 Thank you. Well, it's very pleased to be here to celebrate Jean-Pierre's retirement and the important thing when you go to a conference in honor of somebody is that he or she should understand the speaker lectures. So you should make it easy for them. And since I'm the first speaker, I'm going to set an example to help everybody else will follow so that he can follow the lecture easily, he should enjoy it and he should if possible be able to ask a question at the end of the test. And to set the tone right, let me start with telling a story. My favorite story, you may have heard it before, but it's relevant. The story is about the high school physics teacher who had a class. He noticed that somebody in the front row, perhaps called Jean-Pierre, was not paying attention. So he turned to him and said, Jean-Pierre, what is an electron? So the boy scratches his head and said, oh, well, I used to know, sir, but I've forgotten. So the teacher said, ah, what a pity, he killed homage. Only two people in the universe knew what an electron was, you and God, and you've forgotten. So replace electron by spinner and you get the tone of this lecture. So what is a spinner? Now I spent most of my life working with spinners in one form or another and I don't know. Only God knows. Maybe you're acting as he's no longer with us. So, um, now Herman Weill was one of my favorite mathematicians and he also wrote beautiful poetic language English. And here is what a quotation he makes in his book on the classical groups, which I'll read for you. He talks about starting off with the spin representation. It says, only with spinners do we strike that level in the theory of its representations and we Euclid himself, flourishing, ruler and compass, so definitely moves in the realm of geometric figures. In some way, Euclid's geometry must be deeply connected with the existence of the spin representation. Now that is typical Delphic Weill remark at the end. Well, you have to think deeply to what he means. By the way, Weill, you know, in that classical groups, he has this introduction which he apologizes that he's writing in a language not sung by the gods that he's crazy. You know, he's German, but you know, beautiful English, as you see. So now spinners, to be very mundane, remind you the basic algebra of spinners. And this is a way to understand it. You have the orthogonal group. Let's take an even number of variables, special orthogonal group. And in general, it's not a simply connected group. It has a double covering, which is called the spinner group. So double covering, now the group, the complex, if you have a complex structure on two end spaces, n dimensional space, if you consider the special unity group, the unity matrices of determinant one, this is a simply connected group. So this maps into the orthogonal group, because it's simply connected, it lifts to the double covering. So you can regard the SUN and subgroup of spin 2n. If you make any representation of spin 2n, you can restrict it subgroup and see what happens. Of course, an irreducible representation, when you reduce it to a subgroup, in general, will become reducible and break up. Now the spinner representation of SO2n, spin 2n really, is a space, always talking about complex spaces, is dimension 2 to the power n. And it breaks up into two irreducible pieces, S plus and S minus, half the dimension. If you pull this back to the SUN, then you find that the spin representation becomes the same as the complex theory algebra of the complex vector space Cn. And S plus becomes the even part of the exterior algebra, and S minus becomes the odd part. And you notice that although S plus and S minus are irreducible representation of spin 2n, their restriction to the SUN breaks up and has degrees. But in the spin representation, there are no degrees. It's irreducible. Only under SUN, you do see the degrees appearing. Now at the bottom, I've also written the fact that if you take the tens of the spinners, then what you get can be viewed as the exterior algebra of the complexification of R2L. Not assuming an underlying structure, complex structure, but also just forming the tens of product with the complex numbers. So these are the fundamental algebraic facts about spinners. Now, of course, algebra is not the whole of mathematics. And Herman Waal, you mentioned, talked about Euclidean geometry. And so it's geometry that I want to focus on. And the algebra just gives you the framework. But it's by no means explains what the underlying nature of spinners is. It just says, here's the representation of the group. Now, I want to focus on the comparison between spinners and the complex numbers. Complex numbers, no, it took mathematicians several hundred years to really feel they understood what I was. I, the square root of minus one, was called an imaginary number. It did not exist. But by finally very useful. They used it in formulas and gradually, step by step, it became accepted. And finally, the formal definitions were given. So I became, the square root of minus one became an accepted part of mathematics. But it took a long, long time. Now, I want to suggest that the spinners are analogously a square root. But they are the square root of geometry. Now, think about that. In geometry, the fundamental elements in geometry are undoubtedly to do with measurement, lengths, lengths, areas, volumes. And those are described mathematically by the exterior algebra. And it's used by Ali-Kartan, the differential geometry, as a fundamental tool to describe integrals. So that we understand very well. So what is the square root? Why do I say it's the square root? Because you remember that the tensor product of spinners with themselves is the exterior algebra. So the spinners are, in some sense, the square root. Not of one form