Geometry & Physics — A Marriage Made in Heaven?

0:00 This lecture was established through a generous gift of $100,000 from the Ford Motor Company in 1999 in connections with multi-million dollar commitment from Ford to the University of Michigan. The GIF recognizes Ford's continuous relationship with our department, a relationship that is fostered through collaborative research, online presence in Ford's management structure, and a common belief in the importance of study of physics. The 2002 Fort Motocommunikist English lecture is entitled Geometry and Physics, A Marriage Made in Heaven, and this lecture will be delivered by Sir Michael Atea. Professor Atea was born in 1929 in London. His early education was parted in Cairo at the Victoria College and parted in Manchester at Manchester Grammar School. His undergraduate degree as well as his Ph.D. are from Trinity College in Cambridge. Following his postdoctoral training at Cambridge and at the Institute for Advanced Studies at Princeton, Professor Atiyah was appointed a lecturer at Cambridge. He remained at Cambridge till 1961, at which time he accepted a leadership position after the University of Oxford, and shortly afterwards in 1963, filmed the highly prestigious civilian chair of Geometry. In 1969, he was appointed Professor of Mathematics at the Institute for Advanced Studies in Princeton. After three years at Princeton, Professor Atiyah returned to England, becoming a Royal Society Research Professor at Oxford. In 1990, he went back and became master of Trinity College and director of the newly opened Isaac Newton Institute for Mathematical Sciences in Cambridge. Sir Michael retired from these positions in 1997 and is currently an honorary professor at the University of Edinburgh.

2:30 We are very pleased that Sir Michael accepted our invitation and will spend four weeks as a guest at the Michigan Center for Cyrillical Physics. The work of Professor Atiyah spends many branches of mathematics. His unrivaled ability to cross mathematical boundaries allowed him to reveal important interactions and connections between areas of geometry, topology, and analysis. His first major contribution was the development of a new and powerful technique in topology called K-theory, which enabled to tackle many difficult mathematical thrillers. Shortly afterward, joining with Ian Singer, Professor Artilia established the celebrated index theorem that led to important new links between differential geometry, topology, algebra and calculus. For these early achievements, Professor Artia was awarded a Fields Medal in 1966. Many of these important developments in mathematics have subsequently proved relevant to theoretical physics, particularly in the area of gauge theories of elementary particles and in quantum field. The theories of super-space and super-gravity, as well as string theory, all greatly benefited from the ideas Professor Atiyah has been introducing over the past nearly 40 years. Apart from the Fields Medal, Professor Atiyah has received many distinguished awards and prizes, simply too many to list here. Let me just note here. The Royal Medal in 1968, the Democratic Medal in 1980, the King Faisal International Prize for Science in 1987, the Copley Medal in 1988, the Benjamin Franklin Medal, and the Nehru Medal. Professor Atiyah was knighted in 1983 and made a member of the Order of Merit in 1992. He has been elected a foreign member of some 20 national academies and received honorary doctorate from over 30 universities. Sir Michael served as president of the London Mathematical Society between 1974 and 1976

5:00 president of the Royal Society between 1990 and 1995. Since 1997, he has been president of the Pagbosh Conferences, an international organization dedicated to reducing the danger of armed conflicts between the nations. Ladies and gentlemen, it is my great pleasure to introduce Sir Michael Akia as the 2002 Ford Motor Company Distinguished Lecture. Well, thank you. After that introduction, I can't hide my age. One of the advantages of living to mature years is you can look back over your past and you see a substantial part of history. My working life, mathematics, has covered essentially the second half of the 20th century. went back to the first half of the 20th century so really I've been in touch really with the whole of development in the 20th century and it's a really very fascinating story and particularly the last 25 years which is what I will really be talking to you about now I'll start with a personal anecdote and story, it goes back 25 years approximately when I came over to MIT just at a time when we math physicians were discovering that the work we were doing was very similar to the work that the in physics were doing, and we got together to discuss this common interest. So I came to this room in MIT with a theoretical physicist, and I talked to them, and I said, well, why didn't you talk to Professor Singer down the corridor in the mathematics department? I said, well, there's a door there. It goes through. They say, ah, but that door is locked. I said, why do you keep the door locked? He said, well, we have a nice new carpet here. We don't want the mathematicians walking over our carpet. So you see the problem sometimes there are Now, at that time, Romar Jakib, who was one of the physical physicists, he said he raised the question whether this new interaction between mathematics and physics was, let's say, a short-term relationship, or whether it would develop into a long-term marriage arrangement. So now, 25 years later, I can tell you it's a very firm and solid marriage. Another passing fancy. Moreover, the mathematics and physics were not active when they met in 1975,

7:30 young teenagers. They actually were very mature individuals. So mature they go back several thousand years. So the history really is a long one and I want to start at the beginning before I bring you up to date. So let me start from the very early days. Now I think somebody is going to switch the lights off above so you can read what's on the screen. That's right. I can't read it because I've got a screen in my neck. So let me remind you that In the early days, geometry was closely related to important things like astronomy, and that's where the heavens come in, and navigation, which were related to astronomy. These were practical purposes which geometry was developed. And, of course, the Greeks applied all this, and they even got as far as calculating the radius of the Earth. Ah, yes. Thank you. Well... Right on. Good night. On the wall, by the way, excuse me. Ah, I didn't squeeze it. Ah, well done. OK, so, the Greek got as far as calculating the radius of the Earth. I never trust modern technology. Do you have an experimental thing? Hold it. Keep it there. Thank you. And of course, from that time until the recent time, the great names of mathematicians and physicists were already more or less interchangeable. I've listed here some of the names to remind you. Starting with Archimedes, ending up with Hamilton and Maxwell. These were all people who could be claimed to be both mathematicians and physicists. There was no distinction between the two. This was one body of people working in a common enterprise. And by the time we come to the 20th century, things begin to change. And the 20th century, the first half of the 20th century, there were of course some very famous mathematicians

