 Okay, so let me remind you, so this is the basic inequality which I want to explain, but I want to remind you the basics about von Neumann entropy, which is one of the parts of the formalization, but von Neumann suggested for quantum mechanics in context with Hilbrich spaces, and so which we turn somewhat, it presents me differently because we emphasize not so much operators, but rather quadratic forms on a Hilbrich space. You should adjust functions, right? They're quadratic, they're special kind of functions. And this is a, and this function can integrate, so the measure or density state is something, so this quadratic form that allows you to evaluate subsets. So measure is function on subsets, and so the subsets it's being applied to because they're in Hilbrich space, and this is being applied to subspaces, yeah. I'm not sure the meditation can read to what's written there, yeah. I think the S and the rest of space is G. This would be my Hilbrich space, yeah. And when we have a subspace, you want to make sense of this quadratic form attached to G. And what we do, we take this quadratic form, we restrict it to G, so it become now quadratic form in G, and we take the trace and trace, it depends on the Hilbertian structure, right? So in operator language, when you have a self-adjoint operator associated with quadratic form, you can take trace, you can take trace without appeal to the underlying Hilbertian structure. On the other hand, what you cannot do, you cannot restrict. So subspace, you just have to do some tricks and it's not a function, right? Restriction is an operation requiring, depends on the Hilbertian structure anyway. But in this terms things are kind of easier, so it's a function, it's a trace, and so what is the trace of quadratic form? There are several ways to define it, and one of this kind of pure algebraic, you choose the normal frame, and you submit the values of the quadratic form and this unit vectors of this frame. And again, the answer doesn't depend on, doesn't depend on the frame by Pythagorean theorem. Or if you don't want to choose a frame, you can say, aha, I just restrict it to the sphere, look quadratic, the quadratic form as function of this unit sphere, and integrate all the sphere in the harm measure, which is more symmetric somewhat, but the answer is the same. There are many ways to do it, and again, this is a remarkable, the circumstances depending on the Pythagorean theorem. Without which, of course, there would be no quantum mechanics in this book. But then, of course, the point is that when you work in this Helbertian setting, there is no kind of simple category where you can do it. It's unclear what is the correct category. So what corresponds to maps between sets? There are two classes of maps, of course, there are embedding of Hilbert spaces, there are other normal projections, but they're done quite categorical. Actually, in the measure theory setting also, there was something not quite right in what I was saying, because besides the category of measure spaces, there is a particular relationship between the category of measure space and this topology, the category of topological spaces, right? Because you can consider the kind of natural map from measure spaces to topological spaces. But what people sometimes call random variables, which is extremely confusing language, when you suppress domains and the range and just everything to confuse you. But it's not, so this category has, it's not quite categorical. It's kind of, it's not that it's, because beyond true categorical language, it must be somewhat ornamented with this particular condition. Yeah, I don't know whether that kind of abstract expression of that. So it's kind of two categories, interact with each other. And for finite sets, for finite measure spaces, you can, your axis is also finite topological spaces. There are two points in that. Now, what is entropy? So, and so it may be the entropy, which it's a function. It's a function on a sphere, it's quadratic function. And the entropy, the way maybe you want to define it in the most general term traditionally, and this corresponds to the formalism of phenomenon quantum mechanics, what you do, if you have a quadratic form, and it's density state. So the density meaning that trace is one. So the whole space has this measure, so to speak, and this sense is one. But you can always write it in diagonal form. It will be sum of some constant and here form of rank one. And the standard form, there is projections on the axis. Right, this by the way again quite general point. So the basic form on the Hilbert space, which you can call one. It is, if you take this form itself, it's not a density state. Because the sum, the trace of the form to give dimension of the space. And this way they will already make the whole thing slightly more delicate for infinite dimension spaces. You don't have the background space, you can normalize it. But this will be this normalized background state, so it's diagonal form, it's everywhere. And so, and on any Hilbert form, there is preferred quadratic form. On any Hilbert space, there is a distinguished quadratic form. And now we have a linear subspace, you restrict this quadratic form, and when you project it, you can pull it back by this normal projection. So, and these are very special quadratic form of states, which we call sub-homogeneous. They are most invariant, almost full symmetry. Certainly, that you cannot equate value here and value here. Here is zero, right? And here is some, it's maximum value. But apart from that, the s-homogeneous is possible for this, for states which have not maximum rank. And this is a difference, and as I was saying several times, it's essential. In the classical case, you can do the same. You have a measure. You have a big set, and you have some measure, and they can average over symmetries. And what you have, a measure supported on some subset where it will become constant, but it will be zero here. But you don't have to do that. Yeah, in classical thing, you just rate it as zero, right? And so, when we define entropy, you have this homogeneous measures which have equal values at all atoms. And here is kind of not quite so. I mean, just you can do it, but then you will be unable to prove what you want to prove. And apparently, it's physically questionable. You cannot restrict to subspaces and quantum mechanics without losing something, kind of. So, but anyway, there are these distinguished states which you project here. In particular, when you project the axis, you have states kind of, they're called sometimes pure states, corresponding to rank one form, quadratic forms. And just let me remind you that these pure states, you have this euclidean space. It's our Hilbert space, but it's just euclidean space. Or certain dimension, these pure states leave in the symmetric square of that. And this is so-called Veronese variety. It's a projective space embedded into the space of quadratic forms. It has dimension by two less than the space. So, this is, if this is dimension n, this Veronese variety, I think would be like that, yeah? Even so, this space quadratic form would be like that. And Veronese variety, we have dimension minus two. It's lying in this projective. Topologically, it is a Veronese variety. It's a projective plane of dimension n minus one. I'm sorry. It's lying in the dimension n minus one. It's lying in the hyper plane here. So, it's lying in the hyper plane where it easily cannot be. It's not lying in any small linear space. Two m's, I don't know. I'm confused because you subtract one hint, one here, there are two different ones, but they don't fit up. So, and this is a space of all pure states, Veronese variety, and everything else is convex combination of that. And again, this is the kind of most remarkable convex cone. So, if you use your probability theory, concern yourself with this. Convex cone is just positive vectors that this concern itself with this positivity where extremal points are not in discrete set, but they make projective, they make projective plane, and they kind of related and possible relational formal coming that one permutation group is a wild group of the orthogonal group, but I don't know how far you can go with that. Yeah. So, if this is a true relational, it may be just slightly accidental. You can use other series of classical groups, but anyway, that's what it is. So, and then any form can be diagonalized. So, in this rank, one form denies the states. And this, and then entropy is by definition, so these, and these coefficients make a measure. So, this becomes probability measure on the space of index by the spectrum. And it has entropy, and this entropy is declared to be entropy of this quadratic form. But, and this is how you are most always in classical discussions how people do that. But, if you are more geometrically minded, you certainly don't like this definition because you have this sphere, you have this function of the sphere, which happens to be quadratic, but you want to have a definition which would make sense, even if it's not quadratic. Then, of course, you use this quadratic, but still it's kind of, and then, and then, because these are one hands very kind of fast definition, you don't have to think. On the other hand, it's, when you start proving theorem, it's not so convenient. And this is one of the reasons why the theorem, which I stated to you, it was five years, to two, five years to prove. And that this superditivity and, you know, Leib, you know, super strong kind of analysis, it was really, really hard, really kind of, toward the force of this kind of analysis, was really considered a very difficult, kind of great breakthrough, difficult theorem. But as I explained, if you change perspective, you become more or less a tautology, almost a tautology. And now, so that's, so in the way that you find entropy, there are, the quickest definition. Excuse me. That's the question. density states of rank one, so the rank one and then normalized to be really, have trace one. So just dxi squared. But I just hate coordinates, because we need double notation, you see just, you hear, you want to even see triple, and so the, but the quickest definition is solving, you have this function, the sphere, you take any autonomous frame, evaluate this function, you have a, a bunch of numbers, positive numbers, think about this measure, in the case of density state, there's always some one. Even if it's not arbitrary function, you can normalize them. You have some one, take entropy of that, and then minimize over all frames. And the moment you say this verse, part of this is immediately obvious. Namely, it's obvious the function is concave in this space of all states, yeah. Because infimum, because usual entropy is concave, and this is essentially Shannon equality, and we discussed it at length, it's very simple. And once you know, of course, what to do with this, if it, it's not that simple if you first encounter it. But anyway, it's concave, and therefore infimum of concave function against concave, right? It's still concave. And therefore, entropy is concave, and this again was a theorem, it was really achievement