 So I thought I would do something rather different from what the previous speakers have done, and I would concentrate exclusively on Dick's work before the momentous events of 1997, which I remember very well, where he discovered, and so you discover unique to him the hot found your breath of the renormalization by iterated subtraction of sub divergences. And I've chosen two topics, not some numbers and four term relations that you can find in his book, the only book that Dick has published, it's called not some Feynman diagrams. You saw a picture of the cover and Marx talk. It characterizes being not even wrong. It wasn't sufficiently well defined to be turned into something you can say yes this emphatically does, or does not work. But it was extremely important as a heuristic. Without it, we would not have arrived at our conjectured enumeration of multiple zeta values by weight and by depth, and we certainly wouldn't have been able to evaluate the terms and seven and five of the fourth as we did exactly and remaining numerically. But before that, you have to remember I'm talking about things during Dick's first postdoctoral appointment in Tasmania. Before that I thought I would talk about his graduate work you've heard an awful lot of appreciation for the way that he advises students. I use that word, advisedly, you should think him as an advisor, not a supervisor. So I'm going to say a little bit about his time as a graduate student. Now I was a regular visitor to the University of Mainz working with my good friend Carl Shilcher throughout the 1980s. And towards the end of that I think it was 1989 Carl said you have to come. We have this new wunderkind called Dick Kreimer. He hasn't gotten an undergraduate degree in mathematics or in physics. He studied humanities. But somehow he seems to have mastered the whole of Landau Lyftschitz and past all, or bender's examinations. And I've given him this problem to work but he keeps going off with his own ideas. So they come and visit and see if they make any sense to you. And so during Dick's graduate work. These are the summer things that I learned from him. I also was able to give him advice but they contributed to my education. The first one which I mentioned here is a problem of gamma five. This is the matrix which in four dimensions commutes with the four Clifford, the four matrices of the Clifford algebra and you just make it by multiplying the four of them together. But Carl Shilcher had given Dick a really rather neat problem which he finally got around to answering and that is, is the weak interaction at one loop multiplicatively renormalizable rather than subtractively renormalizable. And that's a non-trivial question. But Dick immediately saw that there's something very, very different about the weak interaction from all the other interactions. The fact that it doesn't conserve parity means that you have to calculate traces of gamma matrices with an odd number of gamma fives. And there was a long standing prescription going back to shortly after the introduction of dimensional regularization by a toothed envelopman in 1977, I think it was bright loner and may zone gave a prescription for this. And which they claimed would eventually work. It's extremely inconvenient. It separated gamma matrices into four dimensional ones and extra dimensional ones. Anticommutivity of gamma five was lost. Everness and operators appeared and order by order the fundamental aspect of a gauge theory BRST invariance was not preserved you had to perform finite renormalizations and took my hand. And as a graduate student his first publication if you go to his homepage and his publications is a novel approach to this problem. And he recognized that something had to be abandoned I mean the existence of anomaly was responsible for the decay of the piezeria to two photons the fact that you cannot have a word identity for the one loop diagram that couples to vector currents to an axial current. And he realized that there's something really different going on with an odd number of gamma fives matrices. He didn't want to abandon anti. Anti commutation and he didn't want certainly to go through this nasty separation that was near to felt man screen. What should we abandon and he said well really we're dealing with matrices of infinite dimensional matrices. And for those we know that we shouldn't be insisting on the cyclicality of the trace. So that's what he gave up. And then he had to give reading instructions if I can't in my trace cycle gamma matrices where do I start. And he showed that that was consistent with the anomaly. So quite a remarkably innovative piece of work you see but he'd been given this problem and he'd put his finger on what he thought was the most important thing to address. Now, there was a subsequent paper with Carl Schilke his supervisor, and with your conkerner which and this attract quite a lot of interest and my good friend, you're conkerner I think would agree with me he wasn't the best person to go around giving seminars, explaining Dirk's work and eventually said no actually this comes from Carl Schilke as a young graduate student and if he can explain it better. So I took my dick to show one of his other talents and that is the