Radical Thoughts on the Foundations of Physics / Cohomomorphisms and Operads

0:00 My pleasure to be with Mr. Professor's speaker this morning, Michael and Sia, as all of you know, we just discussed briefly that it seems that we always leave each other outside our premises in the United States. This lecture is, as you know, a radical thought on the foundation. I'm not sure it works. Is it working? Does that work now? Well, it's a great pleasure for me to be here to celebrate Alain Conte's 60th birthday. Of course, I think 60 is still a young age. And I gave him some encouragement when I told him, my son told me recently that since the expectation of life is constantly increasing, each day is only 19 hours. So every day you save five hours and every year you save so many months. So think about this and then it helps to make you feel more cheerful when you pass such critical ages. Now, when you get older, I'm older than 60, when you get older, I think one should attempt to do what younger people do well. It shouldn't compete with the younger generation. So, in other words, hard technical mathematics is not suitable for an understatement.

2:30 On the other hand, some things are not suitable for young people. One thing is not suitable for a young person is to have a crazy idea and spend a lot of time on it. Because it doesn't work, you don't get your PhD, you don't get your job, your career at risk. You can't take big risks, unless, of course, you are sure that you are Einstein or Newton. Otherwise, it's a bad thing to gamble on high stakes. If you get to my age, I got my PhD. And don't need to worry. No, I can follow crazy ideas. Sometimes it's a credible thing to do. Now, this lecture is non-standard. It will be one quarter of philosophy, one quarter of physics, one quarter of mathematics, and more than one quarter of speculation. so it's an unusual lecture in that sense for a mathematician so my title is some radical thoughts on the foundation of physics and conflict of non-community geometry and I should explain that I'm not going to be talking about non-community geometry per se I'll leave that to Alan and other people but my ideas are they have some great relations I just remind you here first of all that non-community geometry has a core of all roots in mathematics and physics, a whole range of motivation, and, of course, multi-immunitative things have appeared in mathematics and physics a long time before, going back to, I mean, Hamilton was the person who really introduced quaternions to their multi-immunitative structure, and that leads more generally to league groups, and Hamiltonian mechanics have, of course, a platform bracket in it, and then, in parallel, there was the work of Grassman, So, the question of non-immunitivity has been with us in mathematics and physics for a long time. Now, let me just start by reviewing briefly what I think is the current state of physics. Well, there is classical physics, starting with Newton, Maxwell, Einstein, Weill, and then you go on to quantum physics, with the current strategies to use gauge theories, and then in particular the standard model of quantum physics, which is experimentally very well-guested and is a great basis for most handy physics. But there are problems.

5:00 There are problems. First of all, there are philosophical problems. I told you there was a bit of philosophy in this talk. And philosophical problems have to do with the foundations of quantum mechanics. And as you know, Einstein was never satisfied with the foundation of quantum mechanics. He had long arguments. He was bored. He went off a long time. And And although the general verdict of posterity was that Einstein lost the argument, nevertheless, Einstein was a good physicist, and I don't mind having him on my side. There are problems also with the nature of the Big Bang and the anthropic principle. Why was the universe creating this precise confidence that certainly exists? There are philosophical questions of that nature as well. And then there are conceptual problems, mainly the fundamental conflict between quantum mechanics and general relativity, which is a source, of course, of a lot of difficulties, and which people are struggling to get around in different ways. Then, of course, there is the experimental side. Everybody is waiting for the Large Hadron Collider in CERN to start producing experimental results, and my colleague Peter Hayes in Edinburgh is hoping that they will find this particle, and they also you know, it takes a long time to build an experiment, it's been waiting for a long time and there's a super synergy which may be discovered experimentally there's a lot of exciting things that may happen in the next couple of years from high energy accelerators, but meanwhile there's a lot of experimental results coming from observational astronomy in a way it's cheaper to tell it's a space to build accelerators The problems in cosmology are moments focusing on the things called dark matter and dark energy. Large amounts of matter in the universe, which is needed to explain gravitational forces, is not explained by standard matter. Dark energy is even more mysterious, which leads to the cosmological constant. These are things that are really quite serious problems, which face experimental astrophysicists and cosmologists. and some people think of them as rather that we have our real theory or we need a little bit of tidying up, but it's a bit embarrassing when you find 75% of the universe is not explained so I think that this is potentially a serious problem and a serious problem may need a serious answer

7:30 so I'm not an optimist I don't think at the moment people are as close as they think they think there are possibly big gaps So let's consider, first of all, what are the current theories, in broad terms. Well, the standard orthodox theory is the best technique for trying to combine quantum mechanics and general relativity, and beyond string theory is something called M-theory, because string theory is meant to be only a perturbative theory. People are grouping around to find some theory which will satisfy all the constraints. Whatever the theory does, it has a very rich mathematics. The impact on mathematics has been extraordinary, and the benefits of mathematics has been very profound, and that will be with us for a long time to come. Whether the string theory is the right theory for physics or not, he certainly... Real physics is more complicated than theoretical physics. The mathematical benefits of string theory are really enormous. And I encourage people who are interested to get involved in the interface between physics and mathematics at this level is extremely rich, and I'm sure will take around 50 years to sort out, will provide probably some of the most interesting mathematics of the 21st century. Now, that's the orthodox view that there are, when the conferences on string theory are held, I've been to one or two, there are 500 people. Now, on the other hand, here in the IHES, we have alternative views of non-committed geometry. Alan Korn has been working hard, and he's explained to me recently his most recent results, which are very encouraging, about how to explain the standard model and other things by non-committed geometry of a very elegant kind. And this, obviously, will be clear about in this conference. Then, a third strand comes from my former colleague Oxford, Roger Fedoros, his twisted theory, which is based, but more fundamentally, on using spinners as a key point, spinners and light rays.

10:00 Let me take a look. I'll shout meanwhile. These are different approaches. This theory is about the minority, but also have mathematics and interest. And then there are people who work on the physics and gravity. The Ashdakar school works through different approaches. There are many people trying in different ways to combine fundamental theories and physics in a satisfactory way. There are crossings between these, there are next between swing theory, non-community geometry, non-community theory, not disjoint, but at the end, the features of non-community come here. And connections experiments are, well, the standard model, and connections experiments, broadly speaking, there are no serious tests in theory that can separate one from the other. So the fundamental question, the director, this is a hands-on director, the fundamental questions really are what is the nature of space-time at the local Planck scale, 10 to minus 33 centimeters or more fundamentally does the continuum make sense at scales which are too small to observe i mean you know this is a kind of fundamental philosophical question what is physical reality what is space-time uh or what is mathematics comes to that and what is the relationship of mathematics to the real world are we making mathematical models uh which then we just compare with experiments where does the mathematical model come from is something we It is something we invented in our human mind. What is the relation of the human mind to the physical world? These are deep, unknown philosophical questions. But I think I mention them because I think they're now relevant again. People thought about them in the days of the Greek. They thought about them later on. Then perhaps in the modern era, scientists became more confident of themselves. They thought they knew the answers. The philosophy was sent off to the library. I watched the classical scholars. But I think when you start to think about the fundamental nature of things at a very, very small scale, very, very high energy, and what it means, what is the relationship of observation, mathematics, you realize that these questions are at the root of all physical science.