by Lanford Robinson in 68, something, five years prior to this big theorem of Liebber and Ruskai. And, but now we should want to look at this from a different angle, and so we don't have to prove it. So first again, explain what sub-editivity means, yeah. So you have a kind of physical picture which equally applies to classical and to quantum case. You have a crystal, and you observe it from some part of it, and see certain number of states. Namely, you just attach something, covering some region, and you see some tic, tic, tic, tic, tic, and how many times the sticking per unit time, that's the number of states, right? There's nothing else. That's what you observe the tic, tic, tic, tic, tic. And then take another piece, and have another tic, tic, tic, and you take log of this number, this entropy, and it is sub-editive. And the reason, of course, sub-editive, if you have two pieces are very far apart, of course they tic independently, and so the number of tics just multiplies, and this will be, this will be just additivity, because they take logs. And again, I explain that logs come, that it's not completely frivolous, it's just why you go from multiplicative to additive group. But if they come close together, some may be incompatible, right? So this tic, so again, you see what you observe, of course, not the number of states. You observe transition from one state to another state. It's another, by the way, point, right? You never observe a state, right? In physics you observe quanta of light coming from you, or eventually this, I think, the only thing you can observe, right? Though indirectly. You have something coming from you, and this results in changing states from one state to another. And then it may happen that some, this kind of, but still you think of them as states. I'm not certain that it changes much. Logic, if you really remember, the traces of movements within the system, you know, you just kind of, there's a process, how you go from state to state by some random process. If you think it's a random, in classical case. So when they're closed, they may interact, so the whole, some simultaneous tics may be impossible, so you have inequality into it. You'll be clear whether it's classical quantum. And when they overlap, you have this and that. It's not intuitive at all that this strong subordinary that tells you the total number. So entropy here is smaller than entropy of this, plus entropy of this, minus entropy of that. This is a stronger subordinary. And this, however, is true in its classical cases. Shannon inequality, basic inequality about entropy. And the same is true in quantum case. Though, of course, many, many things are not true in quantum case. Except you have to now define carefully what does it mean, this decomposition. And in, in case of classical space, there's something concerning measure being sitting in the product of two spaces. So this picture is implemented in classically, kind of union of system, meaning their product. So I have product of two spaces, x1, x2, x3, or maybe my terminology was, it sets, actually, set 1, set 2, set 3. You have a measure on the set, and which you can reduce to 1, 2, to 1, 3, and 2, 3. Right? You can project this measure on three different coordinates. And you have this inequality of these, for measures of these projections here. In the quantum case, what you do, you can see the tensor products. And you have this quadratic form on this tensor product. And then you cannot project, right? The tensor product doesn't project. There is no way, unless you have extra structure in which you haven't discussed it long, so I'm not certain we'll have enough time. It's written in the final version of my lectures. This exactly, this logic here, amazingly enough, can formal, this formalism is very similar to what happened in the genetics of Mendel. Right? You know, Mendelian genetics is basically the same. There's no hybrid space, pure linear algebra, with essentially formalism to say how you project tensor spaces, what are distinguished states, whatever. And maybe in a second I'll say something about it. But so what you can do, how to produce, when you have a tensor product of two spaces, how you produce objects go from one to another. You have S1 times S2. And how you can go to one of them, yeah? Say to S1. This is no projection, but if this quadratic form here, it can give you quadratic form here because you can take trace with respect to this variable. These people always say which I don't get understanding what it means, but it's much easier to say I have an issue with respect to orthogonal group acting on this, but in this factor. So what I get will be something symmetric with respect to this, with respect to that, meaning constant or homogeneous form, therefore it will be tensor product. I'm sorry, if I wanted, sorry. And therefore it will be product of something here, tensor product of something here, and this canonical one form, canonical one, I mean, I'm sorry, this background form here. And this is a reduction, quantum reduction. Okay. So, and this is inequality. But now what is the logic of the proof? Sorry, in classical case, we're reducing everything to homogeneous states. And here we cannot quite do it, but still we can pursue the same kind of logic and say, ah, we can define entropy in two stages. First we do it kind of approximately, and then we do it, we do it sharply. And so we take your state B and take it high tensorial power now. So it will be now on your Hilbert space tensorial power N. And again, just applying the spectral picture here, we say, ah, it become closer and closer when it goes to infinity to this, what we call this quasi homogeneous state, sub-hyngenious states. So almost everything will be very well approximated by projection with a form coming from orthogonal projection, some subspace. And on this subspace, it will be just multiple. It will be multiple all the Hilbert form. And in the classical case, we will, yes, the whole, and entropy is like log of this rank of this projection, and the cardinality is replaced by, repaid by dimensions or rank of this projection, take rank, take log, divide by N, go to the limit. And of course it's just a formulation of the classical thing, but the picture is different. And so to have some, the point now is that it's hard to use these dimensions. So if you try to prove the theorem in this language, so everywhere replace this numerical entropy by dimension of these subspaces, I tried this actually. Before I changed my mind, I tried this. It's become some geometric question about, you know, euclidean space, some geometric lemma, about, you know, 23 or four linears of space and space, how they behave, I just couldn't prove it. It's become, for some reason, extremely, extremely, it was kind of as difficult as the original problem. However, there is another point you can make. Instead of emphasizing dimension of this space in where you project, you emphasize this number, so it's proportional, it's proportional to the background form with some constant, right? And this log of this, log of this number is just asymptotically, it's minus entropy. So it's small number, right? Of course, it's less than 1. Entropy is positive, and this number is less than 1, right? Because it becomes spread. There is this state in the big dimensional space, it's particularly very extremely small. So it's like, this number decays exponentially, the limit properly normalized, it's negative. So, and this and this brought you, and this what you want to exploit. And at that point, I want to make a little digression about entropy in the classical case, because there is yet another point of view on entropy, classically, which is close to what I'm saying here. So, there is another way how entropy can be defined, and this is one of the ways, actually, Boltzmann was defining it. Boltzmann kind of, I don't think he was defining explicitly, but certainly it was inside his book. And this is as follows. This is what nowadays called large deviations. Actually, I never could understand the history of that, yeah. Who invented that, and who proved what, and just, you know, the major figure in the making big theory was well done, who developed general theory, but it's unclear to me who proved what, yeah. So what was known to Boltzmann, it was proven later on by other people, because much of these were done by physicists, by, like, by Boltzmann, Gipps, and then in modern time by Lanford and Ruel, and then by mathematician, like, what are done, and, yes, I don't, and different people give different references, yeah. So, I don't know, but the subject is quite simple, in its simple form, and I'm, by the way, pretty certain it's not the right form, but still I say a couple of verses as follows. You have a space X, it's a biological space, and you have a sequence of measures on the space. So the, the example, kind of, seemingly different from what we have here, and on which is, from where theory kind of, usually people start exposing in textbooks, when you have sums of random, of independent random variables, and some of the independent random variables means that you have a measure on, say, Euclidean space first, and then you take, convolve this measure with itself, n times, and then you apply to the scaling of this Euclidean space to itself by factor one over n. I don't know how it's called, so you kind of divide it by n, by dividing this, and in a sense you scale the space. So when it's, when you convolve it, it spreads, yeah, and become roughly like, it spreads a lot, and you compress it back, and it starts switching in a finite region, and with a lot of large numbers, if it was kind of original, measure had compact support, it concentrates to a single point, so this function depends, measure will be almost all concentrated near this point. And then it was, sometimes, you know, you say, I'm curious, people say, well, big discovery, of course, everybody knew this, the law of large numbers, what happens near there, when people will kind of ignore it, what happens here, which is, I think, completely absurd, I think Bernoulli perfectly understood it, I think from, like, people, or Laplace, whatever, I think they perfectly understood it, it was a kind of minor issue, the measures, they were interested, of course, deviations, and here it was kind of negligible, and so what happens here? And this was essentially, you can say, this exponentially converges to zero, of course, now it's in the point function, because measure could be discrete, so taking values is not very good, so you have to evaluate this measure in a small neighborhood, but still, if you're away from the maximum, where it can give you the full thing, so near here, of course, this measure will be almost all measure with the law of large numbers, but if you go away from this maximum point, take any other point, except, and this corresponds, of course, to the average, center of gravity, center of mass of this measure, take any other point, and this will be exponentially decaying, so you take this tiny little point, tiny little ball, take this n-measure, n-meaning convolution, but what I said applies to any sequence a priori, take it, and