 I'm here with the professor Alen Kohn at ICTP, he's visiting in conjunction with the workshop on non-communitive geometry. So the first question would be sort of an ICTP specific question in this, if you had advice to give young students who want to study math, in particular young students from developing countries, what advice would you give them? Wow, that's difficult. I mean, you know, to study math, of course, it's, you know, they're challenging to say. And somehow, I mean, I always thought that the key step in studying mathematics is to understand that, you know, you don't learn mathematics. You make, you do it, you do it. And until you are really able to take a problem and solve it by yourself or try to solve it by yourself, you are not doing mathematics. I mean, because learning, there are topics that you can learn. I mean, there are some even scientific topics that you can learn. But this is not the case in mathematics. In mathematics, you have to do it yourself. So that would be the best, you know, what I could say in the short time. But so it's a little bit like, for instance, you know, if you try to, if you complain about it, if you try to do a canvas by reading books, it's the same story. You have to practice. The practice is far more important than whatever reading books you find out. So in that way, it's a very democratic subject. And I mean, there is a key step also, which I mean, the manifestation of this key step is when the student in the room finds a mistake of the teacher because he's able to think by himself, by himself, and find out that he is right and the teacher is wrong. So this is something which is very important in mathematics, which is different from what I've talked about. There are other topics that require so much knowledge that some of this will not be possible for people to learn. So the next question is thinking about this quest to find a unified theory of the universe. I think it's interesting the impact that maybe this has had on the interaction between math and physics. And so maybe some had a perspective that at one time, one field said another, but it seems that now there's sort of a symbiotic relationship and I wanted your perspective on the evolution of the relationship between math and physics. Yeah, so I think, you know, it's a very delicate issue in the sense that there is one problem that people are trying to solve, which is the quantum quantum. But you know the quantum quantum test exists, you know that it's quite difficult. But in a way, if you want for mathematics, at least as far as I'm concerned, the issue is even more important for mathematics in the following sense. That for instance when Riemann wrote this game, he was very clear on the fact that the hypothesis that he had for Riemann and Geogathe, he was no longer old at a very, very small scale. And he was so lucid and precise that he was already foreseen developments that become much later, and in particular, because of the fact that the notion of solid or the notion of the day of life no longer makes sense in the way that it's called, whereas it was to have these notions that when Porsche released the function of Geogathe, so there is a symbiosis. There is a symbiosis, but there are also, I would say, deviations. And what I hear from, for instance, when they use some talks, I hear deviations because some people just want to change the rules of physics, because people can say they don't know. So I think one has to be very careful. And at the same time, I would say that there is an intermediate goal to contemplate it. And that goal, which is very precise, is to understand the effect, the impact on the notion of Geogathe, that the experimental physics has provided for us in one century, where the Inward Bound 3 was beginning at the end of the 19th century, in the discovery of the area form of other activities. And we have increased our perception of the small structure of space-time by a factor of 10 to the power of 8 in the century. And that has implications on the geometric model here at space-time. And that implication has been fully understood in the thought of Geogathe. And what happens is that space-time is no longer purely continuum, but it's a mixture of the continuum and the discrete. And so this is a lesson that was understood. And it's a lesson which very, very strongly forced to change the Remain and Paradek. But this change in the Remain and Paradek, of course, Remain couldn't foresee it, because it involved the quantum mechanics. So the new Paradek on geometry is very close to the Remain and Paradek. But there are nuances, and these nuances come from the quantum. They come from the formalism of quantum mechanics which has been discovered by a formula in the 19th century. And it turns out then that the idea of the notion of the derivative space becomes more natural, I think, but more easy to understand in the quantum quantum. So yeah, you described that it's not just the immensity of the universe, but also these very small scales that are bigger point. How would you define a point? Okay, that's an interesting question, because we can ask really a criminal question. And that question is simple. How do we communicate with the extraterrestrial possibilities, the place where we are? You see, if I tell you that we are in three years, that won't help, because first of all, these people don't want to know what