傳播數學知識．促進數學教育

Interview with Prof. Gerhard Huisken

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewers:** Tai-Ping Liu (TPL), Jih-Hsin Cheng (JHC)

**Interviewee:** Gerhard Huisken (GH)

**Date:** October 21st, 2011

**Venue:** Institute of Mathematics, Academia Sinica

Prof. Gerhard Huisken was born on May 20, 1958 at Hamburg, Germany. After completing his PhD at the University of Heidelberg in 1983, he went to the Centre for Mathematical Analysis at the Australian National University and, since 1992, at the University of Tübingen. Between 2002 and 2013, he was a director at the Max Planck Institute for Gravitational Physics and honorary professor at the Free University of Berlin. In 2013, he took up the post of director at the Mathematical Research Institute of Oberwolfach, together with a professorship at Tübingen University. His main research interests are in mathematical physics and differential geometry. He is a member of German Academy of Sciences Leopoldina, a recipient of Medal of the Australian Mathematical Society and Leibniz Prize, and also a fellow of American Mathematical Society.

TPL: So first I want to thank you for coming. I know that you are extremely busy.

GH: But I like to come and it was a very nice invitation, thank you.

TPL: Maybe we could start with this. You lead a group. Could you describe it a little bit?

GH: It’s at the Max Plank Institute for gravitational physics, in Potsdam, in Berlin; there are three Max Plank Institutes, ours is one of them, so we have shared a lecture hall for 200 people, some shared administration. And our institute has some 120 scientists, in three groups, astro-phyiscs, quantum gravity and string theory, and our group is geometric analysis and gravitation. So our central work is mathematical relativity, but also all the foundations in mathematics that you need for that. This includes geometry, PDE and we interact, say, with the numerical relativity at the astro-physics department, and with the staff of the string theory, because there are some interesting mathematical models to be investigated. The total number of scientists now in our institute is about 120 and our group is about 30-33 people.

TPL: I see. So for now you are the director of the whole group of 120.

GH: That’s right. In fact, we have a sister institute in Hanover which is doing experimental gravitational research, based on laser interferometry; they implemented this gravitational wave detector; so at the moment I am also the managing director of that Institute, because it’s one big institute.

JHC: Is that a tradition in Germany that mathematician and theoretical physicist work together?

GH: In our institute, yes, but it’s not a tradition in Germany universities, usually it is separate. So that’s why I found it so attractive to be in an institute where mathematics is mixed with physics, I think that’s much better and we have taken great care to have all the scientists mixed, not one wing mathematicians, one wing physicists, but all mixed together and we have joint seminars. It doesn’t always work, but we tried.

TPL: So you have these two groups; mathematicians and physicists, which one is harder to drag to the other side.

GH: I think it depends on the individual. We have some people who just worked on their own project, it’s very hard to entice them to talk to the other people. But we have some people who really enjoy the interaction and they have some joint projects. So I think it’s really at individual levels and senior people can help by organizing joint seminars, some joint workshops, things like that.

TPL: You yourself, how much a physicist are you?

GH: I am always flattered if people ask the question even. I am a mathematician by education. I came very late to the realization that geometric flows, geometric PDEs, following the work of Yau and Cheng can be used to prove things in general relativity, so my direction comes from the mathematics side, I would not really call myself a physicist.

TPL: But now you are actually very committed and believe that for mathematicians to learn some physics, and in particular to work with physicists is a good idea.

GH: I think definitely it’s a good idea, because if you look at the physics then you are forced to bring different mathematical theories together, if you are in a math department alone, then you may have one group looking at probability theory, another group at phase transitions, somebody else at, maybe mean curvature problems. But if you want to solve the Einstein equations, in particular if you look at different matter models then you have to know in which regime do you need fluid, when do you need elasticity, and maybe some stochastic effects, so you have to understand the relation between the different theories. I think that’s a great aspect for future mathematics to understand the relation between different fields of mathematics.

