And so maybe I'll be in English, otherwise there will be confusion.

And so the point which I want to make is that whenever you look in science, speak in the creative sense, it is always there is mathematics behind it, but it may be sometimes

invisible because this mathematics doesn't exist yet. So when you don't see it, there might be two reasons, either what you look because the mathematics is unknown, or what you look is not a science, but just something else.

And so I look at various examples, and the most interesting ones are inconclusive. I brought with me some preprints, which on subject I tried to pursue, which are not conclusive. I couldn't go far because the mathematic, which I think is needed very far from what

you know, but a couple of examples, you can go slightly further. And I will do in slightly different order from what is suggested in my resume of the lectures because of the subject which I want, so these are subjects which are kind of, I had made

some stats related to Mendelian laws, and now there will be something about entropy,

and a little bit, again, technical aspect, but not so much, it is a theory of space, which of course, as I say a couple of words about that,

and something will be less well-shaped, and I'll put them before I've forgotten, which I have on my website, but I brought some when I try to indicate possibilities of other things.

So in two subjects, which I believe there is much more interesting and profound development, which I will only speculate when I just erode some, and these are related to evolution, and this about, I don't know how to say it, learning, where all three subjects,

all five subjects are rather mathematical, and here I have pretty good, well, you know, pretty good justification. This is, I shall formulate and explain mathematically, though I don't pursue it too far, and these are speculative for that reason, I have quite

rather long articles exactly, because I don't know the answer, and you can only speak, go around it. So the point is that, in mathematics, whether internally or externally,

when it applies to something, there are two kind of faces to that, which is completely opposite, and just even within mathematics itself, and even more so when it applies to something so-called real, yeah, which is, what means real actually is a big point,

because just of those article based on quotation taken from various people, and so about reality, there is very good quotation, U.W.G. Pauli, who says that,

as you know, was working quantum mechanics, and he was saying that exactly the most, and the hardest thing to understand what reality is, yeah, you don't know what reality is, you make various physical theories, but how they relate to reality or to reality is questionable, and it's, and this is also true from many other aspects, yeah, so this kind of,

these two related to entropy, this, I'm sorry, but these two related to genetics, entropy enters in, and in here, essentially, we actually discuss it in the mock, quasi-physical context,

and well, these two, it's kind of biological, and these kinds relate to psychology. Coming to psychology, again, it's an interesting point about psychology, that there is a, if you look at the internet, and look at the section, and look for,

if psychology is a science, or it is pseudoscience, and there is a discussion, there is a very good argument by somebody saying it is not a science in a way science is, but it is true and not true, because psychology is sharply divided, though the division is not always apparent

into what we can call science, and what we can say politics, essentially, this is kind of practical subject, how people behave, how manipulate people, how people can do, and there are lots and lots of that goes on in psychology, and then the scientific psychology, which has completely different origin, and just scientific, but it's less kind of in a shadow,

it has no, and it's purely quite, quite, quite, quite, quite, quite scientific, and so, but indeed, what usually psychologists do, but science, psychologists, how can you, from a certain point of view, has nothing to do with science, mathematics, they are completely irrelevant, but if you look at scientific bicycles, and

this essentially psychology of animals, but up to some extent people, and this, of course, concerns behavior of people, and confusion is these people call themselves scientists, which of course, the question of terminology, and so this about, so yeah,

look at mathematical example, just say, I love two examples saying that, so mathematics has two aspects, and instead of that, one of that is miracles, and one is the kind of negation of them, yeah, and being kind of logical, by logic,

but again, everything is simply confusing in the language, yeah, language is terribly confusing, it's just recently somebody pointed out to me, if you have A and B, and whatever they are, I want to show that A greater than B,

and where I priorly I know B is the best, I know B is the best, but I want to show that A is greater than B, because B is the biggest and the best, and I'm saying say best,

in what, bigger meaning best, and you say, nothing is bigger than best, right, is best is the best, nothing is better than best, on the other hand, A is better than nothing, therefore A is bigger than B, and this is logic used all the time in speculative philosophy,

