 Дай мне помнить, что я описал в последнем разе, чем я уже представил алжебра, которая была, в основном, в том числе из-за мандалов. Так что, важная обзыва, в каком-то статусе, как и в биологическом, это довольно просто, но, как вы видите, там была какая-то ремка behind it, и это было, что, когда есть определенные фенотипы, фенотипы, которые всегда appear in a very sharp and discreet form. For example, you may have two kinds of flowers. I mean, there are flowers that may come in two different types, say red and white, and it's never mixed, there is no mixture there. There's either red or white, and there are other features like that. And this was in a sharp contradiction with all mentality of the types, so this paper of a mandal was written in that about six years after the Darwin's Origin of Species, and at that time Origin of Species was extremely influential, and the main point made by Darwin was that all changes happens continuously. And that exactly, which was violated in this example, this variation of a very sharp, and there is no in between. And another example, which was considered handed here prior to that, actually more than that, was by Meperti, who is reluctant to write, who you know from mathematical physics, and he was observing how was distribution of people with six fingers. Sometimes there are families when there are six fingers, some have five, some six six, and they never have five and a half. There is no, it's very discreet, and he was observing statistics of that. And the boss meant that Meperti was very much impressed by another fact. First, they are discreet, and secondly, statistics of that is a statistic of that kind of, when you start looking in different phenotypes, proportions are roughly integers. So, small integer numbers like this. Certain phenotypes may be distributed like one to three. And this is kind of, it's never, of course, sharp. So, if it will be 1.3 versus, you know, 2.7, Perfect. By all the point of view, it says 1.3, and still it's incredible. Even small deviation, probability of this being random, it's just zero. No matter how small this error, because it appears many many times, no matter how unlikely it is to happen randomly, to happen systematically, it's impossible. This was realized by Meperti, who developed kind of conjectural theory different from Mendelian, which was not quite correct. And then it was done by Mendel, completely missed by all biologists in between, including Darwin, who also observing sometimes these phenomenologists, it was contradicting their philosophy and so it couldn't make any sense of this. As you know, actually the first, I think the first publication of Darwin paper, this actually was quite also, I'm using in many respects, this is when his first original species appeared. First, the major premises there was completely false, and it was pointed out to him that his model of heredity would never give the observable phenomena. Exactly this continuity, whatever variations, advantages variations you have, will be dissipated. And that's it. And then Darwin competed, he missed it, but then he suggested what is now called returning to Lamarckism. It was not Lamarck who suggested that new features you acquired unique life may be inherited, but this now called Lamarckism incorrectly, and Darwin accepted that. So Darwin couldn't figure out what to do. And on the other hand, in the same first edition, he is not an inexplicable point, he analyzes, and this is extremely good example of how things fit concepts of natural selection, because usually examples which brought in books are completely irrelevant, they are completely wrong. They don't show anything, like growing long neck with giraffe or whatever, they just can be explained or rather interpreted billions of ways. But what Darwin observed, that there is one particular phenomenon which is not so obvious and kind of paradoxical, and this is preservation of the sex ratio. Very many species have ratio half to half, despite the fact that their mating habits such that, like particularly for the sea elephants, when usually for one male, there are about 10 females. And the question is, why we still have the same ratio? It will be kind of look absurd, because most of the male will not function, why the adverse, they have the same proportion. And Darwin gave an explanation quite reasonable. And then in the following edition, however, this explanation disappeared. And Darwin was writing that it was, that this was a mystery, как это могло быть. И потом, 70 лет назад, 19-20, Фиша, или, как говорится, статистичный, математический биологист, объяснил, действительно, если вы посмотрите, если вы правы, это очень просто, очень простая логика, что это должно быть так, потому что, если какой-то... альбейт, вы нуждите fewer males, но у каждого males 10-10 times more than female, и у каждого males 1,5-1,5 ratio. Но, если вы думаете о этом, я помню, в последний день, я говорил о Лайде Гусе без этого, и он immediately found the flaw в этой аргументе. Это, это официально, Фиша никогда не писал. Это обычно, он думал. И, evidently, это не очень очевидно, как это работает. И, evidently, это так, что он исчезает из его последней эдиции. Потому что, обычно люди, которые говорят, что это не эдиция, у них был какой-то ошибок. Но я не думаю так. Есть какой-то интересный, логичный, тревелогичный момент, и это, как это может быть? Как это sustains this ratio being close to 1,5. Но, но, в любом случае, это было, это было, мандель, и второй, это был, что это не так же, в каком-то причине, это, феномен, феноменология, не как-то располагал. Это, на Hankereum, был большой мисконсепция, людей, подavorим, הה踢к, и вот это было все эти explosion, тревелогия, ассенсирующаяся, но и располагал, какие-то фичи, 20 лет назад, он не был в стимуне, и так, так, яч и джин были композициями, в деплодируемом организме, в 2, в 2, в 2 аллеллзе, и эти были вредные. Эти были вредные, и в то, что мы обжавляем, так что они были 2 варианта, скажем, А и В, афинотайпс был функцией А и В, но это было вредным, а это А и В, но не Ф. И в таком случае, в результате, это было бы очевидно. Например, если мы миксировали 2 флоры, то, скажем так, вредные флоры могли бы быть вредные. У них 2 флоры, у них 2 флоры, у них 2 вредные, а у них 2 вредные, а у них 2 вредные. И в то, что функция, если у вас здесь вредная и вредная, а и В, и в цвета будет функция, и типичная функция здесь, то это будет всегда вредным, если у вас есть 4 комбинации, у них 1 вредная и вредная вредная. И так же, у нас есть родители вредные и вредные вредные, и их дизайнерство может быть вредным, вредным, вредным. Так что визуальная феноменология не будет вредным в простой мере. И это, конечно же, очень эффективная идея, очень profundная, и почему это не было предназначено за 30 лет, именно потому, что оно будет более на уровне интеллигенции или на уровне того времени. И, по-настоящему, мы перейдем к этому позже, и уже в физическом и физическом кемнистике уже визуальная идея была. Как это быть? Так что, простая прелиминальная обзывация это, если вы делаете эту идею, то есть, есть два компонента в каждой, в каждой, в каждой фенотипике, называемой, представленной бензинкой, это компонент 2, у них может быть больше, когда у вас есть деплодирование, но, скажем, 2 из них, это очень простая функция, и с этого можно сказать, ага, например, если у вас есть белые родители, все деплодирования будут белые. Если у вас есть белые и белые, то у вас есть все, но если у вас есть, long enough, для particulей, они становятся все белые и все белые, потому что, и тогда, когда вы смешиваете их, там есть какие-то пропорции, как они визуальны, то есть, белые и белые, две племена, а потом вы смешиваете их, делаете пять компонентов, но это будет три раза, конечно, рёвее, а вот, это три, у этого есть один, поэтому это 1-3 пропорции, это был в голове мандельная, Again, this, I was saying, this kind of paradoxical thing is that by many, at that time, even later, what Mandel was done was perceived with mathematics, on the other hand, natural selection seemed as biological phenomenon, but it's completely opposite, principle of natural selection has nothing to do with biology, it just cut off exponential function, absolutely pure mathematics, because so trivial is easily accepted. And here is non-trivial biological phenomenon, you discover conjecturally, you make some conjecture about structure of biology in mathematical language, and exactly because by order, more subtle, in most sophisticated people couldn't get it. And even now, yeah, people argue, you know, about Darwin, everybody understands it, agree or not, about Mandel nobody argues, because if you understand it, you don't argue, I mean just. It takes some effort to understand all the modifications. This, for me, actually a mystery, biology is not that excited by Mandel, they lie, they laugh, Darwin, I don't understand why. Darwin was a great philosopher, but not a great scientist. And Mandel, in the country, we don't know what Mandel thought, because unfortunately all his documents after his death were burned, for some reason. When he was in Abbott, somewhere in the new Abbott camp, he burned all his documents for some political reason. So Mandel thought, he was working on another subject, by the way, and we don't know what he was doing, which is a picture. But at the point, this is kind of so far very, very, if you just say possibilities, it's very qualitative and quantitative, you say everything happens as round as conceivably can. And so, and the conclusion from that which was made by Mandel, and this is a hard dividing principle, which I describe here in the second formula. Remind me again what it tells you. In biological terms, it looks completely kind of infeasible. And again, we have a field of red flowers in another white flowers, and they separated first by mountain range, and then you erase it. In the first generation, proportion of red and white would be different. You may have more red and less white. This now color, it's not their positions. But after the second generation, we