12:30 And so, we don't really know the right way to think of it. There are different schools of thought. Now, in string theory, space-time is thought of not as a fundamentally fundamental concept, but as an emergent concept. It comes out in some limiting process. In non-community geometry, Alan Conn is telling me he has similar ideas, and any way space is being modified, to be replaced by some non-community structure. And in Penrose, I told you, he thinks about the fundamental nature of space of being built of light rays and spinners, not points in the conventional sense. And there are, of course, attempts to discretize space-time. If you don't have a continuum, maybe the structure ultimately is discrete. But the difficulty with discrete space-time is extremely difficult to make any coherent mathematical structure. It's extremely difficult how you will combine discrete spaces with the beautiful symmetries and so on of the human beings. If you know how all those symmetries and so on you've learnt about, then you're left with the fact that you know you take no tools. So, attempts to do discrete space-time really doesn't get very far. I don't know of any really successful methods. But, philosophically, people say, well, ultimately, perhaps, space-time is discrete, and one of these aides will know how to do it. So, where do we start? Well, let me give you my own philosophy first. Well, what is my view? Well, first of all, I remain open-minded. I talk to all parties. I'm skeptical, if you like. I don't believe anybody has a final answer yet, but I'm prepared to listen and find out what interesting is happening. Secondly, I think we should remember past history. Many episodes in the past have been where people thought they knew the answer, and then a few years later, a revolution takes place, everything's turned upside down, and the door is open to new ideas. There's no reason why that should stop. The idea that we are at the last stage of closing the door on understanding the physical universe seems to be hubris. Why on earth should our generation be the one that suddenly finds the golden key? It won't happen, I don't think. So, I expect revolutions to continue to occur at a very fundamental level, probably. That's why I think it's worth asking the question. And my third point is, I am a strong believer in intuition and great theorists, or great scientists generally.

15:00 Intuition is something which guides people when logic fails, when experimental results are not available, when they have only guidance is their insight, insight based on their past knowledge, insight based on their aesthetic appeal, and this insight is sometimes of more value than the technical arguments that are present at any given time. So, go back to the great physicists of the past and ask what they thought. And Hamilton might have mentioned, and Hamilton's instruction of Quaternions, and incidentally, when Hamilton discovered his Quaternions, He had, of course, until that time, been famous as a mathematical physicist, and his fame has persisted in that way. Hamiltonian mechanics became the framework for quantum mechanics. And when he discovered quadernions, he thought he'd made the major discovery of his life, and he said at the very moment he discovered them, he felt this was an idea he wanted to spend 10 or 15 years of his life to study, and he did. At the end of his days, he concentrated up quaternions. The orthodox view of the community outside was, the old man was wasting his time. Why did he waste his time on his stupid difference of algebra when he listened to great physics? Great law of physics. Well, as we've seen already, Alan Conn will explain more in his point of view too, non-community things, quaternions in particular, and things they need right to, have turned out to be of fundamental importance long after Hamilton was dead. In fact, Hamilton, well, many of you may not know, actually wrote down the Drac equation. He wrote down, in three dimensions, D with the X and I, D with the UI and J, D with the X and K. He observed the square of that, he said to the class operator, and he said, this process is fundamentally important in physics, along with 100 years before quantum mechanics. So you're very far, and of course, he introduced the same experiment. So, Hamilton has a great insight into physics, so much to pay attention to his thoughts. And similarly, Einstein has a great intuition. And the fact that Einstein was not satisfied with quantum mechanics, and everybody else was, I regard that as an equal value, you know. I put Einstein into a large, large factor of physics. So, I think we should think about the intuition of the past physics. And then I like to use Occam's Razor. Occam was a medieval philosopher. Occam's Razor was a logical tool. He said, don't use unnecessary hypotheses. Use the minimum complicated story to try to explain things. And I tend to believe that. So when the theory gets more and more baroque, I just wanted to use Occam's Razor and cut it off.

17:30 And I think you should also remember what is the aim of science. What is the aim of science? Well, people say the aim of science is to work out how it works, so you can predict the future by doing experiments and checking results and so on. I don't think that's correct. The aim of science is for the human mind to understand what's happening in nature. See, if you were computers, giant computers, you'd be quite satisfied with saying, look, give me the input, press the button, and I'll set it the output. That's a prediction of the future. But it doesn't explain anything. And the human mind in the community, it can only comprehend things if they reduce a simple form. That's all you mean by understanding. So the aim of science, I think, is to produce some understanding. Understanding involves things like simplicity, beauty, and so on, which are sort of intangible ideas, but they really are what guides scientists in their work. And so the aim of science is closely related to simplicity, beauty, elegance, beauty, and things that we can encapsulate in small form that explain larger than other things. And finally, in my list of philosophies, I say, and this is one you might think of surprising for a mathematician to put forward, I should say, don't be seduced by beautiful mathematics. I've just said you should pay attention to beauty, but you shouldn't be seduced by beautiful mathematics. Perhaps something has a beautiful mathematical factor, it may or may not be the right way to look at physics. I'll say a bit more about that later on. It can sometimes be a mistake to be overly impressed with mathematics. So now I have to... Okay, these are my... So, what can we explore in terms of new ideas which might open the doors to some new revolution? Well, if you think about it, in the past when people like Einstein came along, you know, he got away from the idea of space and time being separate things, you made a big philosophical step forward, and so every idea forward involves going backwards and throwing out some previously

20:00 strongly held belief. You don't make progress unless you discard the more privileges. So So what can you throw out that, you know, we've become attached to, but maybe it isn't holding us back? So here's the idea to explore. So, orthodox physics, ever since the time of Newton, or the beginning of the modern era, has built on the assumption that complete knowledge of the present, position and velocity of all particles and mechanics, determines the future. This is meant to be principle, even if you can't actually carry it out. It still remains true in quantum mechanics, with some qualifications. The Hilbert space of states is meant to describe the universe at a given moment, time, and then you have a time evolution that determines the Hilbert space evolution in the future. And these have been a remarkably successful formula to adopt since Newton's time, formula, why keep it up? And this leads, of course, to the fact that physics is based on partial differential equations. Partial differential equations embody all the fundamental physical laws and enormous amount of work has gone into developing the analysis of partial differential equations precisely for their purpose. But, however attached you are to this dogma, maybe we should question this at times. It's a pretty radical step, that's why my type is radical. is that perhaps we need to have some knowledge of the past to predict the future. The knowledge of the present may not be enough. Why should it be? It's not a... We come here from the past. If you think about it as a biologist, our DNA comes from the past. If you don't know your DNA, you can't predict what's going to happen to you. So the idea that the past in some sense has an effect on the future is not totally crazy, particularly if you think of the past as being the very near past, Why should an infinite time be adequate to predict the future? Even the notion of velocity, for example, is a mathematical fiction. Velocity is based on knowing that the immediate path of those limiting processes is an ideal story. So it's not so unreasonable to start to open that door a little bit. So let us now go to the quantum mechanics. So we'll have one of the figures of the talents.

22:30 Is it sacrifice? Well, it's certainly remarkably successful. It's 80 years of spectacular success, all experimental results predicted by Workout. It's a fantastic success story. From the hand, I find he didn't like the fundamentals. He said, OK, it's very successful, it works well, but maybe it's only an approximate theory. We want to search for the real truth behind it. And if you think a successful theory that has lasted for 80 years is therefore here forever, you have to go back and look at the history. Newtonian theory lasted well successfully for more than 200 years. It worked all mechanics, gravitational forces, fully successful. And then Einstein came along and wasn't satisfied with purely aesthetic reasons to do with the compatibility with Maxwell's equation and so on, and then the devotist theory, and then subsequently the experimental confirmation of general relativity we can now build up to a large body of work. But there was no pressing need for 200 years for anybody to question Newtonian theory. Anybody who questioned Newtonian theory was regarded as a crack. I said a successful crack, but there was probably many unsuccessful cracks on the way. So, in view of that, you couldn't say that because the theory had been remarkably successful, And, of course, Newtonian theory is successful up to the kind of scale that people tested in their day. So you have to go to much more precise measurements before any deviation from Newtonian theory is noticeable. So maybe we have the same situation with quantum mechanics. In fact, quantum mechanics is fine, but yet we haven't tested it at the limits. So each quantum mechanics might be quantified. Now, here's where I make my comment about being seduced by mathematics. It was, I think, a remarkable fact that at the beginning of quantum mechanics there were two great mathematicians, Hermann Weill and von Neumann, who very rapidly understood what physicists were doing in quantum mechanics, and provided a very firm mathematical framework in which all this theory could be understood. And they were so successful, the math and physicists brought in the line of Kostinker, he's become orthodox dogma ever since. If If you haven't had any involvement around then, it would be much more messy and, you know, the success of quantum mechanics as a mathematical framework might not have been quite so unequivocal. As a remark I make, you may not agree with it, you may think it is fortunate that we had two such great mathematicians that helped us along.