of a given degree, but some of them all. So it's a very deep notion. And it'll take us probably at least as long as it took to understand the square root of minus one, to understand what spinners are. And they've only been around for perhaps 100 years. So we are still in the very early days of the equivalent, perhaps, of the 15th century, in terms of mathematics. And so I don't know the answer to my question. Neither does Jean-Pierre here. I say maybe Dirac knows, but he's up there with God. So we are left on our own. And now if you have a complex manifold, if a manifold you're studying is complex, then it's well known to work with hodge and so on. The differential forms on the manifold can be broken up into the forms of type PQ, where P involves the z and Q involves the dz bars, the mixture of the two, the tensor product. So when you have a complex structure, you see the square root. And the square root is complex geometry. So when you have complex geometry, you found a square root inside the real geometry. The complex structure has given you a square root. But spinners exist without the need of complex structure. So what is a spinner when there is no complex structure? That's really the question. And complex numbers are, of course, not important only algebra. They're important in geometry. And it was known, of course, early on that you could do, well, the theory of Riemann surfaces were fundamentally developed by, formalized by Hermann Weier himself. And so the important thing in complex numbers is the analysis. Algebra, square root minus one, is interesting. We know that the complex numbers of the algebraic closure of the real numbers, you don't need to keep going higher and higher up, just to equation degree two. But the really deep part about complex numbers is their role in analysis. Complex analysis, Cauchy's theorem, and all that. That's a very sort of unexpected bonus. When people started to use ii in formulas, it was thinking only in terms of solving polynomial equations. But then, in the hands of the analysts, Cauchy and others, it became an indispensable tool. So that's where the deep part of complex analysis is. And similarly, you expect the deep part of spinners not to lie in the formal properties of representation theory algebra, but in their geometrical meaning. The question is, what is the geometry here? I identify like Jean-Pierre with analysis. You do geometry on curved spaces, you have to use analysis, differential equations. So spin and analysis has to be found as a substitute for complex analysis. That's the first stage going from Cauchy's theorem in this idea of the square root of geometry. Well, this problem was of course solved for the mathematicians by the physicists. Dirac introduced the Dirac operator in studying the wave function of the electron. And I'll come back later on to his motivation in doing that. But basically, it's a Lorentz invariant first order operator, which plays a fundamental role in modern physics. And the Dirac equation is the equation that he wrote down. And so, mathematicians can learn from that. Now, the first step you have to learn going from physics to mathematics is that the Lorentz metric is a signature with a minus 1, and in remaining geometry, all the sides are plus 1. So they're very different capsicum equations, hyperbolic equations become elliptic equations. But in fact, this step of going from physics to mathematics to replacing the Lorentz metric by, well, it's already made that by Hodge. Hodge's motivation in developing his theory of harmonic forms was Maxwell's equations. Maxwell's equations are the equations of electromagnetism, and Hodge took the same equations, applied them in Euclidean or Romanian geometry, and developed the theory of harmonic forms, which as you know has been fundamental in the development of modern geometry, modern algebraic geometry in particular. So similarly, mathematicians, and here I enter the scene as a graduate student around about 50s, and interestingly enough, my supervisor was Hodge, and his colleague in the next office was Dirac. They were in the same department for 32 years, and they never spoke to each other, not mathematically, that's because Dirac never spoke to anybody. And in other words, the transition from Maxwell's equations to harmonic forms was made by Hodge with great profound implications. The transition from the Dirac equation to corresponding mathematical equations was not made by Hodge or by Dirac, and it was left to people like me and Singer to, if Hodge had spoken to Dirac, I would have had no mathematical career. So we had to be grateful sometimes that our supervisors don't talk too much to each other. Now, so the Dirac equation in Romanian geometry, and it works not only in flat space, but in curved space, again we learnt this from the physicists, you can look at the equation for harmonic spinners, just like you can for harmonic forms. They are the solutions of the equation Dirac, flight of phi equals zero, but equally well you can think of it as a solution of the second order equation, d star d phi equal to zero. The solution on a compact manifold are the same. So these are harmonic spinners. Now harmonic forms play a fundamental role in geometry, and the reason is Hodge's beautiful theorem that says the harmonic forms, the dimension of the space is independent of the metric you choose. As you vary the metric, there are no jumps in the zero eigenvalue of this operator, which in general you expect, but for the Dirac for the Hodge operator, there are no jumps, and the harmonic forms always have the same dimension and give you the better number, topological. So the question was, is something similar true for harmonic spinners? First question one asks. The answer is no. You start off in dimension two. In dimension two, the