 So, yes, I remind what was happening in the last time. It was so brief, so I repeated, namely, how from the concepts of physical concepts, the kind of interpretations, we arrive at, we arrive at a usual measure of spaces and with all kind of packages and where significant for us, really, to entropy, and namely, this is Kolmogorov's dynamic entropy. So the issue which was in erotic theory for about a decade before Kolmogorov entered was as follows. So, in general, so I have abstract measure space and abstract measure space for this purpose. You can think about this as a unit square. Usually, say, unit interval, I prefer square in certain kind of, easier to make a sketch. One dimensional thing is invisible, right? Then you have a self-mapping on it. And this self-mapping is a measurable map. So it's a map which preserves measurable subsets and preserves the measures. Just automorphism. I've forgotten my minutation, which is not f. It's transformation, yeah, that's bg. And then if you have another such transformation, it's one measure space called x, and then I just measure space called y, this is kind of u. So these make a category where measure spaces, as I said, they explain, they can be things dramatically just like squares on the plate, or define abstractly as functors in the category of fine measure spaces. And this equivalence is the only category where measure 30 enters. If you are currently involved in the language of this fantasy, much of what I say is material, I'm just going to say it. All the Magorov thing reduces to one line, one line trivial computation. But anyway, abstractly, so I have one measure space and another, with transformation acting on them. And then morphisms imagine preserving maps, which by no means by ejective, right? But then it's ejective. Such that they commute with this transformation, such that this g and this u one goes into another. Very often people identify a priority, two spaces, because all measure spaces kind of are the same, all isomorphic. But this is the same like all sets are, or even cardinality are the same. So identifying them immediately creates tremendous terminological problems. How usual language of, they have conjugate, semi-conjugate, horrible language and dynamics exactly because they identify beforehand two different spaces, right? This is the category. And in this category, we want to find something from this category. And in particular, we want to know when to object in this category isomorphic, right? So given two such things, x and y, when x isomorphic to y, which means any category arrows and two directions. But again, it's much easier to speak when we have two different spaces. When they're in the same space, x, you lost the notation because they serve different purposes. This one of the difficulties in general, in exposition of dynamical rewarded theory is completely incorrect language. And we have to explain things, I think it's simply people who say it, but actually this you can cut by a factor of two or three, and you have those papers just making language correct. Well, I mean, it may have changed, of course. I haven't followed this for the last 20 years. My collection is somewhat oldish, I must admit. I don't know how people say it, if still it's conjugated, conjugative transformations, still they use this language or semi-conjugate, no? It's horrible, I mean, just horrible. OK, so, and the basic example from abstract point of view of this transformation is how to describe it with the right canonical space, canonical transformations. When you take some group, of course, they define it's nice to have any group gamma acting on the space. Actually, it doesn't have to be transformation. It may be the whole group gamma acting. He and the same gamma acting there. Actually, groups may be also different, right? Actually, you can say the groups are different, and then the whole errors was included, I'm watching between groups. But say groups are the same, and so given any group, there is a simple class of measure spaces where it acts defined by products. Namely, it starts with defined probability space p and take it for the power gamma, which means you can see the whole function from gamma to this p. So, another way is this sets index by gamma of numbers p's between points of numbers and the probability of space and so forth, so on and so forth. And the point you can consider is infinite product of space. As I explained, this is one of the basic construction measure theory and probability taking products. I don't know whether it was first for my life, but Kolmogorov was known before him. So, and second, I return to that and explain how it done. But anyway, it's product and there is a natural kind of measure structure, if this was defined by the space. And natural meaning, it agrees with all of these obvious maps. So, we have this product. And you have a subset delta in gamma. Then this product, p of the gamma, of course, goes, just restrict function to delta. And this structure, the only condition, will be compatible with these arrows in this factorial language. So, you don't use anything else, all of these infinite objects. But of course, you use property which we do know of this final category. So, you have this kind of the most basic dynamical system, a basic action of gamma in a measure space, and this infinite product space. And then the question is given to action. This is a very kind of elementary question. You don't want, and it's unclear to me the meaning of a abstract question for general dynamical system, general actions, when there is a motion. When the two isomorphic. Because here you know the answer might be simple. It depends only on p and q. And each p and q is just finite collection of numbers. So, this is the beauty of this question. It's extremely, it's extreme simplicity of expected answer. So, what is the p? p is collection of numbers. And q is a confined collection of positive numbers. So, looking at these two array of the positive numbers, you want to say, is this isomorphic or not? For given gamma. And the answer is still unknown in general, right? The answer is unknown in general. And it was, nothing was known till Kolmogorov. And so, he was, he's proven. And as I explained, I remind you, in a second, I show the basic kind of ingredients of the proof. Our language become kind of tautology, is that if this isomorphic is the gamma, in fact, proved for gamma equals z. But this is, we integer, but this applies to a large class of groups. Then it follows that entropy is equal for these two spaces. And the entropy, as you know, can be computed by a Boyceman formula for entropy of pi. So, entropy of the space p, so we define equal minus sum. Minus is convention, because we leave 30 minus, there is no rule, pi, among pi, it is summation. And the minus just to have a positive. Actually, in a second, I come up with the similar expression, which could be on the contrary negative. And you again change the sign to have a positive. And this is a quite, quite, quite simple theorem. And then it was, at some point, was reversed by, oh, should I get it again? Probably his name is Ornstein, yeah? This also true in both directions. There is an open arrow as well. So entropy, equality of entropy is implies the spaces are, the dynamics, I suppose. There is some controversy with how much the opposite theorem is relevant. It's by far more difficult. The proof is really difficult. You have to produce the summation. Well, that can stand clear what's the point of this, yeah? One theorem tells you something specific, rather concrete, you observe. So the meaning of that is, physically, when you observe a crystal, no matter how you measure it, you get the same entropy. So you can think about this as one dimensional crystals represented in different ways with a plane cube. You see it in different coordinates. There's p's and q's kind of coordinates that we should see. And we see in a different coordinate system, we see the same result. And opposite theorem says, well, if you always see the same result, the kind of system of the same. And it's unclear, significant, this kind of deep theorem, but I must admit I don't understand the significance. For me, it looks kind of very complicated, very profound theorem. But unlike in Magorov, it has no kind of something, which is a more kind of artifact of the language. But of course, in its own sense, theory is much more, much more than that. It improves much more. It's quite refined theory of that, but this doesn't seem to, yeah. I think eventually the theorem would find kind of a place in something, some categorical approach. But this hasn't been developed for the moment. It doesn't exist. So that's the theorem. So we temporarily forget about that. There is a weaker result, the most kind of Yotushinai, which says, in a way the same, up to exponentially small area, give you this quality before. So, and this was kind of interestingly enough before that, for some time, people had no idea when these two systems are isomorphic. But make them different. So how to look at the system and say something about them. And so, with the case pointing, hence in this theorem is true for all amenable groups gamma. So let me say what is amenable. In this terminology, it says the gamma is countable group for no purpose. So it's represented, it can be exhausted of subsets, that increasing sequence of finite subsets. Such that, if you take this subset and multiply it by any fixed subset and compare this cardinality, and product means I take a product of all elements. And compare that cardinality, that goes to one. So it means there exist this sequence. For example, if you have integers, so you just take a system in increasing sequence of intervals. In Euclidean, in a billion groups, you take three billion groups, you take increasing sequence of finite subsets. So it's a well-approximable in the way by finite sets. In some sense, you can say it, you see most of the product doesn't bring you out of the adult end. When you multiply it by some elements, it kind of basically remain there. And so, this was proven by Klamogorov recently. And maybe I just remind you the ingredients of the proof, how you do that. They kind of, the basic thing involved. So this is how you define product again, this sum. Then, if you want to identify infinite products, some of them, you just stand at interval, we stand in space, you need to just go back, then slimmer, but again, this elementary kind of, have nothing to do with dynamics. You see, you have to reformulate in these terms, it's kind of, rather trivial matter. And the point is that this entropy of dynamical system is defined for any