25:00 That's certainly the view that most people would take, but perhaps, perhaps, if it turned out that quantum mechanics does need modification, perhaps having a kind of respectability, mathematically, may have put people off questioning it. OK, so now we go back to our idea of... We want to think about physics in which the past has an influence on the future. And I said, the orthodox view is that only the peasant has creatures in the future, that it leads to a differential equation. So what happens, what do you replace differential equations by when you have to worry about the class? Well, the answer is you have to use things like retarded differential equations. The simplest way to do that is to just consider a short time interval of the class. that short time interval is s-street to predict the future and if you write it down you will get a different differential equation but it will be your non-standard kind it will evolve you have one point with one variable x at the time t and also x at the time t minus r which is a small positive constant it's a small back history and then you write down equations of that type here Here's the simplest toy model of an equation, dx by dt plus the constant times x to t minus r, a linear retarded differential equation. But obviously that's just a toy model, and you can imagine building a much more complicated equation of this kind. Now, of course, equations of this kind, retarded equations, are not certain new in mathematics, But, interestingly enough, they're mainly used by electrical engineers, things like control theory, think-backing, and so on. In my library at Edinburgh University, I found a book on discarded differential equations and similar things in the library. I took it down. It had been published about 15 years ago. I was the first person to take the book out. So it's not actually a very popular subject in the mathematical world, even if it's used by engineers. But you will find references to it in a very indirect way. There are two references I can draw to your attention. One is in Simon's thesis,

27:30 where he investigates the interaction of charged particles and has a situation where he does end up with a part of the equations and a counterpart to the vast equation as a way of handling his particular problem. so there's a physical model where it turns up another place to turn up in a more philosophical way is in the famous book on quantum field theory the textbook of quantum field theory by Bjorkin and Drell where in the introduction they make some general remarks about physics and they say well we don't know what happens at very small scales and so on and so forth and maybe space-time is in some sense granular. The word is sufficiently vague, but it conveys the idea that when you get down it's like a small pebble on a beach instead of, you know, a nice smooth surface. And if it's granular, they say, well, then, of course, that would lead right to things like the positive differential equations, because each granule is above the other one of them. Information would take time to propagate from place to place. And they say, well, that's the case. But, of course, things are much more difficult to handle, so we won't bother with that. Well, I mean, that's a good reason for not doing something. It gives you difficult. Not only a fundamental reason. A fundamental reason is that you don't do it unless you have to. So they were developing a theory satisfactorily for their purposes, so they didn't go into it. But they recognize that, in fact, if space-time was not continuous down to micro-scale, you would need to do something more complicated. And in principle, the introduction of it opens the possibility that, if you need to, somebody will explore. So I think that is an open invitation. Why not? Somebody should look into it. Now, if you take a simple equation I've written down there, that is, you know, this baby equation, you can imagine, it's a linear equation, one function, one variable, and you can see that this equation, in fact, has a unique solution in propagating into the future to any initial data in the interval between minus r and zero. Given the function of that range, This then will enable you to solve it in the next range, step by step, by successive integrations, so that it has a unique solution to propagate. And the initial data, therefore, is the infinite dimensional space of other functions on an integral. Now, that's to be compared with ordinary differential equations, where the initial data is finite dimensional, just to one value or value of derivative or few derivatives,

30:00 you've got higher order equations, a finite amount of information, this is an infinite amount of information. And it's not even all derivative, because that would be corresponding to expanding the function at the interval at one end. It's actually non-literative, it's a whole interval. So you'll see the potential there for an infinite dimensionality, which is really important. Well, that's us, you know, taking step one of what I've done is to put my toe in the water, and immediately the answer comes out, you have to worry about these things. Well, if you play around with it in a very elementary way, one thing you can ask is, well, is there some kind of spectrum associated with this situation? Well, we're not going into doing any formal analysis, this is not formal. But suppose you just plug in an exponential function into that equation. Then you will get, for the frequency of the variable z, you will get an equation like that. z plus k e to the minus rz, equals zero, that's an equation for a hollow water function, into the discrete zeros, and definitely many of them. And the time propagation is z to r, think of the space based on these functions with those discrete eigenvalues, so to speak. then you will get a flow which is, but this flow is not unitary because the solutions are not purely imaginary numbers. That's basically that bad. On the other hand it's not that bad because asymptotically they are. So, you get two things. You get infinite dimensionality and you get sometimes discrete spectrum. Two of the characteristics, if you like, of quantum mechanics arising at step one. So, in some sense, it's a, you know, maybe an indication. So, the idea might be then the following. Let's say, if I go back to Einstein and resurrect him, and tell him, here's my crazy idea, what is the origin of quantum mechanics? It's the, the initial data that you need in a small interval of time, actually gives you the health of the space state, so it's the mystic quantum mechanics. And once you develop that, you might sort of see how quantum mechanics emerged. It might not be exact, it might be an approximation. By the way, a similar idea of different kind was put forward independently by an Indian physicist called Raju, who is very well known in the physics community.

32:30 I don't agree with all his ideas, but he has had a similar idea. Now this is philosophically attractive, in the first place, because it says, well, there's no mystery about it. this Hilbert space is. Quantum mechanics has this peculiar sort of dogma. It says, here are the axons of quantum mechanics. We live in a world in which state is given by a vector of Hilbert space. What happened before Hilbert? and every all these beautiful laws of self-adoint operate as an evolution. This is a formalism. You have to adopt. Then there's all the questions about expectation values and how you compare the experiment where you get slightly controversial waters. But the basic assumptions are, don't question where it comes from, here are the rules, and that's what Einstein didn't like. Here we have a different situation. The situation is that the Hilfer space is somehow meant to correspond to the initial data which we need in the pre-class future. Small piece of data, the time interval could be very short. So, you know, that's philosophically, I think, much more attractive. Einstein, I'm sure, would have been happy with such a possible possibility. least worth investigating. You know, these are, I'm not doing all the work, I'm just throwing out ideas, philosophical, you see, what I meant by a court of philosophy, a court of speculation, and so on. I'm not sure which court of this is. So, I think it's worth investigating, and what we should do is, of course, try to, you see, there is, having a small interval of time, is some kind of discrete process. It's So you keep the continual, but you introduce something which is sort of semi-discreet into it. Now you try to combine this with geometry, because geometry has been very successful. The beautiful laws, the ones I don't follow with at all, are in Maxwell's equations and Einstein's equations. These look very beautiful to me. I don't like to do with quantum mechanics, which I, the share of Einstein's misgivings about. And so, what you have to do to develop this is, of course, you need to develop what you might call new subjects of retarded geometry. How do you do all the geometry that you've done? After all, as I said, partial differential equations for 200 years, was developed by Lagrange, Laplace, or the big class of the past,