Riemann surfaces have a complex structure, you can analyze everything into the complex structure, and you find the space of harmonic spinners does depend on the metric, but the dimension lies in a bounded range between zero and the genus. So that's already some information. The genus is the upper bound, but if you go to dimension three where things change, then Nigel Hitchin, who was that student in Oxford, I suggested to him, it would be interesting to work out what the harmonic spinners were in dimension three, some suitably good example. And so he took an example of a sphere with not standard metric, but slightly deformed metric, and he computed the space of harmonic spinners for these metrics. And what he found was that this dimension is unbounded, unlike the Riemann surface case, it can go on and on increasing forever. Explicit formulas, implicit values of the metric give you bigger and bigger dimensional spaces. So it looks like there's no particular role for harmonic spinners, they don't give you any information of a topological kind like harmonic forms, so it's to be disappointing. That's the conclusion you get from such a preliminary investigation, but you dig more carefully, you find there are topological significances. So the first result, and this result was what I did with Singer, and was motivated by the work of Hitzel before, and I put Hitzel for question mark, not because any question mark about his work, I couldn't remember the exact date. But anyway, Hitzel discovered early on that if you have a manifold of dimension 4k, then it has all these invariance called Pontriagin numbers. If the manifold satisfies the condition, it's oriented first, it's compact, but if in addition, 2nd Steve of Whitney class is zero, then a particular combination of Pontriagin numbers called the A-Roof genus is an integer, called A-Roof because originally he defined something without a roof, which was an integer, but this was a better one, and his conjecture was that this should be an integer. He proved indirectly that it was an integer, but there was no explanation. Why was the integer? Well, this is really the start of my investigation with Singer, and that conclusion was that using the ideas of Dirac, the Dirac operator, you take the Dirac operator, acting, it interchanges the role of plus spinners and the minus spinners, it's an elliptic operator from one to the other, adjoint goes backwards, and so this operator has an index, which is the dimension of the harmonic spinners plus minus, the dimension of the harmonic spinners is minus, and this number, the index, which is the difference of two harmonic spaces, is stable under perturbation of the metric, whereas the individual dimensions can keep going up and up, they are equally on both sides, and the difference remains constant. Now, we're familiar with things like Euler characteristics, which have this great stability, whereas individual homology may change, Euler characteristics is much more robust, and in the case of Headsdenberg's Riemann-Roch theorem, if you have a complex manifold, and particularly let's take the particular case where the first term class is zero, so called Kalabi-Yau manifolds, then Headsdenberg-Riemann-Roch theorem says that the Euler characteristic, the chief of holomorphic sections, is a topological invariant, that's the arithmetic genus, and that agrees exactly with the index of the Dirac operator in this special case. So the index of the Dirac operator is given by a formula, which generalizes the Headsdenberg-Riemann-Roch formula, the top genus essentially becomes the same as the A-Roof genus, and the reasonably A-Roof genus is an integer, it's the same as why the top genus is an integer, it is the index of an operator, the difference of two dimensions. This is a very satisfactory answer explanation of Headsdenberg's theorem, and it requires the hypothesis that the second Steve Whitney class is zero, which is where they spin as a hidden, because w2 equals zero is the condition that the manifold structure group lifts to spin, and therefore the Dirac operator can be defined globally. In general, the Dirac operator can only be defined locally, and then shortly after we, shortly after we proved the same year, it was observed by Lechnelevich, who was a differential geometry in Paris, who was closely interested in relativity theory, and he'd known about the Dirac operator in relativity theory, he knew that he could define it on a curved background, and so he was in well good position to make this application. So the theorem of Lechnelevich is that if you have manifold, now ordinary Romanian compact manifold oriented, if w2 equals zero, and if you assume that the manifold has a metric of positive scalar curvature, it's a very weak condition, only a scalar curvature, positive, then this number A Roof M is zero, topological consequence. For example, if the manifold is four-dimensional, this is as the first Pontiagin class is zero, and here is a rather strong surprising theorem, and the proof is one line, if you know the geometry, you can compare the Dirac operator at times adjoint with what you call the covariant derivative followed by its adjoint, the covariant derivative involves much more than all the derivatives, the Dirac operator involves part of them, and the difference between the two in generalism would be completely expressed in terms of the Riemann curvature, it involves only multiple of the scalar curvature, and so if the, since the covariant derivative times adjoint is always non-negative, if you add a positive quantity, that strictly positive operator has no solutions, therefore the conclusion, the Dirac operator has no solutions rather plus or minus, the index is zero. So it's a very beautiful, simple consequence, this by the way is an idea of going back