have the same proportion. It stabilizes on the first stage, and very much against intuition of people who believe into selection. So I can say, it shifted toward red, because red were advantages. And then it must be keep moving there. It will not happen. And people here, by all this, were arguing about that, they don't understand what was happening. And then, let me show you a formula in a second. Yeah, so this is a formula. And this exactly corresponds to a certain square of symmetric to itself. It's important, and this is actually a formula. And this was incomprehensible, if you speak kind of in biological terms. It's a kind of trivial formula anyway. But on the other hand, if you say it's inverse, it's really difficult to absorb it. And this was hardly done. He just proved this formula. And as I said, he looked, however, so he was saying it somewhere differently. And I personally, what he says, hardly. So you just can read. And the point, of course, is not proving this formula, but he realized that this biological kind of discussion translates to a very simple arithmetic. But then there is the next level of this arithmetic, which he overlooked in this, which I now want to explain. And this kind of amusing, in a way, it's extremely instructive to think in this kind of biology, suggest what happens. And just generates mathematical objects quite simple, but still not quite obvious. And so let me remind you, so what it is. So we have linear spaces, they are distributions, we have this kind of function, which corresponds to just non-trivial linear function, when these are distribution of some entities, it's just summation of them. And you can normalize having one, if they are positive, it will probability distribution. But so all you have to know, this is just this vector space of reals and this is a linear function. So what in this kind of abstract term will be this random arithmetic. So this is just general notation. When you have this deployed case, you have A and B two spaces with this kind of functions. And you take the tensor products. And in this tensor product, we read the self-mapping. So you have a mixture. So the point of mental, when you have A, a level and B, you write it in this form. This kind of symbolic writing. A and B just abstract symbols. But whoever you think there is independent variables, and you write them together, and they become quadratic polynomials, and they behave like quadratic polynomials. If you interpret these numbers as probabilities. And this kind of kind of remarkable. Formal manipulation. But mental we are doing also statistical meaning, and you can check them by experiment statistically. On the other hand, you have the kind of formal manipulation. So what will be this self-mapping? So in naive terms, which I find quite transparent, however, mathematically unsuitable. If you think about this tensor product, a space of matrices, which is a tensor product of column by rows. So you take any entry. You take column and row. You submit elements in the column. You submit in the row and multiply them. So column coming from some entry. Some IG. Then you take a row coming from IG. Again, I hate this notation. You take summation of this column, take the summation of this row and multiply them. So sum, sum, multiply. So we have a space, a map of matrices into themselves. And the point is this map square is equal to itself. So it's the next generation map square. And this is what happened there. Another point, that the moment you make the description, you have pretty high symmetry in the picture, which was not before. At most, you have symmetry of permutation of your features, for the deployed organism, permutation of two elements. But immediately what enters here is a linear group preserving this core vector. But, in fact, in a second, you shall see, it's a bigger group, it's a full linear group, and they are operating there. So there is high symmetry, and the symmetry explains this and also some other things. So what come next? And this is the formula in terms, in again, this I described, in a kind of algebraic term, this is extremely kind of easy to see. But if you come back to this distribution picture, so you have the distribution of genes, but corresponding distribution of these alias will be A plus B. And so it's kind of bizarre map from polynomial point of view. You have a polynomial, you replace each product, each product you replace by sum. And then you square it. And then the separation will repeat it twice and become it important. And this is kind of a amusing thing, except for the scalar problem. Of course, it's not a polynomial map, it's a rational map, because you have to normalize, all the time you have to normalize, by a total sum. And this is rather amazing, when you have a polynomial, and you compose with itself, and it's equal to itself times constant. If you divide by this constant, it becomes square. And typically you expect degrees of polynomial grow when you compose them, and it doesn't happen here. And in the next level we shall see that this is exactly the same phenomenon, which brings in importantly groups, importantly groups, importantly array groups, exactly of this kind. When you have group of polynomial transformations, which transform certain space, and degree doesn't grow. And this exactly is what the importance comes in. Phenomenon is already all in this kind of mental description of genes. Okay, this is just a simple formula, a simple algebra, which is how things are being done. This is more or less almost as edited in Mendel's paper, almost his notation. And again, but these kind of formulas look a little bit kind of strange, but now if you turn back to this tensorial product picture, it's extremely simple, because if you want to show, because when you have tensor product of spaces, which is functional, there are projections to A and to B. So the fundamental thing about tensor products, you cannot project them to the effectors. And this is a kind of a what makes quantum mechanics so kind of tricky. There is no, you cannot see part of a system. Things are kind of entangled. If you mix two things and don't direct some, then you cannot separate them, you cannot project back. However, if you have this thing, you can do it easily, because for each monomial you just go to A times B. And this gives you linear projection to A and to B, and similarly linear projection to A. So if you have this extra, whenever you have this structure, you have this bracket functional. So in Hilbert space you have brackets between two, and this is the kind of mono operation. Then you can do it, and then you can multiply the two. And whenever you have this projection to A and projection to B, you can take it back, you can just multiply them back, take their tensor products. And this kind of obvious square will be when you square it, you just multiply by this. And if this normalize, you have for being probability space, so these are ones, so it becomes square, it becomes one. So it's extremely simple. And this is a hard divinely principle. In just a different moment, I was making a different estimate of this on Google, coming again to hard divinely. So what I did yesterday, gave me the following. Actually, even before I remember I was doing it, and this ratio was not 1 over 8, but 1 over 30. And I don't know why. I was writing this article 5 years ago. And of course this Google doesn't give you kind of a fair estimate. So hard divinely was extremely with respect to applied mathematics. And this is just playing golf with his friends for a matter of seconds. And this was the most significant contribution he made to science. In science nobody cares about anything he's done, except for this kind of computation. Which is in a way, it's not relevant to true genetics, but still it gives you kind of frame for thinking. And always being referred to. Because the problem is that in biological system really this never happens. We never have this pure mixing, this random mating, because there are extra factors. And one, of course, is selection which is the superposition of that with environment. Environment kills somebody. And not that it kills more advantages, less advantages, but somebody being selected. Just this happens. It's another, of course, misconception of conception of selection. Things are being selected, because it just so happened. In most examples advantages being lucky. And then, but still even then, for example, if you have flowers of different kind, we prefer particular flowers and always just choose seeds of flowers of certain color. Still, however, the map you have will be by linear map. So this map you have in these places will be still quadratic. And so you have quadratic map on a product of simplices and they have very tricky dynamics. They never stabilize on the first step, but they stabilize asymptotically. They often converge to equilibria points. And this equilibria usually vertices of the simplices. And these are kind of kind of significant mathematics behind it. And another factor, of course, is that the space geometry is involved. Because the mating is not that random. It depends if you are far away, you cannot mate. By the way, I am using a point that just for me is mysterious, the fact that if you have separated population, they cannot mix. And this seems no big deal. However, there is a biologist called Ernst Mayer who is very, very famous essentially because he is famous for saying that. It is a great kind of emphasis. It is related to the concept of species. Again, it is very confusing who first made this definition of species. And apparently it is not definition, but there is a phenomenon which was emphasized by Buffon quite a while ago, of course. And the phenomenon is that you may have descendants of kind of different species, like donkeys and horses can give you two types of animals. Mule, or I forgot how it's called, another one. But then it stops. They are not fertile. They have no great children. And that was kind of remarkable Buffon kind of realized that. And he