Roundtable Discussion

0:00 If you want to do a physics, as you said probably several times earlier, there are a few ways of looking at things in this diary, where you either formulate questions and leave a few things, but I mean that's different from formulating new, if I could put it that way, these meetings, the first three of probably, and I made a comment, you know, I started... Thank you very much for your time today, and I hope you enjoyed it. I think it's one way of addressing the issue of being able to talk about properties of theories in general. I mean, you came up with one of the questions here. I mean, to separate different theories in terms of bring on the various information processing capabilities in the recent foundational research of the signaling and broad counting itself. I mean, I think that's one aspect. This is a very difficult theory, but as you started off by saying, there should be some input and data for all of that. No, it's not clear to me, but I have in mind the fate of quantum mechanics, and I think the scientists know it very well. Quantum mechanics is really easy. People really sculpt it. People try a number of things with quantum mechanics. Not the mathematics of general physics. It really could be used as a new type of knowledge. I can maybe answer that. Yes, sure. I think the fate of quantum logic was sealed by the fact that it needed to be used.

2:30 And that they didn't simplify any method or... It was just too poor. Also the sort of structure of the services people took was... What is not true in quantum mechanics which is true in classical physics, and it doesn't help you anything, saying that something is non-distributive just doesn't bring you anywhere, you have to say what it is. Now I think in this sort of program we are doing, we are actually trying to put by hand features of quantum mechanics in. So we are actually building, in a way we are building spectra. What Samson is trying to say is we are not going yet for one specific one, but these are broad sections which we are filming. Like one example, which I recently built, is like a toy theory, which is like a little toy theory, which has a lot of complications in our globe, and it's like a simple, just a category. That's what I was meaning, that's what I was really getting at. But there are examples, not just all of them. Oh yes, yeah. Because it's not just either, it's more like anything. The more you see the scope. Everyone's question was in that direction too, to build more general theories, just to understand, always just to understand quantum mechanics better, not necessarily to build it in theory, but also to understand quantum mechanics better, to do the spectrum more clearly. I've spent so many years, I mean being like halfway between mathematics and physics, learning both disciplines. I've been really astonished at the extent to which things have been really fundamental in one discipline never being crossed over. And one of the things I've felt for a long time is that the natural way to say a lot of what physicists do is using the categorical language. Let me just turn it down. First, the quantum mechanical theory is a skill. The quantum mechanical theory is good. It might be worth it. There's an attempted category of versus an attempted specificity. Attempted product means when you combine these spans, you know, direct sum is going to have to first position a different state in a single system. And all of that, I mean, the axioms of an attempted category are, you know, they're down in the physics books, there's the alien-beam articulation, which is the same as the stash of the pentagon itself.

5:00 Um, and, um... Having learned that, I then went and said, okay, so we can generalize all these spin vector notions into a large class of things, and that would have made it possible for me to try to write down these models for quantum gravity. I mean, Penrose had an aptitude of a student who tried to do it in four dimensions, but using the same category, so he had something very, very unsatisfactory. So that's one example. Another example, which I think is still very important, is the assignment diagrams. They're not really something that the medical students do. When you go and look at them, if you take a modern course, you know, for people who know about particle physics, they tell you that the way to identify knowledge is first to identify the citizen. They need to figure out all the interactions, all vertices, as they would put it, which are not forbidden by the symmetries of the theorem. Okay, what they've just done is defined the category of representations of some causality. And then they say, we begin by summing over all these things. We rerun the renormalization group, and most of the vertices become irrelevant. So at the fundamental level, they are exactly summing over organisms in a intrinsic category, and that's what a finite diagram is. And if you go in the diagram for these things that took development, where they did the thing that was necessary to make... To make what the business system would poorly term a renormalizable period of gauge. So that's the foundation of the whole understanding of nature. What they did was to show that they could regularize the things uniformly in such a way that they were still intertwined. That's the thing people do. So it's all category theory. It's just that the physicists don't know what category is and the mathematicians don't know what type of diagram is that. And it's such an extreme intellectual gap. I think it should be closed on its own right. Once you know the right mathematical formulation for something, you have to generalize it. You have to look at larger classes of models.