10:00 who had a strong interest in physics and made contributions to physics but they were fundamentally mathematicians Henri Poincaré, Helen Weill and John von Neumann then of course there were famous physicists whose work had also mathematical components such as Einstein, Dirac and Schrödinger but they were fundamentally physicists so already at this stage the paths are beginning to diverge a bit and if you follow that through My picture of what happens, the history in a nutshell, as Stephen Hawking would say. So this is the classical period that's put in a straight line where mathematicians and physicists were more or less the same people, including Newton. And then about 1900, things began to diverge a little bit. So in the early years of 1900, there was a bit of divergence. This is the physics side, this is the mathematics side. New ideas came in from general relativity and quantum mechanics. In the 1950s, quantum field theory came in. By 1975, the divergence between mathematics and physics was quite large. Over the preceding 25 or 30 years, the techniques used by physicists in quantum field theory were highly specialized, very far removed from what mathematicians did, and vice versa. that what mathematicians did had no interest or little interaction with physics. And suddenly, in about late 1970s, this is where I met with the people at MIT, that's the point in history. See, that's me. And we met in this room with a closed corridor at the end, and then the question was, what would happen after this? Would the path cross, like the dotted line, indicate and carry on separate paths, and form a unified time forward, and that's in fact what happened. String theory, as I put down here roughly speaking, is the sort of starting roughly at this time, and has continued to the present, which is where we are roughly speaking, 2000. Of course, what happens after this? In the next century, this present century, still remains unknown, but over this 25-year period there has been a remarkable interaction and unification between mathematics and physicists, which is really the theme of my lecture. Now, I'll go back a little bit, so the geometry, what happens here, I'm assuming that most of you here are perhaps physicists, perhaps not all of you, but anyway, while physicists

12:30 were doing their own thing, mathematicians also were doing their own thing in the 20th century. The geometry in the 20th century, not the whole of mathematics, but the geometry, had a number of roots, had roots from the work of Riemann in the last century, introduction, of Romanian geometry. The work of the Italian school, Levi-Civita, who developed the tensor geometry, calculus. So was Li, who introduced Li groups, continuous groups, all of which were put together by Eddie Carter into what we now call modern differential geometry in the theory of Li groups. This was a large part of the development of mathematics in the 20th century, building on this. From there onwards, it moved in other directions. Topology became an important component, large-scale behavior, structure of spaces, which the name of Solomon Lefschitz is primarily associated, and then ideas of homology and linear analysis of manifolds, the introduction by the work of Hodge of the Laplace operation on manifolds, theory of harmonic forms. This was a big development in the 1930s, and then moving on further still into the 1950s the 60s, the work of the French school of Serre, Carton, Gritton, Deacon, and so on, where complete algebraic geometry was being attacked by all the modern methods which had been developed in this century. This is an old subject being treated by new methods. This was a continuous development of mathematics from the beginning of the 20th century until the period we're talking about. And then the index theory which was mentioned by the chairman was a thing I myself got involved with which comes to the end of this chain of developments here. Now, in the 1970s, this first interaction between the physicists and the mathematicians of this recent period, as I mentioned, that meeting in MIT, was concerned with something in physics which were called anomalies, which had been discovered by physicists developing quantum field theory, Geographically, what anomalies were, you had a situation where you had some classical system, which you tried to quantize, and the classical system had some symmetry, which something was preserved, and when you go into the quantum situation, that symmetry was destroyed, and the conservation law broke down, and the thing which made it break down was called an anomaly, they didn't like it, or maybe you liked it, that showed there was a difference between the classical and the quantum regimes.

15:00 an example of the kind of symmetry might break down is what's called chiral symmetry the symmetry between right-handed and left-handed systems and some physical systems have this some don't and so the study of anomalies became a rather important technical tool for physicists doing quantum field theory it came in a whole variety of different models and this was then shown to be related to the work which I was doing with Singer in physics in mathematics which was concerned with what index theory and this is a quick summary is about. Basically, you have a differential operator, which I've written here as capital D, a linear elliptic differential operator, and it's adjoint, D star, and you look at the solutions of the equation D5 to the 0, you have a certain number of solutions, independent solutions, the mouse space, you take the dimension of that space, the number of independent solutions, and you do the same thing for the adjoint, and these are not necessarily the same. And the difference between these two is called the index of the operator, and turns out to be highly stable under all continuous perturbations, and therefore is a topological character. And the index theorem simply gives you a formula, an explicit formula, for this quantity for any differential operator under any geometrical circumstances in terms of the geometrical data you start with. So a large range of possible applications, because you can vary the data in all sorts of ways, and the formula is a very explicit formula. This was the formula and we discovered, say, in the 1970s, this formula and the work of the physicist on anomalies were very, very closely or intermittently related. And so immediately, a bridge was built between what we were doing and what they'd been doing. But not only between what we were doing at that moment and what they'd been doing, but really all the work in the past, because this was the, like, culmination of 50 years of work by mathematicians, developing lots and lots of ideas and techniques. Suddenly, we realised that all these techniques being developed by the mathematicians useless waste of time where there could actually be used for something. And so a door was opened. The physicist went into the library and started to read. That's being unkind to them. Now, that's a very quick summary of what happened in mathematics over that period. Now, what happened in physics over this period, or a more modern period? Modern physics, by modern me, I'm meaning here the last 25 years, are centered around gauge theories. Gauge theories are the

17:30 generalizations of Maxwell's theory when you consider more complicated groups than the circle, when you consider non-community of groups like groups of matrices, then you get gauge theories. And gauge theories are the bread and butter of modern particle physics models. They all models are based on various choices of gauge groups and various particular formulas or Lagrangians associated with them. So gauge theories are the framework in which particle physics takes place. It took physicists a long time to get to that point. Once they got it, it's a very simple statement. Then there are, and of course gauge theories are very intimately, are geometrical in character, because gauge theories are for geometers, basically the theory of differential geometry and of lead groups that were developed in the early part of the century. Then there are the Klein theories, and I'm glad to note that Haskell Klein was here in the department, and Mike him. Calusa Klein theories, these are theories of higher dimensional space-times where space-time is now possibly not to have just ordinary four dimensions of space-time but has some more dimensions, up to perhaps ten or eleven. And these other dimensions are called internal degrees of freedom. They are thought of as being very, very small dimensions you can't normally see, only come in at very, very high energy scales. And these Calusa Klein theories of higher dimensions are becoming become more and more popular as a way to try to explain and unify all the different aspects of particle theory. So these internal geometries, the additional variables that you have to take beyond the four, to climb up to ten or eleven, these are supposed to form some compact manifolds like a circle or a sphere, higher dimension, of very special type. They're not arbitrary but they are very special ones. They are special geometries and these special geometries are very close to the classical ones studied by differential geometries in the past. So a lot of the contact between mathematics and physics came in at this point, because these new degrees of freedom were ones which were kind that had been studied by mathematicians. And then when you have gauge theories you get into quantum theory, and you get in particular into quantum field theory. Quantum field theory goes back, of course, to Dirac's work, in the early cases but now is a great developer in the 1950s culminating in quantum electrodynamics and subsequently in the work on the standard model. So quantum field theory has two aspects to it.

