by Rob Kirby
Let’s start up again by talking about your experience of studying physics. When was that?
Let me set the stage for some of the vignettes from my foray into the world of physics. There was a special year at the Institute for Advances Study in Academic 1996–1997. The purpose of that year was for theoretical physicists to explain to mathematicians how physical ideas were leading them to fantastic mathematical conjectures and statements unlike anything seen before in mathematics. The hope was that we mathematicians would get some understanding of how their “art” works, so that we might be able to perform it as well, or at least understand where it comes from.
As part of this program, there were various series of lectures. In particular, Witten gave a few lectures in the fall and then an entire course in the spring. In addition, he even gave us homework. Witten acted both as professor and TA, for he wanted to see how we were absorbing the material he was presenting. David Kazhdan, who had a head start on the material, also acted as a TA.
At the beginning, the problems were not too difficult and we could do them. But by January, we were all getting beat down because the material was harder and harder for us and we understood less and less.
So who was “we”?
As I said, Kazhdan was there. Others included Dan Freed, Lisa Jeffreys, Pierre Deligne, Pavel Etingof, me. Dave Morrison was in and out. Younger people — Dennis Gaitsgory and Roman Bezrukavnikov — joined in the spring.
In one problem session, Deligne is at the board, working out a solution in the usual “Deligne” way, starting from first principles. Witten has his head down, getting more and more frustrated. Finally, he says, “In about six weeks I am going to give a series of lectures about Donaldson Theory. It will rely on all the material covered by the problems I am planning to give you between now and then. But if you continue to proceed at this pace, you will not be ready.” Deligne meekly puts the chalk down and returns to his seat. No one else dares to go to the board. Witten, in complete exasperation, says “You know this is…this is not…this material is not that hard. We teach it to first-year physics graduate students in a year. What is the matter with you guys? I give lectures that I feel are pretty good and then I give you homework problems where I make a minor change from what I lectured about and then you can’t do the problems!”
Never in my life have I had more sympathy for my calculus students, since this is often exactly how I feel about them!!
Still, Witten’s question is a fair one, right? One partial explanation is the difference in background and training in physics and math. But it goes beyond that. We have different cultures, different ideas of what type of answer is a satisfying one, different starting points.
I have heard many stories of how a year at the Institute changed the career trajectory of a mathematician. Of our group, only Dan Freed seemed fundamentally enriched by the year. Certainly, I struggled with the material both during that year and to some extent later, to no real benefit to my mathematics.
Anyway, back to the special year. Witten was building up to the study of \( n=2 \) supersymmetric gauge theory in dimension four, a supersymmetric Yang–Mills theory. Witten drove home the point that quantum field theory was the study of questions and answers. A question is posed by a fundamental quantum field theory; that is to say, one that is well defined at arbitrarily high energy scales. To answer a question is to give a description of the relevant theory at low energies — energies that we see in every day life or in our accelerators. Now, \( n=2 \) supersymmetric gauge theory in four dimensions is a fundamental theory, and the moduli space of vacua is the Donaldson moduli space. The question is what does this theory look like at a low energy. The answer turns out to be Seiberg–Witten theory, which is not a fundamental theory but rather one that is well defined at low energies.
You were supposed to deduce that?
No. Witten explained in his lectures how he and Seiberg did that.
And your homework problems were steps along the way?
Yes. The problems were teaching us how to deal with supersymmetric theories. The fields in supersymmetric theories come in multiplets which are various representations of the symmetry group, vector representations and spinor representations. We did easier cases, including \( n=1 \) supersymmetric cases, Sigma models that are easier to study than gauge theories, and gauge theories in dimension 3. All of these simpler examples were aimed at giving us a feeling for how things worked. With all that under our belt, Witten then explained in several lectures the \( n=2 \) supersymmetric gauge theory fields to us before deriving the Seiberg–Witten answer to the Donaldson theory question.
But speaking for myself, I really did not fundamentally understand what was going on the entire year.
One last anecdote along these lines. Toward the end of the year, I was sitting outside our apartment on Einstein Drive at the Institute with my wife Ellen and her cousin, who had come up from Philadelphia. He asked me why I was spending a year at the Institute. I briefly told him about the special year, the physics lectures, etc. He said, “Well, how’s it going?” I said, “You know, I’ve always been able learn new mathematics quickly and easily. It is one of my strengths as a mathematician. But I have been trying to learn this material for almost a year now, and I have no FUCKING idea what is going on.”
Ellen said she had never heard me curse before.
I just could not get it. I don’t work on it as hard now as I did then, but even now, I keep trying little bits and pieces, and it still escapes me.
One problem I have, and I think many mathematicians have, can be illustrated by another anecdote from Witten’s lectures in the fall. Witten would introduce a term, almost in unison we would say, “Give us a definition.”
