 No, thank you, Riff. So, thank you for the invitation to give this public talk, and this is a public talk, so, if you're an expert, well, bear with me. So, the title of the talk is The Shapes of Spaces and the Nuclear Force, and here's the outline for the talk. So, we'll begin by talking a little bit about physical mathematics, and then we'll talk about a problem in mathematics about topology, and then a problem in physics about forces, and then the rest of the talk will be about the relationship between these two things. So, let's begin. So, what is physical mathematics? Well, it's a noun, and that's how it's pronounced, and physical mathematics is a fusion by definition of mathematical and physical ideas motivated by the dual, but equally central goals of elucidating the laws of nature at their most fundamental level, together with discovering deep mathematical truths. Now, that's quite a mouthful, so I want to take a quasi-historical viewpoint of explaining a little bit about what I mean. So, I'm going to show you just a few snapshots from the great debate over the relationship between physics and mathematics, and it's going to be, this is a huge subject, so we're going to be tourists, so I'm just going to share a few snapshots. So, if we go back to the beginning of a modern scientific era with Kepler and Galileo and Newton and Leibniz, we could ask, were these mathematicians, or were they physicists? And the answer is neither. They were natural philosophers. Now, nowadays, in our universities, we have, yeah, there's something wrong with this. Yeah, just move out a little bit. Okay, it's bothering me a bit. How's that? Can you hear me? No? Yes? It's not attached now. How's that? Okay, so now, nowadays, in our universities, we have departments of mathematics and departments of physics. We don't have departments of natural philosophy, and indeed, the mathematics and the physicists don't always talk to each other, so we could ask, when did natural philosophers become either physicists or mathematicians? And even around the turn of the 19th century, if you think about figures like Euler, Lagrange, and Gauss, they made major contributions to both math and physics, and they probably would have considered themselves natural philosophers. But sometime in the middle of the 19th century, that situation clearly changed, because, for example, if we look at J.J. Sylvester, who's definitely a pure mathematician as president of the British Association, at a meeting, he said, what is wanting is a discourse on the relation of the two branches, mathematics and physics, too, and their action and reaction upon one another, a magnificent theme with which it is to be hoped that some future president will crown the edifice. And that future president was James Clerk Maxwell, undoubtedly a physicist, and I recommend his response. It's very interesting, and in the course of that response, he actually recommends to the mathematics community his somewhat neglected dynamical theory of the electromagnetic field, another theory of electricity, which I prefer, and so on, and according to Freeman Dyson, the mathematicians did not pay attention, thus constituting one of the greatest missed opportunities of all time. Now, that situation began to change around the turn of the 20th century at the first international congress of mathematicians in Zurich, Henri Poincaré chose as his topic the relation of physics to mathematics, and he had a lot of interesting things to say about Maxwell. At the second international congress of mathematicians in Paris, David Hilbert famously announced his 23 problems for the 20th century, and if you haven't read this, I highly recommend reading this article. It's a great read. The opening is magnificent. Who of us would not be glad to lift the veil behind which the future lies hidden to cast a glance at the next advances of our science and at the secrets of its development? You can hear in the background the opening bars of Alzo Sprachs de Ritustra, and unlike Strauss, the rest of the article actually lives up to this opening. This is the most famous and the greatest problem set of all time. If you think doing those problems under the tent is challenging, well, try doing some of these. Your homework will be a little late, but it's still worth turning in. The second problem, sorry, the problem that concerns us here is the sixth problem, to treat by means of axioms those physical sciences in which mathematics plays an important part. Well, little could hair professor Dr. Hilbert know on that cold and rainy Wednesday in Paris that within two short months on a warm and balmy Sunday in Berlin, Max Planck in an act of desperation would guess his formula for the energy density of black body radiation, thereby introducing two new constants of nature into physics, H and K, and initiating the upheavals of the first 20 years of the 20s, wife years of the 20th century physics. And that was a great period where lots of major ideas of mathematics came into physics like differential geometry and group theory, linear algebra, and functional analysis. And we, with our hindsight, can see that Professor Hilbert lacked some prerequisites when we teach. We know that you need to have proper prerequisites, quantum mechanics are equivalent, Lorentz group, relativistic wave equation, and so on. So at the end of this period, you'll find statements by people like Dirac and Einstein which are very exuberant about the great interaction between physics and mathematics. My favorite of all of these is the opening of Dirac's paper on magnetic monopoles where he introduced the idea of fiber bundles and connections on fiber bundles into physics. It took a long time before the physicists really understood that's what he did, but that's what he did. And