20:00 There is what's called a perturbative aspect where you expand things in power series you have complicated combinatorial methods of computing it which are called Feynman diagrams which you calculate very high orders of accuracy and give very very good results comparing theory with experiment. This is the standard technique quantum field theories of interesting theories are not linear they can't be solved exactly you expand them in parameters and you put term by term approximations that's the perturbative theory but physicists have got increasingly interest in the fact that the fundamental theories of structure of matter have a non-perturbative aspect something you don't see by perturbation theory something beyond the range of the perturbation regime which is mysterious hard to understand and so you have a question mark what are the non-perturbative aspects, how do you get information about them, this is one of the main challenges which physicists meet in studying quantum field theories, particularly for field theories where the perturbative expansion is not too good, where the parameter which you have to expand in terms of is large and not small, as happens in connecting with strong interactions. So the perturbative regime works well sometimes, but other times it's not, and the question is, what do you do then? Now, it's at the level of the non-perturbative aspects of quantum field theory that physicists gradually begin to realize the role of topology. Very likely speaking, topology is concerned with how things happen in the large, how things happen in the small is given by small bits of calculus where you do go to pass these expansions. But what happens in the large, the difference between the circle and the straight line is topological. I'd like to say that Christopher Columbus was the first experimental topologist. He went round and found the Earth was round. That's a bit of topology. Well, it is topological aspects of these gauge theories that physicists found gave an insight or a clue to the non-perturbative aspect. And so from this point onwards the topology became more and more of interest to physicists in various ways. I've already mentioned in the index formula which Singer and I obtained where the topology comes in and links to the physics but is part of a much broader context. So I think classical, the way to understand the role of topology is you can go back and observe that in classical physics fundamentally you're studying forces.

22:30 Forces are things locally. You measure forces, you examine them, how big they are and so on. And in broad terms, you make a comparison between that and things like curvature. and in Einstein's theory that's the exact statement the curvature of space-time is meant to correspond to the gravitational force and a similar interpretation applies to electromagnetism so curvature is measuring how things are curved and not straight in space and those measure the strength of forces and they are given by real numbers which vary continuously and they're described locally quantum theory is concerned with other aspects first of all quantum theory tends to be global you need to know where the wave function is everywhere It tends also to give discrete answers. Things come out to be quantized. Sometimes they come out in integer values. And these aspects of being both global and discrete are characteristic properties of topology. Topology is not local, it's global, and it tends to break things up into discrete types. You know, the circle or the sphere or different topological types come in discrete families. And this link between quantum theory and topology, between the global and the street, goes really back to Dirac's work on the quantization of electric charge, which, if you examine it from a modern point of view, is the first time a topological argument was really used in physics in a fundamental way, and links directly with the quantum theory. Because Dirac studied the quantum behavior of an electrically charged particle in the background of a hypothetical monopole. now let me let's try to summarize how the ideas went from the mathematics to the physics what happened once the slinks had been established and suddenly developed there was a flow of what do I call techniques and formulae of modern geometry especially associated with these internal degrees of freedom all the geometry that had been worked out in the first half or three quarters of the century was now being brought bear on the problems of physics. And as physicists learned more and more of them, they became more and more skilled in using these techniques and these ideas. So this was the payoff of physics. The fact that mathematicians had been developed in these quite independently was, of course, a bonus. It means that physicists didn't need to do it all themselves. Some of the work had been done. all the time physicists were getting more and more sophisticated. These models I referred to

25:00 in higher dimensions have become more and more complicated, more and more and the more sophisticated they got, the deeper they needed to dig in to the mathematical techniques. And so there was interaction between the physicists and mathematicians working together on the geometrical problems arising in this way. Now the other direction. The interaction with physicists and mathematics had a very surprising outcome of a rather different kind. You might think the flow about it is all one way. The mathematicians would work out some things, the physicists use them. That's fine, that's what you expect. But the other direction was very, very surprising indeed, so I must spend some time on that. Here, there were completely spectacular new results. By new results, I really mean new results, totally new discoveries in classical geometry. By classical geometry, I mean the kind of geometry that was studied, let's say, in the 19th century, really going back a long way. And these results were in areas which apparently are very, very far removed from physics. If you'd asked a geometer of the 19th century about a problem, he'd tell you what the problem was, and he'd say, has this got anything to do with physics? And he'd laugh. Nonsense. There's no way in which those could be thought of related to physics. And I'll give you some examples, by which you've seen what I mean. So this was a very spectacular, surprising result, and I think, in a way, unique in the history of mathematics. And over the last 25 years, there's been an enormous growing amount of the every year passes, is more and more results in mathematics get inspired or suggested or even proved by methods coming out of physics in a very, very subtle and interesting way. Now here is a list. I could have taken this, I could have given a larger list, but I picked a few examples just to show you what I have in mind, because you want to be concrete. So here are some problems which have been solved by methods coming from physics. So the first thing is the following. You take an out-of-break equation. Here is a polynomial equation in five variables. It's simply the sum of the fifth powers. It's a homogeneous polynomial equation. The solution of these forms complex numbers. Give you something of one dimension less. There's one constraint. You ignore the fact that two points which are multiple to each other, you think of them as the same, so that cuts down the dimension by one more. This is effectively one equation for four degrees of freedom. It gives you something of dimension three, three complex dimensions.

27:30 On this, which people call algebraic variety, you look for curves. A curve is simply one-dimensional complex object, for example, a straight line. And you ask how many straight lines there are on this complicated surface. And having done that, you ask how many conics, things of second degree, or how many cubics. And you ask for an extreme number of curves of a given degree, degree is one for a line, two for a conic and so on, and you require only if the curves are rational, that means they can parameterize by a rational parameter. And for each d, there is a finite number of possibilities. And this number, if you know enough geometry, you should be able to find. And for the number of straight lines, this has been known in a long time, it's quite a big number. For the number of conics, it's a much bigger number. but beyond that the classical techniques of algebra geometry from the last century even of this gave no answer the answer got extremely complicated and the physicists came along with a formula not only for the first few degrees but for all degrees they got a universal formula in terms of generating function they calculated these numbers and for the first ten numbers you get answers turned out to be millions you know really big numbers so this really impressed the mathematicians there's a very concrete problem answers are very concrete numbers and the physicists got them by a very beautiful formula and they could just turn the handle and get more and more numbers out. And then mathematicians had to learn to live with this. They struggled to prove the first few cases. They disagreed, but then eventually they found they were wrong. Eventually they agreed with the physicist. So this has been one of the big surprises, a very concrete problem solved by matters coming from physics. That's algebraic geometry of a very classical type. Second example is algebraic geometry. It's about curves again, complex curves. Algebraic curves which are not rational are labelled by what's called the genus. I'll say a bit more about that later. Genus 1 curves are elliptic curves and so on. And for fixed genus they will have some parameters that describe how many there are. For rational cases there are no parameters. with other elliptic curves that are one parameter called the elliptic modulus in general there are more moduli of parameters and these parameters don't form a linear space they have a complicated topology of their own understand the topology of the space of parameters of algebraic curves is a very classical problem