again you want to go, instead of writing multiplicatively, so what would you, of course, right thing to do, but it's common to take log and divide by n. This is how it was done traditionally, again, it has not obvious meaning why to take log. I don't quite understand the meaning of this physically, so I say, oh, because one number is big and other is small, it's not true, of course, there is some logic in that, as we know, more profound that these kind of convenience of notations, and we just seen, in particular, when you go and you look at this matrix, like, like a fissure metric and its variations, and so, and then you observe this, and then you take this thing, shrink to the point in the end, and then you make second limit, so first n goes to infinity, and then this little thing shrinks to a single point, and then you have a function, if you are lucky, if the limit exists, you have a function r of this point, which is called rate, or it's kind of, it's a logarithmic rate of vanishing of this function, function goes to zero over the first exponential, if I take log and see how fast it goes, if, kind of, it may go, of course, with a different rate, and, but that's, kind of, that happens so often, and this is what was done by Boltzmann, and Boltzmann computed that, in fact, for independent random variables, and Gibbs computed that, and they followed all these formulas, which now, kind of, assigned to other people in this large deviation, particularly, Boltzmann proven, but if you measure, well, just sitting on the frame, an autonomic frame, right, it was, so it was a finite set, Euclidean space, and there's a measure sitting on the frame, then this r at a point q, so I would, I would want to think so, so I have here this measure p, I'm sorry, sorry, sorry, p i, so I have one measure p i sitting on my frame of i vectors, and then I take, on the simplest, take another point, and want to know where you converge here, so we point q, and this will be what I remember, I, I define, it will be entropy of this measure with respect to this p, but yes, it reminds me what it is, a priori we defined entropy when we had p being just standard measure on integers, on finite sets, to each atom weight one, right, but now a measure they put here another kind of p, any other measure, in particular this will be probability measure, and this by the way is, these measures are always non positive, because when p is probability measure, so what you do, first you observe that, so what you, what you do, so you have this now obituary measure, so continuous measure, so you divide it in small pieces, and then becomes approximately, becomes homogeneous, p measures become homogeneous, but with different scaling, and then when you multiply it by, by constant c, of course this will add a subtract, this log c, so, so you have to normalize it, then to, to this measure being everywhere, above the computer, so another way to say it, this exactly actually was defined by Boltzmann of course, and another way to think about that, you have this measure on, basic example was, you have measure probability measure on the Euclidean space, this is always probability measure, this doesn't have to be, so I have probability measure on the Euclidean space given by this function, on the, on say, on linear space, l, you take this to the power n, this goes to the power n, becomes almost everywhere, approximately constant, take this level with this constant, and take the volume of this level with respect to background measure here, and take log of this, that then, so it's exactly the regional definition of, of Boltzmann, what the most general one, and then it was specialized by other people, and then it was unraveled back by, by non-people who apparently didn't know about Boltzmann, it's unclear to me, I don't understand how it was, why terminology was like that, yeah, because it looks kind of there, it was this kind of loop of development and came back to Boltzmann, and many, many names are actually in the way, yeah, and okay, and so with Boltzmann proven that for this particular measure I just have this, and this if you look at the definition, which I gave with kind of methodology, it counts number of states here and there, but when in practice happens, if you think about the meaning of that, for example, when this is measured with all atoms being equal, it's formulated the following thing, yes, you can see that this formally sum of, yeah, take this polynomial, here is i terms, take it n's power, and look at this by multinomial coefficients, so it will be n factorial divided by product of n i factorials, where sum of n i equals n, when n goes to infinity, so how distributed by polynomial coefficient, what is the rate, how fast they would normalize them, when you properly normalize them, how fast they converge to zero, so they're all big, you have to bring them to one, the one simplex, so they all say for binomial you have two coefficients, n factorial and here is n1 factorial and 2 factorial, they spread over interval from one to the n, this function, you compress it to this function of the unit interval, they still of course go to infinity, you normalize them, you take log of this, you divide by one of n, go to the limit, and so you have certain function of the interval, and it will be just entropy of this measure, so when this is atomic, it will be just entropy of this measure plus this constant term, yeah, so it, and this is was proven by Boycevon, of course he couldn't kind of, if we're proving this was all his business, yeah, he was counting how many states of this configuration of atoms, you divide them this and that, and certainly I guess he wrote billions of these formulas, yeah, and for him they were so obvious he hardly kind of thinks so, I mean I may be mistaken, I read him very long time ago before I understood anything of that, yeah, but anyway the point is that now entropy is not defined via some cardinality, but actually via the value at particular point, and that is what is, we shall be, shall be doing, shall be doing, if you're understanding this stronger activity, so now let me, and now may I say in other words about the strong deviations, so the, so the strong deviation again we have a sequence of measures on a fixed space, and, and but we want to know the asymptotics, away from the point where they concentrate, so typically they concentrate, all the measures concentrate here, and then something happens outside, and however for, in examples which immediately comes to the one's mind like a geometry, you never have the fixed space, space changes along with the measure, and this is very typical, simplest example would be, we have spheres of dimension and with a higher measure, so how measure is distributed there, right, and if you took for example any ball inside of radius strictly less, some radius fixed radius, but strictly less than pi over 2, then this measure approximately, measure exponentially decays, and there is some red decay, but on that hand if you take any band around the equator, no matter how narrow this measure converges to one, so almost all measure will be there, and so if you think, look at this limit, yeah it's kind of hard to say in the, in my understanding it hasn't been done so far in this legislation, you cannot say what happens, but of course it's not so difficult to say, you can develop certain language and say what happens, and just describe what happens, and this is commonly used in, in, in, in, in analysis role in this in geometric study of, especially in bunny spaces, where that language however is not used, and more subtle example, if you look at simplices, and so you want to know how, so when you look at this sphere from very far, you kind of, what you see depends on the kind of your point of view, what you measure, so you have kind of class of constructive functions, which you measure, you go to the limit, and this is of course in algebra now associated with what is called tropical limits, and another basic example is the simplex, yeah, so how simplex looks like, if you look at the simplex and go to the limit and where measure is sitting, yeah, how distributed the measure, it depends how you look at it, you might say it's kind of concentrated on there, or concentrated on center, and just to have good language for all that, you need a proper, proper large deviation theory language which is not developed, for all I know, at least if you look at the internet, I haven't, I haven't found it, and it's, and well, much is known about that, of course there are lots of specific results, how these things are being distributed, but now we come back to the, now picture and try to prove this, try to prove this reduction, strong, strong subjectivity, and that's how the picture, just you have in mind is just, I think there are many, maybe there are, you know, is that entropy is defined when you take this very high Cartesian power, so entropy of anything of this P is seen when you take this very high tensorial power, and then in the limit it becomes this quasi-homogeneous, sub-homogeneous, so it becomes proportional to something induced from orthogonal projection on the plane from the background measure, and so it's either rank of that or it's rate of decay of the value of this quadratic form in here, and now the dimension is something global, hard to control, but the value is very easy to control, and so this is how you define entropy, but then again in this language it's not very convenient, as much more flexible language is so again I spoke to people, so this was conjectured here by, and proven by liberal sky, and I said during the summer there was a meeting dedicated to the, oh maybe not this summer, it was previous summer, 17, so it was a 19 in 2013, so there is another inequality that, which here it means the following, when you have a group, compare group actually on a space, it acts on quadratic forms, and you can every show this group, if the group is, the group is x-biasometers by linear, it's linear group, if the group acts by preserving your background form, then of course it doesn't change traces, so density stays, remain density stays, by concavity of the entropy, of course when you smear it around, it becomes, it becomes an, entropy goes up, like you know if you have single point, and this is your space, and you move it around to average, it becomes constant everywhere, of course it spreads, so entropy becomes bigger, so entropy how much is spread, what minus entropy, what happens at every point, in the, in the, of course when the things are quasi homogeneous, one is reciprocal of another, as log, reciprocal, before log reciprocal, these log become minus, and then you have that, and actually this is, again it's kind of, in this way, I spoke to Skai, at some moment somebody arrived at that, and that from point of view, this people was a kind of a big step, and all computational for me is incomprehensible for them, was obvious, but that was a difficult step, and on that hand, if you're a geometry, immediately you come, generalize to that, because you don't understand what trace is, yeah, so, so immediately you replace trace, but what you understand, the average, here the