he began his paper by saying the steady progress of physics requires for its theoretical formulation of mathematics that gets continually more advanced. It seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalization of the axioms at the base of the mathematics. And then something happened because, well, a lot of things happened. There were major political, violent political, geopolitical events, and there was nuclear physics with its cascade of amazing experimental results. And over the next few decades, the physicists and the mathematicians drifted apart from each other so much so that in 1972, Freeman Dyson gave a famous talk in which he proclaimed that as a working physicist, I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce. Well, well, I'm happy to announce that mathematics and physics have remarried. The relationship has altered somewhat. AC change began in the 1970s when some great mathematicians got interested in aspects of fundamental physics. And at the same time, some great physicists started producing results requiring ever-increasing mathematical sophistication so that after these decades of ferment, I would say a new field has emerged with its own distinctive character, its own aims and values, and its own standards of proof. One of the guiding principles is certainly the discovery of the ultimate foundations of physics. Hilbert's sixth problem generously interpreted. And that quest has indeed, as Dirac predicted, led to ever more sophisticated mathematics. A second guiding principle is that physical insights can lead to surprising and new results in mathematics. And such insights are a great success. They are just as profound and notable as an experimental confirmation of a theoretical prediction. So in the rest of this talk today, I'm just going to explain one beautiful story of a remarkable convergence of physical and mathematical ideas, just one vignette out of this whole subject of physical mathematics. It's a story in which I played a small role, just a small role, two of the people who played starring roles or at the Institute for Advanced Study. One of the most important stars in this story is unfortunately no longer with us. And that's Michael Atia, who is one of the greatest mathematicians of the 20th century, and definitely one of the most charismatic. You could not talk to Michael Atia without getting excited. He was always bubbling over with ideas and enthusiasm. He was always cheerful, witty and very generous and we're going to miss him quite a lot. So the beginning of our story starts with two basic questions, one in mathematics and one in physics. In mathematics, the question is, what is the shape of a space? How can we tell when two geometrical objects can be deformed into each other? And in physics, the question is, what holds stuff together? How can we describe the forces that attract and repel matter? Now on the face of it, these two questions have absolutely nothing to do with each other. Well, we'll see. So let's begin with the mathematics question. That's a question about a topic called topology. So take a geometrical object. I'll just call it a space for the experts. I'm talking about a compact oriented Hausdorff manifold, but just a space like the surface of a coffee cup, okay? And imagine that this coffee cup is made of malleable material, so you can continuously deform it. You can squish, push, and pull, but you can't cut, rip, or tear. Then as we saw, in fact, in yesterday's cross-program talk, you can continuously deform this coffee cup into the surface of a donut. So in topology, we consider the surface of a coffee cup and a donut as the same. They can be continuously deformed into each other. So now supposing we have different spaces, like the surface of a coffee cup, a donut, and a basketball. We can ask, are they topologically the same? In order to answer that question, we need to find a property that does not change under deformation. Such a thing is called a topological invariant. Now one such property is the dimension of a space, and let's make sure we're all on the same page with regard to dimension. So the screen here is a two-dimensional space, and there's a point on the screen, and why is it two-dimensional? Because there are two independent kind of motions we can use. We can have down and up, well, up is just the opposite of down, so that's one kind of motion, and then we can have right and left, and that's another independent kind of motion. So with those two motions, we can specify the point on the screen, X and Y. We live in three special spatial dimensions. You should think of that X axis as coming out of the screen, but you could of course imagine what would life be like in two dimensions, and that idea goes back at least to Edwin Abbott in a beloved book called Flatland. On one level, it's really a biting satire on the social inequities of Victorian society, but it's also a very good mathematical read, and it's told from the point of view of a square that's confined to live in a two-dimensional space like the surface of that screen. In the course of the story, the square meets beings from line land that can only go left and right, and they start comparing notes and find interesting things. So in that spirit, let's consider topology in different dimensions. So let's start with one dimension. So what can we have in one dimension? We can have a line segment, and we can deform it, and it can get very complicated like the path in a labyrinth or a piano curve, but it's still one-dimensional. Now another thing we can do is we can have a circle, and we can deform that, and it can get very complicated. So now if we have one-dimensional spaces, we can ask how can we tell them apart? So again, a topological invariant is the answer A of S to a question you