30:00 going back certainly to the last century and the early parts of this sorry the 19th century early parts of the 20th century and this information came out of quantum field theory too miraculously very concrete results again. Here's the third one. If you look at knots, ordinary knots in three-dimensional space, closed pieces of string, the entangled apples in a complicated way, then studying knots is a classical problem of topology. You might like the typical problem of topology. Knots are things that, it doesn't matter what shape they are, measurements, you just pull them around, only question is, is one knot equal to another knot by moving it around, can you disentangle it? Those are the questions of topology. And the classification of knots, or the study of knots, is an old subject, again going back into the last century. Interesting enough, it was given a big impetus in the 19th century when knots were thought to be models of atoms. When Kelvin was quite interested in the idea and put forth a theory that knots were really atoms. Well, that theory failed, but knots were studied by themselves. but only 20 years ago essentially that about Vaughan Jones produced an entirely new way of trying to distinguish knots by what he called polynomial invariants and subsequently these were shown by Witten to arise in a natural way out of quantum field theory so here quantum field theory quantum field theory in two space and one time dimensions interpreted as three space turned out to give a marvellous insight into the classification of ordinary knots in topology classical problem, and you notice, apparently, very different from the other problem. These are algebraic geometry, this is about topology of knots, very far apart, and yet the same techniques in physics apply to all of them. And the fourth one, was probably the most surprising of all, there were new results about four dimensions, four-dimensional manifolds, smooth, no complex algebraic geometry, you look at four-dimensional manifolds, the analogue of surfaces with four dimensions, and you want to know something about them. And Simon Donaldson, who was at that time my graduate student, discovered fantastic new results there, which came out of ideas of physics, and subsequently those ideas of physics have been further strengthened and applied by Zyberg-Witton. And so this was a very spectacular application of quantum field theory, which generated a whole industry of work by mathematicians. Now I'll say a bit more about that one because it's such a spectacular case.

32:30 So, let me clarify what I said about, first of all, algebraic curves. So let me just start off with a polynomial equation in the plane, complex in three variables, thought of as homogeneous coordinates of the plane, at a degree d. For example, just the sum of the powers will do. Then this gives you one complex dimensional object, which is, to take a real imaginary part, It's two dimensions, it's a surface, it's closed, it's a surface, and that surface has a genus. The genus is given by a certain formula in terms of the degree. So, if you take the degree equal to 1 or 2, then you see the genus vanishes, you get a straight line or a conic, those are the number of rational curves, where the genus is 0, and the topology of that is a sphere. And it's parameterized by just a complex number, including infinity. If you take the case when d is 3, then you put it into a formula, you get g equals 1, that's called an elliptic curve, then the topology of that is a torus, and this leads to the theory of elliptic functions, what are called double periodic functions, which are what is worked on by Arbel and Jacobi in the beginning of the 19th century, and a big industry in the 19th century. Theory of elliptic functions. That corresponds to curves of genus 1. Then if you take genus bigger than 1, as soon as D is greater than or equal to 4, the genus goes up, beyond 2, and what you get is a generalization of the torus, which is a surface looking like that. So this example 3 holds, and this is what we call a connected sum. You take a number of tori, you glue them together with little bridges like that, and you get the general surface. And the topological classification of surfaces is they all look like this. Given any closed surface, orientable surface, it is of this type some g. g is the only topological invariant in the classification of surfaces. And one way of understanding that this is this goes back to Riemann's work, really if you take a surface, well let me keep I should have, if I take this surface back here if you look at what we call one cycle, closed contours on that surface. But every time you have a torus, you have two circles that go round. One going round the big circle, one going round the little circle. These, and every

35:00 other closed loop is a combination of those two. You can go round a certain number of times this way, and a certain number of times that way. And these two circles have what you call intersection numbers. You can take the big circle and the small circle meeting one point. Two copies of the big circle are sort of parallel, don't mean at all, and two copies of the small circle at all. And so you, in general, can form what's called the intersection matrix. The intersection matrix of the one-cycle on a surface is simply a matrix whose size is the number of independent cycles and you write down the intersection number of the i-th cycle with the j-th cycle in the position i-j. The main excuse of the metric, because you take two cycles and work out how many times they meet, there's an orientation convention which means that you do them in you get minus 1. And a 1 by 1 block, 2 by 2 block like that, that's the little matrix you get for a torus. This is the one you get for another torus. And a general, a general schismetric matrix of this type by linear algebra can always be put into normal form which looks like a sum of diagonal blocks, or the off-diagonal terms are 0. And so the classification of surfaces really says the following. The algebraic classification of matrices, by linear algebra, corresponds to the geometrical classification of the surfaces. Not only is it true that every matrix is a diagonal block of little two-by-two matrices, but correspondingly, the surface is the connected sum of little torus. Each torus corresponds to two-by-two block. So whatever you can do algebraically, you can mirror that by the geometry. The geometry in the algebra are fundamentally not different from each other. Very satisfactorily, very simple. now the question is what happens when you go up one dimension you take not three variables but four variables and you take a similar polynomial equation of degree d in four variables then the solution space now has one dimension more than before one complex dimension more than before and so it defines a two dimensional algebraic surface complex numbers taking really imaginary parts you get something with dimension four so you get examples of four dimensional real manifolds equations. Okay? And you can take a few examples and see what you get. Now I'm going to tell you a little bit about them, and for simplicity I'm only going to talk about the case when the degree d is odd, there's a difference between odd and even, life is short, so I leave out the even ones.