the kind of core of philosophy, this kind of logic, language is extremely kind of, and this is a sort of mistake of the language, it's kind of, it's the power of the language being so ambiguous and so tricky, yeah, but you must be very careful of that, on the other hand,

this is, we want to use it, yeah, so, but coming to, to mathematics, so what are miracles and what kind of logic, so this part is not of course accepted, not common, it just from outside mathematics, even inside mathematics, that mathematics is about impossible things,

they start from impossible, and then you have to misunderstand them, and two examples, many, all, and I maybe even make three examples, they're purely mathematical, yes, people sometimes smile at them, but, and because the same attitude, I look at this mathematics the same, I would like

look at this kind of phenomena, it starts with very naive things, and they're in a way, they're not at all naive, they're much more profound, everything, and historically, I don't know, maybe, as this example, like I was saying many times, yeah, yeah, two plus two equals four, second it will be the Pythagorean theorem,

and the third one will be Archimedes theorem about what area of the sphere is related to area of a cylinder, and this is, this was understood,

kind of, so what is miraculous about that, and this is rather recent, yeah, I would say, but actually some theorem emerged, yeah, I would just, in 20, only in the 20th century,

we fully understood implication of that, so, so what, what, so this I postponed, yeah, it is the trickiest one, yeah, right, so what's so special about two plus two equals four, yeah, it's very different, I'm saying, it's very, it is similar, actually, so already, yeah, and here is actually this related to this, yeah, yeah, this was

Archimedes theorem, and there is, this is follows, and this Pythagorean theorem, and of course, if you have something like three plus three equals six, nothing interesting here, but at least you don't know anything growing out of there, all right, so start with Pythagorean theorem, right,

and Pythagorean theorem, if you look at school, and you give a proof, and you think you're finished, and this is extremely, I think, confusing way being presented, because proving the theorem is a trivial aspect of that, and certainly the way it's written is completely, it's completely, completely misleading, because as you know, the basis of the theorem is

the essence of the theorem is Hilbertian geometry, shock breaks down, that you can define with the sum of squares, Hilbertian geometry,

and there will be fantastic unitary group operating there, and as we know, this unitary group is the essence of quantum mechanics, which actually linked to this theorem, and this theorem of Archimedes is corollary, by the way, of Pythagorean theorem in the second as I explained, and where the quantum mechanics enters in a more significant way,

which is a collapse of the wave function, so this theorem tells you about something about the collapse of the wave function, how wave function collapse the probability distribution, so this is what, if you think about that, not in a high school terms, or not in the 19th century mathematics, but in terms of the second half of the 20th century, right,

this is one of the way I want to emphasize, so the way you do, you relax, you think, as Eve said, Grothendieck would think about this, not as like somebody like Hardy would think, I'll come to Hardy in a second, this way I'll mention his name, from the modern perspective,

to a certain point, people in science were thinking in terms of multiplication table, and that's not so bad, and just great men were making, had made great contributions, but we can think in a different terms, and this has not been done systematically, you think in terms of modern mathematics, about very simple thing within mathematics and

outside mathematics, and you come immediately, you go somewhere, we don't know exactly where you come, so now do you know this theorem, why so great, but it was actually, this was explained to me once by Michael Ache, is that it is so remarkable about the theorem and what has

done to mechanics, it is a momentum map, this projection, if you project sphere to the interval, this map is area preserving, preserves measures, namely up to scaling constant,

if you take any band here corresponding to this interval, take the same interval in any other place, corresponding band on the boundary of the sphere will have the same area, and this follows from Pythagorean theorem, some squares being invariant, and this is a, so where it goes next, so the next way immediately related to quantum mechanics,

when you have n dimensional simplex, I'm sorry, so I have to push the buttons,

and these are the map from complex projecting space to the n dimensional simplex, so I put n here,

actually n you have to be, I hate to write down, this is where the n is, n is not indicator of power, and there is such a map which is, goes via the sphere, you start with