system of transformations. It has nothing to do with group, obviously, billion groups, whatever. It always well-defined. The question is, if we give you useful answers, right? So if you do it, say, for our free groups, whatever, we get always infinity or zero or something, and you are nowhere, you get infinity actually, yeah. But once you have it, you make this computation and using this, it's kind of the basic formula which you use. You use this relative Shannon formula, and so let me remind it in a nice language, relative Shannon formula, or Shannon inequality, I'm sorry. It says we have two morphisms between finite measure spaces. That entropy of this, it's very simple to remember, is less recall, that entropy of F1 plus entropy of F2. But you see, it's exactly where it's nice to have, to have things for errors. And this is a kind of operation which corresponds to joining partition in measure spaces. So secretly, if you stick to the old-fashioned measure theory, these finite measure spaces are obtained by subdividing your interval into finite many pieces and assigning to each piece weight of this interval. And when you have, so it's better to see in the square because you have this partition, and then you have this partition, and then dividing into squares corresponds to this join operation. This vice square is much better. In fact, the full picture, this is equal to even better to be seen in the cube that I was explaining in the, this Lumis-Wittner theorem is very demonstrating that. Okay. And so, this has nothing to do with groups, with ever, it's a very formal property which you have to verify, and then it would come, ah, it goes too high, ah, ah, ah, ah, ah. Then we'll go this way. And then, but when it comes to actual proof of Kolmogorov, you see, so what happens, you just exhaust, indeed, this, you just apply this for exhaustion by intervals and the only, this limit on equality is the only thing which you use. So anyway, specifically what Kolmogorov uses that this n plus i divided by n converges to one when it goes to infinity. It's kind of maybe a little bit caricature on the proof, everything else has nothing to do with the subject, just generic stuff. And here it's amenability of the group which enters. And this was also narrow neck in this argument, and people at some point started to believe that it's maybe completely false for other groups. But then, as I said recently, it was proven by Louis Boyne that it is true, and it's true for so-called Sophie groups. Let me say a couple of words to them, because these are quite amusing concepts. And which proven to be, we see, ah. So another point, of course, why mathematician couldn't prove it, you know, 100 years earlier, theorem, that's for me a mystery. Why, you know, it was going through it, there was a tower, a Birkwald theorem, all this kind of abstract nonsense, and kind of the most elementary stuff which physicists always were doing, yeah. You're looking crystal and measuring the entropy and showing that entropy doesn't depend how you measure, and they kind of knew that in a way, mathematician could never pick it up, unless it was, went through Shannon, who actually rewritten Boltzmann in discrete language. So right from mathematical point of view, the difference between Shannon and Boltzmann that Boltzmann used in natural logs, and Shannon was using log 2, right, interpreting them as bits, which is, of course, nice. And by the way, even now, you can find discussion in the information that people, they're proving various theorems using this language, and it's got high bar means. It depends on what kind of, it appeals you to stock market. You say, you buy this stocks, you sell this stocks, and there things, it helps to understand, right. Of course, natural log is much more convenient for computational, but for any formulas, much better. Log 2 is nice for purposes used for Shannon, but Mathematic, of course, was known to Gibbs and to Boltzmann and Gibbs, and they manipulate with completely freely. However, mathematician couldn't do that. That's because, and then Shannon did it, and then Magorov understood Shannon, and then he actually done it, right, because already it was in Shannon, of course. All the formulas, all formalities were in Shannon, and Magorov just put it into this framework. It's rather bizarre, I don't quite understand that it was a big gap between mathematicians and physicists, yeah. But then, there was this, so this problem is still unknown in general. So, is it true that this is isomorphism implies and to be equal, and it's nice, so it's a simple question. You don't have to know, my match about general measure space is just, it's not even true, it's clear even for continuous maps, whatever. So, this is widely open, and apparently the issue, of course, is not about that, but about notion of a group to which it applies. And Bohr improved, say, for three groups first, and then for so-called soffie groups. So let me say a few words about them. This is a concept which is kind of ideologically very close to what I'm saying, because much of Boltzmann's writing and many other pieces implicitly used non-stand analysis, and again, non-stand analysis was known since Leibniz. Leibniz actually introduced it, but then mathematician kind of be unhappy without that, because, you see, because there was kind of a concept of Riga, which was not grasping, not encompassing all mathematics of the time. And it's partly, could have been influenced by Berkeley, yeah. He wrote like, by Bancher's influential, extremely stupid things about calculus foundation. Calculus was kind of sheer nonsense, but however, very influential one. It wasn't influential till the middle, beginning of the 20th century. You can see, I count the history of mathematics saying it was great text. If you read what's written there, it's gibberish, yes, sheer gibberish. But apparently it has influenced mathematicians. And so, instead of thinking, they were just trying to be kind of closing their eyes to obvious things, like non-stand analysis. But of course, for fees it was obvious thing, you manipulate with them, it certainly was not. It's trivially, reading no contradiction. If it doesn't make a mistake, and so what are surface groups? What's not standard about them? So one of the definition is, which is, must be in the, see the point, because I'm coming in the second part of my lecture using philosophy on non-standard analysis and more, you operate with infinite objects. You, in particular, when you speak about measure spaces, we already had it. And not to understand one space P, we take this n and n goes to infinity. But it's convenient to have this n, this infinite large number, not to make it variable. So otherwise, your notation become much more flexible. It's just one number, but infinite large. At every moment, you can kind of change the concepts of large. It's fluid. And this is one of the possibilities of the language, mathematical language. Now we know, you just check our language, you know, it's relaxed and just can go freely with this. You don't have to stick to epsilon delta, whatever, particular language. And now I want to refine Sophie groups in this kind of language and say, yeah, so gamma is, the group is gamma, if it can be represented by a subgroup in the permutation group, symmetric group, this n infinity, infinite large. It's not infinity and it's n being uncountable. It's infinity in the sense of, in the sense of being the standard large number. And then you have to decipher it. And when you start deciphering it, you immediately see that there are various possibilities. Right, you have to specify the language. And again, it's the same like in category theory. By itself, it doesn't prove anything. But it tells you immediately what you have to check, where you have to go. And so it gives you a very compact way, make conjectures, a very easy way to formulate theorems. But then you have to see what happens. But this verification is trivial. Once you know where you go, you just go. You don't have to think, right? And in this respect, from the standard analysis, it's similar to category theory. It's convenient language. It allows you to cut off junk of notations. And thinking, yeah. So now we have to decipher this. So another way to say that this group and this class of gamma, gamma is suffice, yeah. And I think it's, if again, it acts on these spaces. In other way to say it, you can see this kind of proper limit. It's called alter limit of finite permutation group. And you go to the limit. And so I have this compact group, non-standard compact group. And this might be subgroup of this group. And this agrees with concept of the group being reasonably finite. So in one sense, I saw this. Amenable groups are approximately acts on finite sense. So it can be approximated. So when I say limit means it's approximation, of course, right? Because it's a limit. So it might be approximated by finite permutations or by finite groups. And so rigidly finite group, gamma, are exactly those groups which can act by summaries on compact metric spaces. So, I just don't want to write because I already wrote it over here. So that definition of rigidly finite. So again, in both cases, compact spaces are the ones which are well approximated, of course, by finite spaces. So it's also action, approximately by finite action in a different sense. And now we want to give this full definition. So I guess I put it there. Then I had to write it because I already written it once. So we want some closeness between objects. And we have to say one action close to another action in a second. So we have this concept of two maps being close. So if they differ on a subset, so two maps apply to a finite set, no matter where they go, called epsilon close, or epsilon agree up to error epsilon if the subset where they're different constitute only one over epsilon of total of the tallage of the points, right? This how we measure. The moment we said this measurement, we can speak about what is called alter limits or limits, or all this non-standard stuff, right? This, because you get a measure. So when you use non-standard analysis, you need kind of unit measure of unity, how to go to the limits. And now we have to say it. What happens? And now once you have it, you say group approximately acts on finite set. So first again, we introduce this concept. What does it mean? There is a epsilon monomorphisms of the group into the permutation group of finite set, right? So you have all any set for the matter. So it must act there, but at every moment when you make composition of maps, whatever, all the qualities, all the maps are equal only up to error