37:30 When d is equal to 1, you have a linear equation, and a linear equation just defines a plane, and so you have a complex-strategic plane, we know what that looks like. If you take d equal to 3, then already In the 19th century, we know a lot about cubic surfaces and the outcome of that is that a cubic surface looked at topologically, what it is, is a connected sum of copies of the predictive plane and six copies of the predictive plane thought of with the opposite orientation, a kind of negative version. And so, that's a very simple, that's a similar result to what you might have got for the case of surfaces. a general Riemann surface or algebraic curve was a cliquey sum of tori here we're getting some cliquey sum of these simple building blocks now you ask yourself what happens for larger d well, first of all you look at the intersection properties this time, and four dimensions the things that are analog to cycles and surfaces are two-dimensional objects and two cycles will meet in general some points and so you get a matrix of intersections again but this time the matrix turned out to be symmetric because when you think of two numbers two and two put them together the permutations are even and so you don't get any sign problem so you get a symmetric matrix a symmetric matrix with a certain size the size depends on the degree d by a certain formula I won't bother you with like the previous case and now you it turns out every symmetric matrix of this type can be put into a normal form the normal form with either plus 1 or minus 1 on the diagonal. And the number of plus signs is 8p, the number of minus signs is 6q, and you get p plus q equals n. When the degree is 3, and the black box over there, the size of the matrix is 7 by 7, you have 1 plus sign, 6 minus signs. That's the algebra. And the geometry says, correspondingly, this algebraic surface, there's a 4-manifold, is a collected sum of 1 copy of the brinkive plane and six copies of the relative plane with the opposite sign. So, very nice, simple story. The question is, does this happen, does it hold for higher degree? So here was a fundamental question. Is it true that the algebraic classification of matrices is mirrored in the geometric classification of manifolds? That's how that happens in the lower dimensional case.

40:00 And I think everybody explains the answer to be yes. and the answer would have been a certain number of copies of the plane and a certain number of copies of the opposite plane and the surprising answer which was by Donaldson was that the answer is no this is not possible as soon as the degree d is at least 5 then this thing fails completely it fails in the strongest possible way every conceivable conjecture you make along these lines is wrong And the surprising thing was that Donaldson proved this by using the Yang-Mills equations, which are the fundamental equations of Gage's theory, generalizing Maxwell's equations. He used them in a very beautiful way and got this very, very powerful answer out. That was a spectacular example, and that was around this time, and over the subsequent 20 years, there'd be enormous development by mathematicians working out the further implications of this new theory. To show you how surprising this theory is, Donaldson proved the answer was no, somebody else proved the answer was yes. Now the difference is that the no answer was when you want things to be smooth. It was differentiable. Nice tangent spaces. But if the answer was yes, if you allowed only continuous functions. Continuous functions are nasty things where they can get edges and nasty points and so on. And if you openly work with continuous functions, which is pure topology, that the same is true as before. So these algebraic surfaces are ones which topologically are as simple as you might expect, but if you want to wonder about the smooth structure, which is where calculus lives, where differential equations live, where real geometry lives, and where physics lives, then the answer is different. So there is a very subtle difference between the continuous case and the differentiable case, which has a weak counterpart known before, but nothing as deep as is. This was a very, very deep distinction between the continuous case and the smooth case, and it all hinged on the fact that Donaldson's work used the differential equations of the Yang-Mills theory, which is not available unless you can differentiate, of course. So they're not available in the other case. And, as I mentioned before, some years later, as I've written, developed a new theory, a new quantum field theory, which is different from what one Donaldson has used, a different version.

42:30 in the same way as Donaldson did it, took four manifolds and gave new results which are more powerful than what had been obtained before. So the interaction between the physics and the geometry went backwards and forwards. Now, there's a very famous statement due to Eugene Wigner, who commented on the unreasonable effectiveness of mathematics and physics. After all, physics is about the real world, use this very sophisticated mathematics which seems to work remarkably well. Large parts of physics. We have very sophisticated bits of mathematics which seem to give us the right answers. Why on earth does that work? Why should mathematics be so successful in physics? After all, mathematicians have worked it out, come back to their own purposes, unrelated to physics, yet time and time again, physics finds that the mathematics is the tool they need. It's a great philosophical question, mystery, not the kind of question that has an answer, of course. Why is it true? You can ponder on it. But I want you to ponder on something else. I want you to ponder on the unreasonable effectiveness of physics in mathematics. I think it's even more surprising. You know, after all, mathematics is a tool developed for various purposes. The fact that it's useful in physics, well, it's a piece of good luck, if you like. But, you know, it's a tool. Hopefully it will be used somewhere. But to use physics in mathematics is very surprising. Mathematics is meant to be the creation of the human mind. We invent things. We invent all sorts of marvellous formulas. pure algebra and so on what should the real world how can that help us why should that help us at all why should sophisticated ideas of physics and quantum field theory be of any use in mathematics and you see from the example I gave you the applications in mathematics had nothing to do with physics by themselves they weren't about field theory they were about pure geometry and yet the physics was the secret to unlocking the new results in geometry so this I think is even more spectacular than the bigness and equally difficult to answer, but let me try to answer it. Why, why could that be true? Well, all philosophical questions don't have a real answer, but here is some sort of answer. Philosophical explanation of the success of physics in geometry. First of all, if you go back to the early days, what was geometry? Geometry was the study of space. You know, here is space, we in space, your name is Euclid or Archimedes or whatever it is you go out and study triangles and things

45:00 like that, they live in space, you're studying things in space and then what do you use? Well you use rigid measurements, you use straight lines rods, compasses you use rigid measurements as a way of studying space that's what you mean by studying geometry but after that people became more They said, we don't want to use only rigid rods, we can use things like light signals. We can send light rays out. We can study things, you know, by long distance. And then you can study geometry by linear waves, if you like. And that was what odd theory was about. I'll come back in a moment. Then you can study it by non-linear waves. The Yang-Milk equations I mentioned to you are non-linear analogues of Maxwell's equations. So you can start to try to use more complicated ways that don't go out in nice or straight lines but do all sorts of complicated things and curl around the space, and they perhaps capture more information about the space because they're more complicated kind of ways. So the way I want to put all this to you is that these are all increasingly sophisticated ways of probing space. You want to probe space to find out the nature of space. And you probe it by rigid rods, by sending light signals, by sending more complicated signals around, all these are ways of trying to grab hold of space. surprising, perhaps, that as you use more and more sophisticated methods of probing, you get a deeper understanding of geometry. And in these examples above, the first level is really classical Riemannian geometry, because that uses measurements, rigid measurements. The linear theory is really the Hodges theory of harmonic forms, and the monobelian gaze theories are the sort of things that Dominson was using. So we see increasing sophistication in the techniques which are used to investigate the nature of space. That's my way of trying to explain how it happens as physics helps to understand what is in mathematics. Now, what about the future? Okay, the future. What is the future of mathematics? Well, I don't want to be too ambitious. I'll put geometry. Geometry is not the whole of mathematics, but the way I understand geometry is a large part of it. It includes algebra, topology, analysis, complex variable theory, and so on. So, not much left out. But, what is the future of this kind of geometry? Well, first of all, the first thing you might hope for is you'd like to understand the significance of the new ideas and methods that are coming from physics. These new methods that have come from the last 25 years