7:30 So it seemed to me that this was just something that was hilarious. It was the problem that people went to graduate school and they were one or the other, but I felt that it was a bad thing here in the United States of America. But, um, the failures, I mean, you know... Physicists use the language of mathematics, but they're using the dictionary that was published in 1920. So I think that the language is just so heavy. And really, even elementary textbooks on quantum mechanics, they collapse completely when you ask what happens when you combine two systems or more systems. The first time I heard the word tensor product in a physical setting was in meeting Everett's work on the many worlds of different texts. He's somehow, he's one of these people that I met that's exposed, and he said, oh, this is a tensor buffer, you know. And all those things about how to look at, at, at, at, well, I think that the right way to write it in elementary text would have been... This includes that when we start combining systems, it is a bit of a carry for me. Physicists know how to get around this and do calculations, but it makes it simpler, so it's not as smart to get around. So that's been my perspective on the subject for a number of years. So part of the initiative here was really to start with the singularity, but since we locally had these two groups working together on these sort of things, which then might sort of spread out and then it becomes like a stronger community, I think that's how you start something like that, and it's actually really scary. So it's very good that you've brought somebody from what we call the N-category of quantum gravity. We didn't expect that anybody from your area would be here, because it was supposed to be a lot of work from the beginning. I have a question. Because it's just following the... I mean, I've spent the life of my categories in other areas of physics, so these comments sound like music to my ears. But the big question is, why is the market of ideas such an irrational place? Markets, by definition, are not irrational. So, did you do anything to cause this?

10:00 Well, I think we invented a system for funding people to peer review and negative feedback groups. I was always interested in doing business. I stayed away from mathematics or physics in America because I saw what they were doing to all those poor people. I mean, it was just a huge problem set. And you're eventually supposed to develop intuition, which is, in my opinion, supposed to work with big biology and not by estephlogy. So that's not bad if you can sort of go and get into school of mathematics. Anyhow, but why it's developed to such a point, I don't know, but it's... There was a cultural historian I was reading once who said that the early scientists, in the early 1920s actually, always seemed to have something to wrote about. So, he seems to have been, by and large, been sort of discouraged in graduate school, so... I mean, it's supposed to be something practical, right? You're supposed to worry about your career. And so, I mean, you know, I mean, I'm from Japan, so I'm too used to getting taught five years after he was in graduate school, so he almost didn't get a job, so... Because, in a sense, you are preaching to the converse already. People coming here do have the twistor knowledge and the overlap. What did you say? I was trying to ask you if you were on the street. I keep walking past you. I think you're leaving. I'm happy to. I'm happy to. It's wonderful. It's my pleasure to speak to you today. Thank you very much. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. My topology is very good at writing in a way that you can't do anywhere else. And so, they may well be large enough that a person has to be able to do it once. But they don't do it unless they've done it before. So, after that they can't do it. So, they can't do it. So then, what do we do?

12:30 We can't do it. And so then, divide. And I'm going to give you an example. I would call myself a computer scientist, and there are computer scientists who use quantum physics more than category theory, which is a very interesting way, but I think there is a possibility for a very interesting interaction between computer science and physics which may be facilitated by category theory. So, like, I'm not sure that computer science is science, because we don't do any experiments, I don't know, it's called computer science, but maybe it's something else, whatever it is, but it's kind of an experiment, but, yeah, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, That's something like an experiment, but something different. But there's this little thing in relation to the twistor, you see, and in what sort of program do you write? I mean, if you know a program like Nico's Speed, you know, where you drive a car around, and the car is positive, and it's a physical model of reality which can be simulated on a computer. So that's an experiment. And you can look at the structure of these programs which are supposed to simulate reality, or simulate physics, and I think that's quite interesting to do. And the use of computer science, the use of all physical realities, and one called the real number, and I promise you it doesn't use any of that.

15:00 From reversible ones. So, the reversible computation in group reads and examples would be basically the projection on sets or final sets and the unitaries on a Hilbert space, but from this then you want to have maybe a reversible computation that basically just functions and in a quantum setting comes like super operators. What I wonder is, is there a structure that you can construct the irreversible as derived operations from the reversible, from the reversible as more fundamental. What would be an example of this sort of drifting interaction between computation and physics, facilitated by computer science? I'm very complacent about that. Well, I think there are... I think you're right, but coming back to what Chris was asking about the research program, I said something about deducing observations from information, something that may seem sort of small. I think it's true, as I was saying earlier, that the work on communism is usually incompetent, but maybe the pre-energized work on the foundation is incompetent, I think, because it's asking kinds of questions that open new kinds of windows on the subject. So maybe that's what we actually need in order to make progress.