epsilon. Error epsilon meaning outside of a set whose relative measure is epsilon. It's only epsilon percential points. It's quite reasonable notion. Of course, you can introduce other topologists, other measurements, and you get different class groups and it's unclear if all of them are equal. There is a very close class groups called, I think Hypalinia or something you can introduce by Ellen Kahn earlier. And it's unknown if the class is equal. We say it's again, you use some limit, but with different topology. And we can probably, there are many different classes we can introduce, but this happened to be a kind of, it's an extremal case anyway. It's L1 metric is extremal. And these are usually very special. For n to be non-surprisingly. And so if the group admits such an approximate action for every epsilon on, and for any subsets, a finite subset on somewhere, on some finite set, it's called suffix. It's not so clear, of course, so what class of groups you have, but it's kind of clear that amenable groups and this is defined groups and this category. So in particular, all linear groups or a billion groups, all manipulation you do with them, there's subgroups, whatever, they're all in this category. So it's a big category. Big class, well, category meaning class. But apparently they're very special. So we don't know. This is a little bit embarrassing. So we don't have counting example. I think no matter what you look, kind of you have counting example. But we cannot prove it. And this is annoying. And I guess once you have it, you will have also counting example to this property that two power spaces to gamma power, it's called Bernoulli shifts in time of a group. They are some of it. Probably they will not give you, don't have to have the same entropy and they all probably will buy some of it. So that's naive conjecture. That being suffocating is less insufficient to have a going theory. But for the moment because we cannot say suffocating or non-suffocating, it reminds problematic. Okay, so this was more or less repetition of what was last time. And the point was that starting from this kind of physical perspective, you have a crystal and you measure its entropy of the number of states, how you measure it. You take some machine, you move it around and measure only with this machine. And sometime you can put this machine in two different positions and measure with this. And just doing that, you'll find this entropy. And in fact, no matter how you position them, you get different results. So a priori you may have many entropy, a posteriori it's not the case. But given any systematic measurements, you get the same result. And this entropy doesn't depend on how you measure it. It's intrinsic to the system. When it is, what is this? We call it system without interaction. It's non-interacting particles and so this kind of very trivial system. But, and in a way, of course, if you make a particular kind of coordinate system, this PI is the service coordinates. And if things change from coordinates to coordinates, not that you say, ah, everything is wrong. You just, you have to change your concept. When you measure certain quantities in the space, you take a finite transformation. Of course, the length changes. It doesn't mean that geometry become faulty. It just means you have to introduce different kind of rules of coordinate change. Therefore, even if some groups think you will not turn out in such a simple way, they probably turn out in a more interesting way. When you change, you apply transformation between two things, the change will depend on some invariant of this transformation. So nothing wrong. But we don't have this theory because there is no objects which can be applied. But now I want to turn to different subject because I want to give kind of, and this is about geometric kind of way to think about probability, and entropy specifically. And they were kind of two motivations. One, we're introduced by Fisher, geometric thinking, and then in context of mathematical biology. And because you see because there is kind of this feature metric in the space of probability distribution on a set. So when we fix a set, we vary these PI's. So this is the simplex, right? And so it become kind of geometric. Now we speak of probability as we have the simplex, and just talk about the simplex. And this point, I don't know why it first was emphasized but definitely developed by Fisher who was interested in mathematical biology and it was proving various theories about this metric. And it's kind of funny because this metric sometimes called by people in biology, mathematical, it's still called sanshahani metric who introduced it. And some people called row metric, which was before sanshahani. And sanshahani somewhere in the late 70s and they say, okay, all these properties of this metric were proven by Fisher. And many, many properties. And they're proven, it's unclear. So I think the word metric, Fisher probably didn't use the word metric in a second. Just I explained the reason. He was a little bit reluctant to use the word metric. However, so we want to find natural metrics in the space of this probability distribution. So what you'll be, I start with the last, I actually am not certain. I couldn't, at some moment I found the references of the paper by Fisher, but now I couldn't. People all say, Fisher would never mention the paper. I think it was in 20s. In 20s, and at some moment I found it with Spaniard one day searching on the net. But then I couldn't find it. People don't refer to that, which is, I think, unfortunate. Because quite, I will be not surprised that in this paper by Fisher, there is more than usually extract from the paper. I know this happens for more modern authors who kind of have this kind of, who introduce new concepts and people refer to them and then they missed half or two, or two thirds of what's written in the original papers. But it's maybe true also about Fisher paper. He was certainly a smart guy. But apparently I think quite nasty, yeah? I'll write about him a little bit. So I start with the last kind of definition, because most nature is probably the one. Ah, well, maybe first we discuss a little bit. Prior to that, something about the Simplix. So first, this category of method spaces may be seen as some documentation of the category of Simplices and Simplisher maps. So category of finite sets can be thought of and not the right word. It's canonically equivalent, but it's not quite, I'm not certain here again that there is the right terminology. As a category of Simplisher, it's simply so much because of every finite set, uniquely defined Simplix and every maps between finite set induces unique Simplisher map between Simplices. It looks like there's nothing. However, just to explain why it is highly on TV operation, this change of perspective is. So we have a category of finite set and this looks something extremely discreet and nothing just typological. Now we take Simplices. So we interpret this category in this language. It's the same category, right? It's canonically, I don't know how we can say. They're not a thermophic, it's just kind of the same, except for interpretation. And then you say, ha, consider covariant, I'm sorry, contravariant, function from this category to category of sets. Take this. And then you just realize, you think in Simplisher terms, they become Simi-Simplisher spaces. And so the theory of this space, all algebraic topology. So if you reinterpret that, you immediately have all algebraic topology in your labs. Of course, you have to prove through them. But that's kind of trivial. However, if you say in terms of sets, you never, you never see that. In fact, by the way in there, when people speak of Simi-Simplisher sets, they will make some, for some reason don't understand the introduction of, no, simply see the generation, you know, just completely irrelevant. What it is, it's a function, this very simple function. If you have Simplisher complex, so your bunch of Simplish glutes together, and for every Simplish, you can consider all, the set will be all Simplisher map from the Simpli there. That's your function. And that's all, and it's functorial. Because when you go to your face, it composes, it's a gracious projection. That's it. And all algebraic topology can be expressed in this language. You can't prove maybe too much, but you have immediately the language. And we can question this. So there are other interpretations by the way of that. Of this category of finance set, you can implement them by different, the different physical, so to speak, models, and you have different, completely different kind of, heavy news of thinking. And for us, so it is a measure, it's a measure space from this point of view is such a Simplish, plus a point in the Simplish. It's a measure, right? It's given by the centric coordinates. And again, morphisms will be, again, this Simplish map, but such that this point goes to this point. So you can readjustify a little bit this category, so we have essentially the same morphisms, but you have more spaces, yeah. It's still sufficient. So that's what happens here. So this is a category, and then so you think how far you can go which term of Simplish is what? What kind of geometry? I'll say a couple of words because we shall come to this later if there is a time in the context of Mendelian dynamics is what? Yeah, so one amusing point, which kind of always, of course, implicit name of books on probability, but never said explicitly, that the Simplish can be seen, is a limit of purely combinatorial objects. In the combinatorial object, you take this underlying set, take it as Cartesian power and divide by the permutation of the coordinates. When you go to the limit, you convert to the Simplish because this is explained by the sum. It's kind of trivial, right? Because you see the actual, this looks big space, this exponential power space, but this group of permutation much bigger. It's n factorial much bigger than power. So essentially it acts almost transitively. So what remains is very little. And this, of course, this symmetry is everywhere in probability theory and statistical mechanics and just in my following lecture, explain how to exploit it in a much deeper way. This is very, of course, it's rather superficial. But still it's kind of nice and entropy I will not explain today may be also seen in terms of this symmetry. But so is the symmetry. And then there are other operations which we may Yeah, and so you see the dimension. Cardinality goes tremendously down from exponential goes in n. It becomes polynomial. So polynomial is nothing compared to exponential. So