difference is that average makes sense for all kind of measures, all kind of action anywhere, you don't need linear Hilbert structure, on that hand, if you leave in this structure, you just, it never comes to your mind to do that, because it's not there, it's just, you work with respect, that's exactly, we have different context, right, so when you, when you start to think about the term, it translates the language, you know, and for this language, it's always there, but anyway, this is a, yes, a formulation of that, but the groups you have, so, let me show you what about the group, and this in this form I want to explain it, so to what you reply, yeah, it's easier to look, right, so you act by a thermal group on each factor, and because reduction corresponds to exactly that, and this is just you, this reformulate, you have to be slightly careful, yeah, because it is extra factor, because you multiply sometime by standard, with canonical form, which add log term to the entropy, but have to be sure it makes sense, so and again, in the case of the usual Shannon equality, when you have some measure here, you project it here and project it there, the reduction, you push forward the measure, but what you can do is pushing it forward, you can average it, say about group acting on these fibers, and then become constant, each of them, and this has the same effect as projecting it, of course, it's one functorily determined, so now that I did slightly different quantities, right, because you have to add up before you go, but they're now equal, so this adding camp corresponds to multiplying by a fixed constant, and computation, it becomes kind of very simple, however, there is a little difference, and that, this is what this averaging does, and now, so let, I'll describe the proof, and so the point of the proof is as follows, so you pretend, so all entropies defined by taking height and zortial powers and going to the limit, so in the limit, your states become approximately projections, induced by projection by orthogonal subspaces, and so, so it has infinite sequences, and in some generalized sense, in the limit, they become these sub-homogeneous objects, they're not homogeneous, they're sub-homogeneous, meaning they're induced by projection to these subspaces, but of course, subspace now, infinite is kind of very high, large dimension on every step, and the value of your quadratic form become very small, right, so product of these values doesn't change, so some logarithms doesn't, one is kind of reciprocal of another, and this, it's one, yeah, but, and this is how they kind of, what specific value here, give you minus entropy, and the dimension of this give you entropy, if you take log and non-dy by n, but now you think in terms of kind of, kind of non-standard analysis, you say you just take this, say, d limit, and pretend the limit actually exists, and then everything reduces to the case when all states are of this type, and so, and when these are there of that type, the proof become absolute trivial, because you just to have, she checked sub-cycle I a plus b equals c, so let me write it down, it's so trivial, it's impossible to say, I just show it to you, it just is absolute, absolute triviality, and here it is, I even don't know how to explain it, just you, it, it's literally I a plus b equals c, so, but the ingredients that I wrote down, because, yeah, so this is, yeah, compared to the Bernoulli and what they're speaking about, this argument, also this argument which I described, and this power is, it's of course extremely old and goes to physicists, and then it was taken by people in formation theory, written in absolute incomprehensible terms, yeah, I remember you say, actually, I understand, understood it, trying to read one of these articles, which people think of, they speak about stock market, it become more transparent for mathematicians, okay, so, so that again we have these states, and there's some language pictures to have in mind, right, so every state has some rank, quadratic form has some rank, so it always is used by projection on some subspace, if it's maximum rank, the whole space, otherwise it would be subspace, and so there is a, here's maximum non-degenerate position of quadratic form, any other induced by orthogonal projection, and this is support, and this is a kernel, so this support is perpendicular to the kernel, so zero is kernel, right, and so what we have to know about that, more, so if, in order to have the full measure, you have to take a subspace which contains this support, so all measures sitting in this support, if it is subspace which contains only part of this, value of this density state as a measure of probability, it will be strictly less, again, just, there's nothing, absolutely kind of trivial, there's a whole point, everything just, I listed just to see what was being involved, so what else you need, right, secondly that entropy now is defined by a single number when it is sitting in support, just this classical homogeneous states characterized where they are being projected, and this automatically gives this number, and this number is, which we keep track of, we don't keep track of dimensions, right, and this amazingly now is difficult, if you try to do it that way, to look at dimensions, I just, for some reason I couldn't do that, maybe I was kind of stupid when I thought about that, but once I look at this number, things become absolutely transparent, and then they observe the following thing, yeah, there are two, they not separate properly here, so if you have two density states, two quadratic forms, and one kind of both homogeneous homogeneous, but one strictly big, not bigger than another, but has bigger support, usually bigger support means it becomes smaller almost everywhere, right, so this must be some confusion in words, so I have some form support on this line, and then I average it, and then it becomes spread everywhere, it trace doesn't change, but the value, specific values here on this surface, we're bigger than everywhere here, so it becomes smaller on this support, but outside of support may become, it's not true anymore, because this was zero here, and this one will be not zero here, right, so it will be say one half and one half, so this was one on here, after averaging it will be one half and one half, right, so it was one here and zero here, and then it averages like one half and one half, so on this support it becomes smaller, but outside of support it may become bigger, right, so that's, and accordingly entropy, entropy is minus of that changes, again take Logan who is changing, it's quite, quite, if you say it inverse you are lost, but of course computation absolute trivial, and this what I said here, of course you can consider these ratios only if one kind of support of one is kind of one kind of engulfs another, so it's bigger in a sense of support, but smaller maybe typically will be smaller in terms of, in terms of specific values, and then, so this what we use, another property which we need say when we average a form, its supports become bigger, right, but it's kind of clear, yeah, you move it around, average it becomes spread, yeah, I explained with example you have these points, you have a measure somewhere here, you move it around and take this average of all of them it will be distributed over a bigger set, and so what this double average inequality says, it says when it says average something twice by one group, then by another group, and they commute by the way, first actually I wrongly said they don't have to commute, don't have to commute, they still have formula, but not exactly like that, and then on the second round the way by how much it becomes smaller or by how much it spreads is less than the first step, the first step was crucial, and this is not surprising, because first was most concentrated, so it's easier to spread it, and when it becomes more, bigger spread with the other group, it will spread less, and because spaces, these quasi-homogeneous, it just become some statement about numbers, essentially the inequality we should use in the end of the day is that you have a quadratic form of certain rank, then its maximum value achieved on its support, everywhere else it's smaller, and it's just all kind of algebra we should use, but then for general inequality it enters in some average sense, so this is, the average goes up, and then you conclude to the proof that this inequality, if there is sub-homogeneous, and here is the proof against pure, it has kind of shame to write it, it's just so trivial that you wouldn't write it if it were not considered, yeah, this is the inequality you have to write by how much thing changes, this is the value, but how it change, they change by this ratio, and on the second round it's smaller, you just have to want to choose, you fix it, you choose any point where it's looking at the average, and then becomes exactly what I said, but this again just, if you look at the algebra, it's quadratic form of certain rank achieves maximum on its support, but now the point is, this is, again yes we have to look at that, it's just kind of really, kind of absolutely, provided you know definition of quadratic form is just completely obvious, but the point is how to use it for general case, this we use this homogeneous thing, there was only one number to look, yeah, we do look at one number what's happening in this number, and you just become this computation at a single point, yeah, about two numbers, the question is how to make it a general, here there is some, well I don't want just you don't know, if you say it's obvious it's much much easier, much better to swallow, and then we have to prove it in the general case, when space is not homogeneous or they have lots of values, and when you go to this ideal limit they become quite homogeneous, but this operation do not quite commute, kind of, the way you go to the limit, the way you average, not commute only up to error which goes to zero all the time, and then I don't know how to do that without kind of really, kind of really working little bit, so because the corresponding formalism of the formalism of, of this non-standard analysis in Hilbert space, so it seems there is a kind of this kind of, this kind of non-standard Hilbert space, and there is a kind of large deviation theory, but this large deviation theory will be even more elaborate than the one should be in the classical case, even in the classical case, in my view it's just, it's not adequate for the purposes, and in Hilbert space you have to look at simultaneous limits of several, of system of operators which may not commute, or operators quadratic form, whatever you find there, you go to the limits, and then when you think of these terms you realize entropy is a tiny little part of what happened there, you have lots of geometry going on, how things may go to infinity, and in different situations, exactly in Hilbert space, the limits will be of more elaborate nature, and some people do something there, but in different terms, that this is, I don't know, I haven't tried to do that, but you can do it, just, just, let's do it by hand, deciphering all I said, and