 Thank you. I admire your stamina. Everybody, I didn't think we'd have so many people at the end of the day, but the organizers asked me to talk about motives. And so I thought I would give a fairly relaxed, it's an awkward subject because it's a subject that Growth and Deek was very interested in. If you look at his letters to Ser, for example, they come up frequently. But he never published anything, at least to my knowledge, he never published anything about motives. And I suspect the reason was that you run up against some difficult conjectures that nobody, and in fact, I talked with Growth and Deek once in my life as he was leaving mathematics I was coming in. But I did talk with him once and I asked him precisely about what are called the standard conjectures, which are, so to speak, the obstruction to pushing through his program of what are called pure motives. And he told me that he had high hopes that they would be proved before he died. But it was not to be so. So what I thought I would do is give a fairly relaxed talk and I want to emphasize the progress in motives. So the first is what you might call Growth and Deek motives. Those are intimately linked to the Bay conjectures. I think it's fair to say that Growth and Deek was inspired to develop the notion of motive by his attack on the Bay conjectures. Then, well, after Growth and Deek had left mathematics, the subject of mixed motives grew up. Primarily, I think, well, Deline certainly developed the notion of mixed hodge structure and Baylinson, my memory is, and I could be wrong, but my memory is that it was Baylinson who kind of ported Deline's mixed hodge structures to the theory of motives. So here, one of the important motivations was the Baylinson or were the Baylinson conjectures. And in particular, you could argue you might think that it replaced the Bay conjectures. Here, the conjecture is concerned values, a hodge of Bay. So mixed motives, at least one of the primary reasons to develop such a theory is arithmetic. I mean, you'd like to understand these very arithmetic objects. And then the subject, which has become kind of dear to my own heart, has not been really developed very much, but I want to say some words about it in any case, is the role of motives physics. And here, there are two things to say, one of which is true. So certainly, it is the case that Feynman amplitudes, which are the essential numbers that one needs for a perturbative quantum field theory, Feynman amplitudes are periods of motives. I have to explain what that means. The second thing, which very probably is not true, but that's all right. Probably I won't have time to explain it anyway, is that, how shall I say this? Let me just say, it's an Akian theory of quantum field theories. Essentially, I think it's fair to say for Grelten Dieg, one of the central ideas involved in studying motives is that they form a Tanakian category. The pure motives form a Tanakian category. And so we'll have to see a little bit what that means. And the question then here is, what happens if we try to apply the same kind of Tanakian calculus to the study of quantum field theories? And the answer is, I don't know, but there's at least a little bit one can say, and it's kind of amusing. So I hope I'll have time to say some words about that. The stick is there. Oh, well, that's a little. OK, so motives. Yeah, I mean, I've kind of reminded, I taught undergraduate calculus for 40 years. And there were certain things that would always happen. And one thing that would always happen is that after the first lecture, some kid would come up to you and say, the first lecture, I would always devote to sort of talking about numbers, I mean, real numbers, where we were going to do the calculus, the properties of real numbers. And some kid would come up and say, but professor, what is a number? And I never got a good answer to that. But the one thing I can say with statistical certainty is that that student was going to do very poorly in the course. So I mean, it's in some sense the wrong question, right? Math doesn't really tell you what a number is. It can tell you properties of numbers. So for example, it can tell you if you have five pairs of shoes, you will need 10 shoelaces. But it doesn't really tell you what a number is. So I think similarly, if we want to talk about an algebraic variety, you have this phenomenon that is kind of like, I mean, there's sort of two things, and they're close together, and I'm not enough of a philosopher to be able to say whether they're different or not. But one are the sort of manifestations of this mathematical object. And the other are the kind of interactions of this mathematical object with other mathematical objects. And these are two kind of things that we can hope to say concrete things about. And I think that's very much growth index approach. So how do I want to say here? Well, the manifestations, so how to see an algebraic variety, one way to see it, and I think, as I say, it was important because of the vague conjectures, was to look at solutions to polynomial equations. So solutions to polynomial equations, everybody kind of knows how the game is played. If I have, for example, x squared plus 1 equals 0, there's a simple polynomial equation, I can say that this has no real solutions. It has a solution because 2 squared plus 1 equals 5. It has a solution, so we get a solution mod 5. But on the other hand, there's no solution, no solution 3. So the vague conjectures are