in a way, this action is transitive or ergodic. And that's quite essential, by the way, in this, in this Suffolk, Suffolk kind of theory. I must admit I haven't thought for myself the definition of Suffolk to explain it effortlessly. It obviously requires a little bit effort. And I'm pretty certain it can be brought also to purely categorical language. If you put the nice setting, you don't have to write a single formula. But the moment it's not so, you need computation. But then this computation, this map. So we have operation of Cartesian product on probability spaces. And algebraically it means the following. It says, it's not so trivial map from product of Euclidean spaces to the tensor product. And this is just, you take tensor, tensor, tensor, and multiply units and you have this map. And this is highly nonlinear map. It's kind of complicated polynomial map and just natural algebraic geometry is associated to this map. Yeah, it's just where nonlinear of algebraic equation come from. And for simplices, so we have this particular map also. So you embed product of the small simplices to the big simplices. And an instance of that is called Sagra map. And we shall, we will be rejoining with this important instance of that. Or a special take part of this is called Veronese map. You take all factors being equal and you restrict everything to the diagonal. And then symmetry responds to what? That you go to the symmetric power, symmetric tensor in the power of Euclidean space. And this is just, and this is just the space of polynomials. And the map is very simple. So how it works. It again, it looks very inocular but it's not at all. And the geometry of this has many, many remarkable features. You just have, you think about your vectors as linear forms, variables M. So it is summation Li, I prefer API, Xi. And you take it, power N. And then you have now polynomial of degree N. So you map space of linear polynomials to the space of high degree polynomials. Euclidean space goes to high degree in space. A particular instance of that, just to explain, it's highly, it may look at it or nothing good so to know this, it's a completely trivial but it's extremely non-trivial and rather poorly understood yet. Just a simple case where equals equals two. Right? So you map, you go from R, M. If I'm not mistaken to A, R, M plus one. This is the number, no? So from R, three, you go to R, six, yeah? Is that, get you right, yeah? Right. So this is, my numbers are right. From R, two, you go to R, three, right? So it's quadratic. So this is quite a remarkable map and just property of this map, we're essential, this kind of maps we are inside of this, you remember, we had this Mendelian discussion of the Mendel map which was this kind of map. And they have kind of remarkable properties, some of them are so obvious and because now I actually hardly, you know, it was a formula, but I don't think he has a picture of that. Like he didn't mention that, let me just say so, what interesting about this map, just for the case, R three goes to R six and you look what happens to the sphere and it goes here. So first, the sphere will be contained in some refined five dimensional space. So you take linear form, take it square, you have quadratic form. So this is space of all quadratic forms, six dimensional space of quadratic forms. However, the one which, and you can see that one on the sphere, this form has trace one. And therefore they're like in the subspace case point trace one. So they're like in the five dimensional space. So it's a kind of, and secondly, it's a sphere will factors that goes here, factors are a projective plane because the quadratic form is symmetric. Plus minus the same, right? So what you have, you have this kind of particularly embedding of projective plane to five dimensional space. And so it's kind of one of the most kind of fundamental object, exactly for this dimension, or maybe for higher dimensions, one of the most fundamental objects in mathematics. And particularly in quantum mechanics because she saw what happened. So you have this high dimensional space and here there is a sphere. And the convex cone over there give you all positive definite forms. So this is an extreme point, this is a set of extreme points in the space of positive definite forms. Or as the fees will say, self-adjoint operators, positive self-adjoint operators, which corresponds to statistical, physical, state in quantum mechanics if you go into the complex numbers, which for us in material. Yeah, we will be working with real numbers, mathematical part, we will discuss, not dependent complex numbers. But so if in classical probability, this is a cone, yeah, just positive number, it's kind of only invariant under the permutation group of coordinates, so it has these corners, yeah. This is much smoother, see. This is a set of extreme point, is now not finite set, but projective, projective, projective plane. So kind of formally, you started with a final set, and permutation group, say, of a cardinality M, acting on that. And now it starts with a projective space of a dimension M, and orthogonal group acting on that. So you replace this group by this is. I don't know the formula just because it's a wild group or something, yeah. Why the theories kind of, this transportation of theories, much of what you were saying remains true if you say it correctly. But this is the fact, and this is the kind of, this kind of game and mathematics you play very often. You just make something similar, they look removed, very different, but eventually, but they're basically kind of many things. But of course, there is no kind of unique approach to that, yeah, and you don't know what happens to other symmetries or other categories, whatever, but of course, it's suggestive. So in both cases, this is kind of a maximally symmetric object with respect to the action of the group, right? A permutation group cannot act on any smaller set, and also orthogonal group cannot act on any smaller manifold or smaller dimension. So they maximally symmetric object, and out of them, this thing arises. So, but about this projective plane, just let me ask a specific question which I think is an answer to that, yeah. So this is a projective plane, same dimension M, sitting in the Euclidean space of dimension, I think M minus one or something. So for two, it must go to five, right? Two, three, oh, it's M plus one. So it always embeds projective space to this Euclidean space. The question which is unknown for large M, so you won't understand geometry of this, yeah. And the question you might ask is as follows. So it is fully homogeneous, yeah. It's orthogonal group X there, fully transitively. So it's everywhere the same. But the metric you have embedded, it's not the intrinsic metric, it's kind of curved. So you kind of, this projective space, curved in certain way. It's on the same shape, it's not so easy, and we'll hear you the question. So what is the minimal number? Minimal number, call it K, such that this space is induced Euclidean metric, may be covered by K subset of diameter less than one. When I say one, I normalize the size to have diameter one of the whole thing, okay? Because the subtler point, each point has very big complementary set, which is maximally removed from them. And the question, for example, if it grows exponentially, polynomial, whatever. And I think if you think about that, it provides conjectures, account example, but now it's known to conjectures of Borsig and just many other conceivable conjectures. So its geometry is not at all, not at all transparent, yeah. And it's related, of course, to a property of matrices and analysis of matrices, et cetera. So these maps must be expected, despite the fact that they look quite simple-minded. Okay, so we stop here and then we, on the next lecture, explain more about that. That's how this got the symmetric test, so they look kind of, not big deal when it changed it, but of course, tremendously, like in the previous example, now on the level of linear algebra, it changed pictures significantly. And of course, this must be relevant in probability of the theory, but our definition of entropy we're not exploiting the symmetry. And actually it is becoming more significant in, as we see in the second, for norm and entropy, because there is not many visible invariance of the symmetric group as it actually sets, but there's a lot of structure and action on the linear space, but because, you know, the representation theory of the symmetric group. By the way, that's what people say on the same representation theory of finite group, they say just the only theory which exists in the theory of limitation group, everything else is just words, but this is amazing, quite amazing theory, which I have no idea over it, but it certainly must end. Now I want to discuss some metrics associated to entropy, and so I want to start with the last one because it is the most natural. It's called Fisher metric, yeah. So it's a metric on the space of probability distribution on a finite set, so it's a simplex, and we just want to define some metric there, and what will be the most natural metric? And the one which, and this is a common differential geometry, and apparently I don't know in what form it was introduced by Fisher, but the logic is quite simple. So you have some geometric objects, and you want to introduce metric, so it's more dramatic on the tangent space at every point on the tangent space. So they're defined by, they spread over some spaces, so at every point you have a metric, or you can integrate this metric, and then you have a remaining metric in this space, and this example, the remaining geometry is common, like the space of all the remaining metric on a smooth manifold, right? And at every point, the quadratic forms, this metric, so it takes distance infinitesimal at every point of value of this quadratic form, and then integrated with the whole manifold over the very form, over the measure defined by the metric, and this is not work for all structures, you need measure there, but here it's extremely easy. So what you do is, this is the kind of formula of what you do. It's kind of trivial, but it's easy because you have to compare different points, and the different points, you have the metrics are pretty different, for example, you have a real line here, and real line here is the fine space, the additive group, you only can compare them up to your constant, and so you just don't have consistent thing everywhere. You know there is one, where I write real numbers, there is zero, there is also one, but one is an artifact, it's not in the group structure, you