to Bockner, who applied it in the case of differential forms, but easier in the case of spinners, and then a few years later Fritz and I discovered another rather striking geometrical result, which was again you assume that W2 is zero, the manifold spin, now you assume that a non-trivial circle action, some compact group, just a circle with enough acts non-trivial on the manifold, non-triviality means it's not all points fixed, then the conclusion is a roof is zero. Now this uses not just the index of the Dirac operator, but what we call the equivariant index, the index including the symmetry of the circle, in other words, when you have a metric invariant under a circle, you always assume it's invariant under a circle by averaging, then the eigenspaces of the plus spinners and the minus spinners will decompose further according to the representation of the circle Fourier series. So for each Fourier component, you have an index, so you have a much more refined invariant, and with the conclusion of the theorem is that the whole index is zero, and in particular the constant term is zero. Now this is a very strong result, and interestingly enough, the hypotheses seem to be exactly relevant. If you take a nice class of manifolds which have symmetry, for example, the complex predictive space with its natural invariant metric, then you say, well, how about this, it has big symmetry. Why doesn't that contradict the theorem? Well, the answer is you see that the first term class is n plus one, if complex predictive space has dimension n, n plus one times the generator, and this is, if this is cognitive zero, mod two, the same condition as the W2 equals zero. So you see this happens only if n is odd. When n is odd, the dimension of cognitive space is not developed by four, so trivially the A roof genius zero. When it is developed by four, it's on spin. So you know, you can't either one way or the other, it escapes the theorem. It's very nice showing the theorem is sharp. Now the third, well, the next deep result about spinners is a different order. This is the result of Zeibergen-Witton. Now, I have to summarize here, but this followed the earlier work of Timon Donaldson, my student, who became very famous when as a graduate student he discovered the remarkable things about dimension four. Fantastic news. Something opened up. Four-dimensional geometry of suddenly became much richer field and uniquely dimension four. Few years later, Zeibergen-Witton had a different approach to the same invariance of Donaldson, which was more physically motivated. Donaldson's work also had physical connections, but Zeibergen-Witton was more physical and it involved looking at not only solutions to the drag equation, but in dimension four you can write down an extra term. The drag equation is one equation for spinners, but you consider in addition to the spinner fields, you consider what we mathematically call a line bundle with a connection. Physicists will call that an electromagnetic field. So if a physicist you couple the drag equation to an electromagnetic field, and this follows just from the algebra that you can get this coupling in dimension four, you can square the spinner related to the curvature of the line bundle. And this is a quadratic equation, not linear, not linear equation. And the question is, can you solve these equations, simultaneous equations, the linear equation plus this additional nonlinear constraint? And this is Zeibergen-Witton equation, this pair of equations. And the Zeibergen-Witton showed a very beautiful result that if you want nonzero solutions for this equation for a given metric, then that puts a constraint on the possible churn classes of the line bundle. The line bundle must take a churn classes, only have a finite set of possibilities. And those possibilities are then invariance of the manifold, differential topological invariance. These are new invariance of a manifold. They are the ones for which the Zeibergen-Witton equation has solutions. So if you dig deeply into the drag equation with this extra condition in dimension four, you get some fantastic new results. These new invariance really are translation of Donaldson's work in other language. In particular, it was shown by Witton later on that if you have a complex manifold, a scalar manifold, then these classes are essentially reduced to the chronicle class and it's negative. Which somehow generalizes the notion of the chronicle class of complex manifold to real manifold. So this is where you go without assumption of complex manifold, you get something analogous too, but much deeper. This comes out of using the spinners in a very deep way. So this is a very, very deep result and in some sense gets most use out of the drag equation. Now, a few minutes I have left, I'm told the timetable is very tight here, but I think we took five minutes at the beginning. So I wrote a paper a few years back with when you get to my age, you don't try to write trivial papers. You leave that to the youngsters. You write tight, write crazy papers, which might be, so our crazy idea was this. We wanted to introduce into physics the notions of not just differential operators as usual, but retarded and advanced differential operators. One should depend on the past and retarded and depend on the future advanced. And if you have one variable, it's easy to do. You just write down, for example, you consider a differential operator, you consider its value and its derivative, and then you write a linear relation. But in an advanced equation, you take the derivative at a different point. You translate by some amount. Now, that you can trivially do and write down advanced and retarded equations like that, operators. But you can observe that translation is an infinitesimal generator, which is differentiation. So