corresponding sphere of dimension n plus one, maybe I put, I hope I put the right dimensions, I'm sorry, sphere of dimension two n plus one in c two n plus c two, I'm sorry, in c n plus one, and so this is given by equation sum g i g i bar equals one, and you replace each coordinate by

this, and then you have, this will be a p i, and then you have sum of p i equals one, which give you simplex in the Euclidean space. There is another, and this is a, this moment map,

it's obtained, another way to put it, is the action of the n torus here, right, torus of dimension n plus one, x here, this is a quotient under this map,

it factors through the simplex, sphere goes here via Hopf map, it goes there, and the map preserves measures, measures preserving map up to normalization, and this exactly it is, this map is what is called a reduction of wave function to that, you forget the phase to the probability measure, so you, and this is of course extremely remarkable powerful thing,

just relating convexity and complex geometry, and this is inside, but this is inside with the theorem of, so Pythagoras theorem is the first instance of that, and it contains one gradients of the proof, so if you think about this in these terms, not like stupid formula,

but you understand what it is, it's only the first step, and then you look, you have toric varieties, and they have much more than that, immediately, well do mathematics come to you, so before you had this Pythagorean theorem, then that, and then there was two plus three plus four, and here I'm saying it's one plus one

equals two, because it's complex numbers which are involved of course, so complex numbers are here, and of course another way to think about complex numbers is to say that when you have whatever you mean being minus, and just you have a real line, that minus meaning rotation by 180 degrees,

which was this very simple thing was not understood too recently, for example, I think earlier I didn't quite understand that, you know, this was a figure y minus by minus gives plus, it was really big discussion, yeah, this geometric interpretation is relatively recent, it's post Illyrian, and so, and the fact that you can stop 90 degree, you have square root of minus

one, and this is a kind of quite recent, then complex numbers come, and just the whole incredible incredible thing come, but realizing that is not that obvious, I mean, it's too simple in this setting to be miraculous, but this is miraculous,

right? So what's miraculous about that? You can say how this form, you're so used to that, but if you think in terms what it tells you before you write an explicit form, it tells you, and I prefer it in three-dimensional shape, right, and this how it generalize, okay, you have a segment here, and you project it to three walls, so you have this

piece of stick here, and here projection of the shadow on three walls, you have three numbers of the length, and the sum of the squares doesn't depend on position of this, of the stick,

and this is a miracle, I mean, why they have squares, but it would be, and this is existence of, of, of symmetries within real numbers, and so it's unclear what is more fundamental, I was saying there were some respectful people were saying something important about numbers, and the definite statement was made Maxwell, who kind of said that the whole mathematics and

science are about numbers, essentially what he says, but within numbers there are the symmetries, and Maxwell himself implicitly contribute to the symmetry introducing this Lorentzian group, yeah, which was inside of his equations, but already this, this orthogonal group,

the fact it is there, it is in a way rather inexplicable, yes, of course we can say how you understand it, but then of course you have to learn, but what about that, so this is all kind of preparation for what I'm going to say about relation to sense, so what's, so, so miracle is there, and the point is that if you

think about this in combinatorial terms, it two plus two equal four is not just this identity, it decomposition of four elements set in two subsets in cardinality two, and you can do it this way, this way, right, and this way, there are three ways to do it,

if you take for one it will be one, for two it will be three, but if you take say three plus three equals six, here it will be something about ten, I guess,

in how many ways you can decompose, and so essential thing that three is less than four, it's here is about complex numbers, it's invisible, it's too simple to see it in this model, it is of course there, but it's invisible, but these are, it's visible, because it means

if I have four element set, I can canonically associate to it three element set, namely the set of this decomposition, which means there is a homomorphism, permutation group of four elements, the permutation group of three elements, which

was used in algebra since, I think, Ferreira, who proved it, or Cardano, I'm confused, who proved the equation of degree four reduced to equation of degree three, because this group is not simple, instead of the, well, if you divide by centiply

in evolution, right, this, this is non-simple group, and the only one, all other groups here are kind of simple, kind of, yeah, but this is what has this homomorphism here, but then if you look at the next level, if you think about these groups as a wild group of compact