just choose an element, and if you don't have it, you don't have it, and now our group is a multiplicative group, right? The space of measures on a finite set is naturally acted by multiplicative group. Probability is kind of numbers, they're kind of additive, but the symmetry is multiplicative group. I was mentioning before, this scaling, we look trivial matter, but it's fundamental in probability, it's always there, even if you're not aware of that, and so on the space of metric you have action of this group, but when it acts there, so you just have bunch of PI's, it's kind of trivial, but otherwise we don't say what they say, people write some formulas, and fish are awesome formulas which I'm unable even to read, yeah, I mean, because what I'm saying is worse, they're formulas, they have no kind of, anything computational about them, right, there are formulas like, written by a manugian, they're formulas, but there are things which just, should be said in words, if you write a formula, you just use the wrong language. So, you have these weights, P1, P2, 10 at a time, and you can multiply them by lambda which is positive, you put everywhere lambda, so they go out of probability space immediately, of course, right? So you have that, but when you have it, at each point, in particular, you have this action of the same group, so whichever left invariant metric you take on the multiplicative group, of course accidentally, upper stereo, you know, it's isomorphic to additive group, but this, it comes later on, it gives you this metric at every single point, and this has the same magnetic, so it takes the same magnetic at every point, and then some mate over all piece, it's metric understanding as quadratic form, so it gives you on a simplex, at every point, it gives a quadratic form, positive definite quadratic form, so that, and then there is a metric attached to it, and of course, it's very easy to write it down in additive coordinates that will be the metric, if you'll be, this is a kind of formalization coming from, from this, because it was multiplicative group, and we use left invariant metric on the multiplicative group, and this is a metric, it's remaining a metric, you see, from that you can make a distance, of course, so it's one definition, and it gives you this formula, so I'll give you a kind of three definition giving the same formula, and I don't know, I guess, all you know on to efficient, I don't know which one to use, because when you write formulas, the old guy was the same, you write this formula, they're all the same, but they have completely different meaning, and when you start generalizing them, you're on in different worlds, yeah. Second definition depends on what is called conditional entropy, so this was disgusting anyway, now lambda now is a measure, an immune measure, and this actually, condition is my look, of course it was okay, but of course this was how original entropy was introduced by Boltzmann, it was not introduced on a discrete set, it wasn't introduced like that, they have a background measure, and with this end probability measure, and then they define entropy, and a specific example was Euclidean space, or usually it was, even dimensional space, have a phase space, but anyway, Euclidean space, this is lambda, it's standard measure, it's even infinite, and then you have a function, F of X on Arrhenia, such that, positive function, so negative function, such that integral of F equals one, so when you take this measure mu equal F times lambda, this will be probability measure, and for this, Boltzmann defines entropy, so let me make some little modification, what I was saying, because before that, what you were having, our background space, so this is kind of background space, and then there is probability distribution, in our previous example, basic example, our background space was essentially just, it was a finite set, all atoms of unit weight, so we could have countable set, all atoms of unit weight, this was your lambda, and over this, we were pushing measure, which was probability measure, indefining count, but the fact that this is background space with discrete was never used, and in a way, it may be, so how you can make general definitions, have Euclidean space, or for any space, you just divide the small atoms here, in approximately equal weight, in put, renormalized measures where it becomes one everywhere, and then we have this function, it's say function, say it's continuous, otherwise, yeah, you have to partition slightly carefully, replaced by these values, so you immediately go into the previous situation, and go to the limit, one way, and on that hand, you can say, how I can use the same definition we had before, and this may be even more advantages, here, maybe as clear as it was before, imagine function was constant on its support, and this is an important case, so we have support of the function, it was equal to certain value, and this value, of course, f of x, where it's not zero, equal one divided by, this is support of fs, this, you know the lambda measure, so it's volume of this set, and the entropy of this will be, of course, log of this volume, right? A constant function, and now you can see this may be positive or negative, for atomic space, we always positive, this may become negative, s may be smaller than kind of unit, and then you do the same game as before, you, this is the only kind of thing you have to know, then you take this part of the n, and this measure, and take everything to the n's power, so we take here, n's Cartesian power, and here is tensorial power corresponding to this measure, and you say I have the law of large numbers, it becomes eventually almost constant on its support, so. And this, so we deduce the preview definition, so we take entropy, we take volume of this subset where it's approximately constant, take log and divide back by this number, by this part of the problem, and of course all properties just say, I will repeat the proofs, all you observe we can approximate this continuous measure by discrete, but there is a little bit tiny point, if you have arbitrary measures made, and there is a big kind of atom inside, and this atom you can't approximate by discrete, well so you have to, well, it's a material, of course. It's clear that once there is such a proof, it works in one context, it's completely universal, it can be made even more general, and I said but of course there is a law of large numbers, and of course in order to make this thing, work in general you have to say that this integral of f log f converges, probably have to put here, log f, yeah? Of course you integrate only on the support, outside of the support you postulate this being zero, and if it doesn't converge you have problems, and then unsurprisingly you have the Boyceville formula where it was written exactly for this case, and but then I'm using point is that if you apply it to two probability measures, so it has entropy based on lambda of a measure mu, and this also was probability, this entropy instead of being positive becomes negative, and so if people are high, change the sign, you won't see people like positivity, you see? And then this called Cool Bake Liabla Divergence, if I'm not mistaken. Some people introduced it, it became famous for that, because it changes sign, I don't understand this history, but it probably looked upon from different perspective, everything is of course in the concept of entropy isn't Boyceville, so Boyceville was done, of course much more general, his context is more general than of Shannon, and his mathematics was also more profound, because he was attached to deeper questions, but anyway, that's interesting for property, and so it's function is like that, and it's vanishes if and only if lambda equals to mu. If the two measures are equal, this is zero, and this is kind of, yes, completely obvious, of course, and once you have such a function in two variables, now it's function simply sent to variable, right, and this is much better than just one entropy function, two variables, so we have function p and q, and the function is positive and vanishes on the diagonal, then it's always defines the remaining metric on the space, which may become degenerate, because function may be identical, zero after all, but this function is sufficient, it's mostly smooth function, it's smooth function up to the diagonal, and this is quite simple, how we do that. So, because it vanishes where two are equal, if you look at one particular point, so it looks like that, so it's positive and vanishes in this point, because everywhere positive, first differential is zero, right, it's tension, and then the second differential, because it goes up, it's my positive semi-definite, so second differential of this function with respect to p at every point, p is the quadratic differential form on the simplex, and you see, we don't use any structure in the simplex, except for its smooth structure. We say it's second differential, it's not Hessian, in the second Hessian with the p, which depends on other structures, but this applies to any smooth manifold. This, by the way, applies to usually remaining manifold. If you forget, concept of remaining manifold only remember distance between two points, and you take a square xy, this is a smooth function at zero, and remaining metric defined, but that's a regional remaining metric, right, so from remaining metric to the distance, it can go both ways. Of course, distance, a priority, carries more information, distance function in two variables. Reminding metric is several functions, but in one variable, meaning one xy, and for that reason, in fact, it distances per set as function in two variables beyond grasp, yeah, it's two general objects. You cannot work with them, they provide you with a language, but you cannot prove anything about them. You cannot verify, given function in two variables, it's a twice triangle in quality. It's just inconceivable how you can check it, right? It's absolutely given this concrete function, like matrix, I guess, is what they call NP problem. It's kind of transcendentally complicated problem, but the but, if you can localize the remaining metric, that's the beauty of that. It reduces the subject very simple, there's finitely many functions, the manifold defines the metric, right? There's function one variable, and these can well do a difference. Function one variable, even many of them function two variables. Everything is function two variable. Any mathematical object can be described as function two variables, but not as one variable. Okay, but anyway, that's the point. So I think, and if you