suggested this is what divide different species. And then it was saying so very well. And then came Ernest Mayer and well, but even if they live in different kind of continents they also cannot have children. So we have to correct the definition. And he became very famous Ernest Mayer and just actually everything Ernest Mayer was saying on the same level. And Buffon just loved it. He's unclear to me what the point. Because this is again phenomenon emphasized by Buffon. Why it happens on the second generation? Not on the first. So what's so special about the number 2? Number 2 is has something to do with DNA having two strands which deployed. It's really on these two shows on molecular levels. Reflection of something happening on molecular level. Not just being a different continent. Okay, so this is the point. And so so what we go next. Now just again about this formulas which I wrote before just to see that there is a kind of algebra. Just look at this. So it happens for deployed organism how the thing would behave. You just can read I don't want to repeat it. There are some little formulas. But in a second they kind of rather tricky. These formulas are rather tricky what happens. So I want to show how you can see them in a very simple way along the lines I indicated which leads to a class of very interesting dynamical system. This was what I was mentioning that that space may interfere and on trivial point unlike what Maria was saying was done by this equation of these people that you can write differential equation. It is a very interesting differential equation in corporate space. And you have quite interesting differential equation and again mathematically it is quite attractive whether it has biological significance hard to say. My experience from what I know even highly developed in a kind of sophisticated mathematical means done in this kind of biology usually happens to be wrong. One of those some person called May I believe probably one you know him mathematical mathematical theory of for ecologists and he was very hailed he didn't receive Nobel Prize but was close to that but then everything happened to be wrong when data were collected more carefully it was all based on not sufficient data. So all these equations I'm using but actually what I'm doing I'm only speaking about the equations because they're amusing not because they really have biological weight. Another point which I want to bring in however this is that Mendelian law similar to this law of mass action in chemistry and this kind of again big mathematical world usually unvisited by mathematicians and I want to say a couple words about that. So when you have a chemical reaction and there are several ingredients and this molecule must come together probability of them coming together is product of densities with some weights of each of them, right? The more of them have to come together the less likely it is which means that you immediately arrive at class of polynomial equations differential equations polynomial coefficients. And however these equations and the way to look at them is absolutely not the way mathematicians speak about differential equations. Of course it should be noted I think it's correct somebody told to me I haven't checked myself that say two thoughts if not more of all articles on differential equations written by chemists not by mathematicians they have a huge amount of equations they want to understand and secondly their philosophy is nothing to do with classical what you know now there is stability of dynamical systems because one of the major issues in industrial chemistry that even when you have a simple process like burning of hydrogen the number of intermediate products it goes into hundreds and so one of the phenomenon so this what I said give you immediately differential equations however if you look actually deeper at this chemistry you see it's not like that there are lots and lots of intermediate products yet the equation is if the process is if they are not there remarkable phenomenon purely mathematical you have a process where there are lots and lots of ingredients but this of how being erased and only you have a shadow you have this naive equation this equation by the way was discovered by Gulbert Nibagy in Norway a few years before Mendel but that paper was completely kind of went unnoticed and then was discovered about 15 years later by somebody else and it was kind of interesting also this is a major equation of course in chemistry like in biology ideal chemistry but mathematically it seems a kind of interesting class of equations and there are two points they are very different from what mathematicians do so one, so it runs in a high dimensional space but this number it's not a number it's a high dimensional space it's an actual object because there are many different chemicals have different properties, have different input so and in many cases you can reduce at this dynamic to understanding combinatorics of this set representing dimension right so just one work which I know here and again it's done by applied mathematicians actually I know two both by applied mathematicians but in fact one of them probably pertains to what is now called tropical geometry that there is a kind of tropical limit of this equation because these parameters spread in a kind of large way and so there is a limit of many of these system differential equations they become commutorial and this corresponds very even this has been done in a model on the equation linear and again it's quite amusing that you look at the system of linear equations with constant coefficients and you believe you know everything about that just linear algebra however if you think about this linear algebra so they have this space of matrices and what happens to this equation depends on how the matrices degenerate and so this depends on the stratification of this equation and this is highly a kind of non-trivial phenomenon the way how it can happen and there are these limits but for realistic equations it still has not been done and another instance of that of this kind of differential here is the point which I want to partly make with these lectures that dynamical systems motivated say by biology chemistry infinitely far from what 45 years well developed theory of differential equations partly supported by physics but completely kind of missing the problems in chemistry just completely kind of oriented in different way so one of the point is the dimension is not a number never dimension is something understanding which is kind of fundamental we cannot change the kind of coordinates and not symmetric at all we cannot create different symmetries etc and another point is that people love just having how you have this kind of chaotic equation you have hyperbolic system etc which is however practically never happens in biological system the moment you are say hard genes network start working in this chaotic region you are dead the whole point is that amazingly despite the fact that hyperbolic system kind of typical the system chosen both by chemistry and by biology opposite they usually tend to have simple equilibrium point and this is a mathematical question under what conditions what classes of systems would behave they are very complicated very high dimensional system one of the point they are all very high dimensional and why they behave in such a simple way and this is extremely again controversial issue because for example the following point is completely kind of not settled say in cell the regulation function of the cell there are varieties of enzymes and the point is is it essential that enzyme have certain parameters very finely tuned or they are very robust and that is unclear people have very opposite using that so and it is one point secondly concept of stability all very simple examples does not fit stability stability of of how to understand the dynamical system classical dynamical modern dynamical system theory and again if you think in simple terms it is kind of obvious to you we are rather stable with respect to very many temperature you can eat various kind of food and you function but you have tiny little molecule or something poisonous and you are dead so you are stable in particular range so this stability is very kind of specified stability and there is no mathematics but again this is the version this is instance of this of mental when things are quite nice so this I am describing what I told you why this square is is it important yeah what I wrote before yes in tensorial terms just this formula which I wrote before you see it or so this formula is a generalization of this what I wrote before this formula of Hardy where it was yeah here is a formula which Hardy wrote essentially and just he said well this is multiplication type table mathematics however this formula if you write it properly it uses to what I wrote before and which yeah what is written here a times b equals c or something this implies that so there is no addition in this formula it's pure multiplicity formula in a way it's multiplication table but only multiplication and it's not about numbers but just associativity of the product and this I mean is unclear to me why Hardy I will look that because you don't have to write any formulas to see that it's not computational fact it's just coming from symmetry and then it becomes more interesting when you look for diploid organisms they are not very common but mathematically you can become kind of more amusing and this brings in the next level of mathematically picture so what have how you think about that so a diploid means that now you may have formally speaking many parents not two but two different parents and that you borrow from DNA from them which means that you have spaces you take the tensorial product and I runs from from 1 to i equals I'm sorry from i to 1 to g so your object you mix now not two like a b genus combination maybe a b c like g already used I cannot use letter g because g was the number of these terms so that is tensor