group, it tells you that the group, or rather universal covering of SO4 splits into the sum of two SO3, I mean universal covering, so otherwise you have problem with spins, yeah, and from that

it follows that, or rather of the corresponding property for the algebra, follows that certain operators split, and which implies that the existence of the self-duality equation,

if you, if you look at the equation, variational equation that minimizes square integral of the curvature over bundles over four manifolds, because they're four manifolds, this corresponding equation similarly to the Laplace equation and two variables which splits into the

square operator, this splits and, and, and give rise to the first order equation, which is gauge equation, and then there is Donaldson theory, and Seiberg-Witten theory, and the whole world of mathematics opens, because this absolutely impossible happens, yeah, this equation not supposed to split, you have general polynomial to just the chance is being square of a smaller

polynomial, yes, need, however it happens in this case, it happens to be squared, right, because of this identity, which come from this identity, because this group is outgrowth of the permutation group by, by, from kind of construction of the group, and then kind of,

well, just talking, I don't have to advertise much for that, now I want to use similar logic when it comes back to science, but I have to leave this thing up here, all right,

so there may be the kind of miracle you think, and then you want to understand them, to put them, develop them, and put them into proper context, and hoping something happens,

right, and as I said, there are, so this example just I want now to look at from, from this perspective, now if we forget about this, now start, this is kind of most

pronounced, because it's exactly, has this flavor, so what's, why is this a miracle, yeah, and this is one of the reasons why kind of Mendelian logic was not, was not accepted for

quite a while, so the history of this is a kind of, kind of amusing, there were two points in history, or three points in development, right, so one was, yeah, yeah, I hate, so when I write French names with mistakes, so I just say, yeah, so it was M1,

and then it was Mendel, M2, and then there was a third moment, I know many people involved, say Morgan was the most essential, I could hear Morgan, okay, involved in, in, in, in development of, of genetics, so he first observed, he was looking at people who had six fingers,

you know, sometimes people go with six fingers, and have, in certain families, and he observed proportion, and I think he observed that their ratio is close to one to three, and then he was trying to make mathematical theory out of, out of that, which was, was erroneous, but it was certainly in the right direction, and then Mendel was doing something for people, and so he also, this

proportion appears, and he developed what I'm going to explain in a second, and then there was a next level, yes, how Mendelian dynamity breaks, and it was Morgan who developed kind of

background for that, essentially there were some people who discovered, but essentially it was Morgan, and then his student makes kind of remarkable discovery, again remarkable logically, who discovered new way to think about geometry from mathematician perspective, right, there's a linkage of genes, and, and they're all, and so this is, the fact that this

two numbers appears is rather incredible, you can see the certain features, and observe that there are proportion which appears one to three, sometimes, and it's, of course, it's nothing in biology is like that, it's probably 1.3 versus, let's say, 2.7, but for biologists it means one to three,

and these are quite, quite kind of right way of thinking, yeah, so there are these integers appearing, and that's, you know, quite, quite, and, and interestingly now that this similar thing observed by, say, by Darwin, who overlooked it, and that's again, just as I say, it's quite amusing,

amusing, so, so what was happening, so you see that this is at a formal, but this is a biological phenomenon, that these numbers appear in real systems, and in many kind of biological systems, when you look distribution of certain features, and they appear in proportion one to three,

approximately, but amazingly always kind of closer to one to three than something else, and biologists of certain, of the time, who looked at this, except for Maparti, who was not a biologist, Mendel, who was not a biologist either, by the way, but, say, Darwin himself looked at this and just couldn't make sense of that,

actually rejecting it is something irrelevant, and interestingly enough, say, in Russia there was who also rejected it as algebra and not, and not, and not biology. On the other hand, selection principle, this evolutionary championed by, by, by Darwin has absolutely nothing biological,

it's pure mathematics, but because so simple, I mean, because exactly it has no content, biological is so easily acceptable, right, just kind of trivial regardless whether right or wrong, and this is something incredible which needs, needs explanation, and so

