Newer developments

Can we move to newer developments with impact from the Atiyah-Singer Index Theorem? I.e., String Theory and Edward Witten on the one hand and on the other hand Non-commutative Geometry represented by Alain Connes. Could you describe the approaches to mathematical physics epitomized by these two protagonists?

ATIYAH: I tried once in a talk to describe the different approaches to progress in physics like different religions. You have prophets, you have followers - each prophet and his followers think that they have the sole possession of the truth. If you take the strict point of view that there are several different religions, and that the intersection of all these theories is empty, then they are all talking nonsense. Or you can take the view of the mystic, who thinks that they are all talking of different aspects of reality, and so all of them are correct. I tend to take the second point of view. The main "orthodox" view among physicists is certainly represented by a very large group of people working with string theory like Edward Witten. There are a small number of people who have different philosophies, one of them is Alain Connes, and the other is Roger Penrose. Each of them has a very specific point of view; each of them has very interesting ideas. Within the last few years, there has been non-trivial interaction between all of these.

They may all represent different aspects of reality and eventually, when we understand it all, we may say "Ah, yes, they are all part of the truth". I think that that will happen. It is difficult to say which will be dominant, when we finally understand the picture - we don't know. But I tend to be open-minded. The problem with a lot of physicists is that they have a tendency to "follow the leader": as soon as a new idea comes up, ten people write ten or more papers on it and the effect is that everything can move very fast in a technical direction. But big progress may come from a different direction; you do need people who are exploring different avenues. And it is very good that we have people like Connes and Penrose with their own independent line from different origins. I am in favour of diversity. I prefer not to close the door or to say "they are just talking nonsense".

SINGER: String Theory is in a very special situation at the present time. Physicists have found new solutions on their landscape - so many that you cannot expect to make predictions from String Theory. Its original promise has not been fulfilled. Nevertheless, I am an enthusiastic supporter of Super String Theory, not just because of what it has done in mathematics, but also because as a coherent whole, it is a marvellous subject. Every few years new developments in the theory give additional insight. When that happens, you realize how little one understood about String Theory previously. The theory of D-branes is a recent example. Often there is mathematics closely associated with these new insights. Through D-branes, K-theory entered String Theory naturally and reshaped it. We just have to wait and see what will happen. I am quite confident that physics will come up with some new ideas in String Theory that will give us greater insight into the structure of the subject, and along with that will come new uses of mathematics.

Alain Connes' program is very natural - if you want to combine geometry with quantum mechanics, then you really want to quantize geometry, and that is what non-commutative geometry means. Non-commutative Geometry has been used effectively in various parts of String Theory explaining what happens at certain singularities, for example. I think it may be an interesting way of trying to describe black holes and to explain the Big Bang. I would encourage young physicists to understand non-commutative geometry more deeply than they presently do. Physicists use only parts of non-commutative geometry; the theory has much more to offer. I do not know whether it is going to lead anywhere or not. But one of my projects is to try and redo some known results using non-commutative geometry more fully.

If you should venture a guess, which mathematical areas do you think are going to witness the most important developments in the coming years?

ATIYAH: One quick answer is that the most exciting developments are the ones, which you cannot predict. If you can predict them, they are not so exciting. So, by definition, your question has no answer.

Ideas from physics, e.g. Quantum Theory, have had an enormous impact so far, in geometry, some parts of algebra, and in topology. The impact on number theory has still been quite small, but there are some examples. I would like to make a rash prediction that it will have a big impact on number theory as the ideas flow across mathematics - on one extreme number theory, on the other physics, and in the middle geometry: the wind is blowing, and it will eventually reach to the farthest extremities of number theory and give us a new point of view. Many problems that are worked upon today with old-fashioned ideas will be done with new ideas. I would like to see this happen: it could be the Riemann hypothesis, it could be the Langlands program or a lot of other related things. I had an argument with Andrew Wiles where I claimed that physics will have an impact on his kind of number theory; he thinks this is nonsense but we had a good argument.

I would also like to make another prediction, namely that fundamental progress on the physics/mathematics front, String Theory questions etc., will emerge from a much more thorough understanding of classical four-dimensional geometry, of Einstein's Equations etc. The hard part of physics in some sense is the non-linearity of Einstein's Equations. Everything that has been done at the moment is circumventing this problem in lots of ways. They haven't really got to grips with the hardest part. Big progress will come when people by some new techniques or new ideas really settle that. Whether you call that geometry, differential equations or physics depends on what is going to happen, but it could be one of the big breakthroughs.

These are of course just my speculations.

SINGER: I will be speculative in a slightly different way, though I do agree with the number theory comments that Sir Michael mentioned, particularly theta functions entering from physics in new ways. I think other fields of physics will affect mathematics - like statistical mechanics and condensed matter physics. For example, I predict a new subject of statistical topology. Rather than count the number of holes, Betti-numbers, etc., one will be more interested in the distribution of such objects on noncompact manifolds as one goes out to infinity. We already have precursors in the number of zeros and poles for holomorphic functions. The theory that we have for holomorphic functions will be generalized, and insights will come from condensed matter physics as to what, statistically, the topology might look like as one approaches infinity.

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