products and I want to think about this tensor products as polynomial in these variables because tensor products of course you can think about them as just polynomials having degree 1 in each variable and I just embedded into the space of all polynomials because this immediately gives you much more transparent way to think about everything now what was this it was summation of entries now when you have polynomial in sum of all coefficients it corresponds to polynomial some vector 1111 however polynomial is all the same so it can easily take 0 and this too there is just free term of the polynomial so yes this means polynomial take it value at 0 and immediately all formulas disappear yes all these formulas are genetic immediately disappear because there is free term you don't have to make summation just because the space of polynomial is in variant of the translation of the base space so this is what you have now how we describe now all these maps and this become quite nice quite nice description in terms of polynomial so I want to describe some interesting I have a space of polynomials they have some variables divided into some g-groups corresponding to different alas and a priori polynomials have a degree 1 issue of them but we don't have to bother about this anymore we don't have to say it they just polynomial how we can map space of polynomial this kind of style so what these maps are now they are as follows so let me now describe some class of maps in general terms so I had this projection remember a times b I have project to a and to b provided of course I had this operation so what it mean in terms of polynomial so each polynomial in many variables I want to assign polynomial few variables but if these variables split I just project my space to this coordinate so I have very similar projection from polynomials on the whole space they come to polynomial here then again extend to the whole space so I have endomorphism in this space of polynomials just induce the projection to the coordinate space right and then what I doing so these are these corresponding to these projections but then I just can multiply them and then these are my maps so I have endomorphism in the ring of polynomials this endomorphism corresponds to projection to this subspace and then I multiply them the moment I multiply them they are not endomorphism anymore but they are multiplicative endomorphisms so and these maps in general may be rather complicated but we consider projections to mutually independent subspaces and multiply this and so this to them this principle apply square of such a map equals to the map itself times a scalar scalar product where the scalar product comes so let me explain so what will be the formula so you have this call these subspaces and I just use the notation which I have here and I have to repeat the formula yeah this was written what I was saying and now there is this formula so I have collection I have a collection of these subspaces a project when I compose them I hold intersection and multiply them it's very simple formula which in the case when you apply where essential is which I suppressed but it's had to be set even if we have an empty subspace operation makes sense it's just taking the free term constant term in the polynomial so empty sets empty coordinate sets or zero ones still give you free term of your polynomial like when you evaluate polynomial 0 all coordinate 0 all remains a free term so and we have this one given by Ki another by given by Ki prime J so what you do all the intersection and take their product but the constant coming exactly when they have empty intersections so there is this formula again it's simple algebra but it has again this quasi biological meaning that you compose when you multiply such a map with itself you come up with the same map but with a coefficient with this free constant term in this degree corresponding to how many intersections are empty and this is this most general hard dividing formula if you look actually from where I started because I was trying to read this in Wikipedia and I didn't bring this formula there huge formula describing that and it's so huge because you evaluate polynomial at the wrong point and then it has all these binomial coefficients just tremendous mess coming in but in this is just this what happens here ok now so what but these are not the real maps another interesting feature may be again about this kind of maps so the point is here that you start with some linear spaces and there are simple maps with linear projection on them and out of them you construct maps in spaces of polynomials and these maps have the kind of remarkable feature that the degree doesn't grow grow only on the constant term when you want to compose them so they behave like transformational in the morphism of importantly groups interesting feature of them and then there is the next level and they so in another point this map nonlinear map rather complicated map if you think about them and I was mentioning just to be respectful of these maps just consider the simplest instance of that when you go from linear space of linear form to the quadratic form just take square this very innocuous operation you take linear form the sum of chi xi and just square it and you have quadratic polynomials and and this kind of the kind of things you do here but just to be respectful of this map just think how it looks like amazingly on trivial geometry so if you apply it to the union sphere in Rn say Rn it goes to the union sphere in this space plus minus 1 over 2 and it's identified opposite point so we have a projective space embedded there and this is kind of simplest instance of veronazov so it is a projective space lying in the sphere in a kind of extremely symmetric and well balanced way and as I said before it give count the example probably to many conjectures you can imagine about sets unexpected property as I mentioned before for example give count the example immediately a burst of conjecture about partition of sets into something just immediately people who make conjecture never looked at this set this is one of the basic set another feature of that if they convex hollow it it gives you if you do it in a conical way now you take the whole map not only on the sphere image is something and the convex hollow of this is a cone of poise of definite forms so you have poise of definite forms an extremal point exactly image of this map and this is of course the most kind of significant cone in mathematics poise of definite forms so people say usual probability cone is just poise of coordinates but this has it's another cone having maximal symmetry so permutation group and this is fundamental for quantum mechanics so cone of poise of definite self-adjoint operators it's kind of geometry extremely just to think about that you see you have a convex set of dimension about n squared over 2 and extremal points make this perfectly symmetric thing so and all these kind of segments of variations by no means simple but the theory doesn't genetic goes to the next level now and this kind of amusing suggests what it suggests but here I wrote something so we have this kind of maps so again I repeat they are very simple maps you take a polynomial you restrict it to certain subspace and in a particular direction in other way you project you take linear normal projections to these linear subspaces and take induced transformation polynomials and then multiply several of them where this space is just joined I call them equilibrating map in the second part of my lecture when I come to the entropy I explain why and and then I wrote something actually just when I was preparing this lecture today I have some problems with my time but I just couldn't figure it out I wrote this obviously fine but now I must say I don't quite see it must be obvious but I couldn't figure it out when coming here because there were some problems where my time was travel was not so smooth so I just came to explain why it's obvious I wrote it obvious I remember it was quite obvious when I was writing it but this is an obvious situation it's obvious and not obvious this is a kind of linear algebra simple algebra but now when I look at the example it doesn't look to me quite right so I shall see that but then the point for the moment we live it as it is but so what is good about this map they commute they commute with a full linear group operating on each of those components so my linear space coordinates divided into the group we have linear group operations on each of them and the whole picture invariant under the action of all these groups which is bigger than the original group because the picture is extremely symmetric in particular you can scale polynomials you just can multiply each variable by constant and which brings kind of thing from afar to localize them which shows that these maps actually linearizable of course there is another kind of linearization which I had to mention before because these maps have all very simple dynamics but another reason for that was as follows these maps preserve degrees of polynomials so polynomials of degree in each variable less than something remain these properties for example have linear form you multiply them total degree grows but degree with respect to each variable and you can think about them as transformation the space of truncate polynomials which is a ring and in this ring there is an exponential map an exponential map almost on to so if any polynomial which starts with with positive free term constant term is admits a logarithm and this again shows and this is by the way present in many work we write and take a formula if you write this explicit formula for the exponential it will be extremely extremely complicated term but this again quite remarkable that this non-linear highly non-linear transformation however linearizable except of course by exponential map but another reason for linearizations which doesn't depend on that in more general is that these maps are invariant commute is very