then you have the proof, so it takes a couple of pages of two lines, I don't know how to do it in one line to make this non-standard language adequate, you just decipher, look carefully at all definitions and see everything fits, if you do it in the right order, again I was spending some time, if you do it slightly wrong order, because things don't, different limits don't pet commute, you don't get the right answer, but I'm pretty certain if you formalize it properly, you have this abstract formalization, everything comes by itself, you don't have to bother about that, because there are about five points you have to adjust, and there are two variations in each of them, so there are about 32 variants, how you have to organize the proof, and when I was doing it, the right was the last one, so I knew I have to do it in certain ways, and systematically checking one after another, it didn't work, the last one worked, but it's absurd, because I didn't understand the, you didn't get the right logic, if you know the logic, you immediately arrive at the right order, it's written in my article, but then to pay, and this is, once it's written, it's kind of rather tricky, and it's kind of obvious, but finding that was difficult, because it was stupid, but the question is if you can, it's worthwhile pursuing this non-standard depletion geometry, it will be generalization in a way, it's kind of hard-eaching generalization of large deviation theory, but I don't know where it should be used, because I don't understand even where large deviation properly used, and what is needed for that, so it's such thing definitely exists, but it's unclear what to do with it, so, and now we make an interruption, and then we say something else, but again it's not conclusive, yeah, because it's, the proof is, if you just, even what I wrote, it's a little bit kind of technical, it's not as transparent as it should be, though it's very trivial by kind of, there's no technicalities inside, I mean, I repeat all you use, the quadratic, poise of quadratic form achieves maximum on its support perpendicular to its kernel, right, and I don't know how, how to say it better, I think one of the points, of course, it would be nice to, to prove something by far more general, and then become, of course, by far more obvious, yeah, it's used, something is used, and because you have to, it's not incorporated into definitions, okay, so I want to say a little bit more about algebra involved, yeah, and so when you, in, in, in go to Hilbert space, you find everything in terms of linear algebra, so on one hand you generalize everything, and you replace, saying here specifically you replace the permutation group by the orthogonal or unitary group, if you go in complex case, however it's, by the way it's unclear what happens with this category, so it's an, you might say, the category of finite sets replaced by some kind of category of Hilbert space strata, but there is no such nice category, but on the other hand there are construction in linear algebra unavailable, unavailable in, in combinatorics, which allows you to reformulate some of our thermos follows, yeah, so when you have a map between spaces, Hilbert space, you can induce quadratic form from one to another and look at the entropy, the entropy of this induced quadratic form, and so you have, speak about entropy of a map, or linear map, once you do that, you can reformulate the usual adji which of entropy in the following form, so you give now three linear form, which doesn't have to be positive for anything, any three linear form, and then it defines you two bilinear form, so any bilinear form give you a map, any bilinear form in Hilbert space can be thought as a map from your space to the dual space, because the Hilbert spaces, you can induce this Hilbert form, and then there is this inequality, oh it doesn't quite fit, yeah, yeah, oh it's okay, it's just brackets, and then, yo, yes, and this certainly reminiscent of what I was writing in this, in this algebraic terms of isopermatic inequality of, of, um, Whitney, and Lumi's Whitney inequality, so these inequalities in turn, these basic inequalities between entropy are in fact manifestation from certain point, from a geometric point of view, are kind of isopermatic inequalities, and they relate it to them in the case of, in the case of, um, usual, usual, so let me remind me again this, which we had this, um, Lumi's Whitney inequality was geometric inequality, and now I am going to formulate something linear algebraic, which is more, more general than that, by far more general, seemingly more general, right, so this inequality in the simplest case says we have a subset in three-dimensional space, and you can see the three projections called one, two, three, you project it to plane one, two, two, three, and one, three, and then the volume of this square of the volume less than product of these three areas, so it's very simple inequality, and it's stronger than, up to your constant, it's qualitatively stronger than usual isopermatic inequality, quantitatively it's not that sharp, and it implies all, at least in three-space, all kind of so-called sub-alive inequality, including log sub-alive, and this in turn is consequence of, of, um, of this Shannon inequality for entropy, which is kind of weakening, weak form, and the proof is instantaneous with this definition of entropy, but there is exactly example where you can think in these terms, you become obvious, and if you don't have these pictures, it's really kind of a trickier to prove this, yeah, because after all, isopermatic inequality even, even non-sharp one isn't on trivial theorem, it's kind of basic theorem in partial differential equations, you know, 90% of theorem follows from that, yeah, plus, maybe even more, there is basically almost no any other non-trivial inequality there, always just straightforward integration, but then this has this amusing algebraic counterparty formulating there, and this is as follows, so we have now linear subspace and the tensor product of subspaces, so I have y, let me say I am running a little bit ahead of myself, yeah, so I have this geometric space, I have linear space, it's linear space, take the tensor product, so notation, yeah, she just, I have the subset, it's essentially, see if I, this tensor product of the part of this, you see this is the whole point, you cannot put index, if you put dimension, the rank of that, like you immediately get lost, yeah, so this also tensor product of the power n is not correct notation, yeah, and so here is the situation, have a linear subspace in this tensor product of spaces, and then you can kind of project it to subspaces, and again you cannot project it, right, tensor product don't project, however, you can say how, you take minimal kind of subspace in this factor, which give you this space, tensile by everything else, so it's very similar to how we're defining this reduction, reduction in quantum case, in phenomenon case, but here we don't use, we don't use unitary structures, purely in algebra, so this is a kind of key definition, so we have a tensor product indexed by some set, we take a subset and then we can speak about this reduction to this subproduct, it's not a projection anything, and then we need the following inequality, I think actually I didn't plan to write this, so I see it's here, now this was some preparation, I just kind of justification with myself, I get them, so we have this subspace in this product, we project it to each j, and so we have the following simple formula, that is also this rank, rank meaning dimension of my field, which plays a role of the volume, this terms rank of this projection kind of areas, and this product of this areas bound the volume, and here is condition of this, this constant must be certainly not very big, and this, you remember the square of the volume was product, or if I take some volume, I would have to take square root of the areas to multiply them properly, so this alpha i might be properly adjusted, and this is called partition of unity, which is function on sets, on subsets with this property, so this is this kind of formula, and this is in the combinatorial case, it's called Sherer lemma, which again is that kind of, I don't understand exactly terminology because certainly everything follows from Schoen equality, or Schoen equality are very trivial cases of Boyceven inequality, and in the course of computation, Boyceven contains all these inequalities, they are implicit I think in every line of he does, but this is called Sherer lemma, and if you look at the textbooks, it's impossible to understand what it says, yeah, it says something about hypergraphs, I don't know, some kind of fantastic language, yeah, but this is generalization, and this is kind of cheap generalization, it's very easier to derive it, on the other hand as much kind of indicators built in, built in linear algebraic, linear algebraic construction there, and so this I have a logical way of sitting, and this probably the end of this linear, but what is unknown, and this is also unknown for classical entropy, and this is unknown for Neumann entropy, what are the full set of relations between entropies of different sets and different projections, et cetera, what is the simple relation between them, and I think it's known that Schoen inequalities and this superdiversity inequalities don't give you the full convex hull of all possible values of the entropy, they are linear inequalities on the entropy, so they define some convex set in the space of entropy for different subsets, and here there will be ranks, instead of that I can see the logs of ranks of different reductions, so what are all these relations, this is a question of polylineal algebra, and I think if you go deep into that which I couldn't, you arrive at really quite interesting thing about, about, about algebraic geometry, and so in this respect, so what you know, why you think so, that if you look at the following question in algebraic geometry, very much in the same spirit, you have algebraic variety in the product of two projective spaces, yeah, so it's of any field, yeah, for that matter, so you have product of two projective spaces, and you have algebraic variety, so irreducible, which by the way is essential, and you project it to A1 and A2 for two factors, and of course this projection is pure kind of, what involved, then the, say over the complex numbers, and look at the degrees of these varieties, what is degree here, what is degree here, what is degree here, properly understood, this for standard embedding probably, for, well this of course not a sum, and then there is an equality between the two, so this degree bound by these two, and this inequality is exactly parallel, and it kind of implies, it's equivalent in some trivial way to the Brunmin-Kowski inequality, because if you take this, you take this variety, yes, you divide this by action of maximum torus, they go to this kind of variety, sometimes it goes by moment map, they go some way, look at this volume, see how many monomials are there involving this degree, and this become kind of, kind of simple, kind of tautology, it become