ability to stay calm under pressure, because one of the two people that I've mentioned that have this 1977 scheme was not well disposed of an unknown graduate student advancing an alternative. So that was rather impressive. The thing that impressed me most was Dick's ability as an analyst. The conference is algebraic structures in quantum field theory but dick is also very talented when it comes to the analytic structure of it. And I would say, perhaps this is to not not very modest thing to say but I would say that in 1989 I was one of the leading practitioners in doing to loop calculations with with different masses. And then Dick came and gave a private seminar to me and he said I have a new way of doing to look two point functions, three point functions of the full factors and I think I could even do it for double boxes. And Dick's method, which he explained to us very simply was to split the integrations of the loop memento into those components that were in the span of the space of the external memento and do the integrations transverse to those. Now the two-point function with five arbitrary masses of function of five variables that could be gone using skillful use of cautious theorem taking in Mankowski space really taking account of where that I epsilon is in the propagators could be reduced to a double integral of a beautifully logarithmic expression. We know that when it's poly logarithmic it's try logarithmic but we know in fact that there's an elliptic obstruction that down there as soon as you try to do the next integration. And then that was pushed to three point functions at there for the two loop three point function. There's a planar and non-planar diagram. Everyone saw the planar diagrams much more difficult than they at the non-planar diagram is much more difficult but it it yielded to Burke's method. So he was really deft analysis and carried through a good programming. Now I also thought that I knew really quite a lot about special functions, general hyper geometric functions. You know but this young student said well David yes I know you know this but it's all miscellaneous knowledge you know you got it from a Brown vitz and Stegen and you got it from the bankman manuscript and the Russian book Brechnikov marriage of Prudnikov and there was a manuscript of a very rare list of results of the Mainz library he said but what you really need to do is go off and read this book David and he gave me Carlson's book which was a systematic approach to generalised hyper geometric functions I think Andrei Davidich have mentioned this. So that was quite interesting I mean when it actually came to doing calculations and so on I had things to offer but here was a graduate student telling me go off and read this book and I think you'll find it useful. And finally I mentioned it was including this this is when he was looking at form factors with with different masses in certain situations when some of the masses go to zero or a codenium and mentor, then there are singularities and he had to start thinking how to regulate those I'm not talking about renormalisation at this stage. And so he asked himself now, how would it be best to regularize and he said oh David, I'm going to use Hadamard's finite part. Essentially what that means if you have a lower limit of an integral goes to zero things behave badly you you call that low load epsilon and you throw away, not only all powers of one upon epsilon, but also logarithms that multiply those parts. And it happened recently that Francis Brown was proposing a way to calculate the quasi periods of modular forms. He wrote down an integral that was completely undefined and it wasn't exponentially divergent and the way that he did it and I said did you use Hadamard's finite part and I couldn't get an answer. So, Eric and I looked to see whether or not how it will be regulated and I said well it must be Hadamard's finite part. And then Eric said, what's that and I said well go and read your supervisor's thesis, it's quite interesting. And he did the calculation and said yes it is regulated by what I now know it's Hadamard's finite part, but what I realized said Eric, is that hyper ends this program you've heard so much about uses precisely that that prescription. So, I think you can see a picture there, can't you, independent mind clear sight and all done in a calm way. Dick is a very good listener. He listens more than he speaks and he's good at telling you what he thinks he's clear about and what he's still struggling to understand. Right, so let's go to Tasmania. What Dick thought was that the momentum flow from diagrams might be related to knots. Now this is a rather strange thing because the great thing and not is something that is a one dimensional structure in, in, in, in three dimensions. But this made some sort of sense to me because I could see that there was a big difference between planar and non planar diagrams. Something that's not immediately apparent if you're thinking for dimensional space time. And he told me he could maybe work out which diagrams involve Riemann's etra three and which didn't because he could look at the momentum flow and see the trefoil not which he wanted to associate with Riemann's etra three. Now, the, what