do that, and you make it to the one line computation, you know, I have it somewhere. It was here, and you see that you get the same, the same, the same metric. So this whole information divergence with these two guys, which you never forget. So this is just a repeat what I said. And then by, you know, what is this Boyce-Muffin formula? This Boyce-Muffin formula, the discrete case. This Boyce-Muffin case, he applies Boyce-Muffin formula and makes computation. And you have the same, well, we have this metric we had before. But now I'm going backwards more than all. And now we come to the original, the single kind of, which may look, now we define directly in terms of the entropy. This were more elaborate points, they're kind of more natural, but in a way more elaborate. But now we're coming back to the entropy as is written in this PI log PI form. And take the Hessian, now Hessian, unlike second differential, see it depends on the defined structure, not defined in general, so I have this function. So second derivative, you have a function like that. It's second derivative, so to speak, intrinsically defined only here, but not here. It depends on the parameter. However, because the simply defined space, so it still makes sense. So Hessian is a quadratic form, makes sense. You cannot say what a second derivative, but as a form is defined, because space is defined, you see? So Hessian is defined. And again, you have this, you have this, you have this quadratic form, and therefore you have every magnetic on the simplest, and they all have the same form, yeah? They all, as I wrote, because you have the Boyce-Mann formula for everybody, second to, they always will be like that, DPI squared by PI. So what is this metric now? So it's just, it's easy to write it down, because everything is diagonal, yeah? It's so easy to write because it's diagonal. And the remarkable thing is, if you don't know it's kind of unbelievable, that this metric has constant, constant positive curvature. And I see it as remarkable in many, in variety of respects. First, I probably try to compute this curvature. You'll die. People say curvature is good for improving something that is symmetric or not. Curvitch isn't computable, usually. It's horrible, yeah? It just, you say this word, but you never compute it. In a way, of course, it's a very special case. When it's given by formulas, it's completely hopeless. And actually, I will mention that today, maybe, in next time, we'll bring some examples. For some instances, when people computed and just die, they were trying to compute curvature, very simple metrics. You know, pages, you know. Take two blackboards to write, which are very simple metrics. Incredibly complicated. However, we can prove it. But the point is, the simplicity of this metric is just isometric to the spherical simplex of the sphere. This is pointing part of the sphere inside of this positive cone of the metric. And the map is extremely simple. You go from the point of the sphere. You go to the point S squared, right? Because S squared is one, so the sphere goes to the simplex. You just take squares of coordinates. And this is quite a remarkable map. It's certainly mostly remarkable in the complex domain. So, maybe I mentioned what is true about complex domain. It's just for complex numbers. You take complex number Z, and then you go to norm Z squared. So, you map complex numbers, you map positive real numbers. And this is a kind of a basic, basic moment map. And the map has a remarkable property, which is, it was discovered by Archimedes and written in different form, apparently on his tomb. It says this map is measure preserving, which is kind of an unbelievable, right? It's completely different space. It's kind of very complicated map but measure preserving in a sense, of course, that if you take any subset here and pull it back, you have an N use. And the measure of this N use equals the measure of this interval. Well, maybe there is a rescaling at the point of normalization, of course. But this is the kind of the property. And the same is true in high dimensions. There is a map from complex space to positive cone of measures, of measures, right? Which is measure preserving. And this is a quite, quite, you know, instance of a, so-called moment map, which has lots of, to do with algebraic geometry and which suggests that there is something, something profound algebra geometric, which is parallels measure theory and generalized measure theory, but I must say I could never work it out, but it's good indication that there's something like that. But from our point of view, what is relevant in this map? We use it on a real part of the space on the, so you have the map from Rn, Rn plus. Plus, and then the usual metric, dpi squared goes to this metric. I think it's a factor 104, because when differentiated, something, some four happens. So this map, square in coordinates, so this is, you have computations from way before, yeah. So this is a map, this is a moment map, and it's induced metric. So this is a metric associated to Hessian. The one which I wrote before. And this agrees with the metric induced by that. So this is a kind of tremendous, simple, but it's rather suggestive. It says that there is something in the entropy and all these things which are priori were not so symmetric, right? They were not so symmetric. There is this orthogonal symmetry. And so you wonder if you can define entropy without coordinates, right? So from point of view of the simplex, measure was a bunch of weights attached to these points. And this coordinate system can compute entropy. Now imagine I change my coordinate system. And so what remains, what can I see of these measures in order to compute the entropy, right? So we want to somehow extend these measures from points to something in the space so it will be in the language of Euclidean geometry. And so this is exactly what was done by von Neumann in his definition of quantum entropy. But he was working in the complex space, but it's completely material for us. Because they formally, the two theories are up to some point. Here I mentioned something more about this moment map. So this is a fish metric which I was leading backwards. Now we come to von Neumann's entropy. So the logic is as I already mentioned before. We replace symmetric group by orthogonal group, finite set by projective, by real projective space. And this symmetric object will be corresponding to the simplex. It will be, vertices of the simplex will be this Veronese variety, yeah? The projective space embedded into that. So for any linear space, we have to better analyze the map, take linear form, take it squared. It goes to the space sphere, goes to the projective space. And so we have this some linear space, Euclidean space, in fact, and image of this projective space, orthogonal group action. From where we start, yeah? We don't know anything. And we take the convex hull of that. And convex hull will be poised with definite quadratic forms. And so instead of measures, we have poised with definite quadratic forms on Euclidean space. And the background space, our Euclidean space, and which is understood as Euclidean space, not as an linear space, a little Bertian structure, which is, for some reason, forced by quantum mechanics. This, by the way, if you just a little digression. So you know, there are these three laws formulated by Newton, yeah? First, there is inertia, the first law. And secondly, that f equals ma, right? Mazma acceleration. So why to do this? Because this is a special case. Why you would isolate them? And mathematically, the point is that you can't forget about the second law. But then what you have, you will have a fine geometry of the space. The first law, kind of, you move along the line. And you move with constant speed. And this gives you a fine geometry of the space. And the second one gives you the metric. And actually, the metric comes because you have square velocity energies. This energy is positive in square velocity. And this is where Euclidean structure comes from. Physically, it's energy, right? Because it's quadratic in this bit. So it's unclear what was, of course, logic the Newton's had. But it was not stupid, this separation. And of course, you know, this is a big discussion of what it means, you know? Lots of silly people who say lots of nonsense about what it is. And actually, the only place where I found lucid discussion explaining the point of this law is by some writing of Bertrand Russell, who kind of was. And Stephen, you can find lots of kind of philosophers and not physicists and physicists. Sometimes I say, he's very stupid things about that. So what's the meaning of this law? Why is the law not a definition of force? Why you cannot say this is just a definition of force? So it tells you nothing. Granted, you know, concepts of acceleration, you know, what is mass, this definition of force. So it tells you nothing, not at all, not at all. And the problem, of course, is misuse of the word definition. It has nothing to do with physics. This is the wrong word. It's usually a nonsensical word. Whenever people say, I have a definition of something, except for technical purposes, nonsense. Yeah, OK. But anyway, this is why we have this Euclidean structure. OK, as a mathematician, you don't want extra structure. So you want to kind of try to live without it. So we have the Euclidean space with this background quadratic form. I don't know how we call it. P, put here number n, because the trace equals n. Standard form defining n. And then you have other forms. And these are forms being quadratic forms. I think about the measures, and they're called states in physics. So it's changed to nons. And probability means trace once. In a second, I justify it. Linguistically mean probability or density state. Again, language study is different in quantum mechanics. It's convenient, so you know every moment about what you talk. They formally the same as for abuse. So we switched from measures to quadratic form. And now the language is amazingly kind of adequate. And the reason, of course, is a Pythagorean theorem. This makes sense only because there is Pythagorean theorem. This is also certainly the number one theorem in mathematics. Namely, how do you think about this measure? You think about this measure as a sign, something like a sign. So a usual measure is we have a finite set. And we assign weight to this set. Measures is a function on subsets in a set with certain properties. And this is set of set, because we are in linear algebra. Instead of subsets, we have linear subspaces. So this quadratic form gives you a number to every