we're talking about. They want to understand our language, and so on. And then people would tell you, because there's no general identity, we just have to give our coordinates, it's a coordinate system, but that's also foolish, because which coordinate system do you take, and which environment do you have to communicate through a coordinate system? And it turns out that what I was talking about before, it means this re-understanding of doing everything, is exactly for why the universe answers the two questions. The first question is, how do you communicate the space in which we are? Just global, not by giving a picture. How do you communicate the space in which we are? And second of all, how do we define a point? So how do we communicate the space in which we are? It turns out that the best way is to give the music of the space. So if you take a shape, I mean, this is a random metaphor, you know, which goes back to Mark Carras before. So if you do the shape, like the Brahm process, Brahm's shapes, it turns out that each of these shapes has a special scale, musical scale, which is assigned there. I mean, which frequencies are the proper frequencies of the shape? And it turns out that if you want to give invariant in the space, you have to give a list of quantities which are assigned to the space in an invariant manner. Now the scale of the space is invariant in the form, because you can rotate the space, you can do whatever you want. You will not change its scale. You can take a Brahm, you can move it towards the place, you can do it on the same scale. So this is an invariant of the space. And it turns out that, well, Helmholtz found the so-called Helmholtz equation that gives you the scale of the space. But it turns out that there is a small refinement in that, which is that Helmholtz was taken by the square root of the glasian and you have to replace this by the Dirac denominator. But this is a small nuance. And once you know this small nuance, then you can actually reconstruct the space that you need to know a little more. So you need to know a little more than the scale of the space. You need to know precisely what are the points. And what are the points? Each point is defined by a chord on the scale. So the point in the space, technically speaking, how do you specify a point? So technically speaking, what you do, you take what are called the eigenvectors for the Dirac derivatives. There are sections of some boundaries in those space and you evaluate them at a point. Now, when you evaluate them, you cannot just get a number. So to get a number, you take the main page, which are the scalar products of these various sections at a point. This is with your matrix. And it turns out that modulo is in the variance of variance of variance. This matrix is exactly what you need to do with that. So the picture, the meta picture that you should have is that in this understanding, in this understanding, the space is understood by a musical scale and possible chords. And the possible chords are the chords. So in a way, what happens is that you reconstitute the space by kind of fully transformed. And I believe that this is exactly what the brain does when we see, because when we see, we have the photons which are encoded in moment of space, eigenstate. And the brain reconstitutes space like we are used to see. But what is even more important is that this is exactly the way we perceive the distant universe. Because we perceive the distant universe by looking at spectra of galaxies, spectra stars, spectra of the regulatory system. And it is thanks to the spectra that we have the kind of information that we get from the distance of the universe. So in this parallelism, we find out that not only it's useful for microscopic distances, but it also reshuffles and changes the point of view on the large distances, but in a way which is perfectly coherent with our perception of the universe. For instance, typically what happens is that we know that things are very, very distant. You have to remember that there was some time when people didn't even know that there were things outside our galaxy. They took very bright astronomers to find that. But now we know that things are very, very, very, very distant just because of the redshift. And this is again the spectral problem. And here there's a concept of distance or unit of length in terms of wavelength. Sure. That's also a very, very important step, which is so much fun to explain because it relates to very concrete stuff. So the story starts in France more or less during the French revolution. There were more or less, there was a unit of length per city. There were at least 1000 units of length in France, which means that when people were selling for instance tissue and traveling from one place to another, they had to measure with respect to the unit that was at the entrance of the village. Of course, because the revolution was an idea to, of course, unify things and they had current purposes and all that, they decided to, and they had very good scientists. So they decided to try and unify the system by defining a unit of length that would be. So what did they do? They took the largest available object, which is the earth. And they defined the unit of length so that when you multiply this unit of length by 40 billions, billions, you obtain the circumference of the earth. So this is what they