he didn't know was that I had done calculations back in 1985 of counter terms paper called scalar Riemann diagrams five loops and beyond, in which I had identified counter terms involving zeta three and zeta five and zeta seven. But also, some numbers I couldn't identify but I strongly suspected with double sums. So what we were able to do was to test his intuition, the rules that he was developing against case law which I already had where it behave very well, but also to use them to try and work out what analytic expressions to guess analytic expressions using my numerical data. So now what I want to do is to tell you what he did. I'll read this carefully and then we'll go if we can to look at the figures. Dick decorated the braids of positive knots and obtain Feynman diagrams with trial and trivalent vertices. He shrank enough edges to obtain some divergence free counter terms forms that I was able to evaluate so let's let's find, if I go to new share, the theory is that what I have to do is to get out of this loop, because they disappear. I have to, I think do quite a complicated thing I have to go down here and find this so can you now see, is that working. Can you now see some pictures. Hello. Very good, very good. Right, so let me go to the first picture. This is a knot. It's one little piece of string, and it's got three crossings and you can't undo those crossings. It's a unique knot with three crossings. It's called the trefoil knot, and I've written it down here as a braid. This is just two strands running parallel and at the end of the day, I'm going to join up these things. So if you see more complicated braids, you can turn them into knots they might actually turn into links. So what they wanted to do is to relate this to Zeta three, that three loops and quantum field theory. So we decorated the braid with chords here, and he had a prescription which he explained to me for turning it into this diagram. Now this diagram. We already knew it's logarithmically divergence in four dimensions it has no sub divergences there's a period unique period associated with it we didn't use the word period in those days. It has the value of six remains either of five. In fact, the fourth theory you'd have extra vertices, extra edges external edges associated here. And to communicate between Tasmania and the Open University all he needed to tell me was the the braid word here, and I asked him for the angular diagram if we remove. So let's say this is the origin and coordinate space which I used almost exclusively and Oliver Schnetz I think inherited a taste for coordinate space from my calculations, then we just remove this. And this is the angle of diagram and so you complete the diagram just by connecting the origin to these points and this is the wheel with three spokes, which gives six remains each of the five. Let's go to the next one. So, now what about six reminiscences with three. Now there's also a counter term in five to the fourth that comes from the wheel with four spokes you just attached external edges to the four points on the circumference of that wheel. But in general, you're not going to find the wheel with five spokes in five to the fourth theory. You do find zigzag diagrams which you've heard about for which they can I had a conjecture for all loops later. Francis Brown and Oliver Schnetz. But how did you give me something starting with his not well he is. He still is dealing just with a two braid. But with more crossings. He ended up with this trivalent diagram and to give me something that I could relate to a counter term. A propagator turn this into a four point vertex to end up with the wheel with four spokes. Now, this is all plain sailing. But it gets really interesting when we look at three braids where the break group has two generators saying so this is saying, Sigma one is happening here and Sigma two is happening here. Sigma two and Sigma two and Sigma one and they're all positive knots that means that I have that these are all over crossings. Huge number of knots expands as prolifically as does the structure of quantum field theory, but those were restricted number of positive knots. And here's an example where using his prescription. He ended up with something which I had suspected to involve what we now call multiple zeta values I call them a double sum, and we were able to establish a dictionary between the knot whose braid was is here and when there were only four of these blobs account determined quantum field theory that involves the multiple zeta values zeta five three. But then he was actually able to take four braids and turn them into diagrams here for which I was able to actually obtain analytic answers going to so I could find multiple triple sums and multiple zeta values associated with the four braid. Okay, so now let me get out of there. I need to remove myself go down here to new share. And I should be back in my talk so I'll be back in business Karen. Yes, it's not full screen though it's. Yeah you'd like full screen wouldn't you. Well you might. So, let's give it a picture of me. And now I'm on the slides. So we associated families of positive knots with combinations of multiple zeta values. And our dictionary between knots and numbers