TPL: I can see it’s a very satisfying position that you are in.

GH: Scientifically it’s very fascinating to see all these different aspects.

TPL: So geometry, fluid, elasticity, physics, I am just repeating what you said, that’s great, a very wide spectrum.

GH: It’s a broad spectrum, but still in educating the young people, I think you have to strive for the right balance, you also have to find hard problems where young people can learn to prove a deep theorem, not just stay on the surface and do a little bit of everything, right? It’s good to have the broad perspective but also one has to learn to identify interesting projects, in particular for the mathematicians it is important to identify those projects where new mathematics can really help the physicists.

JHC: Do you have graduate school?

GH: Actually, at our institute we have two graduate schools, one more on the physics side, gravitational wave detection, and one more on the theoretical side, string theory, geometric analysis, gravitation, which also involves Potsdam University, Humboldt University and Free University in Berlin, so it’s a shared project. I think it’s important to keep contact with the universities, it is not good if a research institute is completely on its own.

JHC: Is the college system in Germany similar to the US system?

GH: No, I think it is different. They have moved now also to the bachelor master system, but I think there are still differences, there is no tuition for example.

JHC: When students enter the university, they belong to one of the departments or have some major?

GH: Yes, they would either join the math department or the physics department, physically.

JHC: From the first year?

GH: Yes. It’s changing slowly, but at the moment we don’t have, say, a science bachelor degree like you often have in the Anglo-Saxon system, but it’s a math degree, or a physics degree, or a chemistry degree.

TPL: When you talk about this joint project with physicists and this group working together, this reminds me in the old German system, which was adopted by Japan also, I think, namely, underneath a professor there are some associate professors, and then some assistant professors, in other words, a professor is really a group leader, but now what you described is a new thing, it really has not much to do with the old German tradition.

GH: Well, you have to distinguish between the universities and Max Plank. They are different, at the universities you have departments where you have a certain number of professors, say in math department, maybe 12 professors, who have all the possibility to do research, and they have the right to do research independently and they don’t have to listen to the senior people so much, but they may have very different resources, some of them may have a lot of resources, but others far fewer resources. Whereas at Max Plank, I am afraid to say it is very hierarchical, the directors have a lot of power, in fact most of the power is concentrated into the directors in the Max Planck Institute.

TPL: The President of Academia Sinica and some of the senior members at Academia Sinica have been asking this question why Max Plank Institute has in the past done great things, and how could we learn a lesson from them. And one of the questions that we ask is the following: in Max Planck Institute, very few of you are permanent appointments, right?

GH: That’s true, certainly in the theoretically oriented Max Plank Institute. Out of 120 it would be about 10 permanent scientists.

TPL: So the question we had was the following, how could you attract these talented people without offering them permanent positions, what makes them come?

GH: Well, the non-permanent people they come for maybe post docs to my colleagues for 2 years, some of them for 5 years, it’s very rare that we offer them to come for longer than 5 years. Though they don’t have to teach, but they can do voluntarily teaching at the university. That’s why I said it’s important to have good contact to the university, they at least voluntarily can learn some teaching and do something for their CV. I think the other thing is to have critical mass, to bring together a good group of people. I think it’s the responsibility of the director to create the right mix, to make appointments such that a group is created which makes research attractive for young people.

TPL: So, it has a lot to do with the tradition and also a lot to do with the leadership.

GH: Yes, I think it’s important that the director has direct responsibility to appoint the right people according to high standards of quality, that’s the key thing.

JHC: How about other people, post docs, visitors…

GH: Right, so we have post doc positions and then we have quite a large visitor program, only for visitors, and we can use this money to invite PhD students from other countries. For example, we invited PhD students from Australia for 6 months. They do their PhD in Australia, but they came for 6 months, because they work on a project that is interesting to us, and we are interesting for them. We invite senior people sometimes up to 3 months, and we can pay senior salaries for these 3 months. And again, I think it’s important to have the right mix; if you just invite young people and you don’t have enough senior people to supervise them, it wouldn’t be good; if you only have the senior people, and they are sitting in their offices, don’t talk to anybody, it’s no good either, so you have to find the right mix between senior and young people so they talk to each other and benefit each other.