tried. And in other, of course, they couldn't go to the pole or you know, go to the field of the median. But what they did is they looked at the stars and they measured angles and so they just needed to measure a certain angular portion of the median. And they chose the median portion which was between Dunkerque, which is in the north of France, and Barcelona, which is in Spain. And in 1792, so this was a full, you know, revolution and so on, they sent two people, the lamp and the shen, were sent out to do the following. The idea was that they would, they would first of all have a base, what we call a base. So they had a light down on a sufficiently long distance, some bars if you want, and they had taken that as a base. Now they were only measuring angles, which is a very smart idea. So what they were doing, they were putting telescopes on top of hills and so on, measuring angles. And by doing triangulation, they were comparing the base with the distance between Barcelona and Dunkerque. And out of that was defined the minute of length, which was actually a metal bar. It was a very interesting story because there were all sorts of developments in this story. One of them was that one of the guys, I think it was Michel, I don't know, had to make measurements in Spain. And of course, so he was measuring angles with his, you know, by putting a telescope on top of a hill and so on. And of course, he had lots of trouble because there was a war between France and Spain at the time. And he had to explain to the Spanish army that by putting his telescope on top of the hill and looking for his telescope, he was not a spy. But he was trying to define the unit of length. So there were all sorts of very interesting developments. I love to tell these stories, I don't know why, but... And then what happened was the following. So this unit of length was actually deposited near Paris. And when I was a kid, I learned the unit of length is the meter which is deposited in Paris near the port near Paris. So I was thinking, and I'm sure many people were thinking, you know, this is not very practical because if you want to measure your bed, you have to. Of course, they made duplicates of this meter and all that. So that's what was the situation at the time. But then some very interesting event happened. So there were periodic meetings of the metric system people. And these meetings have been going on very periodically, for years and years and years. I'm not sure that the period was one year, perhaps was three years, or something like that. But around the 1970s, what happened was that they noticed that actually the platinum bar, the defining unit of length, was changing length. That was very bad. And how did they notice that? Because they noticed that by actually measuring its length very precisely, by comparing it with the krypton wavelengths for a specific transition of krypton. So that was very bad. And gradually they took, they decided to take the right step. And the right step, of course, was to take this wavelength as a definition of the unit of length. So that took some time. That took some time. But what is very interesting to know is that now there are instruments which are sold in the, you can buy them in the shop. And these instruments are based again on the wavelengths. It's no longer the one which is used is not krypton, it's cesium. Because cesium turns out to be a valuable, there is a really solid. And however, the wavelengths of cesium which is used is a microwave. So it's like when you put something in the microwave oven, it's a wavelength which is of the order of three centimeters. And it is an instrument which allows you to measure length up to 12 decimals. So I mean, it's absolutely incredible. And this is now what is taken as a unit of length. Of course, people will tell you it's not a unit of length, it's a unit of time. But because of the constancy of speed of light, I mean, the speed of light has been fixed to a very specific number. So things have evolved. And now what you see from that is that there was a complete change in the body because the unit of length is no longer a localized object, which is somewhere, but it's a spectral data. And it turns out that the new body which comes from quantum mechanics, which is the body of a combative geometry, which is called spectral geometry, is exactly parallel to this change of the body in physics. So it's a very concrete, it's something which is very, very concrete. And the advantage, enormous advantage, is that if we add to, for instance, unify the metric system, not on Earth, but in the galaxy, for instance, you know, if you tell the people, okay, come to Paris and compare with this unit of length, we have defined that. I mean, they would laugh at you, there would be wars actually because people would say, well, we have our unit of length and so on. Whereas, if you tell the people, take a chemical element, of course, cesium is a little bit complicated because... For robustness, wouldn't it maybe have something very common? Yeah, exactly. Like helium or hydrogen. I would vote for hydrogen because hydrogen is essentially present anywhere. Whereas cesium or heavy elements of that kind, in fact, what is to do is that they only come not only from supernova, but from very, very exceptional supernova. So, their abundance in the universe is not so clear. But if you take hydrogen, for