linear subspace and how it works. First, if it's one dimensional space, we just take unit vector in there and take the value of this quadratic form at this point and this vector. And this, of course, agrees with the dimension. I just immediately, in a second, they say what happens when it happens, how we relate classical quantum mechanics, so to speak. Not mechanic, quantum probability. And then when you have general subspace, what you do, you choose orthogonal bases. Take this value of your function, these bases, and add them together. Like in the space, we take separate points. You take the value of mass at each point, add them together. And by Pythagorean theorem, this doesn't depend on which base to take. And that's quite remarkable. Otherwise, it would be nowhere. It would be just blah, blah, blah, blah, blah. But this is a remarkable thing which allows you to develop. Indicate is very profound. Something happening. And so it has a kind of natural properties of a mass and separate sets corresponding to orthogonal subspaces. Don't talk to each other, and then this thing is additive. And so this is what we call states. Usually, in physics, people prefer to speak about operators, self-adjoint operators. And I don't know. Mathematically, this is much smoother. With operators, you have to, all the time you have to express it. You have to go back to forms. But of course, I don't know what is actually mathematically the right language in this context. Because this background space is the same. The two things are isomorphic. But they are priori, they are different objects. Quadratic forms, of course, make sense even without, even without background mattering. But of course, entropy will not be defined. So this entropy is by nature's relative entropy. It's entropy of p modulated background state, which I called n, because n was dimension. And in fact, given any two poison-definite quadratic forms, you can define this. Well, this must be respect to q have trace 1. Because if it's positive definite, it's the same if they're all the same. So this makes sense. But somehow, this is not common. People have quite different relative entropy, which I believe is different from that. The right formulas and the formulas give you something else. Because here, things become, for many reasons, become more subtle. Now you want to define entropy of this thing. And so what we have is function of this sphere. Look at the union sphere. The rest, it is extended. You have weights attached to it. So it's quadratic. We should be using proofs, but not in our definitions. And we want to define entropy. And so what we do, we can see that if you take any frame, it gives you, say, n dimensional space. As well, you have n numbers. To take a frame of the normal vectors, n vectors, you have n numbers. So this is something we know what to do. It's the sum of them equals 1. I have a density state. So the sum of this vector doesn't depend which base I take. So it's always sum 1, so it's probability space. And I can take entropy. So if this was state p, I call it p. And here I put my base is sigma. This probability space, take it entropy. So it makes sense. But this, of course, depends on the frame, right? So given this kind of observational system, I have, of course, different answers. Now, how to get number out of function. So this depends on sigma. So this one can be used as the definition of entropy of my state, but this depends on the basis. Now you want to take it out. So we have function number. And this kind of logical thing, yeah? Actually, there are two things to do it, or three things to do it. You can take in, you take the soup, or you take the average. This is the basic operation. If you have kind of open formula in logic, yeah? And you want to close it up here, either at existential or universality, right? Functional, this is kind of play the same role. You want to go from function to numbers. Mathematicians like that, because you always can do that. But this usually meaning this, physically meaning this. I mean, infimum and supremum are not computable, right? And this, by the way, is I keep repeating it because mathematicians don't know that it was explained to me in Mandelbrot. He was explained why his dimension is not at all the same as Hausdorff dimension, you know? Fractal dimension in Mandelbrot and Hausdorff dimension, it seems he's a special case of Hausdorff dimension. However, it's not at all, Hausdorff is nonsensical, because it involves inf in soup, it's computable. And he requires limits. And therefore, his dimension is not always defined. But when it's defined, you can verify it experimentally, you can do it, you can prove theorems. Hausdorff dimension, that's language. It has no substance. And that's, so sometimes he uses infant soup, you're convenient, but you must be understood in just notation. They're not kind of real things, yeah? In physics or whatever. Average makes sense. Infant soup, you just, people love this theorem, the infant always exists, soup always exists, they exist because it costs nothing. They just junk, yeah? But as language, you accept them. So anytime now, when I take infant soup, I feel guilty. But fortunately then, epistheore, you know the things, limits, they exist. They can be, because limit when you know, you have a limit and you go, go, you converge. But if you just soup, you go, go, go, you know when you have to stop, yeah? Maybe never converge, conversion, then conversion will well be beyond kind of a reason. Okay. And then, so what you're taking this example. Of course, soup is not very clever because you always can find a frame where all values will be equal. This is a little theorem. For quadratic form, probably it's obvious, right? So we have, there is still for any continuous function of the sphere. Given continuous function of the sphere, always there are full system of the normal vectors where function takes equal values. And that will be maximal entropy to be log n. This is a Cacotani-Sanbadels theorem. Actually, I don't know if a quadratic form of the reason kind of instantaneous, it's not difficult. Actually, interesting thing about this theorem is it's not difficult, but it's proven by high school or maybe essentially high school geometry not by algebraic typology. You think you have to use a lot of fixed point theorems from group X and CMT. You cannot make this proof, but you can make proof but you have to continue it. It's kind of so strange. I don't quite understand the thing. I was shocked when I read the proof, it takes you two lines, but it's not the line that you expect. Algebraic and geometric, topological proof you expect, you know, the things symmetric under this group, group like that, computer homology and you get it. No, it doesn't seem to work, but you can do it by hand. And quadratic forms probably quite easy. Probably it's through algebra or any field or something. But anyway, it doesn't work anyway, so we don't care. So we take infimum, not supremum, but infimum. So what's good about infimum? Now, but and to be what you know, because you know, of course, this also follows from, I should stop by the chip, it's entropy, this p log p is convex functionality, we keep forgetting convex or concave. Entity is concave, so this one is convex. So this is concave. Is it concave? So concave, let me go there. And if you take infimum of concave functions, again concave. Concave trivial, not so bad. So you know entropy, so define is concave. And however interesting that this was proven in 1968 by Lanford and Robinson, it was a really big kind of step in statistical mechanics. It was really, at that time was quite a theorem because they used different definition. That whole point about definition of the use. Analytical definition, from where it's not so obvious, and the reason definition of von Neumann was, of course, analytic, was coming from quantum mechanics. You have quadratic form, or quadratic, sorry, quadratic form. Any quadratic form could be essentially uniquely diagonalized. It is some basis such that it become some PI, some PI XI squared or that. It become diagonal and some, after normal basis, right? If you have an ellipsoid, yeah, it has this principle axis. So it become diagonal. And then, of course, in this basis, you use this as a normal basis. You just use this as a normal basis. And you call this, you call this entropy. This is, of course, quite nice in a way because it corresponds to what? That you have this, on the, for this base, you have boys of an entropy and when it changes, it just don't change the way of the entropy. So it's kind of nice definition. On the other hand, it's not so general as ours because we apply it to any function and this applies only to quadratic forms. And it's clear that the two are equal. And the two are equal, but it follows more or less by a wide variational principle. That from any basis, you can go by any basis, kind of rotating it and making them kind of closer and closer. And then, in a second, yeah, I'll show, cause this little computation which I just can't reproduce so I wrote them beforehand. And then, there is a definition and the spirit of what we had before and this will be quite essential, so I'll show you this definition. Yeah, by the way, there is something I didn't say but this will be quite important. There is significant difference, yeah. So what is the relation, of course, from classical and quantum? So I have this way, it's Pi, you always can write this form. So it becomes a quadratic form on this Euclidean space on the index set. We have index set in miraging elements in your probability space. So this is a quadratic form. And then entropy in one sense becomes entropy in another sense. So that's clear. But the significant difference is that we here allow same, not necessarily positive, but semi-positive definite quadratic forms. And then, when we're speaking about probability spaces, we're preferring to have all atoms of strictly positive weight, completely ignoring atoms of zero weight, which was not quite right. Yeah, and but in combinatorial discussion, it would never cause any problem. But here it is essential. And the reason is, the day we had this homogeneous state and then homogeneous spaces were automorphism group was transitive. And for that, you didn't want to have zero atoms because you cannot go from zero to non-zero atoms. And here, this is impossible. So when you say homogeneous, you have to allow zeros. So zeros quadratic forms look like that. So you have some subspace and there you have definite form. But then you project everything orthogonally here and take pullback of this form. So this form vanishes on this kernel. So you