that. They ask for how many solutions, or how many points, in our variety with values in Fp, or more generally, Fp to the n. So I remind you that Fp is the simplest kind of a field. It's Z mod pz, where p is prime. And Fp to the n, it happens if you study the Galois theory of this field, that there is one extension of degree n, so Fp, n, p, there's one for each n. And so we could ask to study the function, which associates the n, the number of points. So that was the idea. Well, I actually don't want to get into the history, but certainly, Vey studied this problem and others before him. But Vey came up with the basic intuition. And the basic intuition was that you have a Frobenius, so call it, and it's a ring homomorphism from Fp to n, or let's say more general. Well, it sends F of x is simply x to the p. And the fixed points, so Fp, n equals 1 is just Fp. And more generally, the fixed points of Fp to the n, well, the fixed points of nm. And if I look at F to the nth power, then this is Fm kind of thing. Hope I have that right. Something like that is true. And with that in mind, if I have a tuple, let's say x1 to xn, oh, too many ns, x1 to x, what should I say, d, and I apply F to it. So this is x1 to the p, xd to the p. And so if this point, x1, xd, if this belongs to V of Fp, then, or this will belong to V of Fp if and only if. So this will be true if and only if this point, x1, xd, viewed as belonging to a point of the variety over the algebraic closure, is fixed under. And more generally, if I take F to the r, then I can put here p to the r, and F to the r, and I get such a statement. So V's beautiful idea was that we could, using the topological construction of left shits, that we could turn this into a problem of linear algebra. So the point was that if we have, let's say, capital F, an endomorphism of vector space, then we can consider the generating series, the trace of F to the n times t to the n. And with appropriate additional structure, so if we have a duality, if V is given as isomorphic to V dual, then we can work, so I don't want to get too much into it, but we can work in V tensor V dual and write the F and also the identity as elements in V tensor V dual and do linear algebra in V tensor V dual. And this left shits observed made sense if we could talk about topology. So in topology, we take on being a little vague. V is the cohomology of some appropriate manifold with Poincare duality. And then if F is a map from x to x, which has fixed points. So we're interested in the fixed points. We want to study the fixed points of F for his powers of F. And if we're fortunate that the fixed points are in good position, then we can study this by looking at the intersection of the graph of F to the n intersected with the diagonal. And lo and behold, this is exactly in the topological context, the trace of F star to the nth power, where F star is the action on cohomology. Super trace. Sorry? Super trace of 1080s. Yeah, you have to, right, I'm going too fast here. We really should take the ultimate sum of these things. But people are familiar with these ideas. But so the point is that with this mechanism, the intersection here, so in our case, because of the geometric fact that the point is fixed, if and only if it's fixed under the ferminius. So in our case, we take our variety V and we view it as, or we view F as acting on V. You have to be a little careful. It's kind of confusing because Frobenius, you have to work with the relative Frobenius. But again, that's a technical point I don't want to get into. The intuition is that the fixed points of various powers of F exactly count. So the fixed points of F to the r are in one to one correspondence with the points. And granting that, one may hope to apply the left shits machine and to relate our counting problem. So our function, we want to understand our function, which sends n to the number of points of V, F, p to the n. And we relate the generating series that we get from this problem to the sum of the trace of F star. So again, yeah, now I'm starting to cheat because as V observed, we don't have a good comology theory to apply. But if we could apply the thing, well, we do in fact now, but we didn't in those days, if we could apply the thing in the sense of left shits to some nice comology theory, then this we would exactly get. And then we could go further because of course we can write this as a product of what is it? 1 over 1 minus alpha i t, right, where the alpha i's are the eigenvalues. So they had developed this beautiful program and he applied it in the case of curves because in the case of curves, he could use the Jacobian and the points of finite order of the Jacobian to build an analog of the comology and then he could apply his machine to that. In your forward, I should be a sum instead of a coordinate. Sum, no? Sum's what is that? It's a trace, it's a sum. No, it's not a determinant. It's a determinant, yeah, one more. No, no, this is... Yeah, you're right, you're right. So in other words, it's the sum, whatever it is, it's certainly the sum over i, of the sum over n of alpha i to the n, t to the n, right. So it should be the sum, you're right. The total function is different. But I mean, yeah, once you're here, then all the other manipulations are just familiar, just like in other combinatorial games. So anyway, this is where Gorothendijk started. And so he developed the, in particular, the theory of... So Gorothendijk developed the theory of analytic