Bose, actually Maparti and, and Mendel has right ideas, but Mendel was much close to the truth, so, so what is coming behind it, just, I just want to say the consequence of that this is already kind of strange, but then if you follow in the logic of the Mendel paper,

I read quite a while ago, I don't remember if he said explicitly, but the conclusion is as follows, so let me give specific example, you have two fields of flowers of different size, and here everybody white, and here everybody red, and generation after generation, here they're

only white, only red, and they're the same species, but they separated by amount of all, and at some moment it disappears, and there's lots of insects pollinating both ways, and so at the moment it disappears, something happens, and you have here both white and red,

and here both white and red, and proportion changes, you have certain proportion times killed, different from the original one, of white and red, and then people like Darwin around him, say aha, there was national selection, sound and beating, for example, you have proportion was shifted toward white, or it was shifted toward red, whichever,

it changed, there was a selection, you continue, you take next generation, and nothing happens, it remains the same, this process stabilizes on the first step, and then can continue being stable, it's not exactly that, because there are all the fluctuations, of course, but this

essentially what happens, and this was prediction made by, by explained kind of, by Mendel, for whom it was kind of obvious on the basis of his understanding, and he was developing some algebra, but then till 1908, so Darwin was discovered in about something 1988, or something, yeah,

people discovered his work, but they, not his work, they repeated his experiments and came to the conclusion, and then they just started referring to Darwin, this also, and also interesting, why they were referring to Mendel, I mean, they discovered them by themselves, and they pretended, kind of, thought they didn't know about Mendel,

though his paper was kind of published in a quite readable journal, but the point, there were three groups of people who discovered that, and they start arguing who was the first, and then immediately, of course, they looked at the predecessors and found Mendel, this is how they, this was how Mendel, they told, this is the only reason we know about Mendel,

because people were fighting for, for, who was the first, and, and then they were just doing that, and then a red Mendel, and realized, which they, they were, of course, not that smart as Mendel, that there is a strange thing, that it formally follows this kind of thing, and of course, observing in the nature is not so easy, but this certainly

is counterintuitive, why? You have some natural transformation, and y square will be equal to itself, yeah, so somebody is important, and then there is a probably apocalyptic, maybe not story,

about Harje, that was playing cricket with his biological colleague, and this, I've forgotten the name of this person, who was asking him about that, and Harje immediately figured out what happens, and so, and then he wrote half a page in nature, and essentially what he was saying,

that if you kind of, you just can't read it, basically he was saying it was kind of mathematics or multiplication table type, essentially saying the discriminative quadratic equation equals a a b minus c squared, or whatever, and he wrote these formulas,

and he was kind of showing his disdain for triviality, what he said, because he didn't understand the total mathematics of that. In my view, he just, he wrote, you know, just people of that time, as we like, probably nowadays, like Harje, they were so smart, they immediately see the answer, and then he would never think next, because they were so clear to them, but,

and so now let me explain a little bit that before we go further. So what is mathematics behind it? In fact, I would say it's quite trivial, dynamics involved, and it is extremely paradoxical to exist, it's indeed very strange, because the map we shall

describe will be essentially polynomial map, and you have a polynomial, and you substitute polynomial, polynomial, degree goes up, so to have again the same p is kind of rather impossible, right? When it has a proposition of polynomials, to have polynomial of the same degree is impossible, you may say, because degree must go up. However, this what will happen in this

example, and this means that it's something rather exceptional, and rather remarkable must be happening, and if you even before, you just again, you, yes, this, whatever Mandel was writing

it was algebra, so his algebraic formulas, so they were essentially polynomial, they were normalizing constant, which is certainly quite essential, we come back to that, but now it cannot happen in principle, the composition of polynomial will be polynomial, the degree will not go up, just again, imagine, I want to emphasize asking this question before you understand anything,

and where they would bring you, so let's ask this question in the following terms, so imagine you have space, linear space, say real complex whatever numbers,

and you have a, we have polynomial maps, such that when you imagine also, yes, now we are in the context of groups, so imagine they just act, the group generated by this map x transitive here, and they all get make finite dimensional space, so degree doesn't go up,

