 is two and the remarkable thing is that if we scale up this cube then it will still only be two ways of putting the instantons in there. The plus or minus signs that we saw before in counting drains and spigots also applies here and are important for the accuracy of that statement. We can also start deforming the cube in various ways and still we'll only get two ways of putting three instantons into the cube. So that's a taste of what the Donaldson invariants are like. Now using this new in 1984 new topological invariant Donaldson could prove some dramatic new results combining it with results of Michael Friedman he could prove that there are exotic four-dimensional spaces where you can't do calculus. You might be thinking well of course I can't do calculus on an exotic four-dimensional space but not the point is even the best mathematicians in the world can't do calculus on these four dimensional spaces because it's logically impossible to do so and for this and other remarkable results Donaldson shared the 1986 Fields Medal and he shared it with Michael Friedman. This is the statement here is really a synthesis of the results of Donaldson and Michael Friedman who is also producing very deep results about the topology of four-dimensional manifolds. So all was sweetness and light except except that Donaldson invariants are extremely hard to compute and interpret. It took a lot of effort to compute a few special examples. We have experts here in the audience who were leaders in this in this endeavor and in following up on the Donaldson invariants like John Morgan and he knows and he could tell you that it was it was difficult it was a hard slog so in a sense mathematicians hit a wall. Progress was being made but it was slow and difficult progress. Well that's an opportunity and the opportunity came when Michael Atia asked a very important question in 1987 he asked the physicists what's the physical interpretation of the Donaldson and Jones invariants the Jones invariants are in variants of knots which I haven't talked about but he asked the question about both and both are important. One physicist that listened was Edward Whitten and he went and thought about it and his answer was that the Donaldson and Jones invariants can be computed within the framework of a Yang-Mills field theory so for the Donaldson case it's technically a not quite the Yang-Mills theory that we use to describe the forces between quarks and in our world but it's it's essentially mathematically the same kind of thing the only difference is that there's a different set of quarks and electrons from what we see in nature and in particular there's something called supersymmetry. So in the course of answering this question Whitten wrote a paper called topological field theory and topological field theory is a vast simplification of physics it was an important new idea where you could still do physics but at the same time length scales and time scales do not matter topology is what matters so there's no difference between yesterday last week a hundred years ago a billion years ago there's no difference between your commute to work and your extra galactic vacation trip. What does matter is topology so these theories sense things like linking circles we can scale up those circles so that they're light years apart hundreds of millions of light years apart billions and billions and billions of light years apart and so they can't communicate but still they're topologically linked. So this has had a huge impact in both physics and mathematics there are thousands of papers on the subject it opened up fresh woods and pastures new and it might even be practical people at Microsoft station Q are using topological field theory as a potential road to quantum computation. So all was sweetness and light. Except except that Whitten's answer involved a quantum field theory. Computing Donaldson invariance all a Whitten requires computing probabilities in a quantum field theory. Quantum Yang-Mills theory how hard can that be? Well in the case of a billion field theories that's the Maxwell theory it's hard but it's solvable it's a solved problem it culminated in the formulation and by Feynman-Schwinger and Tom and Aga of quantum electrodynamics right after the war and so then there are very efficient ways of keep computing in these a billion field theories. The non-Abelian case is much to the nth power harder in fact it's not solved yet. So now recall that Whitten reformulated the Donaldson invariance as probabilities for certain events in Yang-Mills theories and four-dimensional spaces but computing probabilities in Yang-Mills theory is very difficult so we seem to exchange one hard problem for another. But that opens up an opportunity the physicists had a trick up their sleeve called effective theories. So recall topology is unaffected by distance by scaling things up like those two loops. We can scale things up to bigger and bigger sizes and we don't change topology. What happens to Yang Mills theory when we scale things up? In Yang-Mills theory one asks questions about advanced events at larger and larger distance scales and what can happen is the answers can simplify. The answers can be the same as answers to analogous questions in a different but simpler theory. Such a theory is called an effective field theory. It's a little bit like pointillism in art so if you take a pointillist painting and you look closely at it you don't really have a picture of what's going on but if you step back from the painting then you see a picture of what's really going on. So it's like that with effective field theory. So if we look at nuclear distance scales well it's all very complicated this theory of Witton and Natia and Donaldson and they're all sorts of complicated mathematical compactifications and complicated Lagrangians and it's just a mess. But now if you step back and look at I look at it on the scale of this room that's about 10 meters well you don't see much but persist. You now go to really really long distances like 10 to the 500 meters well clearly something's happening and now you scale back even further and something's going on and even further and it's clear that there's a picture that's emerging and the picture is an Abelian field theory. A generalization of Maxwell's theory it was written by Zyberg and Witton and these are called the Zyberg-Witton equations. So the F here is the field strength of an Abelian Maxwell like field that's the important point notice that it is nonlinear that will be important in just a moment. So the Zyberg-Witton paper said indeed that viewed from afar the Yang-Mills theory used by Witton simplifies dramatically. It's a field theory based on an Abelian group and therefore it's much easier to work with. Now because the Zyberg-Witton equations are nonlinear they still do have soliton like solutions these are very very closely analogous to vortices and a superconductor. And because we can measure topology by counting that's a theme throughout this talk you can define new four-dimensional invariance called Zyberg-Witton invariance by counting these vortices just the way you counted instantons now you count vortices. Okay now these are not anything close to a complete topological invariance so there's no clean mathematical argument that one has to be a function of the other but of course they came up in the same theory