17:30 One very important thing is that it actually has given jobs to people in the foundation of physics. It's not just like, quote unquote, so most people are interested in research and start getting jobs now. Actually, I have a question for you because I know you're a member of a group that is heavily instituted. How does the topic of category theory die down in mathematics and mathematics? It's good. We have some exceptions here. We have some exceptions here. We have some Doris exceptions. No, there's only two departments in the room that you have the opportunity to meet. Because there are nothing more than three or four. It's not a major interest of your annotation. I will. Sorry, I go back to what you said earlier on, good mathematics, what good mathematics you do, what good mathematics you write, what good mathematics you do. Could I just qualify what Peter has said? Some very eminent British mathematicians have apparently decided that. I think one takes a slightly broader view. I mean, right now, it's my very strong impression that after a period of perhaps 20 or 25 years in which the subject has been relatively fallow in France, it has actually started to grow again quite impressively in the last five years or so. So, the subject is... one shouldn't perhaps focus too exclusively on the British scene in making these generalizations about the position of category theory. Yes, well, I don't want to do a complete tour de Riese, but just going to our nearest next-door neighbor, I think the scene is perhaps not quite as black as... Exactly. Very much so. Well, maybe one point which could be made is, I've heard Sammy Allenberg explain the destiny of category theory by saying that mathematicians consider good math, math which solves difficult problems.

20:00 Now, if you solve a difficult problem by category theory, it's very likely that in order to present your solution, you won't remove as much category theory as you want. From your solution, from your way to it. So category theory helps maybe think of a solution, but it can very often be eliminated from the writer. This is my interpretation. He said it for mathematics. But there is that thing that if you want to write a paper, it helps with thinking, but if you want to write a paper without it, then you usually... Isn't there a general point here about the tension of the mathematical culture, it would be very interesting for Peter's reaction to this, there has always of course been far more prestige given to the problem solvers, to the people who have scaled the great heights and proved the great theorems, than to the people who created the tools at the necessary level of natural generality that enabled them to do so. I mean one of the things is just how much of fundamental category theory was created by Grosendieck. And so on and so forth, along the way, just as part of the machinery that was needed to solve the vague conjectures. But the people who created the tools, and they turned out to be the tools of the right natural generality to prove extremely deep theorems. But as you say, the tools were very often effaced in the presentation of the theorems themselves, and there's always been... There has always been more prestige for the people who, as I said, prove the theorems, and for the people who create the conceptual machinery that requires them to do so. And perhaps we need a cultural revolution in mathematics. I completely agree. There's a very interesting point about the way that generality, unity, and what one might call kind of natural and mathematically fruitful generality is operated in.

22:30 In and around ideas in category theory and it certainly operated in extraordinary fruitful form in Grotendieck's hands and it's not clear that it did so in anything like as fruitful a way in the generation afterwards. I think that is beginning to change a little but yes I mean again I'm interested in Peter's view on this. I mean how far do you still hear the well-known accusation that topos theory is It's general abstract nonsense these days. Well, people don't say it for my pleasure. No, considering... Well, yes, glad to hear that, but... So one of the ideas of this initiative is to sort of nourish more interaction between mathematicians and physicists, and computer scientists, because actually there's a lot of... So are there any suggestions of things we could do? This was just our first initiative in this bag, and I can imagine people buy them on the idea of thinking, so yes, it would be nice if you could bring it to the right of the time to explore it. I should definitely note that we have a number of physics centers. I should also note that we have a number of physics centers. I should also note that we have a number of physics centers. You should appear in a few books with that kind of physics. Oh please. 42 I think you're talking. It's going to be more than three books. It's going to be a series.