Brunmin-Kowski inequality, which in a very generalization of isoprime American equality, so it's another kind of similar in spirit generalization of isoprime American equality from the one which I described here, and one understand this kind of the full extent of that, you know if you just define this inequality, and it becomes so-called Alexander-Fengel inequality, which essentially amounts to Riemann-Roch theorem for surfaces, or for photoric surfaces, right, so this relation between the ranks, I'm pretty certain, if you go deeper, they reveal lots of interesting algebraic geometry, which I was unable to do myself, right, so this way I bring them up, so I think there is extremely, extremely kind of interesting field there, which I couldn't penetrate, and I want to make some, before I come to this subject, it will be my last lecture, I want to retrieve my steps and explain some, I haven't explained enough, namely how similar this algebra, so this algebra of this associated probability theory, enters in classical genetics of Mendel, which I found extremely kind of surprising when I understood it, because it's a mathematics as sophisticated as the one said von Neumann, which was certainly not known to anybody and not understood by anybody, and still not understood by all the scholars who do that, and he's a faster way to move it, no? Okay, it may be, sometimes you manage to make it faster, maybe move it back, yeah, and then it can, so I just, I thought it was too boring to move it, but I don't see how, just one second I want to come back to Mendel and dynamic, and we see, because I'm using similarity with this formalism of von Neumann entropy, but purely algebraic, one of the previous lectures said a little bit about that, now I want to say more, but of the same patterns there, but in somewhat different context, okay, now, some reason I cannot do it fast on this, so I already was citing that paper by Hardy, and he wrote in 1908, it's about called Mendelian Proportional Mixed Population, and so what he wrote, where I just, he speaks about this multiplication table, he just said, yes, I want to, yeah, that's exactly, yeah, so he done it, and he just showing his guy being quite, quite arrogant, saying, oh, this is mathematics multiplication table, and so what's interesting is, everything Hardy done, he just, I mean, immensely less than this, I would remark, because nobody knows Hardy, except for few mathematicians, everybody knows what they've done in biology, it was when he was attached his name to really one of them, maybe five or six greatest discoveries for science, which was, when he really was a combination of thought and simultaneously having some implication, in the 19th century, there were only one comparable discovery, that was electricity, it was Faraday and Maxwell and some Ampère who were involved, there were several people involved, and that electricity was discovered, and our life changed, and another discovery was the Mendelian Genomics, and our life changed somewhat later, because partly because nobody understood what he was doing for about 30 years, and even Hardy, he didn't understand mathematics behind it, biologists certainly were completely perplexed, and the story was Mendelian, for me it's obvious, kind of, he was, he knew mathematics, he was kind of mathematically minded man, and he wrote something, and he's shown to biologists, and these were great people, you know, scientists, famous people, leading bowers, when he assumed, they understood multiplication table, none of them did, actually, we know very well, say Darwin, he couldn't add numbers, when they discovered geometric progression, for them, it was, oh, you know, it grows fast, it was revelation, yeah, they didn't know problem of Fibonacci Rebis, apparently, yeah, for them, they never heard of that, yeah, they referred to Maltus, Maltus, never by the way, they were claimed originality, now people are saying, Maltus said about exponential, it was known, certainly before Fibonacci, if Fibonacci never discovered that, of course it was known to ancients, yeah, and there are lots of literature written about that, and one of the most, the most kind of, interesting was here written by Condorcet on the lies before Maltus, all the situation, and said what should be done, and then Maltus argues with Condorcet, and everything so Maltus done, he says Condorcet was too realistic, so he came with control population, we don't know who was right, but mathematically, of course, no science involved, and then it was taken over by Darwin, and exponential function, my God, yeah, and these biology skills are very much impressed, because they don't know multiplication table, and inside of Mendel, there is much more sophisticated mathematics which hardly missed, and so I want to show it, it's very simple, very beautiful mathematics, inside of this work by Mendel, which Mendel perfectly understood, of course, I think so, I mean, of course Mendel was not writing about mathematics, and the major point of him, he had done this tremendous amount of experiments, he had very rough statistical data, and he used on the other hand extremely fine mathematics, which was completely kind of, kind of perpendicular to this data, because it was very fine mathematics, and it was very rough, however, and he rightly concluded, that gave that conclusions, the existence of genes, and the fact that organisms he was considered were deployed, like ourselves, so there are two set of chromosomes, and he saw that, and this was never ever before, and I think never after, done in science, that you can use on one hand very refined mathematics, and apply it where it apparently doesn't work, but it still works, and interestingly enough, Fisher, who was following Mendel, thought he was cheating, he officially didn't quite understand that, yeah, he was certainly great scientist, but not the same as Mendel, yeah, he has, you know, great mathematician, he didn't understand the whole logic of that, yeah, he was a great admirer of Mendel, and on one hand, on the other hand, he believed that Mendel falsified his data, which was not the case, because you see, statistic is a tricky subject, depends how you interpret this, how you derive data, but you see, if you little bit go out of the average, you little bit, and go systematically in biology, it tells you something happening, it's not like in physics, in physics, you want constant being true up to, you know, nine signs, they're very proud of that, in biology, if something is true, 60% of time, it's already fantastic, because it's systematic, and that tells you there's some structure there, and you can extract the structure, and this is what Mendel done, and then another instance of that was done by Sturgeon, another remarkable instance, but that kind of, of course, it's what he says, it's no big deal, but now, this is, again, as you look at how much, how hard he cited for his mathematicians, for the third one, on Google, you see this factor of 100, at least, then, and he was, by the way, not the first to do that, that was done by somebody castled before him, and then was simultaneously by Vanberg, and it was, somebody told me, actually, I spoke recently, some biologist, and he said, no, no, but Hardy done so simply, it was so complicated before, the paper by Vanberg, when he was proving that, was about 80 pages, it was five lines in Hardy, but it doesn't need nine lines, it needs one line, I should explain to you if you say it correctly, it's line A, A, B, C, how he put brackets on A, B, C, that's all, all is involved, however, formally this kind of, kind of identity involved, but what he expresses is, if you look at this map, and what Hardy says, but he doesn't say it in these terms, right, so I have this map, then this map, if the thing P plus Q plus I normalize your probabilities, so sum is one, it is, it important, so it's very simple map, and so what I'm using about that, and indeed, it's quite incredible, you have polynomial map, you repeat it twice, and degree doesn't grow, right, apparently when you compose polynomial maps, degree multiplies, sometimes things become degenerate, and by the way, many inequalities, like all this Shannon type inequalities, and phenomenon inequalities for entropies, they say something doesn't quite degenerate, but sometimes it does degenerate, some kind of paradoxical happens, and then, you know, biologists have had time to accept Mendel, and again, this big controversy why they couldn't understand Mendel, so it was the problem, but I think it was just complete lack of mathematical thinking on their part, and then, so let's just, again, I repeat what I said last time, so we have deployed organisms, so each of us carries two sets of chromosomes, one coming from father and from mother, you know, it's a G and A, each about two meters, by the way, a G and A, and each, in every cell we have about two meters, and they too, and so, and this is a kind of remarkable thing by Mendel's, is what being inherited is not what you see, because it's function of two variables, but each individual variable being inherited, and that was an absolutely fantastic idea, which was, there was some precursor of that, there was the idea of Meperchi, who was saying very similar thing, but Meperchi was saying it incorrectly, but still the idea was there, but of course, little reason, believe it or not, Mendel knew that, and he described it from there, and so what is the statistic, how probabilities come? First, they're not numbers, so because you can say, how many whatever you have in population, and then normalize by the total number, you never can do that in reality, because you don't know the total number of objects, you don't know how many genes are in population, how many animals, still you can work with probabilities, but they're ratios, and in fact, and this I will not have to time, partly because I don't know this, well, I understand myself, this whatever statistic you make, you never normalize, you have very sophisticated functions, sometimes more sophisticated than the ratio of frequencies, of relative frequencies, and from this you made conclusions, right? And this was actually how hard you was writing very correctly, you have these proportions, they're ratios, you don't have probabilities, yeah, when you apply it to real, and then you look at more sophisticated situation, there's something else, they're not proportion, there's somebody else, and the whole probability collapses, this normalization works in physics fantastically well, it doesn't work in biology, or in linguistics, by the way. And so, so probabilities are not numbers, they're points in the projective space, right? They're only defined up to the ratios, even this is not quite correct, because you don't know set. Set R is unknown to you, and this exam, it's in the cases, it's unknown, and you cannot actually bring it honestly in the game, sometimes you do, and here you don't make mistake, in many cases, like in languages, it leads really to actual mistakes. However, once being done, so what you have, you have this matrix, kind of distribution matrix, how many particular pair appears in population? But so each gene, each