so another way to say, you have a Lie group, you have, I'm sorry, you have a map say from Rn times Rn to itself, which is a, it's a group law which is polynomial,

is it possible to have, and so what will be the answer, and the answer, of course we know it is a plus b, or x plus y is a linear map, are there others, and of course you say how, of course not, as usual it's too much, but no, and there it says, no it's very simple, it's exactly all nearly

important Lie groups, so all nearly important Lie groups, this is conception comes, it's linked to this phenomena discovered by Mendel, and certainly nobody thought about that in those terms, because nobody would think in those terms of groups, but I'm saying this phenomenon is the same which comes to life in the first of the three dimensional Heisenberg group, which again

essential, as you know, in quantum mechanics, now the mathematical structure again, just once you have such little miracle, it just, you know, it's, it's, you know, this exactly I'm saying how do you ever look, for him it was so simple, the proof was so simple, it's like two plus two equals four, it's simple, but not the question of simplicity, but the question of

impossibility, it's something impossible, it means something happens, and you have to go to the bottom of that, and I shall describe, I shall describe, probably not much to do, I start today, this thing concerning, concerning this map of, of, of, of Mendel, so, so what was, so sorry,

I'll start with, I'll a little bit start with that, and then, and then go to the next point,

so what was the idea of Mendel, where this one, two, three come from, again, a pretty similar idea, but I think, I'm not certain exactly, but he's hard to read, yeah, he wrote his text as a shot, and rather kind of generic, and so that is not quite clear to say what he meant, but apparently he meant something else, though in the same spirit, and the Mendel, which is how we read him today,

was as follows, there are these genes, and in that particular, whatever they are, but they represent features, in that certain features represent, representable by genes, for example, if you have flowers, and they always either red or white, and there is nothing, no mixtures, no pink, no mixture,

no, so there is a gene responsible for that, it happens sometimes, again, it doesn't happen often, but even happens rarely, you see, it is remarkable, yeah, it must be understood that this shouldn't be so, you accept from general physical principle, you accept mixture of all things, any purity is there, and this discontinuity, this exactly what makes, made mistake of Darwin

and all people following him, they believed in the principle of continuity, coming from Leibniz, so Leibniz was insisting that everything in nature is continuous, and this is, of course, was moving him in development of infinitesimals, and it was a very powerful, very good idea, and

this idea has kind of an amusing history, but well, in Western culture, and, you know, many things are continuous, but some are not, yeah, and but Mendel has kind of a different idea, he discovered the discreteness, discreteness of

inheritance, and how his point was that if there are the sub-separation, so there is something responsible for that, it's called gene, something quite abstract, and moreover, this gene is function of two variables, whatever this abstract variables,

right, the symbols, but that's kind of, we perceive them as polynomial variables and write this AB, so genes composed of that, and then there is a function,

phenotype given by AB, so it means that what we see is not A, you don't see B, you see function of two of them, and this again, a fundamental point which completely was missed by Darwin and all these people, yes, this was an essential part of inheritance,

which makes everything possible, without that all this evolution, of course, she announces, as it was presented before Mendel, and then in this particular example of flowers, this function is color, yeah, of AB, and this color may take two values,

it may be either red, or it may be white, and in this particular instance, this was again conjectured by Mendel, its function is very special, it may have, first you believe this function being symmetric in AB, and you think about this gene,

this composition AB coming from your parents, one A from father, B from mother, vice versa, you assume it being symmetric, which is again sometimes true, sometimes not, but again, even sometimes very symmetric is quite remarkable, because very strong properties, again many think we assume kind of,

in this, unthinkably, because we're used to them, but in order to see how impossible they are, you have to have more general perspective, and this may only come from mathematics, because there is nothing else, yeah, this exactly difference of mathematics in common sense, in common sense we try to associate things with those which we know, in mathematics,

we try to produce things which don't know and relate to them, in this way common sense usually nonsensical, right, because new things don't, cannot be reduced to what you know, and so, and this function is as follows, that you have this A and B, and the rule assumed by Mendel that BB will be white, this function color,

and for all others it will be red, so if there is A present, then it will be red, maybe it's better to change my notation and call them from the very beginning,

these two components W and R, and say this color equals red, unless we have double white,

so that's the rule, and just if you assume it, then, so why this assumption is easy before you do any mathematics, purely even purely qualitatively, it is quite, quite, quite significant, because if you have white parents,

flowers, all their descendants will be white, but if you have red parents, their descendants may be either white or red, yeah, I switch from A and B,