TPL: So the hiring of the senior people, this is another important part of your job.

GH: Right, at the director level this would be done at the whole Max Planck Society, that is a very serious complicated process. The directors meet three times a year, all the directors, from chemistry, physics, computer science, mathematics, and they set up appointment committees to hire new directors and make decisions, actually sometimes we have to make decisions to close a whole department, because otherwise we don’t have the right to start a new one. So I think hiring the director is the most important thing in the Max Planck Society. And sometimes these decisions are very painful if you have to close a whole department, but if we don’t do it then the politicians will take away our right to make these decisions.

TPL: I see, that’s good. So this is painful, but it is so set up that you are responsible to the scientific community on the one hand, to the legislature on the other hand, so that’s a very healthy situation.

GH: As long as we do this responsibly, the legislature is not interfering, right. There is another level, that’s the senate of the Max Planck Society between this body that makes the appointments and the politicians. The senate approves appointments, and the politicians look at that, but the senate will not interfere in the appointment process, so the appointments are purely made by the scientists, the politicians have no right to initiate appointments.

TPL: But the senate of the Max Planck Society, they are mostly scientists?

GH: The majority are scientists, but there are some people from business, some people from politics, it’s a much bigger body overseeing Max Plank. But they don’t make day to day decisions. They just approve things and in rare cases, they say no, but they cannot say you should appoint such and such person, or you should start such and such project. That’s a delicate issue, we don’t want politics to tell us you have to close this institute and you have to start that institute, it should remain our decision, it is very delicate and complicated.

JHC: Do you get support from private donations?

GH: We get some private donations, but I don’t think it’s more than 10% of the budget. At least 90% of the budget of Max Planck Society comes from the government; one half from the Federal Government and the other half from the States.

TPL: Closing a department, how is it done? All the directors get together and then have review, but when you close a department, it really means that you don’t renew it?

GH: Right. It usually happens when the director retires or leaves, which is very rare. When a director retires, then one looks very carefully, do we reappoint someone in the same area, do we change the area a little bit, or do we make a drastic change and this happens sometimes. I mean when about 8 years ago, there was a little bit of financial problem, and a new president came in and I think we closed 8 departments, in chemistry, physics, computer science, out of between 100 and 150 departments.

TPL: That’s really a healthy situation, but this does not happen once in recent history. It happened at other times.

GH: Yes, it happened at other times.

TPL: It has been not easy in this Chinese society to do this. I think we have much to learn. I would report this to the Academia Sinica.

GH: I should say that of course when we close a department great care is taken not to hurt the junior scientists who are at that department or the junior permanent members, so closing would mean that the junior positions run their course until they expire, and the permanent people would be offered to join say, the neighboring department, or move to another Max Planck Institute, they wouldn’t be fired.

TPL: Of course, the set-up is such that there are very small percentage of permanent positions, that’s important, and the temporary ones had been made clear on this before they came, so everybody has no surprise.

GH: Junior people don’t expect to ever get permanent positions at their Max Planck Institute. It’s almost impossible, very rare cases that a scientist from the same institute becomes the director at that institute. That’s almost impossible, the new director mostly comes from outside.

TPL: I see, so in your case you came from?

GH: Tubingen University, so even from outside Max Plank Society.

TPL: You said you would go back to focus on your research group of some 24 people, and the process to start finding the director of the overall institute, is that right?

GH: No, No, we just rotate, all the directors at the Institute are the same, they have all the same rights, they have their own budget. We just share the work, so we have agreement that everybody has to serve as managing director for 2 years. So we have three directors at our institute, the sister institute has two directors. We rotate, so I do it for two years, my two other colleagues do it for two years each. Then after 4 years it’s my turn again.