35:00 because it was a technique to understand mathematical physics. And the differential geometry went hand-in-hand, part of the differential equations. And so we have this vast essence of mathematics, built in 200 years, which has really been motivated to a great extent by the requirements of mathematical physics. Now I'm suggesting maybe that wasn't big enough. and do a new subject for you. Generalization, which involves retarded geometry, and I put underneath, and also advance. Advance means you change the sign backwards or forwards. That, you may worry you philosophically. It didn't worry me philosophically. I mean, you need the future to pick the future. Well, that's a matter of point of view. When you write down a differential equation, saying you need an information on a time interval, it depends on where you think you are in that interval, the middle, or the end, If not, that's difficult to understand that, so in principle, advanced equations are also that should be part of the story. Now, people have looked at retarded equations, as I said, in lexical engineering, but one fundamental, ambitious, ambitious in the sense of not going to do the work, ambitious in terms of the aims of what you want to achieve. If you can't develop a theory, which goes all the way to developing relativistic theory of quantum mechanics, quantum field theory, and general relativity, gravity, it's not worth doing. We're trying to get everything, not just a basic quantum mechanic to be fine, but we want to go much beyond that. And the very first big problem you face has to do with relativity theory. Obviously, there's a fundamental problem, how you define retarded equations relativistically invariantly, because time is not an intrinsic notion in relativity theory. So, say something depends on past time, what does it mean? Well, think a little bit. There is an answer. See, if you place time by time, shifting in time by distance, minus r, that's a motion in time, which the infinitesimal generator is differentiation. to the derivative. So translation is the exponentiation of differentiation, with a parameter r. Now, that's one variable. When you go to Minkowski's base, you want to do the same thing, and you want to have up here a first order operator, because first order operators may essentially

37:30 exponentiate them, they correspond to motion in some way, and there is only one invariantly defined, first of all, the differential operator, you know, in cosmetry, magic of Dirac operator. So you have no choice. The only thing you can do is to expensate the Dirac operator, and that corresponds to retardation. Now, it's an interesting fact that when Dirac discovered the Dirac operator, I told you it was really discovered by Hamilton, never mind. When he introduced the Dirac operator, he was searching, the reason he led to it, he was wanting to look for a third order equation. Now, Sandler's books of physics will now tell you, he got the right answer for the wrong reason. It doesn't have to be a third order equation, he did perfectly well, according to the mechanical theory of this type, the second order operator, and Drac, somehow, stumbled at it by an angle. He led the wrong reason for the right answer. Well, in some places, I want to suggest that you are correct. The right answer, what he found was very fundamental, because, for example, with this notion of redidation, you have to have a first order operator. You can't expect a second order operator in terms of redidation. They're quite different. It's like the heat equation. So, it's fortunate enough that Dirac was thinking along these lines and got the first order operator. So, this suggests that using the Dirac equation, and this operates on spinners, of course, and so here you take the first step away from a toy model, So something that lives in physics, and this I've done 20 with a physicist, written by Greg Moore, you look at the Dirac equation, which is normally written down, just Planck's constant, Dirac operator, minus mass term, and then you add here this retardation term, k down to exponential minus rd. So this is a modified Dirac equation, a retardation equation. Now, actually, when you look at it carefully, you see that because spinners, as you know, come in positive-negative energy states and direct discovery, that changes the sign. So one lot of spinners are retarded, the other lot of spinners are advanced. So you get a combination of both. Now, when I worked with physicists, they goodly write down exponentials of operatives without blinking an eyelid.

40:00 But if you're a mathematician, even if you're not really an analyst like me, you have a conscience. So what does it mean to exponentiate? Well, I know how to exponentiate a blast operator with the right sign, and I know what that means. But the direct operator in Kosky's space is not a heat operator at all. So what on earth does it mean? Well, again, if you go back to physics, it does make some sense. You interpret this operator as something you apply only to physical wave function. the linear combination of plane waves propagating with loss at less than the speed of light. Things don't happen faster than the speed of light, physically. And if you consider only plane waves which are propagating with loss at less than the speed of light, then each plane wave can be retarded by its own proper time. Then you integrate over all the different components of the wave function, you'll get a definition as this. So this operator is simply retarding each component of the plane, the wave, in its own direction, in its own time direction. So it makes very reasonable physical sense. Now, suppose you go beyond that. Suppose you take any book on quantum. They will give you the following sort of calculations. They will call dispersion relations. You take a stationary solution of the equation, you shove in a term like that, you decompose the spinners into two parts, and then you write down for the dispersion relation. What do you get is an equation like that, with this term being new. Normally you would just get, you see, E equals MC squared, but now you get extra term. And this, interestingly enough, because the exponential has, let's imagine, real parts, and the energy is meant to be real, this will require you, imaginary class, to vanish. That will only happen if the appropriate quantization condition holds. So that leads you to the condition that this parameter R, which we put in mathematically, not having any idea what it should mean, should be an integer multiple or something specific. And that integer multiple, or the thing of which it is, it is the integer multiple, is something which is what's physically well known, is called the constant wavelength of the electron. the minus 12 meters or in time about 10 to the minus 20 seconds so this little bit of calculation we came out sort of you know automatically we didn't have much choice really we have to use the dirac operator and so on um tells you that the the scale the scale called this hypothetical

42:30 short-term memory is 10 to the minus 20 seconds well that's a pretty short-term memory if you can Even when you get old like me, you can remember n minus 30 cents. So, it's actually, most practical purposes you might think of is almost instantaneous. It's a very small change from instantaneous time. That's only satisfactory, because after all, the calculation might have come out with a number being vastly different. Well, this may not be it. This current argument may be very, you know, doubtful, but at least it's interesting that it comes out as something encouraging. This scale of this parameter is quite independent of the other. There are two constants in this equation. There's the retardation. One is the retardation parameter, the length of time. The other is the coefficient outside, which tells you the magnitude of that effect. Both of these are initially of choice. So we've got a quantization condition that sort of fixes the scale, the first one, the retardation parameter. The other one is still something which you equatorize and compare with the experiment or something else. So it could be something extremely small, but they're independent. The two have nothing to do with each other. So, at this time, we leave it as a parameter to be adjusted later. It could be so small that, again, you would notice it in many scales. Well, once you've got the rack operator, this is now common ground with Alan Codd, you can turn lots of handles and do lots of things. The rack operator is a marvellous piece of algebra, if you like, because, first of all, the direct operator is geometrical. That means you can define it in curved space-time, so it makes no difficulty to combine it with general gravitation in that way. And you can write down the target equations for spinners which couple to other gate fields, for example, to electromagnetism, and you can get Maxwell's equations coming out, or curved space-time coupling into gravity. So the fact that the direct operator is geometrical the flat case, the curve cases, or to cases where you have intrinsic fields. So that's the advantage of having things applied geometrically. By the way, this, when I talk coupling to gravity, this just means having a fixed gravitational background, it doesn't tell you how to retard gravity. If you want to be consistent, you should ask how to retard gravity, how do you retard Einstein? I don't know, it's a problem. I mean, it's not the first order equation, it's in sort of way, it's not clear how you do it, I don't think it's impossible, because it's in relation with spinners and so on and so, it ends in question what is the right way to define a retarded set of ancient equations.

45:00 I'm just saying these are the kind of questions that this investigation turns up. I don't claim the answers, I'm just throwing out problems. Now, another thing which is very important in quantum field theory at a fundamental level is the notion of particle creation. In quantum field theory, there's the famous fluctuating vacuum. The vacuum we live in is meant to be, you know, hands are shaking all the time. Particles have been created but there is usually conservation of charge so when I create one particle you have to annihilate it, it's opposite there is a place where I can see particle creation emerging in this scenario I'm describing vaguely, very vaguely for this, I now do I told you there will be 25% mathematics this is a bit of that 25% mathematics so I'll recall to you some firm mathematics index theory if you have For example, in the four-dimensional Riemannian manifold, a compact manifold, with a spin striker, you have the drac operator, which acts on spinners, and the spinners come in two caralities, and match one into the other one. This is an elliptic differential operator, and it has an index, which is the number of solutions from one side minus the number of solutions from the adjoint from the other side, and the beauty of the index is it's topological and it's a topological formula. So this is the sort of mathematical content, the way the index Dirac opera appears in topological contexts. Now, there is a three-dimensional Laurentian version of this result, which is not perhaps so well-known to people, but essentially covalent, and physicists understand this in their own language. So if you consider a three-dimensional space now, then the Dirac opera is there, is itself a elliptic operator, which is self-adjoint. If you imagine your space, c dimensions, varying in time, so you take space times time and you think of it as varying, then you have one parameter family of self-adjoint operators that vary in time. If the situation is periodic, so that outside the box you're in, you get back to the standard position,