exploited something which I discovered which I call the push down of multiple zeta values to alternating sums of lesser depth we'll see an example of that later. So our quadruple sums of weight 12 as multiple zeta values can be expressed in terms of alternating sums, which are only depth to, and it was these depth to things that were coming out of the dictionary between knots and numbers. In situations where with multiple zeta values we would need to quadruple sums. So this is the infamous broadcast climate conjecture for the number of primitive multiple zeta values of weight and and depth K. It's a generating function here involves two variables. y equals one it means I don't care about K. I just take all of the knots of all of the multiple zeta values of weight and then I end up with something that's absolutely intuitive this tells me that zeta three exists and this tells me at pi squared exists about all of the enumeration of multiple zeta values is is produced by this and this is proven of the motive as a conjecture. Don's idea that's proven at the motivic level and Francis Brown has explicit basis in terms of multiple zeta values whose exponents are two and three. But this term. This came out of the observation that at weight 12 and depth to there was some speaking between depth to and depth four. And it was just the beginning of something that continue to always and the way it continues in this generating function it turns into wonderful numbers verified in the multiple zeta value main actually is the same function as enumerates the the cuss forms of the fundamental modular group that we had noses that was never in our heads when we did this was just to fit in with observation empirical observations and multiple zeta values. Now our results included all the primitive contributions to the beta function of five to the fourth primitive means free of some that we didn't sit down and do all the normalization of the of the some of the one until recently as you heard. But we knew that the number content could not be bigger than this you always see the new numbers and the primitives. And we were keen to identify the numbers that could occur at seven loops. Now here's where you see this is not even wrong because it's just a game that takes invented. When I say I'm associating them I'm doing them because he's told me that he's got from the bread word for this positive not to some diagram and I say I can find the multiple zeta values. But what what we were hoping was and turned out to be the case. That he could give me clues as to what combinations of multiple zeta values to try and fit the diagrams of seven loops in fight of the fourth theory based upon our previous experiments. And that's, that's much more difficult. I can't see what I've written at the bottom of the slide but I'm sure you can. And that kind to do was to open the the full valent vertices of fight of the fourth theory and you can do that in three different ways, ST and you channels. And so he had many possibilities for rooting the mentor and he had to turn these into link diagrams and he had to scan those link diagrams produce knots and identify the knots. But it worked. When it was well enough to find that they were sure about something and. And there were times where we ended up not being able to use this thing to identify numbers, but it turned out that there was a good reason for that. So what is the association, you can find this in our papers but they're very nicely summarized in this book. Well, I've already told you that the zeta three is associated with this is a two braid it means I only need one generator of the break group to tell me whether I've crossed over the next one. It's associated with the power two K plus one it's associated with the two K plus one. But now look here this is the three for tourists not it's a three braid. So I need to generate a sigma one and sigma two, I raise them to the power four so it's going to be an eight crossing not. And wonderful to relate straight out of Dix not not knowing that. I had a number waiting zeta five three, he associated zeta five three, this double sum. But something much more specific not zeta five three is can occur in combinations with zeta five and zeta three and zeta of eight. And what we discovered was that these alternating sounds where you now allow assigned zeta five three would just be the sum of m greater than n greater than zero. One upon m to the five n to the three. If you put a minus sign in here which we indicate by a bar and subtract offer this is the not number associated by the counter terms of field theory to the not called 819 the only positive not with eight crossings, which is the not. And now, at seven loops. I could draw seven look diagrams not find the fourth diagrams which he could I, which he could arrive out from the three five tourists not. And we asked ourselves now what happens here you see I'm at weight 10 I haven't yet hit the place where multiple zeta values become mysterious. And two K plus five three cares at K plus six loops. But eight loops or maybe beyond but first at eight loops and the diagrams that I was doing I think Oliver finds has to go to higher loops and fight to the fourth to find this number. We encounter and