25:00 My instincts tell me that we've reached a curious stage in the way things are in academia. I actually think there's a golden opportunity there. There's something really good to happen in this type of bringing together of things. But you have to seize the moment. What normally happens in this sort of thing is young people have good ideas and they work very hard. The first meeting is very successful, as people say it is, you see. The second meeting, two-thirds of them only turn up, then the half again, and the same thing peters out. And after a few terms or so, it's all vanished. Now, the only way really we're going to get anywhere is if people actually almost make a commitment to come, irrespective of what sort of agenda. The academic community really want the academic community to start looking seriously at the overlap between these disciplines, and believe me, the politics of this is incredibly difficult. Peter knows what I mean. It's very, very hard indeed. The only way you can do that is having a real body of people who really are committed to it, and who consistently put the energy that's needed. I really think this is very important. As they say, bring your friends to the party. Because the more we have, the more effective we can be. I mean, I was contacted a little while back by the director of the Newton Institute. He said, well, do you think we should have a session there on top of this? I thought that would be a great idea. But that sort of thing, to get the whole community, or Darwin Symposium, you know, that sort of thing, to get the whole community working together is our fate. We have to do it ourselves. So it really is important that people like this. If you want to do it, you must keep coming, basically. So many times I've seen his attempts to cross over borders, you know, the first one or two are successful. So, if we are moderately successful, maybe we can get some money to help pay people's travel expenses and so on, things like that. But the first six months, the first year, everybody has to keep coming if they're going to work. So please, let me beg you to do that. And so these guys have to come up to a veteran today. Well, the community doesn't like us. I mean, people don't like people like us right now.

27:30 I've lived all my life, I got used to that, I don't mind. What extent does it do to mathematical mathematics? Well, it's all sorts of things. I think it's partly because people are very busy. Particularly very senior people are extremely busy. They spend half of their time in science. People die with just a bit of book time in science. People may always start something safe in the main world. And it dies. I think it's a great shame that this dies, because I really do think that there's a chance... Do you agree? There's something in the air... I mean, I've had maybe one of the, there was something of this kind, but earlier, about 15 years ago, there was an exciting series of meetings, and we found that here, here is what, well, in Ireland's group, We have a really exciting series of joint seminars with our people participating, and that's about 18 months, and it did eventually take a lot from some of the energy we've had. I think you're right, it's very hard to count the ranks now. It's partly a matter of having enough critical class for the community to keep it going. Maybe not everyone's going to come every time, but if enough people come, then there's always a reason for it. One of the interesting things is that we had more people than we expected to be here. Right. So who's gatecrashers? Well, we said gatecrashers. Well, you wouldn't put bouncers on the door next time, do you, to keep out the gatecrashers? We were sitting here talking to 9,000 names, we sent an email out, and there were actually more people in than... There are a lot of people just in the area here, and then we have a lot of people in Lumen and things like that. That's a very good question. I thought you mentioned state theory. But you didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind. I didn't have that in mind.

30:00 My sense is there's something different about this enterprise that I'm not concerned about. The key point there is simply that category theory became more useful among people in computer science. A few areas fell there as well. It's the case, as I said, that people apply category theory. I'll say my impression is that in mathematics there's a lot of people using category theory without anything else to do with that. Well, anyway, my sense is this is a new community that's kind of, in particular, collecting people and possibly all things together as a team to form a body. Britain's right that there is an opportunity to kind of see, but we maybe don't. You should give some thought to what the best format of future meetings is that's going to encourage people to come back and to keep coming. Can I ask one comment on what Chris said? I think one of the reasons for this rather diffuse sense of hostility to enterprise like this was actually fingered very nicely by Lou Crane, who I think is still sitting there behind me, last month when we were at a meeting in Boston. On rather similar lines to this meeting. And he remarked that the problem is that mathematics and physics in the United States in particular, but I think it's true in Europe as well, is very much impregnated with a tacit philosophy of pragmatism. And the attitude towards foundational issues in physics particularly is very much don't look down. I think it is different here, but it's certainly very pervasive in the States, and I think even in the case of, since you raised the point about computer science, it's a very good thing that people are getting jobs in computer science departments because they know about category theory, but one of the things which has worried me considerably in the last few years is the danger that the comp sci applications in the category theory have become