47:30 some kind of back position, then the operator at time one should be equivalent to the operator at time zero. when that's the case you have a sort of periodic situation then there can be what's called a spectral flow namely the cell vanant operator has a discrete spectrum assuming your sea space is compact the eigenvalues move as you move time but they are both positive and negative and you move from one end to the other end some eigenvalues can cross from being negative to being positive but you get back to where you were at the end of the operator's equivalent them are the same but it doesn't mean that the eigenvalues stay constant take as an example all the eigenvalues can move forward one step think of the operating d by dx in the circle shifts everything by one step or two steps so you get a shift or what's called a spectral flow the amount of the number of eigenvalues change sign and of course if some eigenvalues are the other way you cancel them out this number you get this topological invariant of the family If you perturb the whole family in any way, this number does not change. And it is, in fact, closely related and essentially equivalent to the four-dimensional index. And this is the advantage that it sounds like physics, but this sounds like geometry. This is about one parameter family of physical objects. And this number is interpreted physically as particle creation and annihilation. So, for example, if your back operator is, space is coupled to this engaged field, which represents outside force, then the spectral flow can be formed by the number of particles created by this external field. Now, in the context of my retarded story, although I want to do things relativistically, it's very like break symmetry. And I imagine that in the short interval between r minus r and zero, whatever happens there is an initial data, it's free. In that initial data, there's no reason why we can't have interesting topological features taking place which lead to particle creation. So particle creation would be something which is coming out of a mathematically known framework and it would fit in well into the idea of little bits of space, the initial data. And this initial data space could even include gravitation,

50:00 because if the space-time itself is, I mean, if the topology actually varies, if the decomposition of space and time is not going to be fine but has holes in it, then in the middle of that space you would have a gravitational input which would lead to the topological change in the index-like quantity, which is familiar in the geometrical story. So, there is space in this theory for particle creation, which is one of the characteristic features of quantum field theory. You know, if you can't explain that, you don't fit with experimental results. Now, I pointed out the last remarks. Obviously, I'm not claiming This is the right way to go forward. These are initial steps to explore. And the first thing to point out, as I pointed out, is the simplest equation is that the time flow for a time equation is not unitary. Or, if you like, the spectrum is not pure imaginary. Secondly, the equation, the one I wrote down initially, is not necessarily symmetrical between time and minus time. Well, that may or may not be a good thing. The octogamy is a fundamental equation that is half-hand symmetrical. or maybe they're not, for very small values of the parameter k, the deviation of these two things could be small and difficult to detect. So experimental tests, somewhere along the line, these ideas, even in this very crude level, could be some experimental test. Yes, you might be able to test some deviation from unitarity or quantum mechanics by experiment, and then you might help to get hold on the parameters involved and so on. Or, alternatively, more likely, you need some much more complicated models that I've just done. I've written down the simplest kind of model and said we should explore it. So, let's just indicate that this is, I'm not putting forward for you, a theory in any sense. These are just hypothetical, something for investigation. Let me finish with a remark I said that you should do, we should develop something to retarded geometry. And the Dirac equation, Dirac operator, is a very key part of our story.

52:30 Similarly, we should go back and develop what we call retarded mechanics. How authentic mechanics is a very important part of mechanics. It's a geometrical aspect of some parts of physics and played a very important role. So, can we develop a sensible way to develop retarded mechanics? What would it look like? Well, here's a picture. You have a symplectic manifold, and you have a Hamiltonian function, which normally you write Hamiltonian flow, Hamiltonian vector field. And suppose also, the manifold has a remaining metric. For example, it could be a K-ler manifold, which gives you both the symplectic function and the remaining structure. And suppose you have parts in this manifold. A map of real numbers into M is a part. As well as you wanted to look for paths which are solutions of what you might call the retarded Hamiltonian flow. Take Hamilton's flow, store it as an equation and retarded. What would that mean? Well, you see, take your path, that's the black line there, and then you take the function of the time t, you take the time t minus r, and you don't say that the tangent vector is equal to xf, But this tangent render is equal to that one after parallel transport along the intermediate path. Parallel transport is defined because you have a re-manic metric. So, it makes sense to write down the solution of these equations and they will be interesting for the curves. And of course, in all these situations, this parameter R, if it goes to zero, you recover the classical thing. The classical limit comes when R goes to zero. So you can study the classical limit. You can study what happens in the neighborhood of the classical limit. You can do approximations. But this is a mathematically opposite equation one could study. It's interesting to look at. And irrespective of its application to physics, there's no reason why we shouldn't do these things. Well, I've told you everything I know, really. for a few years off and on. I usually talk about them as physicists. And then, of course, I get shot down. Maybe there's a physicist in the audience who come when they come already.

55:00 But I found the physics criticism are always very helpful. They focus on certain problems. I go away and think more about them. I define the ideas and so on. So I've benefited by talking to physicists. And some physicists are more encouraging than others. Let's put it that way. So I think, I hear I'm talking to mainly expositions with an interest in physics, and I think I'm just trying to private the door a little bit so that people should at least consider the possibility that there are avenues to explore which are not standard. and involving the idea of the differential equations are not only without modification to a larger iteration, not the only tool. In fact, that's what we've developed. Let me go back to my initial comments about, philosophical comments about, mathematics, physics, and the real world. You've asked, is the physical world built on these marvelous mathematical structures that we've developed? And the answer is, well, of course, we've developed structures because we think they're useful, but that's the only tool we've got. No, we don't have the tools for them. So here I'm insisting that maybe we need to enlarge our set of tools. Perhaps our set of tools is not quite wide enough, and the future may or may not confirm that we need to go in this direction. And lastly, as we said, all the things I mentioned in the beginning about swing theory, monoclonal geometry, and not being any interesting avenues that are being explored to try to provide unification to the next level, are not necessarily joint, nor need they be totally disjoint to what I'm saying here. There could be links between all these ideas on level. I don't think they're meant to be orthogonal. If they're all meant to have some connection with the of the whole world, they should have some non-strivial sections and overlaps, and ideas going from one to the other could be very useful. So I offer this to you, as if you're a young man, if you've got your PhD and you've got a job, you can work on it. Otherwise, Thank you, Mr. Vaggo. I'm sure there will be... I have two comments. The first comment is about all of the Dirac networks, and I have one way to restate what you say.

57:30 in the current world, what happened then, he says, okay, we believe at this moment that the geometry is defined by the Dirac theory. What you are doing when you replace the Dirac equation by its qualification is in fact a kind of first dressing of the geometry in the sense that what you do is to replace the Dirac theory by a sound functionally, which you then use as a replacement of the Dirac theory. But we know, I mean, we know this for sure, that from gravitational effect and from the, in fact, we know that the re-operator, in fact, the propagator of thermos is actually dressed by the monoclonal, so it is actually a fact, in some way, in some way, in some way, that actually what happens, when you look at the IRN, when you do the re-operator, then the re-operator is replaced by a function, it's not the same as the function you wrote, So what you're going to do is, for example, by d times , which is very good, it's a function of the square over a second scale, not a square, but a square of the scale of the square. And this function can actually be constant in this function, this function. So what it is doing is actually addressing the journey. But what you are computing is a little bit of a specific function, which is like, you know, x plus . Now, why is that function rather than what I will invent from? That I will invent from the general's heart, which is exactly what he described in Germany in Germany. He said, by the Dirac-Berner, regular practices do come up easily. All the blessings in Germany are, by action, following, you know, how the Norwegian process forces this. To replace the Dirac-Berner by something that you don't have to come up like that. So, of course, you get these times the function of the square to correct the problem will not give you a partial equation or partial equation. So, this is possible. Now, the second thought I would say is that the way you are creating a kind of picture of this picture in your hand is completed. That's exactly what, standing at the summit of the year, what we are following in the year. He said, if you want to, you know, we're going to see a lot of small, small, small, small, small, small, small, small, small, small, small, small.