nine three, but also this number and seven five with this very precise term in pie. And nine loops and beyond. Again, we weren't necessarily looking at five to the fourth diagrams. We find a way 14. By the way this number here is not expressible in terms of multiple zeta values of weight of depth to you have to go up to depth four. And nine loops. We. We encountered truly depth for poly logarithms multiple zeta values that don't push down. And we found these very precise numbers down here combinations of pie and he's very nice things can you see. God really loves the odd integers he hates pie squared and zeta five three and zeta. And three three three. That's seven news. They emphatically identified the occurrence of this four braid 11 crossing positive not only to positive knots with 11 crossings one of them is the, the two 11 torus not just sigma one to the power 11. And the only other positive not a thousand knots or so to look at has this braid word. And for this he always got diagrams which I could evaluate to involve triple sums in this precise combination. No, no, no, no, no theta three five three minus theta three times theta five three. Of course you could have any other and in multiple is different level because there's another not to the corresponds to that, but we were able now to find the precise combinations of triple sums associated with the family of four braids we're now talking about a depth three, which gave diagrams which I could evaluate. exciting time. So now we're interested well you know how fast the positive knots grow and how fast the multiple zeta values grows and and can we find families of knots that might be associated with multiple zeta values indefinitely. We identified five families, here you see a two braid, here you see a three braid, and here you see families of four braids. And you know you can write down all sorts of braid words that correspond to the same knots that are righto meister moves that turn one into the other. Now one way of trying to work out which knot you've got from some braid presentation that date likes because it gives me counter terms that I can calculate is to look at some polynomial associated with a knot on the best one on the market is called the hon fly polynomial. These are the initial letters of the six authors whose names have been always forgotten because the acronym is much more memorable and it depends upon two variables. And we were able to do something quite remarkable, namely for these knots with these positive braids defined an expression for all crossing numbers of these families in terms of these two parameters. I've you can see that quite on a formal list. This is extremely useful, you see, because I can then investigate relationships between my counter terms and knots. I don't necessarily have to have them in these presentations. If it comes up with some different braid word, all I do is calculate the home fly polynomial of his braid word and I look it up in my table. So what about the multiplicities? And again, I can't see the bottom of my slide, which contains the really important information. So I'm going to cheat a little bit so at least I can see it. We looked at knots grow enormously. So we said we're interested in positive knots and we went to the best notters in the world and asked them how many positive knots and they said that's an impossibly difficult question for us. We can't even tell you we know that there are only two is 11 crossings. And I said, I work that out in five minutes of CPU time on my 25 year old laptop. But can you tell us here? And they said no, too difficult. So we're interested. We worked it out ourselves. We assume that the home fly polynomial is not faithful. It doesn't always distinguish knots. If two knots have different home fly polynomials, they're different. And if they have the same one, they're likely to do the same, but there are counter examples. And we assume that as positive knots, we're really rather scarcely great panoply of knots that the home fly polynomial was faithful for those. And so I was able to generate a quarter of a million braid words and work out their work out their home fly polynomials and work out how many there were. And eventually the knot theorists, I think 10 years later verified this calculation that I did one afternoon in Tasmania. And I still haven't seen, we put this in the encyclopedia of online sequences, but I still haven't seen whether anyone has validated this number. I'd be surprised if it was wrong. Now, what you can see down here is that these are increasing much faster than the multiple zeta values. But this is what's so remarkable that things started off in parallel. You see, Dave didn't know this. Well, he knew it was about zeta 3, zeta 5, and zeta 7. He didn't know that the first multiple zeta value occurs as zeta 5, 3 at weight 8. And I didn't know that there was a unique positive knot with eight crossings. Here at 9, there's just zeta of 9. Here, we're picking up things with these two positive knots, zeta, giving zeta 11 and zeta 3, 5, 3. But at 10 crossings, we first found a knot to which I couldn't associate a counter term that gave multiple zeta values. So what we looked for, and you look at the enumeration of multiple zeta