instance, there are spectral rays of hydrogen which are very precisely defined, they have a very specific pattern. But then one would have to find hyperfine splitting because the advantage of hyperfine splitting, which is used for cesium, is that hyperfine splitting is a difference of energy which is very, very small. And that would, in the universe law, when you pass to the wavelengths, it will generate microwaves, which is much more practical. Whereas if you take a huge difference of frequency, like for a transition and so on, you would get a very, very tiny unit of length. That would be good. Okay, but what I am saying is that if you communicate with people, to people with a probe, by sending a probe, and if you are able to tell them what is your unit of length, this is marvelous. And you just send a copy of the spectral rays of hydrogen and you explain which one you want to find out. I mean, this is very simple. If they are smart, they will understand. Whereas whatever you do otherwise wouldn't work. And in this description of the fine structure of spacetime, you describe it in terms of the spectrum of an operator, which allows... It's a little bit more complicated. As I said, of course, the spectrum of the operator gives you the unit of length. This allows you, in a way, to combine a discrete concept to the continuous concept. What allows to combine the discrete and the continuum is the fact that, essentially, it's a mixture of the discrete and the continuum. If you wonder what the discoveries of experimental physics have unveiled over the century is exactly what is the structure of this discrete space. So, at first, the discrete space is with my collaborators, Shamsedin and Walter, Vance, and the kind of what we found. At first, we were proceeding as from the bottom up approach. Namely, we were taking from experiment and trying to fit with what was going on and so on. And gradually we found what the finite space should be. But in the recent work about two or three years ago with Shamsedin and Mukhanov, we were very amazed because we were asking a purely geometric problem, which was motivated, of course, by unconvincing geometry, but which was totally disjoint from physics and the standard model and so on. And by developing this problem in dimension four, we found exactly the same finite space and the same algebra that was put in by end before. So, we believe that we are a piece of the truth. But naively, why is it important to include this discrete concept? Okay, why naively is it important? This is easy to explain, so why? But I need a piece of paper. Oh yes, here is a piece of paper. So, it's very easy to understand. You see, why is it important to have this discrete piece? Okay, it's the most obvious problem that you have if you don't have this discrete piece, is that the eggs, most of them, the brown angler eggs, I knew brown very much, I mean, died just one year before the particle was discovered. But particle is discovered, we know it's there. But it doesn't fit with standard geometry. Why? Because in standard geometry, if you take a function on a space, okay, you will differentiate it and you will get what is called a gauge potential, a 1, 1, 4. And why? Because the differentiation depends on the direction in which you differentiate. So, this is why you get something which is called a spin 1, if you want, which is, which depends on the direction. But the brown angler eggs particle is a particle which is a spin 0. So, it doesn't depend on the direction. So, you wonder how can you obtain geometrically a particle of spin 0? Okay. Now, imagine that instead of being just this manifold, okay, there is a discrete element. The discrete element is just telling me whether I am on the top or on the bottom. Okay. Okay, so now I have more information. I know whether I am on the top or on the bottom. And I take a function. Now, this function will have a value, will have a development here, and it will have a development under. They don't have to be the same. Okay. So, I can differentiate my function up. I can differentiate it down. But I can also take the finite difference across. And the finite difference across, it doesn't depend on which direction I take. That's a boson of spin 0. And that corresponds to the brown angler eggs boson. So, the brown angler eggs boson was a completely un-mistakeable sign of a discrete structure which was present. And I knew very well, very much, and he was very enthusiastic, of course, to go by this understanding, which is the understanding of why, if you want the experimental fights that physicists had, you know, because for instance, the brown angler eggs mechanism was obtained from years and years and years of source of how to give masses to the particles. So, all the masses of the particles actually come from this mechanism. And what you find out in this model that we have developed is that, in fact, the main ingredient, which is the matrix of masses and mixing angles and so on, the particles, is in fact exactly as the line element for the finite structure. So, the line element for the finite structure contains exactly this information, which means, if you want that, in this model, we have a mixture of the continuum and the discrete. But the discrete contains the information about the masses and mixing angles. Thank you. Thank you very much for your time. Okay. Very good. Excellent.