of them is a pair of two possibilities of the gene, the gene may be of several different kinds, there are say 10 different types of the same gene, and each of them, our gene has one of those combinations, there are a hundred of these combinations, we saw the probability you say, because of course probability must be understood carefully, and this was actually, how you were saying, that you can interpret these as probabilities, but they are in there, and but then when there is random, random mating, so individual mate on randomly the mixture of that, and this actually was the major, main point of how you didn't say in this language, so what happened to the semantics? What is the operation? And the operation is as follows, yeah, this kind of a segregate product, you have a matrix, you take, through given element column, and you take a row, you sum up and multiply, so you sum mate here, sum mate here, and put back that product, and so we have quadratic map in the space of matrices, and what hardly proven, for a special case of sematic matrices of rank two, is that, so this is the formula, and then by associativity of the product, all you use associativity of the product, I don't think even commutativity is used, that on the first round you achieve equilibrium, if you projectivize this map, this will be the same, and that is kind of very simple, but certainly it's, it's not, there is no multiplication table in there, it's algebra, there is no numbers, it's pure algebra, much more primitive, then on one hand, on the other hand, it has kind of interesting development, and so if you restricted to sematic matrices, assuming that male and female population carry the same set of genes, everything will be symmetric, and so this map is just retraction on the veranoise variety, remember we had this veranoise variety in fundamentally, playing fundamental role in quantum statistics, so there is a cone of positive forms, all quadratic form here, this projective space sitting into there, but amazingly you can retract everything here, there is a rational map, it's retraction, meaning square of this equal itself, we can retract ambient space to this veranoise variety, and the fibers of this retraction I find space to be doing, and this is a map which does it, and this is quite kind of remarkable piece of geometry, it has nothing to do with multiplication table, it's kind of nice piece, but certainly Hardy was not a geometry, so he supported formulas, and this is, his mathematics, actually all his mathematics was about multiplication table, and this is why he, so he is kind of arrogant, I find rather misplaced, and so this preserve, this probability may be one that doesn't have to be a positive, you know it's pure algebra, but what I was saying last time, it's not surprisingly entropy goes up, so it spreads kind of features, but again the remarkable thing is that square is one, and people couldn't accept this, so you mix two populations, and first one becomes dominant, so one was dominant, you have more of particular species on the proportion, on the first round, and they change to them, on the second nothing changes, so it completely kills the idea of evolution, how people understood it by Darwin and other people, says no evolution, because arithmetic was again them, of course there was more thing, more and more elaborate in all respects, and evolution is certainly there, and so, and these matrices which are of rank one are exactly those for which kind of they have this, they have this maximal entropy, so the equilibrium state, and the states of maximal entropy, which is kind of very much in the spirit of statistical mechanics, statistical thermodynamics, and this was all in Mendel, and it's kind of shame, like Boltzmann who actually entered the field in the same year when Mendel wrote his paper, so the thesis of Boltzmann was written in the same year when Mendel published his paper, but Boltzmann never read it, and this if Boltzmann read it would be, I think the schedule of statistic would move by 30 years, so by now maybe we have que of many diseases, because he sent stupidly his article to biologists, he had to send to thesis, Mendel, and it seems he talked to these people, he believed they understand something, and they are idiots, they are kind of very good biologists, they are great people, but the idiots would come to thinking really in terms of mathematics, it's unbelievable, actually Darwin was not this, Darwin never read it, but we know that Darwin came across the same phenomena as Mendel, discreteness of certain features, and this was when Mendel started to observe certain features, not mixed, you may have red and white flowers, you don't have them pink, you have either one or another in the mixture, and then he thought about that, and Darwin also observed that, but Darwin apparently rejected it, because Darwin was obsessed with the idea of continuity, his major point in evolution was not people deciding evolution, selection, this was all kind of known, it was the idea that evolution is continuous, and this was the principle of borrowed from Leibniz, in nature everything continues, and it was completely wrong, and you know the whole point of evolution is discreet, the whole thing of inheritance is discreet, so everything, the key point Darwin's on emphasizing was nonsensical, however altogether he was right, that's interesting get by the point, everything he was saying locally was nonsensical, but evolution was there and selection was there, but now what is mathematics behind it, so this was kind of, you have this linear spaces with extra structure, very little extra structure, which makes them much easier to handle, and this you can, so they typically they function on a discrete set, and you can sum up over this set, and this give you linear function there, and so space between your functions, so and the example just this one, and of course in other example exactly quadratic form operators, where there are traces, so spaces with traces, and when you have spaces with traces, I mean just very very convenient, and I believe sometimes it's called vacuum state in physics, where I'm not certain, there's some distinguished state, Hilbert space, and Hilbert space because dual vector is the same, but there is distinguished vector, and the whole point, the whole genetics invariant and the symmetry is preserving this, and the linear transformation preserving this collector, so you don't have permutation of genes or whatever, but you have the whole linear symmetry, all these formulas of genetics have the symmetry, and which makes this theory extremely elegant mathematically, but I'm saying extremely when it's applied to real life is only a little bit better than nothing, yeah, but this is a little bit quite a lot, yeah on the other hand, and then in particular you can have these maps from from tensor product to each component, which usually don't, so you kind of have reduction, I don't know if it has anything to do with quantum mechanics, so there are the distinguished vectors, if you say something is vacuum, and you can kind of throw away some variables, but mathematically it's very simple, now what is this map, next generation map, in this language it's like the following thing, you project to each component, and then tensor back again, but can't be easier, so this, you project it in this, and you look at this thing back, now how to prove its square is equal to itself, because it is a tensor product of elements kind of pure state, what you just said, it's a monomial object, and because it's a monomial, right, it has this kind of monomial, so if you multiply it, so you just, you only use multiplication table, you don't use addition here, so what's involved, and because it's quadratic, this must be, this must satisfy this formula, it's quadratic map, so you know what the normalizing constant is, so even without that, you know it will be preserving up to your constant, map will be important already in the projector space, it will be an important, and specific constant come because preserved, and so this is his multiplication table, so it's really one line, not nine lines, so now if you look at, so what happened there, and this is quite interesting, that you have maps, so you have dynamic, there's some kind of distinguished maps, so this is the instance of that for quadratic polynomials, that act in spaces of polynomials, and these maps have this kind of remarkable algebraic properties, that they are my square, maybe they may be important, or some maps of this kind invert, it may be invertible, polynomial map is invertible, right, and they all come along with Mendelian dynamics, and you know the maps which are invertible, polynomial map is invertible, generate nilpotent groups, so nilpotent groups, nilpotent groups also here, but this is not essential, but the way you can see it, and this is actually, this formalism that I'm describing isn't in Mendel, yeah, of course not in the rethnotations, that tensor products of spaces naturally embeds into the space of polynomials, right, so you don't have to bother about this abstract thing as tensor product, because you look for polynomial in many variables, and there are very special polynomials when all variables are separated, right, and so there is much more in there, but this is some part of polynomial, but this gives you very very flexible, very flexible language in there, and then this vector which we have corresponds to this co-vector, evaluation of polynomial particular point, and again what they do in genetics, and if you look at the textbooks you see this kind of, they sort of never say it, they write some formulas, you evaluate at the point one, when all vectors are one, and this is inconvenient, because it's much more convenient to use point zero, and then point zero all formulas disappears, and formulas are binomial coefficients, so usually when you write, you see the text in genetics, there's lots lots of coefficients, but that because they evaluate polynomials at the wrong point, but they're all the same, this is extra symmetry there, because in the space of polynomials you have this action of translation of the background, and then this, so I just want to give an example of some dynamics which associated with Mendelian dynamics, liberation of Mendelian dynamics, when you have not one gene, but many genes, so in reality you have many genes in the statistical process, acting there is the probability space is the product of involutions, so you have this plus minus group of two elements and some power n, and on the other hand you have n genes in another set of these n genes coming from father and from mother, and this group permutes them on random, some may be muted, some not, and this is called crossover or combination, but crossover, so when a new kind of organism being created, new genome is being created, it creates out of old ones with some measure on this group, and the point is irrelevantly of what measures there, in so far it has a full support, so it mixes everything, the dynamic of this will be always the same, however again when you apply it to real life it's unclear what it means, I don't know if it has any application, but