values, so we can already see that knots are more prolific than multiple zeta values and primitive multiple zeta values. So we imagine that only a subset of knots could be associated with these multiple zeta values. And actually, these families work very well by associations. When I get up to weight 17, experts might think at weight 15, I would need a five braid. But in fact, because of pushdown, I have to go to weight 17. Then here, we were missing a knot, but we reckoned eventually someone could find a knot associated with these two extra multiple zeta values. But here, you can see why we were led to believe that quantum field theory would outgrow multiple zeta values. Maybe it was even outgrowing it here at seven loops and five to the fourth theory. So what really happens at seven loops? Well, we found that there were two positive knots with 10 crossings. Dick can remember their numbers and the 10, 1, 3, 5, and 10, 1, 5, 9, I can't remember them now. But we couldn't associate to multiple zeta values. And there were three counter terms at seven loops, which we couldn't identify. And so we concluded at least this led us to believe that multiple zeta values would not suffice for five to the fourth counter terms. And I'm saying this in case Francis Brown is listening because Francis is under the impression that we thought that we would always get multiple zeta values as Feynman periods. And that was the origin of the Konsevich concepture on zeros of the semantic polynomials and over finite fields. But in fact, here Francis in the published paper is what we did say positive knots and hence their transcendentals associated by field theory are richer in structure than MZVs. Thank you, David. I am listening. Thank you. Now the great thing is, and I said to Maxime, it's wonderful that Maxime didn't understand what our intuition was because he made this very strong conjecture just on whether or not the numbers of zeros of the semantic polynomial of finite fields was a polynomial in Q. And the commentators found that this worked up to 12 edges for every graph. I'm talking about five diagrams, right? Every graph on the world satisfied this Konsevich conjecture. And they were a bit miffed about that. But in fact, Maxime had already told me, said, I think something goes wrong with the far known matroid. And later, Belcale and Brosnan had a non-constructive disproof of the Konsevich conjecture. Non-constructive, in the following sense, they said, if Maxime was right for all graphs, then he would be right for all matroids. But we know that he's wrong for a matroid, so he must be wrong for a graph. But that's completely non-constructive. It's a wonderful piece of mathematics. But the thing was, we were already precisely at seven loops at 14 edges with unidentified things. So what did we find? Did we find some things that weren't multiple zeta values? Well, eventually, it took me quite a long time, increase of computing power, and I was able to identify two of these. They're called in Oliver-Schnepps's tables, the periods of seven loops, and the eighth and the ninth and the table down there. And they involve this weight 11 combination that we find. Notice weight 11 are not associated with these 10 crossing knots. And it was related to multiple zeta values, but there was a new number. This new number, zeta five three minus 29 zeta of eight, is not the combination that we've seen before, and here it gets multiplied by zeta of three. So we can claim no credit at all of relationship between not theory and here. We have a new type of combination of numbers occurring in terms of all multiple zeta values. And the wonderful thing here is that these two reductions in multiple zeta values surprised, I think, Francis, because what he had predicted was that alternating sums would, well he actually says for all of the seven diagrams, multiple polylogarithms of roots of unity up to six would suffice. And that's exactly what happens. But for these, the diagnostic that was given by something called the C2 invariant was that they should really involve alternating sums. And I had obtained some time and I think about 2010, these expressions from here, which were not alternating sums. Now you've heard of the enormous increase in ability to calculate that has come from Eric Panza's hyperint and Oliver Schnetz's graphical functions. And these are quite difficult graphs. Eric was able to do one but not the other, vice versa for Oliver. And both of them attained in their respective domains, neat combinations of alternating sums, very much as Francis had led them to expect. But then they had the multiple zeta data mine, which Johannes von Lein and Josef Masser and I had developed, which also includes alternating sums and using that they were able to prove the reductions that I found empirically. The remaining seven loop counterterm, number 11 in the census of seven loop periods, was predicted by Francis to reduce the polylogarithms of six roots of unity. And that was done by Eric in the most amazing feat of analysis. And here I comment on something that was mentioned by Oliver Schnetz, namely that at eight loops, polylogarithms of all type fail to deliver all of the counter terms. There's a period at eight loops in five to