[laughter]
As a mathematician, how can you think about something if you don’t have a definition. Without a definition, you don’t know what you are talking about. Witten’s response was always the same. He would say, “It’s too early to give a definition because we do not understand the range of phenomena. If I were to gave you a definition now, it would leave out things I would want to study and include in this context. So it is too early for a definition.”
Right.
That is part of the problem. I have also asked myself what is missing from my background that would have helped? The first thing I think of is a course in statistical mechanics. Statistical mechanics is a Euclidean theory — the Euclidean version of quantum field theory. Understanding the simpler Euclidean case first would have been helpful.
Another issue is the importance of scale. This is crucial to physicists, and we do not appreciate it the way they do. To me, a manifold might fit in my hand, but it is always going to fit in the room and will be big enough to be seen with the naked eye. An instanton is a solution to the Yang–Mills equation on a tube (with time as the direction down the tube) that is asymptotically flat at infinity. I imagine it as almost flat outside a region of a couple of meters in the middle of the tube, where all the action takes place. But the name comes from the physics idea that the nonessentially flat region is an incredibly small interval of time, essentially an instant. At the other extreme, I asked Witten how he thought about quantum mechanical waves. He said, “Think of the ocean. From space it looks like a smooth sheet of water. At this long scale, the waves are so tiny as not to be noticeable. But at a smaller scale up close, they can be very important.”
Then there is the question of what a good answer is. Give a physicist an algorithm to compute some physical phenomenon and he is happy. He will compute away and then (hopefully) compare the answer with what an experiment yields — or at least test whether the answer seems physically “reasonable.” No mathematician is happy with a solution like that. We want to know the underlying assumptions, the extent of applicability of the argument, where the exceptional cases lie, the key idea or operable principle.
This difference in mode of operation between the fields makes crossing over difficult and frustrating. I have seen some advances among younger people, who are more comfortable or more accepting of the physics manner of thinking about things. Sometimes they can use it to find new mathematics.
You interacted with Thurston quite a bit.
I did. One of the great geometers of my lifetime. When I look at many great mathematicians that I have known, I divide them into two categories. There are those who have mathematical insight and power that I can understand because it is very similar to the way I approach mathematics. They are just much better at it. Deligne comes to mind in this category. Then there are those for whom I have no clue where their inspiration comes from. Thurston is in this category. Whatever he did was unlike anything I’ve ever seen in anybody else. His geometric intuition and his ability to connect that intuition to whatever problem was at hand it’s just…it’s just beyond words. Gromov is another one in this category.
Taubes made that comment about Kontsevich very early on: that he’s from a different planet.
I can see that. Anyway, Thurston was a marvel — so geometric. I remember at the Bowdoin conference (in 1979), we were talking informally in his apartment. He was talking about the Poincaré Conjecture. He had an approach: Start with any compact 3-manifold and remove a random complicated knot (or link). The complement will have a complete hyperbolic structure. Think of there being a cone singularity along the knot of cone angle 0. Now begin to enlarge the cone angle. Can one deform the hyperbolic structure to give a compact hyperbolic metric with singularity along the knot with that cone angle? Thurston had shown that one can continue this process until the cone angle reaches \( \pi \). This was his orbifolding theorem. At this point in his explanation he seemed to be staring off into space watching this process unfold and he said, “As you go past \( \pi \), things get very, very complicated.” I thought then that he was not going to be able to resolve the Poincaré Conjecture.
I gave a lecture about 25 years later at Cornell about Perelman’s solution to the Poincaré Conjecture. Thurston was in the audience. At the end of my talk, I asked him what he thought about the argument. He said that he always thought it would sort of go like this. Thurston had understood that, as you deform hyperbolic manifolds, they limit to pieces of dimension \( 0,1,2,3 \).
He formulated the Geometrization Conjecture so, clearly, he had the correct picture.
Right. I think he thought that his deformation method didn’t have enough control. That had to come from some geometric PDE.
Iz Singer around ‘74 or ‘75 remarked to me, “You three-dimensional topologists should think about using analysis.” I don’t know what insight he had, but it turned out to be good.
This was also Yau’s perspective. It started with Meeks–Yau (the equivariant version of Dehn’s Lemma and the Loop Theorem) which was a nice application of minimal surfaces to topology. I used to have debates with Yau. My position was that topology had worked out the higher dimensions and we would get three and four eventually. Yau’s response was, “No, no, no. You are going to have to use analysis.” Twenty-five years later, I concede that Yau was proven right. The low dimensions are too hard to do without geometry.
Maybe Singer was talking to Yau.
And/or Atiyah.
Yau really believed that Ricci Flow would solve the Poincaré Conjecture. Some time in the ‘90s, Yau was on leave from Harvard and spent a semester at Columbia. He and Hamilton (the inventor of Ricci Flow) were closeted in a seminar room every day working on this problem. Yau really thought Ricci Flow could resolve the Poincaré Conjecture. It did. But as it turned out, it needed one more incredibly powerful idea from Perelman to get the result.