mathematically it's quite amusing, quite amusing theorem, which is, so let me describe it, so what are they, what it says, so what are these maps which we describe, this corresponding to kind of random matching, so we have polynomials in many variables, right, space of polynomials in many variables, so variables separated, they are groups of variables, so the maps as follows, I restrict my polynomials to some plane, so it's a product of planes, yeah, two by two by two, so they are dot lines, yeah, and then extended constantly outside, so I give you, and the morphism in the ring of polynomials, so I restrict polynomials onto the subspace, but then they extend it, and that's possible because this linear subspace, of course, is a retract of the big space, right, so if I have a function on the plane, and I have this line, I restrict it here and make it constant in these directions, restricting is easier, making constant along direction requires existence of these directions, and this is possible because I'm seeing this projection to like, so there are these, and when I have many of them, I project to many of them, and then multiply them back, and this exactly what we had before, this picture we had before, square of this map up to a constant is equal to itself, it's important, and it's important because projection onto the coordinates in this space are important, linear projection here, square is one, for that reason this square is one, right, so the computation of hardy again amounts to saying projection square equals one in proper notations, right, but this exactly the whole of course in mathematics, it's not multiplication table, just right notation, and then, yeah, and when you multiply them, by the way, yeah, when you multiply them, it will be not already, and the morphism of this algebra of polynomial, but you multiply it from the morphism, right, because they're in the morphism of the algebra, we multiply them, it still remains in the morphism, but only multiplicatively, and so it preserves degrees of polynomials, and it's nice to think about that, not as a subspace of polynomials, but as it works in the space of truncated polynomials, so what kind of advantage to that, so the ring of truncated polynomials is a very nice ring, certainly it's a ring which has very good exponential function, usually in most rings, exponential rarely defined, but if the truncated polynomial, so it means you add, you take polynomials in all higher degree terms, you declare being zero, all term of degree bigger than d, declared zero, so it's additively the same as polynomial of degree d, but multiplicatively it has a little structure, but the point is it has good exponential, and good, usually formula for exponential works, and it's invertible for all polynomials which start from positive term, if the free term is positive, it's invertible, so it's as good as it could be, like for real numbers, it has perfect exponential, so it suggests by the way that all we're saying about entropy and measure theory must have counterpart where you replace numbers by this kind of ring, this ring is as good as real numbers, and I think it's again one line of thought maybe interesting in this, motivated by this Mendelian dynamics, well there are kind of trivial formulas, yeah, between these maps, but the whole thing which I want to now to formulate where some analysis enters, so far it's pure algebra, just elementary algebra, still it's rather amusing because some things secretly cancel there, right, very simple reason, but you have again polynomials where things happen, yeah, so this is it, so what is the formula, so it is strict, and there's zero levels, and there's zero planes, and multiply and that will be the normalizing factor, and so these are polynomial maps, so in the space of truncated polynomials, these are polynomial maps which are multiplicative and amorphous, and moreover they are important, which is kind of the fact such thing exists in a priori not at all clear, yeah, it's rather, and they all of course, and then that you can linear, these maps are not linear, but they're linearized because of this very simple, this very simple exponential, they say they're multiplicative and amorphous, but because you can take logs, they become additive and amorphous, and for additive things become kind of much more transparent, and so this whole, this map become linear maps, right, because additive and amorphous are just linear maps, so they're conjugate by exponential, they're kind of conjugate to linear maps, and this again, and so this is a nice exponential, so just a little propaganda for these kind of maps, and so they retract, of course each of them retract to their respective verinaceae varieties, and the fibers of this retraction are refined maps, because they have kind of behavior why they're fine, because of the linearization, if you apply this linearized map, everything of course, the fibers always are fine, but because they're linearizable, they will be fine, right, so exponential, this of course you can see directly, but in this way you don't have to think, and then, so I want to give some offshore of that what I was saying about genetics, what is the mathematical that, so each of this, of this map depends how you decompose a set of indices into different groups, so it's exactly how it happens to the entropy, so you have this state, so you decompose in different things, you look at each kind of entropy, but now this, instead of entrepies, which are logs of product of numbers, these are, if you want to say entropy here, it will be logs of all these maps in the space of truncated polynomials, so I want to say the whole algebra is absolutely parallel to what we have in classical, classical mechanics of Boltzmann, but I'm saying the Boltzmann couldn't appreciate Mendel, yeah, because Mendel is more general than Boltzmann, he can develop the same formalism of course in a baby form, needed for him, but for more sophisticated probabilities, they are not numbers, but these rings, objects in this ring of truncated polynomials, and this again, it's in Mendel's formulas, his formalism was exactly adapted to that, of course he writes it very differently, yeah, and so the point now I want to come to this, to the theorem, this is just kind of what goes into the proof of what I'm going to formulate, so I want to formulate this theorem, yeah, so far it was algebra, now comes analysis, and this is a theorem, so I have these maps, yeah, these are multiplicative endomorphisms, which I repeat how you get them, you restrict your polynomial to different kind of normal hyper planes splitting the space and extend each of them and then multiply it, right, so it's a polynomial map in the space of polynomials, right, so you have this, and each of them depends how you have set of indices, how you partition it, in each of them of this map is a retraction of the normalizing haplium, the one which corresponds to probability sum equals one, right, they project to corresponding to an other type of varieties, and so this is a very geometric map, they multiplicative endomorphism and you know the images, and then you take the convex combination of such maps, and this is exactly what happens when, how the genomes being formed for many genes, yeah, right, now it's the convex combination of these coefficients corresponds probability of exchange at a particular locus, at a particular gene, this of course is just, you have to look a little bit at that and see why it's just a formulation of that, and once you do that then the theorem says that this thing exponentially converges to equilibrium state, so equilibrium when kind of all components of rank one, those which are unchanged, those which are unchanged are exactly kind of of another way to say of maximum entropy, but now this maximum entropy, totality of that is not a number, but it's an element of this, of this tan k polynomial ring in a way, right, and this is one fixed point and the reason, kind of this theorem works, if you look at linearized map, it's just contracting map, and because invariant and scaling, if you contracted one point, right, and you scale inverted and contracts everywhere, so it's contracting map, so this contraction near, near fixed point automatically propagates everywhere, and this is kind of a very common thing of course and dynamics is everywhere, and I don't know if Mandel understood this, but this paper, this theorem of 20s, I think, quite old paper by these two people, so what they never heard of their name before, of course, then they will look when they look at the literature I found it, and what people do in genetics, so this is it, so I think it's very neat theorem, it's a map which linearizable, so the point again, essentially you prove the certain map has a unique fixed point, because linearizable by exponential map in the ring of truncated polynomials, and it's kind of very nice, very nice map, and so, so at this point I finish with that, and what I was here saying in both cases in, in, for Neumann entropy and the discussion that you can linearize thing, probabilistic thing, and then mathematics become much kind of more dimension, requires more dimension, which are not at all explored, so for example this Mendelian dynamics absolutely kind of untouched, if you look at people, right, they still after hardly use multiplication table, but it's obviously this kind of an algebraic geometry which is underlying it, and there is a much more general, and if you think about what happens here, that it engulfs many kind of general principles of probability theory, so for example the normal distribution law is in a way of the same nature, but very, very special form of this kind of theorem, right, so this kind of phenomenon we have multiplicative and demorphism in convex, we have algebra, some simple topological algebra, and you have this very special and important algebraic multiplicative and demorphism, and they convex combinations, and they give very interesting class of dynamics, which is dynamics you can understand, it's not kind of this mesh dynamics, which, you know, chaotic dynamics, which, I mean, you say it and then you kind of, it's pure first structure, for application I can't understand, so I can't understand, exactly what it has very specific behavior, such as the normal law, which is again a fixed point of certain renormalization map, so this is a kind of renormalization in a way starts from Mendelian dynamics from this point of view, but then my last lecture, next time, I will be explaining next round of linearization, it can linearize it even further than what I said, and then it comes in topology and becomes associated with measures and anthropists will not enter that because they haven't arrived at this point, but there will be kind of measure theoretic view on certain topological and mechanical phenomena, when linearization goes, goes, kind of, goes further, and you see things become more, more, more, more symmetric, so from certain point of view, again probability is enhancing the symmetry, and you can do it step by step doing it more and more symmetric with more and more kind of consequences. Okay, for today it's finished, oops.