translation can formally be written as exponentiating the derivative. So if you want to translate by a, you exponentiate a times d by dx. Elementary, just formal retranslation of the notion. Now, the question is, how do you define a relativistically invariant retarded operator when there is no preferred time direction? The whole point of relativity theory is you don't know which way, where is time. So now the idea is, of course, that what you need is to use a first-order operator, because that's what you need to exponentiate, which is relativistically invariant. Well, fortunately, Dirac discovered that for us. The Dirac operator was precisely invented by Dirac to find the first-order relativistically invariant operator. And therefore, you could use that to define advanced and retarded equations formally. And we did that in this little paper, and we found by some elementary calculation with physics, that the possible variables you could use, in this equation, there are two variables, I call them alpha and beta, two constants. And, well, the second equation, alpha, seems to become, no, it's a, I'm talking about, a, the amount by which you multiply d by dx. This is related to a physical quantity called the Compton wavelength of the particle. So a quantum mechanical notion is involved in deciding on the degree of retardation. So he thought this was an interesting link, potentially. Now, by the way, here is to make a little digression. When Dirac introduced his operator, his reasoning was that he wanted to write down that the Schrodinger equation was not relativistically invariant. He wanted to find the relativistically invariant. Therefore, he wanted a first-order equation. And he found it, and he wrote it down, and he was successful. But then a lot of smart alecs later on came along and said, ah, well, you see, Dirac wasn't clever enough. He could, in fact, he could, you can do it with a second-order operator. You don't need to have a first-order operator. That was a mistake. He found this beautiful result, but he was mistaken. You know, he could have done it without that. But these smart alecs were wrong. For here, I'm using, for this purpose, you have to have a first-order operator. You can't exponentiate a second-order operator with the same result. You get something quite different. So here, for us, we really use the Dirac equation as invented by Dirac. And then we went on, more ambitiously, to say, well, what about Einstein equations, general relativity? Can you, can we retard those in some sense, write down and retard the Einstein equation? And we had some success, but only partial. And in this paper which we had, we did the, what we did was to use the, you see, the Dirac operator acts on spinners. If you want to act on acting on forms, which is what you need to get to general relativity, you take the operator in, on differential forms, which is d plus its adjoint, delt, little delta. As an inhomogeneous operator, because one raises degree and one lowers degree, it acts on all the forms, like spinners. Now, when you square it, d squared equals zero, delta squared equals zero, you get the Hodge Laplace operator, which preserves forms. And that's what Hodge did. People emphasize the second-order operator, but first-order operator is more fundamental. And, but you have to have the view that it's sensible to consider inhomogeneous forms. And that's not very geometrical. It's the first step to spinners. Anyway, if you do that and write down the corresponding story for, which we did before for the spinners, then you find that the scalar curvature, this time gets replaced by the Ricci curvature, which is known to Böckner. And this is the Ricci curvature is, of course, what enters into the Einstein equations. So this at least gives you a bit of a clue as to how the Einstein equations might be retarded. And then you find that the other constant in the retarded equation, there were two constants. One was the degree of the shift, which we related to the Compton wavelength. The other was the size of the correction term, the retardation parameter, which is a couple of beta. Now it turns out that beta interpreted as the cosmological constant. So the same ideas which lead to quantum mechanics lead in the direction of cosmology. But this is an incomplete paper. If you want to read it, you'll find that we don't quite know what we're talking about there, but it's okay. It's an idea. It's a crazy idea. But not insane. Not insane. I mean, well, I don't know. You know, the line between sanity and insanity is a very thin line. It's my father wrote the book called The Thin Line. And here are some references for those, if you want to follow it up. I've given my paper with his book on the spinner index, my paper with Singer, an index theorem. I've been with Moore on this shifted view, the paper of Liknerovich on harmonic spinners, the paper of Witten, which he really develops the cyber Witten theory. And I should have included it in there, but it's got left out, Hitchin's thesis, which he calculates the harmonic spinners in dimension 3. So anyway, that's my quick summary. What is a spinner? Answer it, we do not know. But we've learned that it has some deep geometrical significance. But quite wide has the significance. What it means and where that is going to go, we don't really know. The cyber Witten equation is a very deep study of the link between, and dimension 4 particularly, between spinners and geometry. And there's another approach, which is also in dimension 4, which I should mention, which is Roger Penerose's Twister theory. That is a way which you can relate four-dimensional geometry for special classes of manifolds to three-dimensional complex analysis. It's a very beautiful theory, which links these ideas in many ways. And there's another