Encounters with Masters by Lizhen Ji University of Michigan From January 4 to January 8, 2013, the inaugural tinually transform it by this remarkable algebraic process, Master Lectures on Mathematics were held in Sanya. it just seemed so exotic and fascinating, and so I fell in The following five masters in mathematics gave ple- love with algebraic geometry. nary talks: Schmid: Well, I’m not quite sure what I should say • David Mumford (Brown University), The partnership here. I suppose that one thing was important for me in of pure and applied mathematics becoming a mathematician was an experience I had as a • Richard Schoen (Stanford University), Minimal sub- high school student in Germany. High school in Germany manifolds in differential geometry started at age 10. My father was professor of Latin, and so I went to a school that emphasized Latin, but I was more • Wilfried Schmid (Harvard University), Harmonic interested in mathematics. Around the time I was 14, I analysis on Lie groups complained to my parents that my school did not teach • Stanislav Smirnov (University of Genava), Discrete enough mathematics. One of our neighbors was Friedrich complex analysis and probability Hirzebruch, then the most influential mathematician in • Shing-Tung Yau (Harvard University), Geometry: from Germany. My father talked to him, and he said, well, that’s Riemann to Einstein and onto String Theory not a problem, I will give your son private lessons. So when I was 14, I got private lessons from the best known These five mathematicians are true masters in every mathematician in Germany at that time. sense of the word. They are deeply concerned with both research and education in mathematics. Three of them Smirnov: What made me choose mathematics? Well, hold the illustrious Fields medal. it happened gradually, so it is hard to pinpoint one reason. In order to follow the theme of learning from masters, I grew up with my grandparents, and my grandfather as well as to encourage the students at the conference, a graduated from the math department. But then the Sec- panel discussion titled “Encounter with Masters” was or- ond World War started and he became an engineer—even ganized on the evening of January 7. The following is the getting a second education in engineering. All his life he transcript of the conversations from that night. worked between applied mathematics and engineering, and, somehow, when I was young I was imagining myself an engineer, building airplanes or spaceships, rather than Question 1 a mathematician. I think one reason was the impression Lizhen Ji: In the most famous book in the world, one that I received in school, that mathematics is a science passage says: ask and you will be given. So I think I will done by the ancients, and that it was finished a hundred start by asking our masters and you guys will ask more. As years ago and there would be no new theorems. Even we all know, examples are important in mathematics and when learning physics and biology in school problems also in daily life. Since examples are important, let me ask they mentioned some modern developments. All that the masters here to give us one example in their lives which changed when I realized that this was not the case, maybe has played a very special role in either their lives or in after some university lectures I attended while in high mathematics. Let me start with David Mumford. school. I gradually became attracted by mathematics. In regards to choosing areas, I think now is actually as Mumford: One thing which made me fall in love exciting a time for mathematics as ever. It is certainly very with algebraic geometry was taking a course on algebraic different from a hundred, or even forty, years ago in the curves and doing a paper about blowing up the projective sense that now to arrive at interesting things in most ar- plane to desingularize singularities of plane curves. Take eas, you have to learn much more. But it is not all that out 23 for example y x , the simplest singular curve. What of reach—there are many open problems that one can you do is to take the singular point, and blowing up re- start thinking about, even at the undergraduate level. And places the point by the whole curve of directions through also what is interesting nowadays is the interaction be- that point, called the exceptional locus. And then the tween different areas of mathematics, as well as with singularity, all of a sudden, is changed into a simpler other sciences—physics, biology, economics, etc.—is on singularity. This idea, that you take the space and con- the rise. I find this very exciting. NOVEMBER 2013 NOTICES OF THE ICCM 85 Shing-Tung Yau: I started to be interested in If you persist, hopefully it will eventually come up. There mathematics only after I studied geometry in middle are many examples when persistence helped after many school. In my second year in middle school, I learned the years. Lennart Carleson proved the famous theorem, axioms of plane geometry. I was surprised by the elegance about almost everywhere convergence of Fourier series of mathematics and started to fall in love with it. I pur- for square integrable functions, after nine years. Every- sued the subject of functional analysis for quite a while one—Zygmund, Kolmogorov, and soon—thought there when I was in college. My view of mathematics changed should be a counterexample, and Lennart spent years when I arrived at Berkeley for my graduate study, where I trying to construct the ultimate one. But it didn’t work, learned differential geometry and differential equations. and he understood that perhaps it means that there is no While I was majorly interested in geometry, I found that counterexample at all, and he proved the theorem. So, one many complicated problems in geometry can be phrased should work hard. But if you are stuck, you probably in terms of nonlinear partial differential equations, while, shouldn’t concentrate on only one problem. You should at the same time, when these equations are interpreted take breaks. Richard Feynman, a famous physicist who correctly in terms of geometry, they can be solved much was also very good at mathematics, used to say that he more easily. The power of analysis in geometry excited liked to teach because when in research you get stuck and me. depressed, then you got to work with the students. Shing-Tung Yau: I found it challenge when inter- Question 2 esting questions are being studied. There are always dif- ficulties to overcome. But fresh ideas can be gained by Lizhen Ji: Okay, so I think that besides having mas- working with friends from different subjects. In the past ters from around the world, I believe we have the best forty years, I have been working on subjects relating students coming from China. So we are expecting the best mathematics to theoretical physics, and I enjoyed it tre- questions from the audience. mendously. But it is not an easy thing to do because I was Student: My questions are for all of the masters. My not trained as a physicist. I need to learn from physicists. first question is: what is the most unforgettable difficulty in The subjects that I started with were general relativity and your studies or research? Secondly: how did you get later string theory, especially questions related to mirror through that hard time? symmetry. It was not easy to understand what the physi- cists wanted to say at the beginning, and I knew little Mumford: What happened to me is that, half a about the subject. But many of my good friends were dozen times, I thought I had proven X and not X both. You doing research in the same topics; hence I can learn more ask, “what the hell is happening?” But then you say, “now quickly. I think one can do research on a subject that you it is time to really figure something out. Time to find out may not know well. One can also learn faster if there are how I could have been so stupid.” This will help me a good friends to help. In many cases, I was shocked by the lot—it is total confusion until you get some clear ideas intuition that other subjects can provide for mathematics. and get it straight after all. Some of the ideas were totally unexpected. For example, Smirnov: If we find a contradiction in mathematics, when my postdoctoral fellow Brian Greene told me about we will all get fired. the idea of mirror symmetry for Calabi-Yau manifolds, I was surprised. Even though it may be difficult, and even Mumford: Yeah, that is also possible. (laugh) challenging, to get into a field that one does not really Schmid: Yes, I certainly had experience of this know well, one can often find ideas from other subjects to sort—every mathematician does—but I want to talk about solve problems that one wanted to solve for a long time. a different kind of difficulty. When you are a mathemati- Fortunately, I have many good friends. After working with cian and think about important questions to study, there them, I could understand their subjects much better. Of will be periods when somehow ideas don’t seem to flow. course, I also provided ideas that can help them. Good That happens to most of us, maybe not to everyone, and communication is important. Over all, to cross into a new maybe some of us are luckier.