large group but this but essential part of the group is just scaling transformation so and because they invariant with respect to scaling any global phenomenon brings to one point and at this point everything determined by differential and then it can go back to the large scale so these maps on the large scale behave the same as in the small scale which means they conjugate to linear maps and then that is a consequence of that Yes, I want to say this kind of another essential theorem essential theorem which is again motivated by by genetic so this will be kind of the last step of the formal genetic then doesn't go anymore and then one can go from there so five was speaking about one locus so this genomic genomic genomic each gene may have many copies but they could respond everything I was saying concerning one locus as if the rest was not there however in reality you have gene genomic there are many parts of this and and there is a kind of what is called in combinations you have a mixture of these features here and there imagine you have two genes and again assume the thing happens as independently as it can be so independence as I said before it's again a very tricky assumptions so probability cannot exist without it yes all probability it depends on something independent or nearly independent or something otherwise you just nothing can be said and also and here if you think what this means it means exactly that there is high symmetry on the system and this is why you can accept such an assumption because symmetry applies to objects prior to any kind of probability for example in statistical mechanics this is especially kind of clear if you think about classical statistical mechanics you see that symmetry is much more relevant than probability so you have a system of 10 25 particles which is how many I have in this space so this more or less number of particles of air in this volume slightly less but then imagine how each of them particle may be only in two states so I have so many particles so each particular state in this system has probability of something like that so what is this physically of course makes no sense these numbers the more we can go I think minus 46 this conjectural plan scale even that of course unavailable but beyond that it just makes no sense however exactly where with which number we operate probability of 3 appeals to that however the point is you don't have to do that and you can accept this system of fully symmetric of particles all particles are the same you can assume permutation group X then you can say this is a number it makes the meaning but the point is the difference such a number this equality makes sense despite the fact that these two don't make sense you can say that they are equal probabilities even that all these as numbers they don't exist and that is the kind of kind of underappreciated so probability of 3 is very much representation of 3 and actually it is getting more apparent in these days when new branches come probability disappears and just representation of 3 and symmetry enter that and it is the same of course applies to many other systems but so this is a huge symmetry which is the reason this genetics so in the case of our genome this hidden symmetry when you have long genomes and you can have switching over so it's a big group which operate very kind of family group which is the same which is throwing a coin you have group Z divided by 2Z and extremely small probability of any event but this symmetry act there and assuming everything is independent you may ask what happens with evolution so to speak there is nothing changes there is reorganization of genetic material so the point that in all these pictures there is no change in genetic material or individual gene change but content remain the same so there are kind of basic units which being inherited and there relative proportion does not change so nothing happens what you see visible change phenotypes run along certain process but the fundamental genetic material in population does not change and this exactly was the principle of Mendel disagreement with Darwin in fact it does because there are mutations but this is secondary effect all this evolution happens on a much slower scale in essentially invisible in nature you see it very very very poorly but what you see is kind of this mixture of rearrangement of aldeals in populations which is for undermating it constant just nothing happens and however there are many of them it does not stabilize on the first step however it is exponentially converges to equilibrium and this is this theorem of this guy which I mentioned and this follows from the fact yeah Robin is getting converges property it says that if you have this now any kind of population and they exchange they have a random random recombination the exponentially fast conversion equilibrium situation where equilibrium means that your polynomial is product of polynomials of respect to each variable so we have remember this variable divided into these blocks so the distribution of polynomial will be this product of this polynomial and this is where we come from and the formula for that the maps which are involved are just which one described they are not this original multiplicative endomorphisms but they are convex combinations but however because each of them invariant at this big linear group they behave as if as linear maps so essentially this convex combination of them behave like linear maps so there is one attractive fixed point and then immediately you may ask so the formalism coming to what you arrive is the following class of of dynamics you have a commutative algebra in this case cancaded polynomials or it may be an infinite dimensional algebra and you have some endomorphism of this algebra so I know so alpha i and this even commutative algebra this endomorphism correspond to self mapping of the space of maximal ideals think about them as function in certain space or some quotient of some space of functions and so there are simple transformations and then there are simple otherwise nothing works then you can see the products of those for certain and then you take convex combination of this right you multiply some of them and then take convex combination and from that you produce often kind of simple comprehensible but still on trivial dynamics kind of outshot from mathematical point of view at what kind of kind of dynamics you want to understand right so you start from this kind of random reading and come to this class of the dynamical systems and you want to understand them and see what are the other examples so one of the examples which are the one I described and the second example is even more more classical and let me see if I have it here written which everybody knows yeah it's here skipping is just different variation of the theorem right so look at the algebra of L1 function the euclidean space with respect to convolution and consider this kind of transformation and and then the kind of fundamental theorem says that it has a unique attractive fixed point which is Gaussian distribution so the normal law is exactly of the same nature as the one which is this mental formula of stabilization of population you have this algebra, you have the morphism exactly of the same nature and then immediately become clear that there might be many many of such normal kind of laws corresponding to different and the morphism of this algebra but yeah I don't careful in the literature in the language of course if there is probabilistic in literature certainly I couldn't extract so on this I want to finish what I was saying about Mendelian dynamics okay I say again a few words about Sturzhevan and then I switch to entropy and return to Sturzhevan in the end of my lectures Sturzhevan so because these more or less kind of mathematics we know in there are many suggestions which are not has not been perceived but they still kind of within traditional mathematics but then there was the next very simple step again biologically relatively simple step simple by description not simple by how it was achieved quite ingenious which brings very different kind of mathematics which has not been at all touched by mathematicians and I was mentioning this last time I want to repeat it again and it was done by Sturzhevan who work in the lab of Morgan and that the same kind of logic similar kind of Mendelian type of logic allows you to reconstruct the geometry of the genome namely that before any kind of understanding of molecular knowledge of genes just knowing that there are these units behind pure phenotypes so if you breed any of some particular feature it may appear in certain forms and then you know it is result of the composition of usually several units usually of two we deploy the organism into all else so there are these hidden units of inheritance which are not phenotypes you see that the whole point but knowing having this idea in mind we can say the genes which are a combination of these are organized dramatically on the line so here we have this kind of abstract of polynomial predata where this line may come by this one dimensional geometry and how we can see how huge one was extremely simple and so he reconstruct I think for him it was I priori clear that it must be linear but what he actually done he determined position of about a dozen gene on one of the chromosome who was positioned where and so what input he had so the input was this collection of these drosophila flies in the lab of Morgan they have bread fly with very pure properties and then we are interpreting them and looking at statistics of appearance of different features and because they were kind of purely bread they could say how different gene we are recombining and appearing there on the basis of this kind of statistical data one could say that these features corresponding representing genes were actually linearly organized there was a linear geometry there and it looks first completely incredible but then it's very similar in my view to what Poincare was suggesting