way of trying to understand spinners in terms of geometry, complex geometry, or complex geometry of not the dimension you expect, but of a differential. Anyway, I think at the end of my career, I like to leave some problems for the next generation. So let me know when you've discovered what a spinner is and I'll be listening from above. Thank you. It's time for questions and comments. Yeah, I'll start with a comment if you allow me. You know, we mentioned the way, rightly, the work of Nigel Hitching about the inboundedness of the dimension of our experiment. The interesting point is that the metrics for which he finds that this is inbounded are the so-called vergemetrics, both the rest of HHS and quantum visor. And these metrics are very specific in three dimensions. They are just obtained by looking at the three-sphere fiber, if you want the real projective line or complex projective line or the two-sphere. And when you shrink uniformly the lengths of the fibers which are circled, then you get these specific vergemetrics. And that's exactly the ones for which Nigel just computed the dimension of the Diracon, the kernel of the Diracon. Now my question is, in a sense, as usually mathematicians do, you concentrated your attention on spinors of spin one-half, as would physicists say. But some physicists have also shown that there's an interesting connection with the Einstein equation, if you are looking at the Rarita-Schringer operator, which is the analogous of Diracon for spinors of spin three-half. Yes. Have you looked into that in some way? I know about this, but I haven't really looked into it. But could I come back to your first point? I should say that this work of Hitchin, which shows that there are particular values where the dimension gets bigger and bigger and bigger, this actually has a topological significance. As I said, you don't get direct topological information out of the dimensions of the spaces of harmonic spinors. But if you look at how the metric varies, how the spinor dimension varies with the metric, you find particular subspaces, the metric space of all metrics, where the dimensions jump. And this you can relate to the topology. It has the jump. In fact, when you know the topology, you can predict that they have to jump, they have to be there, because they carry topological information. What they carry is topological information. Not about the manifold, but about the group of diffeomorphisms of the manifold. The group of diffeomorphisms of the manifold is essentially to study vibrations or all metrics up to equivalence. And knowing when these dimensions jump, this gives you topological homology, homology classes in the classifying space of the group of diffeomorphisms. So there is topological information buried in exactly the sort of thing that Nigel studied, but it's not quite what you expect. It's different because there are jumps. The jumps are interesting. The examples are civic, but they're interesting in the general theory why there are jumps, what the jump mean, and calculating the homology of the jump. So the spinors have topological consequences in unexpected places. You know. Take the square root of Michael Atiyah. That's known as taking the Mickey. No, I think, you know, I like to put it, the spinors are the square root of geometry. It's really sort of, it focuses attention on the depth of the problem. And if you think about the square root of minus one, you realize that it takes a long time to understanding something is a very difficult notion. But you have to ask yourself a difficult question. You have to pursue the geometry in all sorts of ways. And in a long series, you, and of course, I should say that the square root of minus one got its ultimate justification with quantum mechanics. Quantum mechanics showed you have to have complex numbers and probability amplitudes. And so you could say that the drag equation which came before the mathematical application already shows us that spinors have a deep physical meaning. Physics and geometry are my view and nowadays accepted view very closely related. So you don't answer that question just like that. But eventually, after lots of struggles, many centuries of work, body of theory which will emerge and say, ah, now we're building to understand what spinners are. But at the time, the younger guys here in the audience may contribute and will be here to see it. The next is the red director, but one or two or three will be able to share a lecture where the answer is given. If I may follow up a little on Jean-Pierre's question about Rita Schwinger's spinners. Some physicists think that the real significance of these things is only revealed if you go to supersymmetry. That is, if you include all the anti-commuting variables and I would like to know whether you would think they're on the right track, that this is the direction to go to understand them really and deeply or this is the sidetrack for mathematics. That's again difficult. And I myself have never been very enamored of supersymmetry and geometry. I feel supersymmetry is a very out of play notion when you just plug it into geometry. You can do it formally. The physicists do it very successfully and they have good physical motivation and people are still looking for supersymmetry in the real world, you know, the CERN experiments and all that. Myself, I have a slight distaste for them, but that could be personal prejudice. We all have to have prejudices, otherwise we can't pick and choose. I'm prejudiced against supersymmetry, but I recognize that a lot of my good friends use it very successfully. It's a very powerful technique. So I think the question is still open. My view is that supersymmetry is closely related to exterior algebra. Well, I think I postpone answering that question for the next century.