so you would expect that one set of invariance can be written in terms of another. And Witton indeed conjectured a formula for that and Witton and I wrote a paper where we derived very carefully the relationship between the Donaldson and Zyberg-Witton invariance. I remember Edward saying to me you know Greg we need to clear the air nobody believes this conjecture and so we went and we gave a very careful derivation which is now a rigorous mathematical theorem. Now but the point is that the Zyberg-Witton invariance are in an Abelian field theory so they are much easier to compute. It's like trying to do a calculation with Charles Babbage's engine number one or doing it with Mathematica on your laptop. So this was a huge breakthrough and now the response of the community was very interesting. So you see there's an interesting cultural difference between physicists and mathematicians. If a physicist gets a really good idea then she or he will work feverishly over the weekend and then get the paper out the next week. Okay and okay you know the logic isn't quite clear and maybe the steps are in the wrong order and you know the implications haven't been thought out but it's part of the it's out there it's part of the conversation and the physicists rush in because all of the really good and easy calculations are on sale. Now if a mathematician gets a really good idea then she or he will think about it feverishly over the weekend and over the next week and over the next month and over the next year and all the logic is going to be you know perfectly in order and all the steps are going to be absolutely correct and all the implications have been thought through perfectly so that when the paper comes out it's like a Greek temple. Okay so so this is the difference between the physics and the mathematics community but it did not apply in this case. In this case after this breakthrough was the mathematicians that rushed in and there was a mad dash after this breakthrough and it was this whole episode was summarized very beautifully in a masterful review by Simon Donaldson and he begins it with this with this statement in the last three months of 1994 a remarkable thing happened this research area was turned on its head by the introduction of a new kind of differential geometric equation by Zyberg and Witten. In the space of a few weeks long-standing problems were solved new and unexpected results were found along with simpler new proofs of existing ones and new vistas for research opened up. So that's what I wanted to tell you so now it's time to wrap it up so we began with these two basic questions in mathematics and physics which appeared to have nothing to do with each other I hope you now appreciate that they're actually very deeply related to each other. Now to my mind it also had an important sociological implication there was a debate that was raging much more fiercely in the 1990s still to some extent to this day you see physical mathematics is not without its critics there are critics from the physics side and there are critics from the mathematics side now from the mathematics side the criticism is well you physicists you never prove anything you don't even define what you're talking about so we have no idea what you're talking about and you know these these complaints are actually quite legitimate and the most articulate expression of those of that point of view was in an article by Arthur Jaffe and Frank Quinn but the tone of the article was actually quite strident you know they used in the abstract words like dangerous and unpleasant and destructive and this elicited an immediate response from a large number of scientists mathematicians and physicists the leading article was by Michael Atia it's very cogent very well-reasoned but to my mind the best response of all was the subsequent discovery of the Zyberg Wittman invariance so we have this remarkable story where the mathematicians start with the shape the question the shape what is the shape of a space and the physicists start with a question what is the strong force and now they turn that into a more more technical accessible kind of question like how do we define topological invariance and let's let's use the Yang-Mills instantons to try and understand confinement now that those ideas can then be lobbed over to the other side and of course the mathematicians will turn it into something the physicists don't understand like Donaldson invariance the physicists will take those topological invariance and create topological field theory this can be lobbed back to the other side and then the the physicists they like to compute things so okay so there's these Donaldson invariance and some Yang-Mills theory can we compute it well you are now led to long distance effective theory which is the Zyberg Witten theory in this case you can then lob that back over to the mathematics side and that turns into major advance in mathematics the Zyberg Witten invariance okay so there's this remarkable back and forth okay so what about the future so this happened about 25 years ago so you might be thinking well has anything else happened since then and I just want to stress that this is just one story one vignette in this much larger subject of physical mathematics there are many other stories I could tell there are people in the audience who participated in have participated and are participating in ongoing very interesting stories about remarkable back and forth between mathematics and physics you might be thinking well what about four manifolds well let's remember that there are problems in the theory of four manifolds which the Zyberg Witten invariance don't touch so this this paper was written 12 years after the discovery of the Zyberg Witten invariance so there are those of us in physics who who believe that we can still use physics to learn things about four manifolds and there has been progress over the last 25 years I produced a few results myself so is Sergei Gukov in the audience so progress is being made nothing like the revolution of the Zyberg Witten invariance but still there are these major open problems out there so you know that's the nature of research you you have some hunch you have some hopes you have some thoughts some ideas you are pursuing a path but you have no idea if you will arrive until you get there so thank you very much thanks very much great so we have some time for a few questions I must not have been very clear be ineffable so yeah please well I don't see any direct connection what would you what you're saying actually reminds me of you know a a failed attempt to explain the periodic table so the origin of knot theory is actually in a in an attempt to you probably know this as an as an attempt to explain the periodic table using knots of ether so okay so to me those questions seem a priori unrelated but of course you know a good mathematician given given some time we'll we'll start imagining things and maybe maybe another connection between shapes of spaces and forces as I said this is only one story in a much larger story so if we think broadly of the shapes of spaces as as questions and topology then as you know there are many many non-trivial topological statements we can make using ideas from physics and field theory so I guess I would generally agree with you what you're saying anything else okay well let's thank great very much