32:30 To some extent, the tail that has been wagging the dog in mainstream developments in category theory itself. I don't, again, I don't know if Pete has any beyond that. Yeah, yeah, but certainly a potential danger. But I think the general sort of pragmatist attitude towards anything which goes directly to the investigation of their first principles and ultimate ingredients of definition of structures, particularly when they try to take There is a general pervasive hostility to any such program, I think, that stems from a kind of diffuse, pragmatist philosophy. Well, but if Louis is right, then, you know, it's pragmatic. Well, I don't think Louis was endorsing this position, were you? No, no, no, I think we can agree on that. I am one of those that came here without a computer. I just discovered a computer on Google by chance. Actually, I think that there are two communities that should really learn what this is. Because there is one common point, apart from the general goal of understanding mathematics and chemistry, So my general goal is that I want to derive quantum mechanics from lots of different mathematical theories. Simply this was the real goal of the project. Quantum logic, quantum mapping, and simple logical goals. And the problem of quantum physics is that it's a true community, so there is a community which we call operational. Operational, not legal. And one member of the committee was Robert Peck. By the way, he's in Cambridge, and we were discussing two weeks ago about the possibility of organizing a workshop, maybe a long one, but I think it would be a good idea to do it together. There is clearly a lot of stuff that we can learn in class.

35:00 I would say, I don't know if we can do this now, but one thing that I'm interested in is that we want to actually be what is the intersection of chemistry and quantum mechanics, in quite a concrete way, and one big question that I see is... It is likely, actually, that the homology is the description of space-time, and that means physical space-time. And this is something where I think chemistry has come from a lot of other things we were starting to learn in our classes. This is truly something where a lot of projection could happen. It's a difficult problem, but physical space-time is something that everybody in it would quite agree with. I'm going to be basically talking about what else to do. But there are surely other things, so this is one thing I'm very interested in. Maybe we have a few of these points on the blog, something we'll develop. I can tell that it's again funny because there was one or two events, or three events I think, which was really about new steps for space, and two of them were organized by computer scientists. Two of them, I think, Sotkin and Martini were speaking at Google, and they were speaking, and I was speaking, and they were speaking, and it was like an argument. Three weeks of three weeks of drag action. People did a lot of work to compute the science of discrete space-time structure. I've written three works on this. Some positions compute the science, few fit for something like this. There have been some initiatives in the past. We should actually get these sort of single events which are happening in the past, and say, there have been these sort of initiatives, but nobody knows about them because they're all single events.

37:30 This is being pretty successful. People are trying to figure out what that's going to be like. I would very much think that this is an interesting topic. The question is, who would be involved there? It shouldn't be hard to encourage anyone. And maybe the webpage of this meeting is a wiki. Maybe the community could go beyond the talks. There's a notion of, you know, community blog, you know, various people could enter their thoughts and they could be much more than just talks and meetings. By the way, this webpage also requires contributions. Because the people of manatees are not, you know, I don't know if you know that. So we can put stuff, so actually, surprisingly, we already put on your work while we have no food in the room here. So we're trying to keep it open for you. This more structural thinking about foundational thinking, including space, time, all of it can be viewed as some sort of main program. The program should also more help the science and engineering for the physicists where these things, computer science, can be doing, which are really relevant to what they're doing and can be done. There are papers by Lamport, now everybody knows Lamport, I feel like some kind of bad guy, because there's like paper, speech, space, time, structure, and so on. Is that the answer? That's the answer. If you could just replace the name by something, it would be... Well, it certainly makes sense. I mean, it's a very interesting point about quantum models and distributed systems. It's a lot of time and it's extremely important. Go on. Coming back slightly to the earlier comment, I'm not so sure what you see in the front view of the slide, so it's the tail wagging the dog, and the information is by the bottom left-hand side.