ask about a space S, and if two spaces can be deformed into each other, that is, if they're topologically equivalent, then the answers have to be the same. So here's a one-dimensional topological invariant. What's the question? Take a little walk, do you get back home? You always walk forward, okay? If the answer is yes, then S is topologically a circle. If the answer is no, it's topologically a line. All right, so that's a very good question. It's an example of a complete topological invariant. So again, a topological invariant is a question so that if S1 can be deformed to S2, then the answers are the same. Now, dimension is a topological invariant, but we see it's not a complete topological invariant because the circle and the line are both one-dimensional and yet they're topologically distinct. A complete topological invariant is a question so that if the answer is the same, then S1 can be deformed into S2. Okay, so now let's go to two dimensions. So we have the surface of a basketball and the surface of a doughnut, and it should be intuitively clear that those are topologically distinct. What else do we have? Well, we can keep adding holes like that and if we restrict to orientable surfaces, then that's a complete topological invariant. Now a theme of this talk, counting holes, a theme of this talk is that you can measure topology by counting something. So as I just said, a complete topological invariant is counting holes, but there are other things you can count to measure topology. So for example, consider water flowing on a surface, a two-dimensional surface. So because of the topology, there have to be drains and spigots or sources and sinks for the water. And we can count adding with plus or minus one the drains and spigots for the water. So there's a rule for how to decide when it's plus one or minus one, which we don't have time to go into, there's a rule. This also turns out to be a complete topological invariant. Now of course, two complete topological invariants have to be functions of each other because they're complete. And it turns out that the sum over drains and spigots will be two minus two times the number of holes. So let's check that. So here is water flowing on a sphere. There's one drain and one spigot and they both contribute plus one. So I get two, which is indeed two minus two times the number of holes. Now here is water flowing on the surface of a donut and there are no drains and spigots, so we get zero and that is indeed two minus two times the number of holes. And just to show you that the minus signs are very important here, consider this two-dimensional surface. And if you're not distracted by other things, you might consider putting a vector field on that surface as the authors of this picture have done and you will get minus two when you sum over the drains and spigots and that's indeed two minus two times the number of holes. Okay, now before I go on, I wanna take about how you can make one topological space from another by a process known as gluing. So you can imagine identifying the endpoints on this interval and getting a topologically different space. So you see this talk here is run on a shoestring. If I take the shoestring, that's one topological space and I glue together the ends then I get a topologically different space, okay? So that's gluing. And of course, we can do gluing in higher dimensions. For example, in two dimensions, we can identify opposite sides of a sheet of paper with the, identifying the orange sides. We get a cylinder, identifying then the green sides, we get a torus. So let's go on to three dimensions. So take a cube and now identifying, now you need to start using your imagination. Let's imagine identifying opposite sides of the cube. I claim you get a space with no boundary and let's understand why that is. So look at the golden arch. This golden arch because of the identification is the same as this golden arch. Now suppose you try to leave the room. So you walk out that golden arch but that golden arch is the same as the other one so you come right back in. So it's like the Hotel California. You can check out but you can never leave. Also you can see the back of your head because if you turn around and face in then the light rays coming from the back of your head come in through the other end and into your eyes. So you can see that things are getting a little more complicated in three dimensions. So if we take for example a dodecahedron, a 12 sided figure and suitably identify opposite sides, we get a space that looks like that. This also figured in cross program talks both yesterday and last week. Clearly things are getting very hard. So it might seem like it's hopeless to go to higher dimensions and ask about topology. Let's first agree that it makes mathematical sense. So we can specify a point in higher dimensions by its address with a number of, by a set of numbers. So in one dimension we have a position x and two dimensions x and y and three x, y and z. Well four dimensions no big deal x one through x four but clearly it's starting to get hard to imagine what's going on and so you might think, well okay it makes mathematical sense but it's hopeless to try and think about topological invariance of higher dimensional spaces. Well fortunately there's an unusual animal that has the ability to think clearly about these topological questions in higher dimensions. This is the subspecies Homo sapiens topologensis and this animal can find topological invariance of higher dimensional spaces. And what has this animal discovered? Well you might expect that as the number of dimensions increases the problem gets ever harder but there's a bit of a surprise. At least for some questions four dimensions is the hardest. There are many unanswered questions about