1:00:00 I mean, this is completely important. because you're not making a product so it's very, it's very hard to, on the other hand, the size of the product is also the bigger size of the product because, I mean, it's the origin of masses, it's very spontaneous display, and all these masses are elated by some factor, it's not quite a lot of one which is So, in that way, there are quite a similar idea. I'm not surprised at all, given the fundamental role that the Dirac operator plays in the non-humanist story. I think, encouraging, there are some links. Of course, the Irodi's toy model is nothing very... I would go back to say that I repeat what I said about Hawkins' Razor. If you, once you make a step forward to throw away a lot of old baggies to start again, you start at the ground level of simple ideas you can, no doubt. So I've been trying to, from this point of view, and it may turn out that it's entirely equivalent, or we can transfer it into another language which comes in a way that we're quite familiar with, different point of view of being mathematically equivalent. But I think there's something there in having a point which is a, mathematically, philosophically, attractive as a starting point, without too much, you know, there's not so much sophisticated mathematics behind it, but just one little idea, and you see where it goes. But of course, a lot of work to be developed to work out the mathematical consequences and how to modify it to fit data and fit other theories. Anyway, I'm carried by what you say about possible links.

1:02:30 And therefore it's not inappropriate to talk about it. More questions. So... Sure. Well, when I first started, I had ideas along of that kind. I did think of a lot of those lines. But some of those ideas are very broad and you don't have quite nowhere to go. But the idea of using a particular integral power, I then got from this chap, Raju, who did it. And then you see, very simply, a lot of things coming out rather easily. Now, as a pro-model, that may not be a full right, you may need to integrate over a... Or the other idea, the other thing is... Yeah, yeah, yeah, that's right. Then that would... Well, I think it might be other constraints. I mean, I think the... I agree. I originally started off thinking much more generally. Backward influence in an internal way. Although, if you might have philosophically, that gives you a little bit of trouble. Do you need to go back to the beginning of the universe to get all your data? So, whatever you have, you have very short of effect, anything, so therefore it's a very sharp cut off. And therefore, a short interval might just be a first approximation. And as Alan said, maybe you could iterate and get a better one. But it has merit, you're starting off with a radical idea, you want a toy model where you can test some very simple things. At this toy model, you can see some things coming out very simply, you can see some space behind it. So, conceptually, you can begin to, and then you can try to relativize, do it to relativity, because it lends itself easily to a number of steps. Before this, I was just weighing my hands, didn't know where to go. After this, I thought, well, I can see possible directions. But I certainly agree with you. There's only one of the possibly much more bigger range of possibilities. But she opened a new door. Oh, real physicists are here. Speaking of the simple things first, first, is the last estimate of your Dirac-Thorch-Odel regular or circular? And did you try to find the total or the divergence of the actual equation?

1:05:00 You know, by the way, historically, that Maxwell always was done with the habits of Dirac-Odel Well, excuse me, I think the master limit is not a problem, I don't think it's any analytical difficulty with that And secondly, the Oh, oh I see Oh, there we go, okay, but the question about the Maxwell equation, I did say that you couple spinners in anything you like, but the Maxwell equation is really spinners, couple spinners So, it comes out that the right equation comes with something else. So you can do the same, you can do the Maxwell equation the same way too. But that, you remind me of something else I meant to say. There is this, a very, very deep philosophical observation. You go back, way back into the past, before Newton, there was Descartes. Descartes knew the universe was you had to have mechanical interaction of things. You couldn't get to take action as this. And when Newton came along with his laws, Newton said, I don't worry about what this means, here are the equations, just see what they were. And there was a long battle between the schools of thought. And eventually, Newton won. People accepted the law of gravity, even though nobody knew how gravitation could operate. Then the next day, he came along with Maxwell. And you remember when Maxwell wrote down his equations, he didn't get the equations by saying, ah! He had a picture of rotating vortices interacting, rather like their car. And he got the right equations. But when he got the right equations, he said, Well, I don't really need this picture, there's a guy, he threw it away. Well, you know, I have to think, maybe these guys are not so stupid. Perhaps, on a small scale, you know, not forces, but maybe space-time itself is in something that's granular and interactive. So maybe those pictures are not so crazy. So people quite often say that there's a great step forward having the ether, whatever it is, and we got away from that, we wrote down differential equations, and we don't know what they mean, but it's fine. But again, maybe that was actually, long sense of history, maybe that was actually a mistaken, throwing away with these old ideas, going away with non-philosophically based ideas, just mathematically works. Maybe that's too naive. So, I'm just saying we should re-examine some of our prejudices, And the ether, although in one form discredited, some form of study of the nature of space-time

1:07:30 on small scale could not be, in my turn, not to be so very different, in a way, from the ether of Maxwell's vortices or even Descartes' vortices. Well, that's a good philosophical point to end on. More questions? No, no, no. As I say, if you look at it mathematically, there is no difference between them. You know, usually you can distinguish between the derivative and the value. You think the derivatives are now, and we're trying to work out how we're going to go in a step forward, and the past is where the influence comes in. You could change those rows around. The derivatives could be in the past, and the values could be now, and you have a mathematical equation to solve. It's still just a question where you put yourself, where you place the origin of time. These equations are time invariant. So there's no philosophical difficulty about motion. And that's why these equations that Feynman had done, advanced and retarded, he made a comment about that. There's no mathematical difficulty. In fact, they have to be good physical models. In his case, there's even energy conservation. So it's something I thought was a philosophical obstacle, but it's not. Questions? I have a question about your theory. Yeah. I mean, have you talked about general visibility and, you know, are they half-handling solutions and the evolutionists of the trade and the half-handling small billions? No, you're going much too fast for me. First of all, I don't answer any of these questions. But I would like to say this, that if you start with a radically new point of view, then you have to examine every question that was asked in the past to see whether it makes sense and is the right one you asked. You don't just make a fundamental change and then try to copy every step that happened before. So all notions have to be redefined, re-examined, and so on. So eventually, your theory has to be one which will reproduce the orthodox theory of substitutable limits. But the theory itself might be non-unitary. Or re-normalizability might be happening throughout the window.

1:10:00 It may come out. So I think everything is up for grabs. That's why I don't think, for example, in a theory like this, that I would accept anything, except Maxwell's theory, Einstein's theory modified, and everything else has to be worked out from scratch. gatefields, whatever you like. Why should you assume anything? Start again from scratch to see what you require to get. I hope that you get, at a limit, you'll get some reproduced previous results. But all technical questions, I think, are on hold until you decide what the technical questions are you need to ask. And you've got to build yourself up on the ground slowly. So I'm not chickening out for asking a question. Because I think it's premature to ask some technical questions before you have a framework. Okay, so I will thank the sir Michael. First of all, I'm very happy to be able to deliver this talk, and we're here in Paris, and also, for me, this is almost precise, in the 40th century. This is remarkable, and it was in May 1967, and because this was the institute of the and I had a vague feeling that I have seen Alain in the canteen of Maxwell, this is And he told me that no, he was at the Koldavale and he didn't dream there to visit him at that time. He was very, very young as well.