TPL: I see, sounds like a very interesting process. But out of your group of 24, how many of them are permanent members?

GH: Apart from me, 2 others.

TPL: I see. So Max Planck Society, how long is this tradition, the history of it?

GH: Well, we just celebrated 100th years, but I mean of course because of the war there was an interruption. The original Max Planck Society was founded by the Emperor still before in 1911, but then of course during Nazi time it was abused by the Nazis, many people left. There was a re-start after the war, only then the name became Max Plank Society. The name Max Planck Society is after the war, before the war it was called the Emperor Wilhelm Society.

TPL: So we are celebrating 100th year birthday of Chern. I really appreciate your explaining these to us, I am particularly interested in this, because Academia Sinica has been asking ourselves this question how could we be like Max Plank Institute, at least operate in some aspects of it. I gathered from you, one aspect of it is that we have a different culture, so painful as it is. You guy just say that sometimes it has to be done, to close one institute.

GH: This can be very painful for the old professor, he may feel insulted, that is a big problem, because he thinks, “Oh, my research is not appreciated if they close this …” So that is painful.

TPL: If Taiwan wants to move ahead, I guess we need to be able to do that. Otherwise you cannot form another group, right?

GH: Ya, Ya.

TPL: And also to form another group, you have to have most of them temporary positions

GH: I admit this is a difficulty for a pure research Institute, like Max Planck or Academia Sinica in competition with top private universities. It has happened once to me that we made, I think a very good 5 years offer, good salary, associate professor level, we lost the competition to MIT because MIT offered tenure track and we could not offer tenure track. So there was some discussion whether we should also have a tenure track system for very few scientists in order not to lose in this competition. That’s a difficult point - you see MIT as a university, they can take the risk of making a tenure track appointment, because if the person after 15 years is not so great, they just ask that person to teach more, whereas at Max Plank there is no teaching there is only research, so if you make a mistake in giving a young person a permanent appointment it is forever.

TPL: And then nobody is happy. Maybe we could turn to something little bit more personal. After you become director you have to oversee things. You pointed out inevitably one has to start with one given field, with focus and being maybe narrow. On the other hand, you also pointed out that one should have a broad perspective, but not to waste one’s time on many superficial things. This seems to be easier to say than to do, what would you advice to young people when they are junior year/senior year or 1st year graduate students? Do you have advice for them?

GH: I would advise to pursue these two aspects by trying to identify maybe already within the first year some good project where they can do something by themselves, where they can do some calculations, try a little problem, and then read, of course get advice which are the most important papers in the field that came out last five years, study these things, maybe try a project by themselves immediately, as quickly as possible, and then also go to the colloquium lectures, go to seminar courses, sometimes there are offers of a week-long or two-week-long summer schools, attend these schools, learn something different, get advice from senior people how to do a good mixture.

TPL: That’s very concrete advice. You also make me to think an institute should provide such environment for young people. The professors should encourage young people to look for some projects, maybe very small, maybe completely unrealistic but encourage them to do that.

GH: Not just read what other people had done but try something themselves.

JHC: Are you interested in other subjects in addition to mathematics?

GH: I am interested in physics, then of course personally I am interested in music, I play piano as an amateur. I like to read, and my family, the family comes first, then comes science, then come the other subjects.

TPL: What is happening to geometric analysis, a field for which you have made fundamental contributions, in your point of view?

GH: I think it has become possible during the last 50 years, in view of great breakthroughs in analysis, say, De Giorgi, Nash, or the regularity theory for harmonic maps, then minimal surfaces, it became possible to use these structures like minimal surfaces, harmonic maps, constant mean curvature surfaces to investigate manifolds, structures in differential geometry that before then could only be investigated with ODEs, geodesics, measuring angels, but now non-linear PDEs are understood so well that they could be used as tool box in understanding structures in differential geometry. And I think we’ve only started, there will be much more that can be done, in particular I think really elliptic and parabolic theory, Ricci flow is a prime example, has become a fantastic tool in a new relation between analysis and geometry. And I think in hyperbolic PDEs, this is harder and will take more time, but eventually there will be lots of evolution in hyperbolic PDEs being applied to differential geometry and of course, the Einstein equations are the first and very difficult example.