40:00 Yes, yes. So one thing I learned is that in the front view of the slide, the bottom left-hand side is the computer science. But what do you feel is the structure of this series? A lot of computer science uses the structure of these conversations to manage the results from the mathematical results. But it seems to me that you need to be in order to take your work into consideration. And yes, respect, you know, you've been involved in this. Let me just say one other theme of investigations that I haven't really heard being spoken about yet is also this other, this more geometrical way, I mean, to, you know, make a connection with the quantum field theory side of things. That angle on it shouldn't be ignored as well. What kind of work would you be writing? Well, that is the bias school, basically. But not just the bio-school, you know, things that make concrete connection to QFT, not just quantum mechanics. I completely agree with that. It's very important. Time, field, then the next step. Yeah, there will be. Yes, we all have, you know, something like that. Yeah, yeah, yeah. What's the back page? There is a list of conflict problems which makes both of these two possible problems. So we have as an academic problem, as an academic block, no discussion yet, but they are there. Actually, I would maybe like to amplify what Bruce was saying. So one thing that struck me about the talks with Bert was that at several points, I think more or less literally people said there are several approaches, like there's the Sartre approach, there's the Monon category approach, the Thomas approach, and it can't be true that these are different approaches, you know, these must all be different, I mean, of course, in a way they are, but what I mean is there must all be different aspects of one and the same thing.

42:30 And maybe one further focus, which hasn't even been mentioned, is I think what Bruce mentioned, this pictorial picture where the quantum field theory is a representation of cohomology and calibration. And I think one thing that is really very crucial, and is more crucial than figuring out what space time is, is to figure out, you know, what quantum field theory really is. You know, it was like, in physics, there was a time when physicists didn't use differential forms, and there was a time when they didn't use groups, and there was this sort of grouping test, because they were thinking it was just some abstract tool that had no use, but you know, physicists are like this, when the point comes that you'll use powerful things with it, they will jump on the train and will use very, very much, and category theory in physics has to get to the point, you know, it's not sufficient, this has happened in the past a lot, that Category theory, text, and physics are very speculative and always start by trying to understand what quantum geometry is. People will always see, oh this is very speculative, and it's about category theory, so I won't waste my time. It's important that category theory gets to the point that we explain renormalization, that we explain what is Dirac-Witten theory, what is Shannon's theory, what's really going on. You know, what is geometric language? It's something, what category? There's so much category theory hidden there, I think. And this needs to be highlighted. And it's important that we don't... Follow like five different approaches of category five quantum mechanics because there's only one physical reality and it's with all the aspects of the same thing. For instance, I mean, because there were questions like that around the Monoil approach and the C-Sauce theoretic approach because you can't compare them directly. In the Monoil approach, in the Schrodinger picture, we're talking about vector space of states. In the C-Sauce theoretic approach, we have mapped them to their anamorphic models. There's a map going back and forth between vector spaces and anamorphic models of things. It's a duality kind of thing, two aspects of the same thing. And the entire thing should list as a thing emphasized in some provost. I think that was also being, I think, Bruce asked us, I was like, because I've seen stocks, how do you formulate dynamics in the stocks? This hasn't been addressed yet, and it can be a thing, and it's important to do that, that we don't just say, okay, we know how to describe the kinematics of quantum mechanics of the provost, but we really want to do something. We want to formulate, like our Witten theory, say something, you know, we want to have vector space of states associated. Two incoming values, outgoing values, partition functions, all these things.

45:00 It's for sure, but it has to be done. And it's important that this is done so that physicists see this applied to something they have care about. They want to know what the quantum mechanics theory is. If they don't know yet, category theory can help. I hope you enjoyed my presentation and if you have any questions, please feel free to contact me via the email address on the screen. But further, well, I think that everybody should speak from some kind of distance, and looking for the essential structure and what can be only described as a kind of legend. I mean, for example, well, Penrose might be difficult to recall, but basically it might be known. Yes, we are running late. I would say we have the coffee break now. Coffee should be outside, hopefully not cold yet. We have, say, 20 minutes. By the way, most of you were trying to pay already, the fine pounds, and those who might not have done this, they can't come there. Quite right. Let's not swing left. No, I did just want to put in a plug. It's precisely the relevance of broadly appropriate ideas, but more specifically of topology at a great many levels. And the study of space-time structure is this thing dedicated to the selection process, which we've spent the last two years in fact trying to put together in Boston. I just wanted to put in a plug for that, and also just one thing I'd like to say.