have to remember this picture. And this makes amazingly enough, it makes all the world of difference in the proofs that you cannot throw away zeros. And then possibly it has some physical meaning that you cannot kind of isolate your system from the rest of a quantum belt. Maybe, I don't know. This is, of course, I say it, which I don't understand. But mathematically, you see, because the zeros have spaces of big kind of grassmines in many forms. It's lots of them, yeah. And the degenerate forms are everywhere. You cannot, cannot get rid of that. And this we call support of the form and this is the kernel space of the form. So that's our state, yeah. For us, the language state of forms actually. And so this first minimalistic definition which I described, right. So yes, you can see in the writing again. That's concave. So this is what concave, which means. In quadratic form, they have this, they make a nice convex space. Just convex hull of this verandese variety. Those verandese is there all the time, yeah. It's skeleton of quantum mechanics. Of course, in the complex case, to be kind of complex version of verandese, it will be something slightly different. And then come the special definition of entropy, yeah. I still, I was switching everything to quadratic form of emission. Because emission means symmetric, I apologize, yeah. We don't need complex geometry. The way I wrote originally, I used emission. But this completely immaterial. Because after all, every emission form is quadratic. We only concern this quadratic part of this. Yeah, and this is the story. But that here, the spectral definition has the following pleasant features, yeah. If you take tens, we take tens of product of space, if you take tens and multiply them, entropy will be additive. Which is not obvious with general definition, which is not surprising. If you have these general functions, tens of products make no sense. You see this. So it's unclear how to generalize entropy to small general functions with some correct replacement of tens of product, which I don't know how to do. But this, of course, is quite clear. Product, yeah, it's product of entropy, yeah. Some of the mistake, and not some, maybe, how are you, sorry, I think it's some. I'm sorry, probably it might be some of the product. I wrote it a while ago, and maybe it is me speaking here. Yeah, it's, of course, some, yeah. It's not product, it's some there. I'm sorry, because before I have to take logs. Ah, yeah, yeah, you have to. Of course, here I see the mistake. They come back and cannot see them when I look myself here. Because for product of space, entropy is additive. It's not multiplicative, because it takes logs. And now there is another definition, which in the spirit of how we defined, how we defined entropy for classical case, but I want to present it in a more kind of, not in this categorical language. It's rather here, there is no good category for Hilbert spaces. There is no morphisms with respect to which an entity behave in some way. There is something like morphine, but they're not quite. I really don't know so what should be done about that. I think there might be some refinement, some modification of the categorical language. In a way, they're very much like categories, partly, but then they deviate from this language. Therefore, I have to modify the definition. So, first we define this entropy depending on epsilon. So we just kind of approximate it very roughly by constant function, very roughly. And then we go to the high power and go to the limit. So this entropy is very easily controlled by a wide variation principle. It's temporary, forget about that, but I want to come to the definition. So it doesn't depend on epsilon. We take any kind of significant part of the space where function is away from extreme zero infinity, and then this measure will be all the same, and this log of this give you dimension, rather it will give you this number. So see, cardinality is replaced by dimensions. But of course, the logic of that is the one we had before. We take tensorial power of your space. Now it's direct products because products of set case want tensile products of point vector spaces. Your quadratic form in the diagonal basis become approximately constant there. So your limit object, it will be this quasi homogeneous state in the limit. So it will be constant in its support. In support in the subspace, such as this orthogonal complement the form's vanishings. And all properties of the entropy must follow from properties for these objects. It's convenient to have three definitions. They give you three very different simple ideas about what happens. And then we have to solve. Subditivity is rather instantaneous because it's equivalent to convexity. That's easy. But now, we have to prove this strong subditivity, which will be the formulated. But so I explain what a canonical reduction. And this is how subditivity observed. So quantum mechanics are going to have a system depending on many objects that there's a kind of joint system with the tensor product of them rather than direct product. But they interact, which means the state doesn't split into the tensor product. And then people speak about entanglement or something very horrible, which I don't understand. So they can vector in the tensor product or it's entangled because it doesn't have to be pure product. So it's not in the image of the segment. And then when you have this tensor product of two space and there is a state, you can project it to one of the factors. And in the language of operators, it's especially nice because this makes sense for operators even without any background euclidean structure. Because on one hand, operators and quadratic forms can go from one to another because they need euclidean space. When you have a tensor product, you have a purator depending on two variables. You can trace it with respect to one variable. And then you have a purator remaining for the second variable. Yeah, I say it and understand it. For me, it's kind of too complicated. Taking this trace and the fact that the trace for linear space of a purator is invariant and doesn't depend on the basis is something difficult. But what you can do in euclidean space, and this is fine much easier. We have this tensor product of two spaces. And this actually approves more general and more flexible. You have a tensor product of two spaces. And you want to go from this tensor from object on the product to object on the one factor. You see, there is no maps between tensor products themselves. There is no maps for the direct product. It's a tensor product. Then what you do is as follows. You have any kind of object there. You can see with object. You just take group acting on one of the factors, this full orthogonal group. And every region of this group. Now we have object which is symmetric with respect to one group of variables. And then it comes uniquely from the second one. Namely, it becomes tensor product of something on the second factor times invariant object on the first one. But if your object are quadratic forms, and the only invariant forms just send the euclidean form up to scaling. So this is canonical operation. So averaging had this effect. So you have this product. And this agrees, you have to, not today, I cannot explain it, agrees with what happens with classical things. And this is the basic formula. It shows strong subredditivity of the entropy. So you have a state in the tensor product. And you can reduce it. And notation are kind of rather explanatory. How we can reduce it. There are many ways we can reduce it. And read this inequality. So this is subredditivity. So you have many, many particle system. In quantum case, now the quantum. We have this entropy of this entropy of that. And this entropy of the two is smaller. And the sum of entropy is minus this difference. And this is kind of the best we may have. And it is not trivial theorem. It was conjectured in 1968 by Lanford and Robinson, and proven by Leib and Ruskai five years later. In last summer, there was a meeting here in Paris dedicated to 40 years of the proof of the theorem. It's one of the central theorem of quantum physics mechanics. And the proofs, no, it was conjectured, it was proven. And all proofs, which I know so far, depends on some analytic trick. And myself, I don't know how to do that. I think I have a proof which is nothing but just a statological proof. So I will explain to you, it's a kind of a tautology. If you say it in the right language. Because it's an inequality between certain operators. But if you formulate it in this way, and use language of non-standard analysis, it becomes tautology. But you have to use the trouble with this language, if you want to be sure it's correct. So you must admit that not one of these people who do this accept it what they say. I don't understand what they write. And they don't understand what I write here. They write formulas. And this is no formula. It seems to me, I think I'm not mistaken, it's just like we've proven usual Schoen inequalities. We have to write any formulas like logs. They follow from this kind of a funcateriology of the entropy and continuity. And he also, there is no funcateriology in the usual sense. But it follows from some Euclidean geometry and continuity which replaced funcateriology. But I will, of course, have to explain it. How many lectures I have? I have another two lectures or what? I will not have, I will not finish, of course, but I won't. Of course, it will take me one hour to explain that, and then there will be, of course, other subject. Now, this is all I posted. There are little mistakes I have to correct, like sounds and products being confused. Yeah, I definitely, and after that, it's not. So up to this point, it is more or less actually with my own mistakes, but then the two majors and the next subjects are not, yeah. So this I have to explain on the next lecture, next hour, but then, and then the, but then the, something different starts. So this will be next subject and this probably only will be able to study. They will never come to the end there. So this will be, so tomorrow I'll finish, probably with them to be next time and explain a little bit, maybe Mendelian genetics in this context, because some algebra there is very similar to what happens with quantum mechanics. And then the next lecture I'll say a little bit about co-homological measures, which is a, we will take, or you will not be able to, of course, explain them to ours, I think, yeah. We'll explain something, a different way to think about measures, linearizing them in a deeper way, in a co-homological setting. Okay.