the topology of four dimensional spaces. One thing we know for sure is that the space of the world of four dimensional spaces is really wild and we do not know anything even close to a complete topological invariant. And so mathematicians that work on this topic write papers with titles like will we ever classify simply connected smooth four manifolds? A four manifold is just a four dimensional space. Okay well enough mathematics for the moment let's talk about physics. So our question in physics was what holds stuff together? How can we describe the forces that attract and repel matter? Well first of all let's just remember some of the forces we know about. So we all know about gravity and we all know about electricity and magnetism. Now just based on your knowledge of electricity and what you were taught in high school about atoms you can deduce the existence of another force of nature called the nuclear force and why is that? Well here's the iconic picture of an atom and in the nucleus there are protons and the protons all have positive electric charge. Now same sign charge objects repel okay? Now there are many things about this iconic picture of the atom that are inaccurate. One of them is the scale. So if we imagine scaling up an atom to the size of this picture well that's about a meter and if the atom were scaled up to be about a meter then the nucleus would be about the size of a bacterium. So it's really cozy in there. The nuclear distance scale is 10 to the minus 15 meters. Okay by Coulomb's law the repulsion between two same sign protons is going like one over r squared. So the electrical repulsion between two protons at a nuclear distance scale produces an acceleration. Anybody know? Well just for comparison the fastest roller coaster ride you'll ever take will briefly subject you to an acceleration of six g factors. The g factor is the force of gravity on the surface of the earth. A fighter pilot wearing special gear can briefly withstand an acceleration of eight to nine g factors. The acceleration given from the electrical field or the electrical force between two protons at a nuclear distance scale is about 10 to the 28th g factors. That's a big number. That's literally one of those Carl Sagan numbers. It's billions times billions times billions. So why is matter stable? Why does matter even exist? If there's this incredible force trying to blow apart the nucleus of an atom there has to be another force which overcomes that and attracts the protons and the neutrons. And I hope you now appreciate that that's also called the strong force. Now the strong force is very subtle. It's been studied with particle accelerators for decades up to the present day including at the Large Hadron Collider at CERN. And as a result of these experiments and the insights of many brilliant theoretical physicists we now have a mathematical description of forces. The idea goes back to Michael Faraday in the early 19th century, the concept of a field. Charged particles create fields and we detect fields by seeing how they move charged particles so we can see a field by looking at how the test particles move in the presence of the field. So here again from high school physics is the picture of an electric field created by a positive and negative electric charge. And so there's a direction which is the initial way, the initial direction a test particle will move in the presence of the field and there's also a magnitude. So over here it might be five volts per meter. Now this magnitude will be a function of position. It will change from place to place. So five volts per meter here, three volts per meter here. If the charges start moving that electric field will be, the magnitude will be a function of space and time. Okay so physicists use equations to find these fields and that's what's called field theory. Now field theory is involved into something very different here at the PCMI but that is the traditional notion of field theory. The equations that describe the electric and magnetic field were found by Maxwell. You can find them on T-shirts and it was a good day for James Clark Maxwell when he added this last term to the equations because then he found that there were remarkable solutions of these equations. There were wave-like solutions of these equations and when he computed the speed of those waves he found that it was the speed of light and thus he resolved the age old puzzle of what is the nature of light? It's an electromagnetic wave. Now what about the nuclear force? Well we can do similar things with the nuclear force so the proton is made of quarks and anti-quarks and they create a nuclear force field but there's one huge difference between the electric field and the nuclear force field which is that instead of the magnitude you have a three by three array of numbers so it might be one kind of three by three array or matrix over here and another one over here and if the quarks move it'll be a function of space and time. Now numbers are like matrices in many ways and in particular you can multiply them. So three times five is 15 you can also multiply matrices here I have two by two matrices and there's a rule for multiplying the matrices but there's a big difference between the multiplication of numbers and the multiplication of matrices. You see if you multiply three times five you get 15 and if you multiply five times three you get 15 the same answer so the order doesn't matter but if you multiply matrices in the opposite order in general you get a different answer. So this is the distinction between a commutative and a non-commutative multiplication and you're very familiar with commutative versus non-commutative in everyday life. When you got up this morning you put on your pants and you put on your shirt and it