1:12:30 So I'm happy to be here once again. And my talk will be dedicated to explaining some parts of the recent paper, which we have written together with Denise Larison, which is an archive, and I have been explaining the philosophy behind this in great length in my talk in the College de France last December. So this time I will not be speaking about philosophy, but rather present several technical definitions, constructions. Just very briefly, motivation for this story One of the possible basis of non-seolar geometry might be operands instead of associated rings. And one can put it in a different way, saying that it becomes useful and interesting to book operands as representing the kind of geometric object that kind of non-seolar space And nowadays, we hope that it is a very natural generalization of these things. And in developing this idea, I want to stress two points. First of all, I will not consider the classical operands, but a lot of generalized operands as well. The generalized operas that exist in the literature are the names of modular operas, proctoras, props, one half props and things like that. I don't present common generalization of all of them containing much more objects. conference dedicated to the imaginary field one element which also might be a non-community geometry. Nikolai Dorov presented some versions of rings which is probably our generalizable

1:15:00 So, roughly the plan of my talks will be as follows. First of all, I will speak about the category of graphs. Well, graphs are pretty elementary things. So, when I insist on various technicalities, the problem will be not in graphs themselves, but rather in organisms of graphs, therefore graphs themselves should be defined in a rather narrow way, in a fast way, I will say. Then I will introduce something which I will call categories of labeled graphs. So the most important example for using almost all specialization known in the literature are in fact graphs. Then my operas do have components in some ground categories like vector spaces or graded vector spaces or logical spaces or whatever and sometimes in the first third or middle of my talk I will formally define generalized so then there will be a construction which is similar to that of tensor algebra. So tensor algebra is a free algebra in the of associated algebras, and I will define pre-generalized literature in my collection. And then I will speak about and internal . Now, what I call internal ,, usually in the literature is called inner . I am trying to sort of struggle with this term because I feel that if something is inner, an object of a category whereas internal is something which is inside the category itself but I do not insist you can agree with inner instead of internal. Then I'll explain analogies and give some examples of constructions.

1:17:30 So let us start with a category of graph. I will choose a basic small category of cells. I will choose not quite innocent operations, symmetric monoidal structure, direct zoom, disjoint union. To see that it is not quite innocent, it involves choices and various categorical passing, you should just imagine what is a disjoint union of a set within itself. then the definition of graphs it's the same as in our old paper graph consists of sets graph consists of two sets F2 and V2, and two maps E2 and G2, and normally we have flags, we have vertices. No vertices, we always forget the sexism, I think, every published The graph that no black is a vertex and no vertex is a flag. Then the map which associates to each flag its vertex, boundary of this flag, and then involution which tells you that some flags were prepared. Okay, and I will collect a roll, a graph with one vertex and no edges at all. And geometricalization is what usually appear in the pictures in the papers. Using a graph is a picture like that. Now I will try to represent this particular graph geometricalization as obtained by a sequence of operations. I just take all flags and consider a flag as a half of the interval. 0, 0, 1, so 0, 1, half, including 0, excluding 1, half, and then I glue the fibers of the

1:20:00 magnetic tor at the point 0, so I will get some corollas, and then I glue the outer ends And then I need the notion of disjoint union of graphs, and of course the form of disjoint union of features is very easy, but as I said, if you want the disjoint union of a graph in itself, then you should go back to your category of sets and recall how to do all this. Now, quite a technical definition of morphisms of graphs, which I will not give completely because it's impossible, just one should see to look at it. So essentially amorphism consists of three paths. One is injective contravariant map defined on flags. And we say that whatever is not in the image of this map is contractive by this morphism. Then we must have a surjective covariant met on vertices. So I'm saying that if two vertices have a common image, then they are merged by this morphism. And then we have an additional stuff that there is an evolution corresponding to these morphisms, which includes some of the non-contracted flags, the non-contracted flags. I will skip the additional conditions about these maps, and I want only to say that taking into account these additional conditions, one can very easily define what are how to compose and how do they act upon geometric realizations. Again, this is usually what is . If you have such an edge, you can track the subgroup,

1:22:30 the subgraph of consisting of those which are not contrast. And then you produce its quotient of the subgraph, may merge all vertices belonging to each fiber and then delete everything that is contracted. Okay, so there are three classes of special morphisms, contractions which are bijective on tails, and graftings, which are bijective on tails, and flags, and mergers, which only merge some vertices. So if one looks at this story of mortisms from this viewpoint to my initial picture illustrating how to construct, how to produce a geometric realization of a cap, then one sees that the steps from the disjoint union of flags to the disjoint union of corollas is represented by amorphism which is a merger. And then the next step from the disjoint union of corollas to this artificial homunculus, is represented also by a morphism which you only graph, so compositionism. This whole passage is a composition of morphisms two times. Now I will need a special kind of morphisms which would be called total grafting. So this special kind is defined by any graftor. So what we do is we cut all edges so we get a disjoint union of corollas and morphisms. It's the source of these morphisms. And then we apply a grafting which grafts everything that should be grafted. So, this passage like that is a total grafting corresponding to graph 2, and it's pretty clear that it is defined uniquely up to unique isomorphism. This is a categorical function.

1:25:00 But there are two more important classes of morphisms. is you take a graph and contract all ages and then merge all remaining vertices into one vertex so you will get the morphism of a graph onto a corolla and this is very important class of morphisms I will call it on C-O-M, not that close. Then there is an important example, Essentially, you start with sigma, so this target, and break it down into Corollas. Then you take essentially green images of this Corollas, I think this was the condition of this, I'll skip it. So I got a category of graphs, and I will now recall you what role is played this structure by this combinatorial skeleton in the theory of usual operas the theory of usual operas you are not considering all graphs but only trees moreover you are not considering all trees but the trees with additional structure which is orientation the film tree is oriented without backtracking and you go down to a route. Now you are considering each tree like that as a

1:27:30 potential scheme of calculation. You're assuming that whatever flags you have that go inside, that orientation goes inside, so when it is not a route, you can input some arguments there. And when several flags meet in the vertex, you can input some operation there, some notation of operation there. And then you say to calculate the result, you start with inputs, go to this vertex, apply the operation corresponding to this vertex go down until you get at the next vertex again, apply the corresponding operation. And then at the root, you get an output. So you have many inputs which correspond to ingoing flux and one output. But it turns out that there can be loops in such a prescription. So not necessary trees. There can be additional information like saying, OK, this flag is white, and this flag is black. And in the white flag, you can input only such and such things. And in the black flag, you can input only such and such things. So in order to accommodate these possibilities, is I will define an abstract notion, a category of labeled graphs. And an abstract category of labeled graphs is some abstract category together with a functor, which you will call forgetful functor. So you are imagining that you first put some labels and then forget them. And then you get just the usual graph. And there are three main conditions. One condition is that, of course, you can power disjured unions of abstract label graphs, which become disjured unions of usual graphs when you forget the library. The second axiom is that this is fateful. So if two morphisms have gone, source and target

1:30:00 become equal after applying, forgetting, then they are equal. And then we need a condition that if that morphisms which are breaking down the geometric graph onto its corollas is uniquely liftable upon my category of labeled grams. So this is our three main axioms, referring to the categories of labeled groups. Okay. But this is not all that is needed for applications to operas, to generalize operas. and you need the going properties that that and then that optimization diagrams also should be uniquely and you should be able to reconstruct an atomization diagram either from the lower side or from the left-hand side so the classical examples of generalized operands considered in the literature are literally graphs into this additional labeling. And the hunter side forgets the labeling. And the labeling can be of the following type. So we can have oriented graphs. This means that this means that any flag is oriented. If two flags form an edge, then they have opposite orientations in the sense, or compatible orientations in the sense that one is oriented from its vertex and another to its vertex. So the whole edge becomes oriented.

1:32:30 And of course one should see also what kind of morphisms is allowed. on the screen. Also, there are directed graphs. There are such graphs that are oriented and that can be hanged in space in such a way that whenever you go along orientation, it's always from up to down. So it's according to gravity laws. It's directed graphs. And And another way to define a labeling is to mark all vertices by images from 0 to 0. And to impose some restrictions upon morphisms again to be restricted at any time limit them. Colored graphs, if you have an abstract sets, abstract set i, then i color graph is a graph all flags are colored and all hubs 1H are colored by one in the same color. And . Well, there is also a cyclic labeling. Well, there is a lot of various labeling in the literatures. I think about a dozen of different kinds of operations appear, and they are all covered by these axioms. Okay, now I will be considering factors from this category, such categories of graphs into some round categories like, as I said, like categories of linear spaces. So a ground category should be any symmetric monoidal category. So it should have tensor structure, it should be symmetric in the used sense, and associative. And then we can consider a new category consisting of functors.