JHC: How did you start the work on inverse mean curvature flow with Tom Ilmanen?

GH: Well, in the 70s and in the 80s, one has learned a lot how to solve mean curvature flow with level set methods, and how to create weak solutions that can go through singularities, so inverse mean curvature flow was first suggested by physicists Geroch and by Jang, they said that if you can solve inverse mean curvature flow, then they said great calculation that relates area of horizon of black hole to the mass measured at the infinity in an asymptotically flat space describing an isolating system, but they could not prove in general they had a smooth solution, you would not expect in general a smooth solution, so in this work with Ilmanen, Ilmanen had done great work on these weak solutions on mean curvature flow, and my expertise in general relativity came together with Tom’s knowledge of weak solutions, it turned out the techniques from mean curvature flow for level set methods can be applied to this inverse mean curvature flow, to construct weak solutions, and we still have the monotonicity properties that physicists have observed.

JHC: In what occasion did you learn physicists’ view? Seminars?

GH: It was Shing-Tung Yau. He said you should read this paper, there is an interesting flow why don’t you look at that. It was suggested by Yau.

TPL: You mentioned also the fluid and elasticity. Is there a natural bifurcation from the geometric analysis to go to the fluid, elasticity?

GH: In our Institute one of the big projects the astro-physicists are interested in, and our sister institute in Hanover is interested in is the detection of the gravitational waves, so Hanover is building this laser interferometer. They also have a big data analysis group, they have to know a priori what they are looking for, what is the shape of the gravitational wave they want to detect. So the astro-physicists in our institute try to simulate phenomena that create gravitational waves, so this has to be massive major events in the universe, like collision of massive black holes, collision of very dense neutron stars, or neutron stars swallowed by a black hole, so they have to study problems where the matter is present. Like in the neutron star you have a fluid, you have a very complicated equation of state, you have an elastic crust of this neutron star, and you want to understand how it interacts with the black hole, or the other neutron star. Since the fields are very strong, the density is very high, you cannot decouple the matter equations from the geometry, you need both, because the space time is very strongly distorted you need the differential geometric techniques, but you also need to understand of course the fluid equations, you need both, you need understand the coupling between the two, maybe you even have a coupling with the electro-magnetism, there might be pulsars, rotating neutron stars which have huge magnetic fields and Gamma-ray bursts and...

That’s something that the astro-physicists do from an experimental computational point of view, and the mathematicians have to carefully select projects where they can prove something. There is no hope of course to prove existence of long-time solutions to such complicated coupled systems in strong field regimes.

TPL: Analysis could indicate that certain things should happen, certain things could not happen, something like that.

GH: Yes. We can help them to find a good gauge, for example, for their numerical computations, and I think that’s one of the contributions really mathematicians made, teach the physicists how to choose a good gauge, good coordinates, which make geometric sense in order to do their numerical computations. If they don’t choose a good gauge, which is geometrically natural, then their algorithms would not be stable.

TPL: Certain invariants should be preserved numerically, otherwise the error accumulates, and it’s bad. So those numerical guys and physical guys listen to each other?

GH: Well, I think it works in our Institute, but it’s something that one has to carefully teach the young scientists, because it is a very different culture in the physics side and in the mathematics side; on the physics side you do more techniques where you experiment a bit, and you create algorithms by experience, right? You do test problems and so on, whereas on the mathematics side you want to prove error estimates and so on, and people have to understand that both are useful, important, and that is different.

TPL: Are there situations where actually it is necessary for the director to bridge the two?

GH: I think at senior level, all the directors have this conviction, this point of view.