co-homology, which precisely did that, which precisely gave us a theory on which we could compute these traces and prove at least part of the vacanjectures and the history is kind of well known. The missing was still that the vacanjectures also concerned the absolute values of the alpha-i's and that was kind of inaccessible because, well, for problems that we'll see in a minute. But anyway, he developed this ellatic homology and the problem was that there were sort of too many of them. I mean, there's one of these homologies for every L. It was not the case as in the classical topological picture that you had a Q-valued homology. You had a QL-valued homology and it wasn't, in fact, I think it was Sear who observed that you couldn't hope for a homology with values in Q because you could construct, for example, elliptic curves over finite fields. So, Q-valued homology didn't exist because there existed elliptic curves. I think you want to take it over f p squared whose endomorphisms were quaternion algebras. And so, in particular, they had no, so this was a quaternion algebra, call it Q, quaternion, and it had no two-dimensional representations. And growth in deep new and it followed from V that you wanted this representation in the case of an elliptic curve. It should be a two-dimensional vector space. So, it had no two-dimensional representation, sorry, defined over Q. So, you couldn't have Q-comology but you had this elliptic homology. And so, growth in deep asked himself, we should think of these elliptic homologies as kind of manifestations of this abstract category of varieties. So, he asked himself, well, can we think of these as like the smile on the Cheshire Cat in Alice in Wonderland. So, the Cheshire Cat has all these smiles that we can see but we can't see the cat. So, growth in deep said, well, can we develop the abstract category which you call the category of motives which sort of underlay all these realizations. So, we want the category motives lying behind. And the story is kind of well-known. I mean, he had a beautiful program to do this but the program was blocked because certain standard conjectures about algebraic cycles were turned out to be, first of all, necessary. These were needed but to this day, no one can prove them. But unknown. But what, I mean, leaving aside the issue of actually proving it, he was left with a beautiful sort of picture of how things should be. And the picture revolved around this notion of Tanakhian category. So, that wasn't kind of enough. So, the category of motives was supposed to be so-called Tanakhian category. And what did that mean? That meant that it was sort of abstractly equivalent, I'm speaking vaguely here because there are many issues but it was sort of abstractly equivalent to the represent the category of representations of an algebraic and how did that work? The idea was that this category, let's give it a name, call it Z, that there should be some functor, which he called a fiber functor, to the category of vector spaces and the category should have a notion of tensor product and the fiber functor should preserve the tensor product and you should look at the automorphisms of the functor preserving the tensor product and that should be the points of your algebraic group called G. So, this should be G, I'm going very quickly. And then this was a sort of an extraordinary idea because suddenly the properties of the category can become translated into the properties of the group and for growth and dig, the group should be semi-simple. The category should be no extensions and the presence of the elatic phenomenon that there were many different comology theories, the fiber functor should be so a variety should be an object, should give rise to an object in here, well I mean there's some issues there with, but roughly speaking a variety should give rise to an object and the fiber functor should just be the cohomology of the variety but of course we know that we have to take QL coefficients and so in fact there are many fiber functors but they're not defined over Q, they're defined over QL and so suddenly there is a fascinating complication that this category is not actually a Tanakian category, not actually a neutral, I mean there are various words that are used but in any case we have to extend the field and for various field extensions we get various fiber functors and even a peatic fiber functors and it's suddenly a very beautiful and very intricate picture but this picture is blocked in the sense that because we can't prove these standard conjectures about the algebraic cycles we can't actually prove their issues. Okay so now what I wanna do is to talk next about the category of mixed motives because I think mathematics is kind of like a river and it flows and every now and then there comes up a problem that nobody can solve but the river, so it's like a dam in the river but the river, that water is coming at you and it's gotta go somewhere and so if you're not careful it just goes off a different direction. So I think inspired by Deline's theory of mixed high structures, the balance and I hope I'm, probably other people were involved but I say balance and it's sort of the main man. Developed the theory of mixed motives and so let me just explain a little bit why these are, what they are and why they're interesting. Well I suppose I should start with the simple case so growth in the theory of Tanaka, I mean growth in the theory of motives was a semi-simple theory, there were