47:30 have been used spectacularly, but they're used quite often without rigorous methods, no foundations, people aren't too happy about it, So the question is, understand this properly. So, understand it and establish rigorous foundations. Well, in a more general way, you can say, answer the question, what is quantum mathematics? In recent years, because of the new ideas from quantum theory, various terms have been coined in mathematics. There are things called quantum groups. There are things called quantum chromology. almost any part of mathematics you can put the word quantum in front and you get a PAT thesis out of it usually quite a difficult thesis so there's a whole range of new things which are quite interesting things they're not trivial but somehow we don't quite know what we're doing you combine ideas of physics coming from quantum theory with ideas of mathematics you develop some new kinds of machinery and what is this? well here is my attempt to answer what it is you're trying to do what you're trying to do is to study infinite dimensional geometry infinite dimension because these are the geometry of fields basically and not linear fields so they have complicated geometry and the geometry here means as before analysis misspelled topology all kinds applied to infinite dimensions function spaces these are what people in the old days called the calculus of variations but you're trying to apply infinite dimensionals now which is the ways that physicists deal with them. Physicists have learnt to study infinite dimensional spaces and get rid of infinities in a very subtle way, based on the needs of physics. And one example of that is, if you're in infinite dimensions, you could look at small things, like things of finite dimension in infinite dimension. You could look at very, very large things, which are things given by a final number of constraints in infinite dimensions. or even look at things that are in the middle which are infinite both ways they're given by infinite number of constraints and infinite number of variables. For example, the Tech Fourier series takes things which involve only half the coefficients. They have infinite number of coefficients which are zero, infinite number of coefficients which are not zero. And more generally, with more complicated systems you study things which are in the middle. This is the sort of thing that physicists do and what one has to understand is how to treat those mathematically in a whole variety of ways And if the 19th century and 20th century were developing the geometry of finite dimensions in all its complications,

50:00 then perhaps the 21st century will have to work out the geometry of infinite dimensions. And that may be what quantum mathematics is going to be about. What is the future of physics? Well, now here it gets more speculative. I'm not a physicist after all, even though I have an office in the physics department. That doesn't really make me a physicist. exist. Well, as you all know, the two great stories of 20th century mathematics were general relativity and quantum mechanics. And people have been living with that, those two successes ever since. And the question is, how do you combine them together? And the latest ideas go under the name of M-Theory, which is meant to be the ultimate theory that will explain in how to combine gravitation and quantum mechanics and particles in one all-embracing theory. And M, at the moment, stands for mystery because nobody knows exactly how to do it. Or it stands for something else. Now, string theory, which came a little before M-theory, is meant to be the perturbative approach to M-theory. String theory is meant to be something like a perturbation expansion of M-theory. And only glimpses of the non-perturbative aspects are present known. If you really understood the theory, you understood both perturbative and non-perturbative, we don't yet know that. So we are awaiting a new insight. This is the, you know, the great, holy grail of present day physics. We would like to understand this theory and some new insights required to do it. Now, this insight might come in various ways. Maybe some clever chap comes along, just tweaks a little formula, and all of a sudden it all becomes clear. But it might require something much more fundamental. It might require really challenging one's basic assumptions. We need to remember how did Einstein get to general relativity. He didn't get there either by looking at experimental data or any of that kind. He got there by sheer reasoning. He argued that Newton's laws were not compatible with Lorentz invariance. He wanted to find equations, a way of doing it, which was essentially ended up with his field equations of general relativity, which when he finally got there, are actually extremely simple equations in the framework of Romanian geometry. And that's the kind of, perhaps that kind of answer

52:30 is what we need now at the next stage. We need a new Einstein to come along and say, look at it this way, and all will be clear. that is the kind of thing which is usually left for a young man, but, as I told you, I'm not a young man, and so So, I'm going to sink my neck out, you might say. In England, there's a phrase known as flying a kite. I don't know if you use it here in political circles. If you're in a government and you want to try out some idea in public, you float a kite up in the air to see what sort of reaction you get. If you get a lot of flack and it's shot down, you say it wasn't my kite. If it survives, then you bring it in gradually and work on it. If you shoot it down, it's not mine. Now, let's start with looking at modern physics. I'm going to do modern physics or mathematics for a moment. I'm going to talk about pre-physics or metaphysics. Let's go back to philosophy. Think of the great philosophers of the past. They all were interested in natural philosophy, as it was called, before it became physics. Aristotle, and what did he say? There are a few famous quotations here for you. He said, Eureka, you remember he jumped out of the bath. and Descartes if your Latin is good enough he said cogito ego sum and then the other I haven't got all the quotations here but there was Leibniz Popper and I'll bring you up the date and in honour of the forward lecturer I will reflect on the contemporary Greek or not philosopher of science Henry Ford I who said history is black Now, like all philosophical statements, that is capable of a lot of exegesis. You can examine it carefully, see what it meant. And one interpretation would be, of course, that reading history books isn't any help to make motor cars. And that says Henry Ford was having me right. But there is a more sophisticated interpretation, which is this. That means we can predict the future from complete knowledge of the present. Now, put it that way, Henry Ford doesn't seem quite as stupid after all. That is actually the orthodox point of view of physics. All physics is based on their premise. That we can predict the future from complete knowledge of the present. Of course, complete knowledge is never available in practice,

55:00 but the theory says that's the case. The laws of physics are governed by differential equations. That's the mathematical interpretation of that statement. If you know the differential equations, and you know the initial conditions, obligated to the future and find the future of the universe. And that's the Newtonian paradigm. Newton did that for gravitation, and all subsequent science, really, is based on that philosophy. It's even true for quantum mechanics, because in quantum mechanics, although there's the uncertainty principle, in principle, a state is determined by a state in Hilbert's place. The future is given by Stredinger's equation. There's a time of evolution. given by the Hamiltonian, and the future state, u at time t, is given by that formula in terms of u at time zero, is given by a predictable formula. So, the future is always predicted from the present. That's fundamental axiom of physics. Now, if you want to question fundamental physics, you can say Einstein, when he produced relativity, he combined space and time, quantum mechanics can we do? What can we question about the accident? Well, why don't we question the wisdom of Henry Ford? This is my tribute to the Ford Lecture. Maybe Henry Ford, after all, was wrong. Maybe you were wrong. Perhaps, you see, perhaps we need to know the past in order to bring to the future. Why do we think that we can get away without the past? Perhaps the universe has a memory. Why not? Well, I mean, so far we got quite well without it, but we may not be right. Perhaps the laws of physics are not governed by the differential equations after all. Perhaps they're not governed by integral differential equations, which involve integration over the past. Now, of course, you can throw out anything you like, but if you want it to be believed, you must of course make a sensible proposal. Now any requirement in theory must always of course be able to reproduce classical and quantum physics as a very good approximation, otherwise nobody takes you seriously. So in this sense if the past is something you have to know about, we'd have to assume that the influence