47:30 What happened behind this is very important, the unifying perspective behind these four or five distinct programs in categorification and conformity. But nobody's mentioned about the applications of categorical methods in GR. And that's quite a lot to say there, I think, particularly if you're looking at synthetic and practical geometry. We had two very interesting talks in Boston, one from Gonzalo de Reus on Einstein's factor equations. The talk I heard about a year ago in a workshop on GR there, which was by an extremely bright German researcher whose name I've forgotten now, who's about sectional curvature from the point of view of the... But what's the tactical question? Most people don't find most problems interesting. If you want to keep a group of people together, this is the focus of the course. Most businesses can find that goal. Let's talk about GR. Yeah, but GR, we don't need it for GR. See, that's the point. General relativity works out very well. It's sort of like the people we've had over the years. Well... I wouldn't argue it's fine for science, you know far more than I do, but I still think there's a very deep... They're not choosing GR, Richard. That may be so, but they're not going to fundamentally agree with us. I mean, if you take any large... There are very important ingredient issues in some of the debates about quantum gravities They're not really, I mean, they're on the sidelines. Any of the answers, it's my experience, in any one of these subjects, there's lots of little things which you can study, which are interesting, but you have to say, well, keep your eye on the trunk. Yeah, well, I think GR is so much more than a trunk. Let's say, anyway, one of the two is great. These are the main branches. Historically, historically, it's not... I don't know how it is at work, but the people can grab it in a different way, the technical side of it there, using GR, but for the most part, general relativity itself has become a moribund subject. It's not, it's not, it's got to focus it within the people's concerns. No, I take your point, but I... Okay, I'm so sorry. Yeah, and I'm a...

50:00 But, I mean, I originally started by developing a language for quantum, I think, physics and geometry. The representation theory of the Lorentz algebra gives you, the unitary representation theory gives you a kind of mechanics in which there are operators corresponding to mathematical objects, and when I tried to figure out how to piece them all together, so they would describe the geometry of a simplicial approximation to a spacetime manifold, I then later realized that it turned out that I was just studying over functors, so that the functor category, the common category between the The category of quantum geometries and the sort of space category, thought of as a sequential complex, a la street and oriental, that common category has a natural way of being summed over, and that took some doing. So that turned out to give a very attractive model for quantum general relativity. But the problem is that it isn't independent of the sequential complex. You put a constraint on it that destroys the theorem that says that it can take all the maps into any tensor category if you get a topological field. So it's not a topological theory, which is good because gravity is there. But it means, in order to try to define the criterion theory, you have to sum over... All triangulations, so summing over all four-dimensional superficial complex. And this point of view was developed by Carlo Rovelli, it's called the group filter. And it's a very beautiful point of view because it says space-time is a quantum process. I'll explain that. That it's a superposition of different combinatorial space-times. And we have to sum them to calculate physical quantities. This is a very beautiful picture. So it does make a step in the direction of replacing the underlying point set with a categorical object, but there's something missing here, because if you just naively sum overall triangulations, then despite all the wonderful coincidences that make the state some finite in each triangulation, which is actually quite a major step, I mean most people think that it wouldn't work. It's quite tricky how you regularize it, so it does work. But the regularization doesn't spoil any of the symmetries. So, we've gone all that far. We've found a quantum theory of four-dimensional general relativity, which is perturbatively finite.

52:30 Each term in this series, if you think of it as a perturbation series, is finite. And then we still have this problem where we have to sum over again for the number of them, and that whole result sort of gets lost. So this is very sad. But there should be a physical resolution. You see, there's a development within classical general relativity, which sometimes goes under the name of holography, which says that the amount of information that can actually be transferred from a finite system to an external observer has only a finite dimension. All of this is going to be observable in order to get a really fine idea. If you took a simplicial decomposition that had too many simplicies, then the information in it couldn't get out to infinity because it would disappear into the black hole, and therefore the sum is regularized by the physics itself. Holography, it's just that if you have a finite region here, I'm oversimplifying enormously, it's general relativity, so you always have to be very delicate about what the real statement is, there's only a finite dimensional amount of information which is proportional to the area in Planck scale terms. So that's finite. So that means if you have observers in the causal future of this thing, there's only so much information that can get out. And so I've been adopting the point of view that the geometry of this region, All of this information is only an organization of the information that can be observed outside. Now this is a bit counterintuitive. I'm saying you can't have operators inside. If you have operations inside it, it's no longer quantum. You're collapsing the wave packet inside it. But also, in gravity, it's so small that if you try to put the disturbance of energy to know the observations,

55:00 it can come out in a very different way. There are a number of different types of observations that can be used to determine the size of an observable object.