no extensions but if one looks around in real life there are notions of extensions of motives so let me just explain the simplest one, the simplest one is so-called Kummer and this is like this and if I have, so I have my variety V but now suppose that I have a unit so I write OV star so this means U is simply an invertible function so it's a global unit then there are various games I can play with U if my ground field so if V is over K and if K is inside C then I can take the logarithm of U and that's of course only well defined so up to 2 pi I or up to Z times so growth in the theory would say that the set of logarithms is a torsor under the group Z times 2 pi I more algebraically we could take the L to the nth root of U and the collection of these guys is well defined up to the L to the nth root of unity instead of Zeta in the algebraic closure such that Zeta L to the n is one and again it's a torsor so these are torsors and if we, it's sort of a little bit more convenient to say things slightly differently if we look at the set of pairs log of U to the A comma A for A an integer then the set of these pairs well it certainly augments off to the integers by sending this to A and the kernel, well the kernel is the set of choices of log here so the kernel is just Z times 2 pi I and that's what, that's sort of a toy example of an extension so this would be written as an extension of Z or of Z of zero by Z of one the so-called Kummer extension or if you wanted to say it elitically I would have zero goes to mu L to the n goes to the set of all pairs consisting of U A one over L to the n comma A where now A wants to be an element in Z mod L to the n so these, but these are not actually motives themselves these are realizations so these are realizations of a quote motive which we haven't defined and the motive is associated to the algebra geometric object which is the original unit U so we've taken something like an algebraic variety in this case not an algebraic variety but an invertible function on an algebraic variety and we've given the smile or various smiles on the Cheshire cat which we want to think of as being associated to a Cheshire cat which is the okay this all seems rather silly but suddenly things are not so silly because as a consequence of a theorem of Borel we have something suddenly something really extraordinary there exists a motive and I'll just state a very, very small simple case of a great big theory which looks like this Z of three M Z of zero so Z of three means Z of one tensor three so Z of one is supposed to be a motive whose realizations include this one and these and our category of motives is supposed to be a Tanakhian category which means we can take the tensor product so whatever it is this makes sense and our category of motives I didn't say it but our category of mixed motives should be closed under extension and so suddenly we have the possibility of looking at extensions but the miracle that the extraordinary thing that happens is there exists one which yields as a period the Riemann Zeta function so this is the Riemann Zeta function at three so Zeta of three which is a completely fascinating number and now I have to explain to you what I mean by yields here we have to and it's instructive so let's actually do that so how do we get Zeta of three from this abstract guy? Well, we have to look at one of the various smiles on the Cheshire cat we have to look at what's called the Hajj realization so I have to explain what that is that I would write it exactly the same way so I'll write it the same way but now I really have in mind that these are Hajj structures so what do I mean by a Hajj structure? Well, this is just a free oblivion group, it's Z well actually say, sorry? Delain mixer structure, so Yeah, so mixed, mixed, mixed mixed, okay so actually let's look at this one that's more instructive so M as a Hajj structure means what? M of Z is a free oblivion group it's Z two times but it has a little more structure it has a weight filtration M of Q so I'll write M of Q as M of Z tensor Q and similarly M of C or M of R same idea so it has a weight filtration and in this case, the weights are W minus six of M of Q is Z minus Z of Q of three Q contained in which is just Q but we don't worry so this is M of Q is W zero Q so it has this increasing weight filtration and it has a decreasing Hajj filtration so I write the Hajj filtration this is actually the crucial point is that the Hajj filtration is only defined once I tensor with C okay and it's decreasing and so what is it in this case? It's decreasing so I have in fact I have zero contained in F zero M of C contained in so it's decreasing so if I write here F minus three that's bigger it's confusing M of C which is the whole thing okay and what's more the category of these mixed hot structures is an obedient category and the operation of taking the various filtration pieces is exact so I then get zero goes to F zero Z of three C goes to F zero M C goes to F zero Z of zero C goes to zero now F zero of Z of zero is everything this is the Hajj filtration on the Z of I's is stupid so this is just all this is just literally C in the sense that it actually has a distinguished element one on the other hand Z of three this is F minus three and so F zero Z of three is actually zero so in fact this is an isomorphism so in other words this mysterious extension has a splitting at least if I tensor with C so I have if I tensor with C I have call it S sub F which