yau, being Chinese, may not show his affections for his students as effusively as Americans, yet all of us know that he’s proud of us, just as we are of having him as our teacher. I think such a bond between an advisor and students is very rare.

Another student Chiu-Chu Melissa Liu describes Yau’s help to students:

It is impossible to recall all the tremendous help that Professor Yau has been offering me at various stages of my mathematical career. I will only recall one occasion which is particularly remarkable. During the summer after my second year of graduate study, Professor Yau sensed that I felt insecure from a conversation with me. During the week after, Professor Yau arranged a conversation between Professor Fan Chung and me, and an extensive discussion among Professor Karen Uhlenbeck, Professor Chuu-Lian Terng, and his female students. This is something that I would never forget.

Alina Marian, who got her Ph.D. in 2004, recalls Yau’s supervision at Harvard:

Mathematical discussions with Yau are unceremonious, and their point is reached very quickly. He is unusually frank in giving his opinion about a piece of work or about the prospects of a line of research, yet every time I felt encouraged by him. Most impressively, Yau was essentially always available for his students. I was once scheduled to give a talk in his student seminar very late in the fall semester. It was in fact two days before Christmas, in a virtually empty mathematics department, that I found myself ready to give a lecture, on a fairly modest subject. The audience, however, was entirely lacking ... Somewhat deflated, I went to tell Yau (who was a little late) that the seminar will have to be cancelled. Yau persuaded me to proceed with the talk, and listened to it, alone, for a long time.

S.Wolpert, an prominent mathematician at Univ of Maryland and a long time friend of Yau, commented:

Over the years since Stanford I have found Yau to be interested in virtually every question in mathematics. Yau has always encouraged the study of the “big questions” that cross fields and that can open entire new areas of understanding. Also I have appreciated conversations with his students who are well versed on the particulars of current research problems in geometry. One measure of the success of the Yau seminars is the knowledge of his students.

His working relation with his students is perhaps best described by R.Schoen, a former student, now a professor at Stanford University:

I first met ST Yau in 1973 when I was a second year PhD student at Stanford and he

was a newly arrived faculty member. We became mathematically involved through a reading course I was doing with Leon Simon on minimal hypersurfaces. This led to a three-way joint work on properties of stable minimal hypersurfaces. I continued to work with both Yau and Leon while I was a student and was officially their joint PhD student. I spent several hours a day working with (mostly learning from) Yau when I was a student. He was interested in anything geometric, and he had ideas for approaching a vast range of problems. This was an incredible opportunity for me, and it gave me a great start on my research career. We wrote two more joint papers while I was a student.

I left Stanford in 1976 to take up a two year instructorship in Berkeley. Yau came to Berkeley during my second year, and we continued our collaboration. It was in Berkeley that we began our work on scalar curvature and the positive mass theorem. We expanded this work substantially over the next few years, and I remember wonderful times working together at Stanford during the summers of 1978 and 1979. During the 1979-80 academic year Yau organized a special year at the Institute for Advanced Study. This was another formative period in my career since there was so much going on in a wide variety of directions. I learned a lot and did some work that I am still proud of.

I have vivid memories of Yau from the early times: his tremendous dedication to his work (he was in his office day and night including weekends), his amazing breadth of knowledge and technique, his openness and generosity with his time. When I look at Yau now, it is amazing how little he has changed. He still has the same basic personal qualities and the same extremely high energy level. The main difference is that he is much busier with the great variety of things he is doing. I doubt that he would have the time to spend with a current student that he spent with me in the early 70s, so I feel very lucky to have had the chance to work with him at that time.

The famous mathematician K.Uhlenbeck once said:

I also became friends with S.T. Yau, whom I credit with generously establishing my finally and definitively as a mathematician.

Arthur Jaffe, the former chairman of AMS, once said:

Yau's versatility makes him a Renaissance mathematician.

Yau’s classmate at college S.-L.Ma recalls:

It was extraordinary to the Chinese tradition for a sophomore to lecture a class of junior students. Yau has made a striking precedent for the Chinese University of Hong Kong in the late 60’s to have done exactly that. Co-teaching with Dr. Stephen Salaff on Virtual Analysis and Differential Geometry, he has demonstrated his great talent and mastery in complicated concepts at young age. At the receiving end, I am honored to be affiliated with the history in the making of such a genius in the world of Mathematics.

His classmate at college Y.-C.Siu recalls:

Professor Yau was also delighted to meet with young people to promote the learning of mathematics. When I taught in a secondary school in Hong Kong, I invited Professor Yau to give a talk to our students. After sharing his experiences in learning mathematics with the audience, he stayed for almost two hours to chat with some senior students patiently answering all sorts of questions till everyone left with satisfaction and unwillingness. To the young ones, Professor Yau is more than a great mathematician. He is an ambassador of mathematics who brings them the genuine beauty of mathematics.”

On the occasion when Yau received the Fields medal in 1983, Nirenberg commented that:

Yau uses these [minimal] surfaces in the way, previously, people had used geodesics, and Yau is an analyst’s geometer (or geometer’s analyst) with remarkable power and insight.

In 1994, the Crafoord Prize of the Royal Swedish Academy of Sciences was awarded to Yau:

For his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.

In 1997, he received the U.S. National Medal of Science:

For profound contributions to mathematics that have had a great impact on fields as diverse as topology, algebraic geometry, general relativity and string theory. His work insightfully combines two different mathematical approaches and has resulted in the solution of several long-standing and important problems in mathematics.

In 2003, he received the China International Scientific and Technological Cooperation Award:

Because of his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers.

Citation of the Veblen Prize for Yau in 1981:

We have rarely had the opportunity to witness the spectacle of the work of one mathematician affecting, in a short span of years, the direction of whole areas of research.... Few mathematicians can match Yau’s achievements in depth, in impact, and in the diversity of methods and applications.

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