57:30 of the past is very weak. And if it's very weak, perhaps only the very near past has an effect. Perhaps you'll need to predict the future, effectively. And if you forget about the lasting microseconds, you get a pretty good approximation. So perhaps we're making a pretty good approximation, but we shouldn't ignore the past entirely. The second requirement, and this is more a sort of aesthetic one, is that we should preserve as much as possible the geometrical approach of general relativity. Now, in the combination of general relativity and quantum mechanics, there are two schools of thought. do you combine them? One school of thought, which is the orthodox school, says quantum mechanics is the right framework. All you've got to do is push general relativity to make it fit into quantum mechanics. But there's another school of thought that says general relativity is a beautiful theory, which Einstein worked out in great pains. Quantum mechanics really should be fitted into general relativity, and we should make quantum mechanics geometrical. So here I'm suggesting really something closer to that second spirit. We should try to preserve the geometric approach to general relativity. And of course, we would like to explain all these things like string theory and M-theory that should all come out of whatever theory we're trying to produce. We should also try to explain, you see, while we're about it being very, very fundamental, we should try to explain quantum mechanics because, you know, quantum mechanics is a marvelous technique. It works beautifully until you start to ask yourself deep philosophical questions. then well it gets a bit tricky some people don't worry about that they say that's a mistake there was a long argument between Bohr and Einstein about this Einstein kept objecting to quantum mechanics Bohr kept countering objections long correspondence went on and most people think that Bohr won and Einstein lost but Einstein was never satisfied perhaps ultimately Einstein was right and really quantum mechanics is not totally satisfactory as a physical foundation philosophical foundation And finally, of course, anybody who likes a new theory, to convince the people it's right, it should be a simple theory. You know, when the Greeks had, like Ptolemy, had epicycles on any epicycles to explain the planetary motion, and Kepler and Newton came along with one equation and beautiful ellipses, it was simple, and all the complications of epicycles disappeared. Well, perhaps all the complications of present-day physics will also disappear one day,

1:00:00 if you like to find the right kind of theory. So, while I'm thinking flying this kite of mine, if you were trying to pursue this line, what would be a fundamental equation? What could it look like? Well, I'm not suggesting it's a serious equation, but that's the sort of thing you might try. Well, you see, if you're doing gravitation, you saw that with a space-time metric, and then you have the Riemann-Kerbuchar tensor, and I've just written all the indices of curly R. You have the Ricci tensor, which is got by contracting some indices, and you have the Einstein vacuum equation, Ricci tends to be zero, or perhaps a multiple scale, a constant multiple identity, those are the Einstein vacuum equations. Now, how can you generalize those to include the past? To use classical physics, it predicts the future from the present. If you take a time slice to time zero, you can propagate forwards and find out what the space will look like in the future. Well, here's a possible candidate, I'm not suggesting it seriously, but look at that, what's wrong with that? You could take, instead of putting zero on the right-hand else. And something else could be some function of the backward curvature. You take the backward light cone, take some function of that, which you want to integrate over the interior of the backward light cone, multiplied by some number alpha, which is zero, except in this region here, which is very close to there. That's the nearby region. Well, you quite smooth it out to get a bit nicer looking than that. There you get an equation which is geometrical and character in which the past affected the future but only in a very small way but only in a very narrow strip you see so it would be something you could approximate and get classical theory out of that's to show you it can be done I'm not I'm not claiming this as a I'm not candid for the Nobel Prize yet but that's a you know to show you that young people should be trying to look for fundamental ideas and all I'm doing is offering you one possible way question the present question the orthodoxy wonder about it if we were going down this route. What would the meaning of quantum mechanics be? Okay? Well, what would the meaning of quantum mechanics be? Well, look at it this way. Since we don't actually remember all the past, and my memory is getting pretty bad, and if you want to remember the entire history of the universe, you'll have probably two. So, since we don't really remember all the past, what can you do if you want to predict the future? Well, you could take an average over all possibilities of the past and form some kind of Hilbert space and that would be

1:02:30 uncertainty principle and quantum mechanics would come in it would come in out of we don't know the past a certain amount of ignorance that ignorance is what makes quantum mechanics uncertain or if you want to go how would string theory emerge well you see you could replace the past something more sophisticated a finite number of non-linear modes and these might be what things like collusion kind of models where they come from out of some complicated nonlinear structure in the past, which we approximate by a finite number of modes. Now, all the beautiful things that string theorists do, supersymmetry and other mathematical ideas, they're marvelous things. And it's just possible that the universe really was created by God, who says, let us take a seven-dimensional manifold with this particular structure of these equations and let that be. But that seems a bit far-fetched that all these beautiful mathematical structures are actually tools which are used by us to simplify something which is very complicated, much more non-linear. And we, the best mathematics we have, we apply it and it gives us very good approximations. But what we're trying to do is describe something different. This is our best we can do. And you can do it in various different ways. That's why we get different descriptions of string theories. Well, that's at least a philosophical way to try to address the question. And being more specific still, you see, I'm trying to suggest that the uncertainty in quantum mechanics can be thought of perhaps as related to the uncertainty of our past. Now, quantum mechanics has a thing called wave functions. So perhaps this is what the wave function is, in non-mathematical terms. Perhaps the wave function of a particle is a shadow of its past. shadow of his past looks like. Well, I think that's sufficiently provocative. The kite is flying. You're well constituted down. But I thought this is my tribute to Henry Ford. Thank you. Thank you very much for the beautiful and provocative folk. I'm sure all the folk boys will be flying kites and perhaps some of his questions.