is a canonical splitting on the other hand if I look at the the original integral version these are after all just free abelian groups and so I can just choose a splitting so I can take an S sub Z which is defined over Z and then the difference so I can look at S F of one minus S Z of one and those are two elements that map to the same guy and so the difference lies in here so this lies in Z of three C now it's not well defined because even though the F splitting was canonical the Z splitting is not I just simply said there was one but there's an ambiguity so to make it canonical I have to factor out by the Z of three Z so to speak the lattice but this then is just well I haven't explained well but when you think carefully about the situation here what you're doing is you're factoring out by the lattice generated by two pi I cute so there is an invariant of the extension of Hawn structure so what is the logic here I said that Burrell's work gives us an extension of motives and then we take the hodge realization of this extension of motives and associated to this hodge realization is an element in so let's call this we call it B for Burrell but we get the hodge realization so this is sort of called H of B which is an element in here but this is just of course it's R direct sum R I times R modulo Z times 2 pi cubed so if we ignore this part and just look at the real part we find that from this extension of motives we get a real number and lo and behold the real number that we get so the image of this H of B in here is exactly Zeta so that's just a quick sketch of the balancing conjectures which are vastly more general and more inclusive but unfortunately the work of Burrell only applies for number fields so the balancing conjectures remain kind of like the standard conjectures largely unproved but we have many examples and so there's considerable confidence that the picture is correct okay so now let me segue in the remaining 15 minutes to the physical picture actually I don't know what I want to do here so I suppose I should talk about it hurts my heart but I suppose I should talk about the aspect of the physical thing that I know is true rather than the stuff that is probably nonsense but I like so let me talk a little bit at least about so this is physics I want to talk a little bit about the Feynman amplitudes and basically the game is going to be the same physics gives us associated to so this is I should mention my collaborator Pierre van Hove if we look at this graph which somebody had the wit to describe as the two banana graph in physics Feynman's program says that if we imagine incoming and outgoing particles then what happens in the mysterious area where they collide represented by various graphs in this case just three lines and so we have an incoming momentum and an outgoing momentum and we write T for the Q squared the Minkowski length of Q and we have masses here M1, M2, and M3 then associated to the three edges with the masses we have propagators and we are led to try to compute a certain integral so this leads to an integral which is equal to some function which let's call it I don't know I which is a function of T of Q squared and also of course the masses now what is this what is this function it's a multiple valued function so actually the story is in two parts the first part really represents my work with Pierre and that was the case we looked at the case where the masses are all equal M1 equals M2 equals M3 equals M so in that case what we got was an elliptic curve so there was an E which depended on T which is an elliptic curve and there was an extension I should put a T here it all depends on T think of T as fixed for a while M does a dope curve this is again M is my word for motive small M yeah it occurs actually it doesn't you're right that's a good point that's a good point so T is here replaced by so T is replaced by T over M so there's no M you're absolutely right so lo and behold comes this extension which we should think of as an extension of motives but here something's a little funny because when I look at the weight filtration on the Hodge realization this fellow will have weight 3 sorry 3 yeah in minus 3 minus weight minus 3 whereas the two cases we looked at we had weight minus 6 and the first case the extension had weight minus 2 but here we have weight minus 3 but if we have a good theory of motives we can talk about this object not just as a Hodge structure but actually as a motive and we should have extension in this way so suppose that well in any case we have this extension of Hodge structures and we can play the same game we can look at the Hodge filtration and the right Hodge filtration we look at F0 so again F0 of MTC again the same kind of discussion will show that it's isomorphic to F0 of Q of 0 C which is just C so again we will have over once we tensor with C we will have if we tensor with C here and here and here then we will have F we will have an S I should here I prefer to talk over Q so an SQ ok so we will get SF minus SQ which will give me an element in H1 of E T C modulo of E T Q2 I mean I have to it's ambiguous again because the ambiguity of the choice of the rational section ok now we want to get I mean physicists you know you try to explain the Hodge theory and motives to them they are a little skeptical they want a function so Q is we can pair we have omega T which is the holomorphic one form so we can pair with omega T and this gives us a complex number not so well defined