the fourth theory, whose obstruction to polylogarithmic reduction comes from a singular K3 surface. Wow. But it's associated with a cuss form of model of a weight three that's the most beautiful cuss form you can think of was complex multiplication of square root of minus seven. It's just the data can either of z times the data can either of seven z to the power three is related to the symmetric square of elliptic curve with conductor 49. So here's now my very subjective summary is that Dirk's intuition based on our explanations of relationships between knots and numbers that multiple z to values would not suffice at seven loops was born out by later analysis, though the really non polylogarithmic action is here. So now I think I have according to my recollection 10 minutes left for the four term relation. Now this is either right or wrong. So let me say what it is. I asked you to imagine that you have a graph in which you can draw a Hamiltonian circuit. That's a circle that passes through all of the vertices once and once only there are certain graphs called Paterson graphs where you can't do that, but the majority of graphs that you can. So now just take three little bits of those circles down here, these three little arcs and find a cord that connects these top two arcs. And the four terms we will get is by connecting this arc at the bottom to here, here, here, and here. Now what about the rest of the graph? Well the four term relation doesn't care about it. It says whatever else is happening, how these are joined up in the Hamiltonian circuit, what else happens at other points in the Hamiltonian circuit. As long as they're the same, then there could be a four term relation. And what Dick wondered was whether this was the case for the counter terms of quantum field theory. And what we ended up, Dick has a published paper in which he asserts that he has approved for a very strong argument that there is a four term relation of the following five conditions are met. First of all, that each of the terms is free of sub divergences, then it would have a unique number associated with it when we nullify the external momentum and put all the masses equal to zero and find the coefficient of logarithmic divergence. That they should differ only by the subgraphs known, shown, that's just saying, I've just defined what that means. But now there are three extra conditions. And I explain where these came from, they came from in the first instance from experiment and not from pure thought. They should have trivial vertices, in other words, just constant vertices like n phi to the fourth theory or phi cube theory or you can't with theory, but not vectorial couplings, not like gamma mus that you get when fermions couple to photons, the gauge bosons. And they should have no propagator with spin greater than half. So there should be no, you can't have vectorial couplings and you can't have internal vector bosons. And each of these four terms should modify one of the dimensionless couplings of a renormalizable for your theory. Now, the paper which I wrote with Dick on Dick's instructions, we said the necessity of these set of provisors is not established. But it claims that they're sufficient to derive a four term relation. For the counter terms, coefficients of overall logarithmic divergence of each of these four diagrams, which I've labeled in cyclic order, and the counter terms are easily calculated, we nullify the external and then to throw away the internal masses, we cut the diagram wherever we please, to get a two point a finite two point function, which we evaluate to the best of our ability. And there's no problem with the R star operation. There are no infrared problems that are excluded by the provisors. Okay. So what's I able to test this? Well, here's the full test. It's in Yukawa plus five to the fourth theory. So this double line here is a fermion. And here is the Yukawa coupling. There's a loop for the scalar particle. And this X is the coupling to some external scalar particle. And what I've done down here, Yukawa theory, by the way, is not renormalizable renormalizable by itself. It generates a finite of the fourth coupling that you have to renormalize. So you can only have Yukawa theory as in the standard model in conjunction with five to the fourth theory. So I'm going to imagine I have a five to the fourth vertex here. I've connected three of its valences to the fermion here. And the four term relation that Dekas predicting occurs when I connect the fourth valence of this five to the fourth vertex, either here, or here, or here, or here. Now, we were able to do this. Because then we nullify everything for each of the four terms. I've just written them out explicitly down here. And of course, I cut this here, because then it's very easy because when I cut it here, I just attached this one loop diagram that I can do. And I'm really only left with three loops left to do. And here, I chose to cut down here. And you can write down explicit formulas down here, you know, they come from the Feynman rules. But here is an integration measure that I defined for an arbitrary number of D dimensional integrations over internal momentum. To make the numbers nice, I work in what's called a G