our brain reconstructs spatial symmetry out of images so this is a problem that we have your eyes, there is a retina your move your head image moves they have nothing in common for your brain they come to different neurons to different places and the brain in different spots in what sense, how you know they have the same image and that's so Poincare was considering this and he suggested some solution again idea was solution of course which is I think very much in the spirit of what we know today of neuroscience what neuroscience we doing of course neuroscientists don't know of Poincare but in theory we don't know about Poincare either but mathematically it is very similar phenomenon and what enters here about theory it is much easier the idea is as follows so you can recognize genes by features genes are something representing certain phenotypes and then you observe the phenotypes often go along for example you may have you know black black hair and long hands more often than not of course for its different features go for flies flies, their basic features their color, their eyes color their body shape, the body shape of the wings they develop the whole language how to describe precisely the features they observe that some of the features we are going along more often than others and then you say aha this information occurs when you cut these two brands and switch this to this and those which are close together being switched relatively rarely and those which are far away may be switched completely independently actually one of the laws of mental of independence of this corresponds to the features position in different chromosomes and when they are different chromosomes they kind of independent the furthest usually positively correlated they are close and you can say from this matrix you can reconstruct from statistics knowing correlation of appearance between features corresponding to different genes you can say this correlation interprets this as a distance take proper function as it becomes distance and then you say aha it happens to be one dimension of course it was not like that or the relation not the metric but that's the logic of that and in the same way again I say simplifying you can speak about what's done so we repeat again this mathematical question which is not very well posed but so how you construct geometry you have a set and it has geometric structure so here is a set of genes right? or it may be screen with some pixel and geometric structure the one superimposed by images so the screen shows you three space and three space or two space dimension has some symmetry how this reflected here and the point is that you have a measure on the set of subsets in there for example transparent in the case of this of this screen so you look picture after picture after picture and you say aha on the set and something is white and black so you have a measure those which say black so you have a measure on the set of the subsets or in partition into sets of course you cannot truly observe this measure it's huge space but you have samples so you have what you believe representative samples and from that you want to say aha now how we can do that say here or there it is a let's do the same here done done for images so again the way you imagine it you have your pixel and then they enumerate in some adiotic way you don't know how they enumerate it how your brain enumerates if it does it all of course it doesn't do it your cell receptor cell in your retina of course it doesn't do it so there isn't just this set which has no structure and sometimes you see kind of images some of them being black and some being white and this kind of systematically repeats can you say aha that this came from a world with orthogonal symmetry or maybe it was the other symmetry so you have an abstract set which has no structure but all you have you have many instances of subsets there million or trillion copies of the same set it's the same set it's very important to identify elements but you have light or color on different subsets how you can say that this came from our world but not from university it's some kind of very foreign for us kind of geometry and here of course it's the same you have particular manifestation of this you have organism so it's only particularly features being materialized and you see what is being materialized and you want to say what was geometry of this background space and the logic again is extremely simple so here the two genes which are close on the genome have tendency to appear often together because the combination happens somewhere with certain frequency and and being cut of course the closer they are the less small probability that something will separate and the same is true about real images this from where all structures start that if you have and this is of course you have to use in the real world if you have an image then for example if you put here probability much higher than on random your point will be also round rather than red this kind of enormous effect looks not very little because it appears many times it is exponentially dominant it seems very weak because we pass the boundary and color changes we have a very small sport very small domain however because it happens systematically no matter how much you gain in probability it may be different between instead of 0, 1,5 it may be 0 4 9 9 or something like that but this is enormous difference if it repeats many times because it repeats many many times it goes into exponential and become but we have this number or we have this number which makes very little difference they both become infinity or 0 when you iterate them right? and because of that granted that you can say 2 points the distance between them is determined by mutual positive correlation between these 2 points if they often come in the same color or not you have to take any function it makes a difference and if you do it systematically you observe that this function into variables will have symmetry of the orthogonal group it may be not a distance it may be a function of distance but they all have the same symmetry and so but the problem which is this point kind of obvious to him but the issue was if you can make realistic algorithm so your brain would follow this pattern and so I'll discuss a little bit later so what can be what can be expected in this way and this of course we don't know of course still we don't know and we only can guess but unless you understand this mathematically I don't think you may have any progress neurophysiologically quite elaborate process which you cannot see in the microscope ok now I want to make a little switch and come to another point of what I was saying and different kind of mathematics so maybe I summarize where Mendelian dynamics brings up by us that some point remained a little bit unclear and I want to elaborate on this point so one was the following class of dynamical system we have commutative topological algebra you have some family of endomorphisms so they were corresponding in the case of prantankate polynomials you have a linear space decomposing to the product of subspaces and you have projections on this coordinates and this give you an endomorphism of this algebra then you can see the product of some of them there are many, they take and subsets in different products now this will only multiply endomorphisms and then you can see convex combination so here is i and j and here is over this j and so the kind of dynamics is here in population in this ideal genetics so it's very simple I repeat no selection, no geography pure probability theory you come to this class of dynamics and the basic theorem which I kind of explain very roughly says that you converge the equilibrium exponentially fast and now I want to say what is equilibrium why we say equilibrium so far it was there is dynamics and this is quite interesting class of dynamical systems which kind of comes in many occasions and it's it's not quite clear so the point again would be not just look at this in a general case but find condition of this endomorphism especially when this algebra is an algebra of functions so they come from endomorphism from homomorphism or continuous self mapping of the space of maximal yields when such maps behave in a nice way would be fixed point either attractive or at least hyperbolic except many directions where it may be repelling these are dynamics which are understandable and feasible simply as possible so again it's opposite to the view of usual dynamics you can see the simple space and look for the most complicated dynamical systems and here we look at the complicated space simplest system there and this corresponds more or less more or less to biology and to chemistry huge complicated space but dynamic there are rather simple rather robust essentially like fixed point dynamics how this can be and this is a model for that so this is one thing and another thing about equilibrium and related to this concepts of entropy so one can show that yes just again look at the simplest example so if you look at this map describing in the way of matrices you have explaining of course this matrix nobody knows what matrix is it's amazing if you say there is a matrix but what is a matrix what is a matrix what is a mathematical if you need to know a matrix you have always abstract mathematics said third and then they say it's square table written on the blackboard it's not so obvious there is a matrix because in different context maybe something different but anyway this is consider this especially the case I want to emphasize when n is a positive number and the sum total sum is 1 and then there are special matrices representing this veraneza variety which are products of rows by columns so what are special about them and if you know this probability theory you say they are matrices of maximal entropy they can be defined so on one hand they are remarkable veraneza symmetric some variety after projectivizing picture in this projective space of matrices on the other hand