didn't matter what order you put on your pants and then your shirt, your shirt, then your pants that's a matter that's a commutative operation. Now you probably also put on underpants and your pants and if you put on your pants first well that's gonna matter. So another place where we encounter non-commutative and commutative operations is in rotations if you consider rotations around a fixed axis then that's commutative but if you consider rotations around different axes that's non-commutative and I met a colleague the other day on the trail who helps me illustrate this and so this is the x-axis and that's the z-axis. So you see if I first do a rotation by a quarter around the x-axis and then around the z-axis I get one kind of orientation of the bear and on the other hand if I rotate around the z-axis and then the x-axis I get a completely different orientation of the bear so thank you very much. And so what we're talking about here is a mathematical theory of symmetry and this is the theory of groups and groups come in in abelian and non-abelian types the abelian has to do with the commutative operations and the non-abelian with the non-commutative operations. Now when we make field theories of forces we have to make a choice of a group and so the group that you make when you make a field theory of electricity and magnetism is an abelian group so I'll call it an abelian field theory and for the strong force I'll call it you make a choice of a non-abelian group I'll call it a non-abelian field theory. So what are the equations that govern the nuclear force well they were first written down by Yang and Mills at Brookhaven National Laboratory and the Institute for Advanced Study in 1953 and 1954 there's a great story about the first presentation of the Yang-Mills equations at the Institute which you can read about in a nice book that recently came out by Graham Farmilo called the universe speaks in numbers and it's full of interesting anecdotes and stories about people who do physical mathematics. So these Yang-Mills equations generalize the Maxwell equations and so yes they admit light-like solutions, wave-like solutions we could call it non-commutative light but they also have another kind of solution you see because of the non-abelian nature of the group that goes into formulating these theories these equations are non-linear and non-linear partial differential equations can have very remarkable phenomena for example, Einstein's equations that govern the shape of space and time admit black hole solutions because they're non-linear in a more mundane way the equations that govern the motion of water waves are also non-linear in particular water waves going down a narrow canal are governed by something called Kordaveg degrees equations and the Kordaveg degrees equations have a remarkable solution known as a soliton now this might not look like much to you but I stress that there are no locks and no rocks here this is a lump of water which moreover retains its shape as it moves down the canal and that's actually quite remarkable I challenge you to produce a lump of water that retains its shape as it moves so imagine you try you're in a swimming pool and you try to make a lump of water so you try and push the water together well of course it's gonna splish out the other end and it's going to flatten out technically we say it dissipates maybe you push the water towards the side of the swimming pool there will be waves but they will flatten out they will dissipate this wave does not flatten out and does not dissipate and in fact what's being recreated here is the dramatic discovery by a Scottish engineer John Scott Russell in 1834 where he happened to be riding past this canal and observed to his astonishment that there was this wave that retained its shape as it moved down the canal he chased it out to sea and spent the rest of his life studying this remarkable phenomenon now the Yang-Mills equations being non-linear have soliton like solutions they're called instantons but these instantons differ from the soliton I just showed you in an important way here is again a soliton of water and you see it's localized in these directions but it's delocalized in these directions it also persists in time it just stays there so it's also delocalized in the fourth dimension the time direction and so because these but the instanton solutions of the Yang-Mills equations are localized in space and time all space dimensions and the time directions and that's why they're called instantons and they were discovered by Bilavin, Polyakov, Schwartz and Kupkin in the mid-70s and so in the mid-70s, mid to late-70s the physicists were having a great time playing with these and other soliton-like solutions of the Yang-Mills equations and what about the mathematicians? Well, meanwhile, back at the ranch the mathematicians were beginning to take note one of them was Michael Atiyah and his student, Nigel Hitchin and they had a student, Simon Donaldson who said, let's count the number of ways of putting instantons into a four-dimensional space Instantons are localized objects and the Yang-Mills equations being non-linear are very fussy so you can only put a certain number of instantons into a four-dimensional compact space in special ways and the remarkable surprise was that this is a topological invariant of the space so let me try and illustrate that I can't draw four dimensions for you so let's go back to that cube where we identify the opposite ends and think of the instantons as little marbles and supposing we take three instantons we try to put it into this cube well, there might be one way but because the Yang-Mills equations are fussy there only will be one other way of doing it and so we would then say that the Donaldson invariant for three