scheme, where you take out the gamma factors that come from simple one loop diagrams. And here now is the precise four numbers that Dekas is talking about. And he's interested in limit epsom goes to zero down here. And they just involve the two and the three measures, the two ones with this combination of a mentor. And you don't need a fancy computer program to do this, you can more or less do these two loop integrals in your head, because you've got my result for the wheel within spokes down here. And all you have to work out is how that rule is modified, you know, if I have D mu two giving me zeta of three, how it gets modified here and here by scalar products and all that does is multiplied by some thing which goes to a constant as epsom goes to zero. So that was two, that was two of the diagrams, the easy ones, these ones here. The other ones really involve some, some real three loop integrals. And I had written my own version of what was then called mencer and I had it on reviews, reducer, my calculator, but we didn't need that full machinery. We were able to use integration by parts to turn a three loop integral into a two loop integral at the price of introducing fractional powers of the mentor in there. And that's where I'll work with John Gracie, where John was developed afterwards, but discussions with John Gracie before was useful because that final step was accomplished by developing epsilon expansions of South Schuetschen F3-2 series to obtain the two difficult terms and Dx4 term relation was satisfied. So now I've only got two more slides to go, I think. If we replace the Eukala couplings by gamma mu, this was my discovery, this was all done in the morning. Great excitement among Bob DeBorgo and Peter Jarvis and Johannes when Dx4 term relation had been verified by David in the morning. But in the afternoon, I just put in vector vertices and it completely fails. So the restriction to vector vertices was post hoc, but Dirk explains in his single author paper how that is sufficient for him. But then at five loops, I could immediately do, well, not immediately, but with present technology because I'd already calculated the five loop anomalous field dimension, indeed the five loop renormalized propagator of phi to the fourth theory. So here now I'm looking at a non-renormalizable theory that involves a phi to the fifth interaction, but I still have got a sub divergence free counter term for the coupling of two fermions to two scalars. And the four term relation here, it's got just the same as before. I haven't considerably more difficult integrals to do, but the important thing is you can see that the four term relation fails spectacularly. So that's the origin of the proviso in Dirk's paper that we have renormalizable things. So now my final slide, I think more or less on time, which is the next slide is for modern quantum computation. I like this phrase from Oliver's talk, quantum computation. So first of all, I've got a question for Dirk because this is the type of question that I always used to ask him as soon as I had an idea. Last night I drew this diagram, Dirk, and I hope that it's free of sub divergences, but I know that you'll be able to cast your mind over it very quickly and tell me if I'm wrong. But what I've done down here, I want to work in Yukawa plus fight of the fourth theory. And so here I've written a two loop correction to the fermion propagator, but I made jolly well sure to put my Yukawa coupling in the middle so that each of these loops is already convergent and I've only got an overall logarithmic divergence. And then for my fight of the fourth vertex, I've coupled to make this nice Hamiltonian circuit I've coupled here. And the four term relation, which according to Dirk's paper must hold because it satisfies all of the provisors will be gotten by connecting to these four points. So providing, I haven't made a mistake, this is one of many. I mean, the constructions become quite ricks down here. So here's a question for Eric and Michie and Oliver, maybe for Eric, because Eric, I don't know if you know, we should celebrate the fact that he has his second five year fellowship in Oxford. So he'll be in Oxford for 10 years. Now I've been telling him he should be starting supervising some graduate students. So here's a nice little master's exercise for someone who wants to do some applied calculations down here. Can you disprove? Dirk Crimers claim that providing this is free of sub divergence. It's a renormalizable quantum field theory. It doesn't involve vector particles. It doesn't involve gamma mues. And then for Michie and Oliver, what about five cubed and six dimensions? Because that was what Dirk wanted me to work with. All of his knots were related to trivalent vertices. So here's a little exercise. Was Dirk right or wrong? And I come to my summary. It's a very short and a very heartfelt summary. I'm going to read it. Stop myself cracking up. Dirk is a skilful analyst, an inspiring comoratorisist and a deeply influential algebraic thinker. But more than all of that, he combines these gifts with a quiet self-confidence and a great concern for colleagues that has enriched my life and many others. So thank you, Dirk. Thanks, David. Thank you to David, yes.