1:05:00 I'll ask you a question. Somehow in your talk the assumption is that space comes first. And that things live in it are sort of derivatives but maybe it's the other way around. I'm a geometrist so of course space does come first to me well I mean I don't know when you meant space well in Einstein space time are both there I wasn't distinguishing between space and time of course I distinguished between the backward light and that relativistic invariant notion so I wasn't distinguishing space and time I was distinguishing between you know position and momentum for example there are other physical descriptions which are spaced at secondary, not primary. I agree with you. One of the marvels of present-day physical theories is the same physical model can be described by quite different sets of mathematical parameters. And you can get a picture in one language and a picture in another language, one of which looks geometrical one way, one of which doesn't. So I quite agree. Picking one particular way of looking at it is, at some extent, a matter of taste or arbitrariness. And there may be fundamental ways. God may have made a preferred choice, or he may have what you like? Yes? To the extent that the marriage of physics and mathematics is happening, as you described, at the same time, the divorce of theoretical physics and experiments is happening, what is the simple separation? Well, that's the famous triangle, you know. you can't bring two together without leaving somebody in the cold of course that's not me I mean that's internal to the physics community the link between the physics experimentalists and the theorists is of course a serious question and there are many experimentalists who think all these theorists are going off barking up the wrong tree following mathematicians out into the vials I won't get involved in that debate I hope that these things go in cycles. After all, when Einstein was... Don't forget, when Einstein was working out his theory of generative relativity, there was absolutely no need for him to do it from the point of view of experiments. There were no single experiments at all which really required modification of Newtonian laws. He did it for purely mathematical reasons

1:07:30 or physical insight, whatever you like to call it. Eventually, after the theory worked out, now there are large numbers of very, very accurate experimental data generativity out in space. So it is a theory experiment that there came much later than the theory. Well, that same thing can happen with the present day physical theories. They may eventually produce some way of understanding experiments which will be confirmed twenty years after the theory. Or they may not. Other questions? Well, I think, I mean, I'm, I had to be cautious with anything I say about the physics. The idea is that the fundamental constants of the universe can change with time. That's been around a long time. and various tests to be made, various constants, you know, over long periods to see whether there's any evidence that they've changed with time. I think, on the whole, so far, the evidence is no. There's no evidence to indicate any of the fundamental constants have really changed with time. So I think, in that sense, that the answer is very negative at the moment. Although, maybe, of course, it depends on how accurate you want your calculations to be. Previous five minutes of the talk are now going to be reprised. it's important to print out all of Tyre's handwritten overheads for this talk very more sophisticated a finite number of non-linear modes and these might be what things like collusor climbing models, where they come from they might come out of some complicated non-linear structure in the past which we approximate by a finite number of modes all the beautiful things that string theorists do supersymmetry and other mathematical ideas they're marvellous things and it's just possible that the universe really was created by God who let us take a seven dimensional manifold with this particular structure of these equations and let that be but that's a bit far-fetched in some ways so it may be that all these beautiful mathematical structures are actually tools which are used by us to simplify something which is very complicated it's much more non-linear and we, the best mathematics we have we apply it and it gives us very good approximations

1:10:00 but what we're trying to do is describe something different but this is how best we can do and you can do it in various different ways that's why we get different descriptions of string theories well, that's at least a philosophical way to try to address the question and being more specific still you see I'm trying to suggest that the uncertainty in quantum mechanics can be thought of perhaps as related to the uncertainty of our past now quantum mechanics has a thing called wave functions so perhaps this is what the wave function is in non-mathematical terms perhaps the wave function of a particle is the shadow of its past and there's a picture showing what the shadow of its past looks like well I think that's sufficiently provocative, the kite is flying down, but I thought this is my tribute to Henry Ford. Thank you. I'm a geometer, so of course space does come first to me. And then when you meant space, well, in Einstein, space, time are both there. I wasn't distinguishing between space and time. Of course, I distinguished between the backward light code and the ballistic invariant notions. I wasn't distinguishing space and time. I was distinguishing between, you know, position and momentum, for example, yes. There are other physical descriptions which are, they, space that appears secondary, not primary. I agree with you. one of the marvels of present day physical theories is the same physical model can be described by quite different sets of mathematical parameters and you can get a picture in one language and a picture in another language one of which looks geometrical one of which doesn't so I quite agree picking one particular way of looking at it is at some extent a matter of taste or arbitrariness and there may be fundamental ways

1:12:30 God may have made a preferred choice or he may have left them all free will you can choose what you like To the extent of the marriage of what physics and the mathematics is happening, as you described, at the same time, the divorce of theoretical human systems is happening, what is the simple separation? Well, that's the famous triangle, you know. You can't bring two together without leaving somebody in a cold. But, of course, that's not for me. I mean, that's internal to the physics community. The link between the physics experimentarists and the theorists is, of course, a serious question. And there are many experimentaries who think all these theories are going off barking up the long tree following mathematicians out into the wilds. I won't get involved in that debate. I hope that these things go in cycles. Don't forget, when Einstein was working out his theory of generativity, there was absolutely no need for him to do it from the point of the experiment. There were no single experiment at all which really required Newtonian modification of Newtonian laws. He did it for purely mathematical reasons, or physical insight, whatever you like to call it. Eventually, after the theory worked out, now there are large numbers of very, very accurate experimental data which confirm generativity, out and space. there came much later than the theory. Well, the same thing can happen with the present-day political theories. They may eventually produce some way of understanding experiments which will be confirmed twenty years after the theory. Or they may not. Other questions? I read some words of the theory of trying to work the constant, like you can explain it with time. Do you think this might want to go with your idea that you have to integrate along with that? Well, I think, I mean, I have to be cautious of anything I say about the theory, but the idea that the fundamental constants of the universe can tame with time, that's been around a long time.

1:15:00 I read some word of the hearing, trying to work for constant, like, teaching, explaining the time. Do you think it's like what's your idea that you're going to bring along? Well, I think, I mean, I have to be cautious with anything I say about the theory. The idea that the fundamental constants of the universe can change with time, that's been around a long time. And various tests have been made, various constants, you know, over long periods to see whether there's any evidence that they came with time. I think, on the whole, so far, the evidence is no. There's no evidence to indicate any of the fundamental constants have really changed with time. So I think, in that sense, that the answer there is negative at the moment. you want your calculations to be and my speculation about the influence of the past are that they're extremely small in any case only relevant at the micro micro level way perhaps only the level at which things from string theory are coming into play so experiment only operates when you get to the stage where you can do the experiments where the things are within your measurements and so but in principle yes variation of things with time is not that far removed from what i was talking This notion of time, in an Einsteinian world we sort of put time together with space, but now we have a sneaking difference between space and time, isn't it? No, no, there's a difference between forward and backward light turns. It sounds very called the arrow of time for physics. I mean, you know, there are various philosophical problems with ordinary theory. The arrow of time is one of the known problems. how the laws of physics look as though they're symmetrical in time but the real world is not symmetrical, and that's called the arrow of time, and all I'm using are having a theory where the arrow of time is built into it the forward and the path are different that's all, nothing to do with time and space, forward and backwards are different, and that is if a theory is right, that would explain the arrow of time, which is one of the things that people like to explain, so, ok, come on right track Well, if no more questions, let us thank you for your time.