they solve this variational problem and now I want again to repeat what I was saying a little bit more about entropy so this process of random mating it's a very physical process entropy goes up and converges to its maximal value in what sense is maximal it's maximal if you can see the measures here which has given projections it's a measure meaning you have a square table and you put numbers in these places they are positive sum equal 1, let's measure on this maybe I do it like that so you put positive weights in all the squares such that this sums are given and this sums are given to you so the two projections of this measure are given to you which of them has maximal entropy and remind you what incorrectly called definition of the entropy when you have this weights piece of i sum minus sum of pi it's not a definition I want to say this what you find of course in textbooks and of course you cannot take it as a definition why log why not you know x why not cosine it's kind of absurd as a definition however temporary and then because the tradition was again I repeat it was written as if it was written by Boltzmann which was interestingly enough not it was written first by Planck this formula and then it was reiterated many times specially by Shannon in the discrete context and it's certainly fundamental concept but my understanding in Boltzmann though it was not implicit it was irreproven entropy was defined differently and I want to repeat definition of Boltzmann in its modern terms and then this will be out come however the fundamental property of this Shannon what is called Shannon inequality that if you can see that all measures with given projections then entropy of this will be smaller than the sum of these two and the quality is even only if this is product ok so it's kind of this rank one matrix and this is not difficult inequality in a way but still kind of significant and why this PI and log PI and now let me explain what I think is better terms you can understand this without formulas up to certain point but then this formula becomes really significant on very kind of rather advanced level and so this is so we want to what I want is to kind of reconstruct kind of naive thinking of kind of thesis not physical lecture Boltzmann was not in a way a thesis and he was active thesis did not take for a thesis he was considered applied mathematician he was extremely mathematically minded and he made fundamental contribution to physics but still he was essentially mathematician and so what he was doing he was not discovering new physical laws he was really discovering how to put them into a new mathematical framework and now just in modern language I think what he was thinking just I read his look on gas theory a long time ago but I certainly don't remember that the feeling you can get is as follows we have these objects finite probability spaces and they extremely have a simple mind object is just a bunch of stones and you normalize their weights total sum will be one you take the units of measuring and total weights is one and this bunch of these stones so this will ready for me a beautiful notation because you use here numbers and there is no numbers in there they are not ordered by the means just these stones and then you can bring some stones together better think about them not as stones as drops of water for example you can bring this together it will be slightly one bigger one and this together will be this one and this means that there is probability spaces and there are morphism between them so it's a category and in a second I explain how physically it's actually you have it you really have category these objects themselves these spaces you just don't see them but you see these morphisms and this is a the point I want to make I was saying already I keep repeating it you can say how when you have this error it's as if you have just this inequality this space bigger than another because all this map has rejection but the fundamental difference about entropy we have entropy of morphisms you cannot write entropy of this sign they are logically different objects having morphism or putting this sign and this looks very trivial kind of different but this completely changed perspective so I want to think this is an error the moment I have it certainly you might be careful this is a category which is topological category because there is topology inside these are real numbers they are not abstract of course this makes sense when they are objects from any additive group however, even semi-group but for me the real number so I remember topology secretly but then they are saying if I have a category and out of this I want to produce something simple and this simple after that we take the growth in the group of this category without thinking say growth in the group this topological category so growth in the group a priority it will be not a group group is a next level take semi-group which means just when you have morphism f g equal h you say that f plus g equal h in the growth in the group make everything community and then you have because topological category gets in topological semi-group if you don't take topological you have something huge and countable it will be too bad but you take it in topology there is only one topology you can do it and the moment you say we may just think what the semi-group will be and this will be the semi-group of numbers greater than 1 but under multiplication you can compute it and this is a quality and this is called the law of large numbers and then log so for each morphism in particular to each object she is morph to one point you sign this element of growth in the group and this happens to be a number and for some reason you take log of this and this again it's very not obvious why we take log naturally it's multiplicative group but you take log and then you have entropy and you cannot argue with this this growth in the group is not subject to you it's just inside of the structure now let's explain why it is so and I think it's quite greedy just of course there is no new technically or anything new what I'm saying but conceptually I think it gives you feeling much better about entropy when you see this and I explain why and then it follows it's more or less given by this formula you can compute it easily once you know that but it is the law of large numbers behind it otherwise it would make no sense this formula makes sense only because of the law of large numbers and then lots of properties of entropy which we prove by some computation it's followed from functionality because nature, functorial, teratitis in particular the channel inequality which I said namely that if you have measure and has two projections then entropy of the whole thing less or equal than the sum of the two and it is exactly physical which I explained physical reasoning becomes rigorous proof because functorial so everything which was blamed on Boyceman that he was not rigorous and I think the point is that he had no language to say what he was saying and there were two in two points one was from functorial language he had in mind and I think it applies to my young thing he is done and still not transformed to the modern language like Boyceman equation so for example this is a Boyceman entropy and you can say it is from growth in the group and then you are happy of course you have to prove law of large numbers but you know beforehand what it is because if you think about this you know about this equation you don't have to write it but you cannot say it in words and you write it formally because you have no mathematical language to say exactly what it is it also is a functor between certain category but you don't know what a category is and this how Boyceman thinking and objection of mathematicians particularly were because he was saying it in the language mathematical of 19th century and another thing he was using all the time he had analysis always he was thinking those terms again and again this language was not ready as we can say it now because in a way entropy must be understood at growth in the group of non-standard completion of this category the better way to think about that is when you use it is unneeded in the beginning but when you go deeper you see it becomes absolutely essential because you have to take so-called non-standard completion of this non-standard model of this category you have to look at spaces when all this now becomes infinite because it only appears in the limit and this is not surprising what you have seen in physics for example there are rather big numbers ok now let me explain why it is so why this growth in the group enters here and so what is the law of large numbers first in this context this is a essential point actually I don't know certainly what I am saying is in slightly different terms well known but I couldn't find references for some of them I found words people use in probability theory but not for everything so what is the law of large numbers and again we see just mathematics which add to very similar to what was happening in this Mendelian dynamic slightly different aspect of that and so you have a probability space p and this is a bunch of atoms in different ways and in this category and this is essential of probability spaces you have Cartesian product which is a very simple thing we just multiply them as sets and when you have atom here and atom here the weight will be product of two atoms better say you have segment here segment here you have the square yeah it will be unquestionable product you must be careful actually in what sense it is it is a product but it is Cartesian product so what the law of large numbers says when it applies it applies to this high Cartesian power when n goes to infinity and when I say non-standard analysis in fact much of what you do you don't have to be this kind of power it may be just space but this is not the limit of the sequence but just finite probability space but n is an infinite large number it's a number so it's a finite many