1:35:00 one defined only on corollas, defined only on corollas, and isomorphisms between corollas. Such an object will be called collection. Collection will play the same role for generalized operas as linear spaces for associative algorithms. Actually, this whole large set of definitions is constructed in such a way as to achieve two goals. One, to include in one conceptual framework, whatever kind of words in the literature. And another, to keep the basic properties of the category of associative algebras, which are linear spaces can be generalized it can be generated by linear spaces subspaces of them and so there is only much more operations than just one of binary so so it is a pretty cumbersome structure, but it works okay, and it contains a lot of interesting particular cases inside. So a collection is a factor from a category of corollas inside a category of legal crafts and and isomorphism between them after the venues in our ground category okay and on collections you have an analog of tensor structure you just make tensor structure for a point twice you produce the structure point twice on any carola Sigma you just take the other product So I dedicate this talk to the spirit of Groton, which is still here, I think. Okay, I did write the paper with some reluctance and did some presentations with some reluctance But I somehow feel that one should do that. This is a correct way to think in this particular surroundings.

1:37:30 There was a lot of confusion. There is a lot of confusion, especially when it comes to . Absolutely. So I have defined what are collections. So they're defined . And again, if you have some intuition about the usual operas, so I am saying that I am taking the value of this factor on the corolla, all of his, all except of one of which flags are oriented towards the vertex in one outside and then I am taking an object which is a sigma say and a of sigma should be imagined as a space linear space of operations that I can apply to the arguments that I put onto this flex which are otherwise this particular intuition might be not quite helpful but something like that is always true okay so I have defined collections and now operas again the intuition is very simple in the case of usual operas if I I have this elementary operations put onto each vertex and then I glue together, or rather, glue together some flags in order to form the ways when information flows from one vertex to another. Then what I am doing, I am defining the global operation from upstairs to downstairs to the route. But in particular, I can ask what happens if I do the part of calculations inside of the graph. So each contraction, each particular contraction will produce a net from the initial situation to the not quite final, final, but pre-finals, so if one axiomatizes all of this, then one gets a notion of which is a tensor from gamma in code with disjoint some to G with its standard structure and the usual isomorphisms

1:40:00 I'm by the deal formally I will consider them as more or less identical in fact because they are non-realized isomorphism and satisfying So as I said, I should imagine collections as generalized vector spaces and operas as generalized And therefore, I should define analog of tensor algebra, which is free generalized operas generated by a collection. And it's a kind of core limit, yeah. Sorry. Associated rings, yes. Oh, yes, the category of the, if you start with the category of graphs, So, several vertices and oriented and let me see, no, it's a category of trees with vertices of balance in three. So you put two objects in here and the third object C here and then a relation corresponding to associate it okay okay so a form of definition is that I must consider to produce a free generalized operat I can construct and end the fact during the of collections and this and the fact is obtained by the following way I will be considering all graphs that are morphisms to a fixed base with a fixed target Sigma and morphisms of this category should become isomorphisms

1:42:30 identical of Sigma and then starting with the gamma collection I will produce this quality of course if the community is infinite then I should impose respect the restriction and the statement is that this and the hunter has a natural structure the triple and the collection therefore when I apply this after the collection then I get the canonical structure of an operat then I of course can't forget if I have an operat I can forget all high notification and then I get forgetting functor which makes a collection from an opera and it turns out that this one is is a joint to it and so it is a free opera so this gives me a notion of answer algebra in this setting and also there is a white product of operas which is the same as white product for collections you just do it point-wise. Now let me remind you what kind of construction I want to generalize from associative rings to this generalized operands. If I have an operat, an algebra with presentation, that means that I have a pair of an space which it generates. One can consider the case when A1 sits just inside A, or one can consider a more general case when A1 is a linear space which is linearly mapped to A, sorry, to A, and the image generates this algebra. And the statement is this, that there is a kind of non-commutative of all maps from A to B. One fixes two associative algebras, A and B, together with linear spaces of predicates. For example, if they are five-dimensional, you can take for A1 the whole algebra.

1:45:00 So, eliminating this arbitrary choice of the linear sequence of generators, generally, it's useful to consider it. So then you can find a kind of non-community of all morphisms from A to B. It will be a morphism from A to E tensor B. You should imagine E as generated by elements of matrix sending generators of the first algebra generators of the second algebra but this matrix the elements of this matrix itself are not necessarily commuting they are not necessarily numbers they are just non-commuting variables which satisfy all relations necessary in order to make this and the theorem says it's one can find the universal object of this kind and these objects first became quite known in the context of quantum groups so if you consider the case when a and b are the same algebras which are generated by two elements xy such that xy equals to q minus 1yx so it is a quantum field then E turns out to define a quantum two by two matrix algebra and it turns out that the construction is so general that you can just formulate it in this one so it's kind of non-commutative object of symmetries in this particular example and if A equals to B that's kind of quantum endomorphism One can just reinterpret this as an internal or inert object. So you define a kind of tensor products, a white product of algebra . You define it essentially on generators, and the restatement is then, it's this formula, that we can construct a functor joint to this white product, and this functor joint to this white product

1:47:30 is called the internal columnomorphism of A2. So, once again, the intuition, you have two associative you can consider just classical spaces of homomorphisms of A to B. Okay, linear object. But this is, in many interesting cases, the two small objects that may allow the coefficients of the matrix defining these homomorphisms in non-commuting, new non-commuting variables and it goes only those relations which are necessary and sufficient to produce it and then you get this one tight space of one more this and this is what so you get quantum matrices and in the example of the opera for which it will be generalized you will get something more interesting an additional class of examples which are very by the way of course if you want to reconstruct the classical space of a to b which is which is linear on the generator and you just say okay now let's take this internal object and it is an also shown associated algebra let's just take its k points points in the ground ring for example algebraic closure of the ground so you will get geometric points which are the classical object but they sit inside this quantum object and generally a very small part of it. For example, in the quantum group Q2, the classical points constitute one particular torus inside, one particular cartons torus. All other tori become delocalized. You don't see them in the quantum picture. But one survives, and this is exactly the classical space of the quantum physics. This seems to be, by the way, a very interesting phenomena, which is more generally deserves some kind of general understanding for example quantized projected plates which is the space is survived just one member of the entire canonical cells why it is interesting No, I think so, yes, yes. Since quantum groups are interesting, and quantum groups are a very particular case of this story, I think it is.

1:50:00 So we have this list of analogies about which I have been speaking already. So a vector space should be generalized to a collection, tensor algebra to a free operant, general associative algebra to an operant, then the product of algebra is white product of operas and this is a restatement of what I said one any associative algebra a is a classical opera which the classical notation contains one just one component and all other components are zero notation is not exactly compatible with my notation on class okay now I don't just I can formulate this theorem about internal co-homomorphisms by just imitating step-by-step the respective formulation for assorted algebras so we know now what is should be called an opera this presentation so you have an operant a and you have a collection a1 and you construct the free generalized over f of a1 and you consider morphism which is subject then you can define the category of generalized morphisms of the operat need to be in S4, and then you can state and prove the theorem that there exists a universal operat where the monk is generalized morphisms. Morally, this universal operat is a non-commutative space, or its elements are non-commutative coordinates, which classify this kind of morphisms and and what I want to say that I've been explaining it at last in December that whenever we are doing non-community geometry because first initial motivation is to think about say algebra geometrically or