TPL: I feel it is really not just physics and mathematics. I am trying to do in a small way myself to go to kinetic theory, and I can see it is very difficult, it is not so much that we have different languages, different analysis and so on, but also we have different goals when I collaborated with a senior elliptic PDE person, very smart. His interest is in deep analysis while I am interested in physical phenomena. I have the feeling that he does not have enough satisfaction in this project, the analysis is not deep enough for him. I gradually feel that I should not try too hard to get senior people involved; instead I focus on young people.

GH: Right, but even with young people, I think in mathematics we simply have a culture, allow them a lot of freedom to develop themselves. So that has happened to me that I asked a young person to start a joint project with a numerical analyst, say you should learn the analysis, then you should do the numeric and you should do great simulations. And the student started, and after 6 months, he came, said,” I find the estimates much more interesting, I would like to focus on the analytic work and I don’t want to do any programming.” I think in the mathematics culture it would be very difficult to tell this person, no, you should do such and such, whereas in the physics culture, when they do this big computational project, it is quite normal and accepted that they tell the person your job is just to parallelize this algorithm or your job is to compute the initial data, your job is to model the horizon of the black hole, that’s your project and you have to do that.

TPL: This is difficult. It reminds me of the story about Michelangelo. The Pope said to him: Forget about the 50 or so sculptures you are thinking of for the tomb, go paint the ceiling. Michelangelo was unhappy, he’s no painter, he said, “But I am a sculptor.” But look how great the ceiling of Sistine Chapel was. It was not done voluntarily, Pope said, “I have some gold for you if you paint the ceiling.”

GH: Ye, let’s find the balance.

TPL: I have also found out that those who are physics major or engineering major and then switched to mathematics are the ones who are particularly attracted to analysis, and therefore some of them actually abandon physics or engineering completely. What are you focusing on research these days?

GH: Well, actually what I want to talk about in this lecture this morning. I am looking at mean curvature flow, that’s one project to understand the singularities, and a long term goal would be to understand singularities of flows where you have just very weak curvature conditions. So we could do a joint work with C. Sinestrari, flow of two-convex surfaces and show that there is a surgery algorithm, very similar to what Hamilton and Perelman have done in the Ricci flow, and the question is can one relax this two-convexity condition, to allow singularities which look like sheets of bubbles, this in the Ricci flow case would be like relaxing in 4 dimensions curvature conditions and have a sheet of 2-spheres (57:57) collapsing. Can one still get complete classification of singularities, can one do a surgery maybe on these two dimensional singularity sets, but that is a very long term project and requires a lot of estimates, and in this work with Sinestrari we try to get one estimates after the other, at the moment it has led to estimates that allow us to classify certain ancient solutions of mean curvature flow. Ancient solutions give another connection to physics. From the physics point of view, mean curvature flow and Ricci flow are related to renormalization group flow, and smoothing out certain quantum field theories and then for the physicists the interest is time going backwards to minus infinity, because that’s leading to the original quantum field theory, whereas so far mathematicians have concentrated more on time going forwards, because that is smoothing out and making the manifolds nicer. It turns out if you look at these solutions, simple conditions that the solution has existed all the time backwards to the minus infinity, puts a big restriction on the solutions, so you can try to classify these ancient solutions.

TPL: This is very natural, because life does not start from today.

GH: Right, back there is a long history and puts a restriction to what we can see today. It also comes naturally out of studying the singularities of these flows, because if you rescale singularities you get one of these ancient solutions. So you hope you can classify the ancient solutions, you can classify the singularities.

TPL: You are going to give a series of talks next week, thank you very much for coming, I hope when you have a little bit time to spare to come, we won’t work you hard, there are interesting things in Taiwan you can look around.

GH: Thank you very much, I really enjoy my stay here. I really enjoy the explanations you gave to me yesterday in the museum.

TPL: Thank you very much.

- Tai-Ping Liu and Jih-Hsin Cheng are faculty members at the Institute of Mathematics, Academia Sinica.