because I have to factor out so modulo periods so this is the and that complex number low and behold is I so this is ok and actually there is a lot more I mean we can tell you a lot about this I and there are many interesting properties but in any case it's the first manifestation of the sort of more intricate kind of motives involving well for example in this case elliptic curves with the physical amplitudes and the the I the I is the roughly speaking is elliptic and logarithm now it's not the full story exactly because we assumed that m's all the masses were equal and in fact that's that's the work that we did but things are never constant in physics and so 9-0 and collaborators Adams and Bogner worked out the more general case so the case of general and when I saw their work I told Pierre I was completely furious with myself because it's completely obvious what you need to do namely instead of taking elliptic curve if we now insert general masses what we are led to do is to take an open elliptic curve so we have our elliptic curve we still have our elliptic curve but we now remove s where s is a finite set of points in fact it's 6 points elliptic, elliptic, elliptic finite set in fact 6 points depending upon the masses so we have a movie we have a fixed elliptic curve once we fix the external momentum we have a fixed elliptic curve but if we vary the masses then the set of 6 points varies ok so what's the motivic why was I angry with myself well because suddenly here we have a fascinating situation this is no longer itself a pure motive this is an open elliptic curve if you know a little bit about the situation this fellow will be itself an extension of motives and so in fact the m will be a sort of they'll have 3 graded pieces so the m I'm out of time but let me just say briefly the m d will involve h1 e t q of 2 and it will also involve a direct sum in fact of 5 copies of q of 1 and it will involve a q of 0 now what kind of sense can we make out of such a triple right well this is supposed to be an abelian category and supposed to be representations of an algebraic group so the and the algebraic group in this case because we're talking about so just to finish up we talk about representations we talk about the Lie algebra because in this case the algebraic group is unipotent so the Lie representation would look something like this there'll be something here there'll be something here and there'll be something here now essentially when the dust settles and I don't really have time to explain this in detail the fellow here will be a point so this fellow will actually be a point on the elliptic curve so actually a point this fellow here will be so I'll give this point a name let's say a this fellow here will be the theta function applied to a and this fellow here will be some algebraic cycle on E T cross affine two-space in fact a curve with certain properties and with that data we build a representation of a Lie algebra which is the one that's associated to this mixed motive so suddenly physics is giving us an extremely interesting new object it's not a surprise I don't want to present this as some sort of miracle everybody knew these things were lurking there but it's wonderful to see them arise and particularly to arise in situations where physicists can actually calculate the amplitudes and we can study them so I think I'll stop here but I hope I've shown that the mixed motives lead to a wonderful both arithmetic and physics applications so I stop so the first number up there is the elliptic curve do you also get an L value? this is part of the balance and conjectures yes absolutely L value of water yeah so the L value that would most naturally occur would be the L value of the Haase-Ve L value of H1 of ET I guess I'm assuming the T is rational here so I can talk about arithmetic QL at the point S equals 2 one step beyond the usual point you look at is S equals 1 where you get the period but S equals 2 you have this SF-SQ it's a rank one flatbender does it have a can you characterize it? sorry rank one flatbender SF-SQ it's an immediate step here in the whole story it is in H1C divided by H1Q can I tell you what the bundle is? it's an interesting as I stand here it's a kind of a nice store you have to quantify what you mean by what is the bundle in what terms I could it's a flat bundle you know them and I don't know how to describe which one but there is a way to think about it from that point of view I can say that I'm not quite sure what the right question would be from that point of view but yes you can interpret that way and one has a nice such theory but which ones arise in this way I don't know oh sorry you said that this is connected to the elliptic dialog so if you expand the function you have to think carefully to see that from this point of view but if you actually expand out the number or the function then you get the expansion of the elliptic dialog roughly speaking again the common belief some years ago that we should have only not anything well you see here not really because here part and parcel of the game is that we have external momenta right and the conjecture in physics concerned what they call vacuum graphs where there was no external momenta no I think that one expected complexities and this is simply no I mean but that's a very deep business that no one now knows that by applying to Francis as example yeah Francis has the most interesting examples of ones that are not dialogue any other questions