atom but finite understood in a non-standard way and there is now good justification to think in those terms there is some mathematical theory highly non-trivial due to Louis Boyne exploiting this idea you prove really hard theorems with this way of thinking but the law of large numbers concerns this so what is the space behaves when n goes to infinity now among all probability spaces there are some things you can call homogenous spaces and homogenous is of course where all atoms have equal weight this again is kind of categorical notion it makes sense pretty categorically you say it in Dutch but homogenous are objects are higher homogenous and the law of large numbers says those things when n goes to infinity or may we say 2 to p to n is asymptotically homogenous now why this would settle this matter about growth in the group and everything so entropy by function reality definition of growth in the group by function reality must be multiplicative on the products if we take log it becomes additive but it's if you write additive with the group it will be additive so you have entropy whatever it is of p t times q must be for entropy of p times or cross depending again on the annotations of entropy of q that's the kind of consequence of op speak about entropy of objects rather than morphisms in this particular instance it's kind of it is sufficient but of course when you go next level it becomes not the same and on the other hand so if time is homogenous the only category look at the category of sets growth in the group whatever everything you have is just a cardinality of a set right so your entropy must be kind of generalization of cardinality so when homogenous object this equation of normalization as I said in the growth in the group what I said there is some ambiguity about what is one what is one multiplied by a constant by a different base of logarithm so normalization would be when all atoms have equal weight this entropy to be equal to cardinality if it's cardinality it will be multiplicative if it's log of cardinality it will be additive so this is normalization you have it must be understood that there are lots of functions in this property a priori there are enormous numbers for example if you take sum of weights squares this quantity is multiplicative under these Cartesian products right so there are lots and lots of them if you think in terms of Laplace transform it is a whole kind of world we can describe all these multiplicative variants will be essentially the space of function all Laplace transform of all these weights yeah essentially of course they apply to the power lambda in their combinations so huge number of them and which corresponds to Laplace transform or the corresponding distribution functions but the point is that the user entropy is kind of has extra property so these are multiplicative for all weights but the entropy which we construct will be not it will be kind of marginally multiplicative only under the condition the sum of pi equals 1 so in particular this you can check this is simple computation but it's not a kind of a priori obviously you have to check it but this thing is this entropy satisfies this property but only using this condition if you take something like that you don't need any condition you put here any lambda and this thing is multiplicative of course this norm is multiplicative but but the entropy will be additive only under this condition but another feature of that that it is maximally continuous certainly when you say this kind of topological category which topology you use and you have to use exactly topology which guaranteed by the law of large numbers as I said the thing is asymptotically homogeneous so when you take very high power almost all atoms will have almost the same weights and this is the law of large numbers because we applied to log so if you applied formally additive of course you can use multiplicative group but if you use it as log and you formally take log p think about this function this probability space and this function become almost constant additively which means that all atoms become approximately equal what it mean approximately in what sense in a rather weak sense but exactly in this weak topology we should use for entropy so let me explain what is well I have just two words and then we can see next time so I have to compare I want to compare two different probability spaces and say what means they are close and then I can say what it mean to be close close to a homogeneous space so I need to introduce some kind of metric on the space of probability spaces and here immediately when I do that I have to work with non-standard numbers so n will be very very large number it will be not property of individual space it only makes sense but number are huge and only that will be used so we don't care about space being close as they are they must be close when n goes to infinity better to think about this kind of non-standard number non-standard being huge non-specified number and this is the following thing there are two notions of being close and they being kind of brought up together one is additive and one is multiplicative and this is again quite significant because in probability theory there is both aditivity of measure multiplicativity coming from kind of independence you have to use both of them it is very simple the most naive you can do will be there aditivity is very simple if you have one measure space and throw away subset of small measure you agree it is close so if n is infinity to large you throw away infinity to small piece of measure and they become equal so this is why non-standard is very convenient you don't have to specify this number in the limited means when n goes to infinity this n goes to infinity this you throw away small in smaller part and the rest what remains is close so they converge and secondly, what do you mean multiplicatively? multiplicatively is 3k so here you see the number n does not enter just something small this makes sense regardless of number when we take multiplications we have two spaces here and there you look at their ratio you want this to the power 1 over n to be close to 1 so another operation which makes spaces close you multiply them by weights but such that when we take this root this factor will be close to 1 so we mix this two notion of closeness and then you can say well that two spaces depending on parameter n another parameter n so their distance goes to zero now it makes sense when they one obtain by another by two of these operations finally many if you wish but two will do you allow throwing edging sets of small measure and you multiply them by weights such that square root of this not square root of this root of this will be small to 1 and the law of large number says in this sense for this power there is a sequence qn or called hn of homogeneous spaces such that their distance converges to zero so for sufficiently large n every space approximated by homogeneous spaces and then the moment you say it you kind of know anything about entropy because entropy is multiplicative by definition and now we know it also must be continuous with respect to this metric so when you take p to the n because it become asymptotically homogeneous this entropy will be equal to the entropy of these but because you took nth power entropy multiplied by n so I have to divide to take entropy of this screeching divide by n in this kind of growth in the group by law of large numbers but not now but because it became constant it becomes cardinality so everything you have to know about entropy you just you can read it from cardinality you can forget about measures it is the same as cardinality it exactly mentality of Boltzmann so what is entropy of a system is the number of states entropy is log of the number of states so there is no ways ways disappear when system is large enough because always become equal and and so everything reduced to this kind of homogeneous case and anything you want to prove about entropy follows from what you know about sets so next time I explain how you can elaborate on that in the same factorial way you continue because this is entropy about is finite measure spaces and kind of the big big advance of entropy was in dynamical system I think in 1958 when Kolmogorov proven that entropy serves as this kind of entropy serves as invariant of dynamical system he introduces dynamical entropy and then it was kind of polished by C.C. Knight and this entropic entropic theory and then we shall see that if you take this point of view it becomes kind of apparent you don't have to even definition proves because you don't know what automatic it's always definition because it's just extension just purely functorial categorical extension what I said you extend this language and complete it in a categorical way and you arrive at all the theory and it's one and then more recently about a couple of years ago there was another line exactly following just Boltzmann kind of reasoning just take Boltzmann reasoning what he was saying translate to this language and you get all this with all definition and theorems just from words I mean for that just reinterpret what Boltzmann says in a categorical way and what he says is just hand waving there is no kind of really hard common thematic in that but by much more subtle point was done recently when you pursue this another line of thought corresponding to this non-standard analysis and this is more subtle and this was elaborated by Louis Boyne and you have a next extension which is much more sophisticated when it's a more difficult question and brings us to what we don't know and so next time I'll start explaining how you can arrive at all that by kind of thinking in kind of physical terms it will be as removed from physics as what I was saying but you start with what you think it's physical, as a mathematician just translate when a very naive thing into mathematical language partly I've done it and then I can do it in a more elaborate way next time ok, so today let's finish