57:30 There are different types of observations that can be used to determine the size of an observable object. In this situation, we would not expect things to go really well because you could try doing different experiments to probe this and seeing what you saw, you know, shine a light a different way. Something like a Hilbert space operators thinking that there would have to be non-communicable regions in a space which is quantum mechanical.

1:00:00 But first off, you have a state of space that has one particular method. You could define sub-regions by where they appeared for different observances, and if you did experiments over and over, you would decide they were correlated, that if something appeared to be here, and the observer would have to have two eyes so they could do parallax, and this observer would have two eyes, if it appeared to be here in the The imaginary space of this one. It would always appear to be here in the imaginary space of that. And in general. So that's a sort of an operational way of reviewing the definition of mathematics. Think of the observers as defining coordinate patches by where things appear to be. And then you could use that. You could say it's in some ways that we're naively thinking they're absolute, but in fact they're relational.

1:02:30 If you put this all together, it's not a mechanical description, and that would give you an algebra of operators. And then the geometric way of thinking about that is something called a quantal, which is a non-distributive one. You would be piecing all of these things together and discovering correlations, and what you would be doing out of that would be building a quantaloid. So the place where this geometry would appear would be in the category of shapes over quanta. Those are very interesting categories. There's more papers on this. It's a very recent development, most of it is 21st century. The most important papers were in the Duaneham. My ancestors came from the Duaneham, but I'm not sentimentally attached to it. But the light has an additional structure.

1:05:00 And there is one paper on its internal logic. And the internal logic of the category of sheets over a quantum load is quantum logic. So in other words, an operational point of view on experiments would be natural in this larger structure. It's built out of the category of quantalites. Looking at a small region of space-time where climate and gravity were significant,

1:07:30 it would be highly curved and there would be high probabilities of singularities appearing. It would be a very bumpy region of space-time. But we know what things look like when viewed through regions like that because we pointed our telescopes at distant galaxies and tried to look either past a single galaxy or through a star cluster. And so there's a very rich mathematical theory of how a region like that looks, how you can tell it's a geometry by looking at what you see when you look through it. It's called gravitational lensing. Gravitational lensing is sort of joint work between groups of mathematicians and groups of astrophysicists. And there are books that have projections. So can we then understand the general structure, then they know what a movie looks like.

1:12:30 The question of whether you arrive at the same place by two events becomes a kind of mechanical thing, so you don't identify things down here in processes. But these processes have a structure of, in the simplest case, they're more structured. But then there are transitions between them and higher order transitions. And so since everything we know how to do using category theory, we get topological information and manifolds.

1:15:00 So we're doing that, and so we know how to construct categorical expressions with probability attributes for them, and then we have to extend it to include the singularities, transitions to the singularities. But there's more down there. The singularities correspond to the classification of legumes. And really, the stuff that's emerged from our Golden School is incredible. There's a very small number of generic singularities, and there's a correspondence, they have Dinkin calculation, Venus synthesis, different observances in different classes of things, and then we're going to have to have some sort of consistent simulation, which will be rigid enough to define what kind of quantum theory it has to be.

1:17:30 That isn't done yet, but so much of the math is just there, so many of the different things... I mean, I was hoping against hope that I would even be able to see complexes because I wanted something that would select families and complexes, but that's exactly what you see. The view that you see is a complex. Okay, so that's this picture and it's very much not done at all. It's not even written.

1:20:00 I'm sorry, I don't think I heard it over, but I've been closely related to Karloff in several of the things he's written, but he's very prolific, and I'm not sure if you know that much about that. But he's very interested in foundations of mathematics and physics. Well, that's the, okay. Carl Rovelli came up with this formulation of relational quantum mechanics after he and I were talking for a year or so. I was talking about the categorical expression of quantum mechanics. So there are many different observers and then there have to be relations between them. He sort of wanted to de-emphasize the category theory and talk more about directional models. There are many different Hilbert spaces and then there have to be maps between them. But it's just sort of different packaging from the same set of ideas. Yes, you mentioned the whole thing. I don't remember the literature. It's a very strange history. It began from two completely different places. People who were trying to connect the geometric geometry of quantum mechanics. Very abstract category theory. We're interested in the very natural problem of categories in the