I, I, I switch from A and B to white and V and W, that's easier to remember, A and B is a direct notation, and this specific notation for this example, so because I, yeah, I never remember who is white and who is red in terms of A and B,

this, this corresponds to A and this to B, okay, and just change my notations, just not to forget who is who, but this, this annotation, they're not colors, and then they represent colors are function in two variables, like colors what you see, yeah, these are names for genes, for alleles, yeah, whatever they call, but,

so, so there are two, this, this is a function, and this is where it takes, and these are abstract variables, and just I use these letters not to confuse them, and if you have red, if you have red parents, then they may have

descent as both white and red, however, if you keep selecting red, always taking red, and red, and red, eventually you arrive population will be pure red in all generations, and this of course what use people in artificial selection for, for, for

about seven thousand years, yeah, this estimate when people start domesticating plants and animals, or ten thousand years, something like that, yeah, when domestication started, we don't know exactly, but safe order between ten and seven thousand years, say ten thousand years, if you start selecting one particular feature, eventually all descendants will have this feature, if it's simple enough feature, like this red, so, and so this is a preliminary part

of the experiment, so if you want to make this experiment, this is how it's done by Mendel, then of course by Morgan, you select this pure organism, always they have the sentence of the same type, and then you start mixing them,

and then when you start mixing them, you observe this one-third ratio, right, and then the, so, and then explanation again is quite, quite simple, so we have white and red, and you randomly start mixing them, yeah, you may have either white-white, white-red, red-white,

and red-red, so this will be white, this red, all these will be red, and so there is this ratio one to three, so you assume that everything which may happen is equiprobable, all things are independent, so again this is exactly what kind of was the point of Hardy, how he was

arguing, of course his computation was obvious to him and to anybody, did kind of multiplication table, in the second though I see it is not that kind of innocuous as it looks, but the point that you can represent everything by this kind of maximally symmetric situation, right, to say

that some variables are equidistributed, so you have unknown quantities, and fantastically the right assumption inside is to say they are equal unless otherwise stated, you have some numbers coming from some process and they're of the same nature, first conjecture is they are equal,

which is mathematical episode, genetic numbers are not equal, but the right conjecture they're equal, and then just say they are independent, which means that the appearance of them in pairs also equal, so extend this symmetry to the next step, and again this is where probability theory starts, it starts from making assumption of something being equal, which is, again, looks

opposite to what may happen, more likely to numbers to be not equal than being equal, but in fact we assume they are equal, and there is some reason for doing that of course, maybe not common sense but mathematical reason, and once you make these assumptions

you arrive at this one to the three ratio, and then instead of observing if it actually happens and how plausible it is, and then these interesting controversy which I don't follow, I haven't followed quite but would be interesting of course to look more carefully, there were

Fisher, Robert Fisher, who was a kind of major figure in statistics and in genetics, mathematical genetics of the theory of the beginning of the 20th century, who on one hand was a great fan of Mendel, but he believed that Mendel falsified his data, that it didn't follow

probably in the real world, the approximation of course with this noise, and Mendel kind of selected good examples and just his conclusion was not justified, and certainly he was professionally, technically much better equipped than Mendel, he developed really

modern statistics very much, depends on him and later on I indicate some one remarkable discovery he made purely mathematical related to the entropy, on the other hand recently I read an article saying that Fisher was wrong and he was making mistakes in mathematics which Mendel didn't do,

because he was statistician, no matter how he was good he was statistician not quite mathematician, that probably there is, I don't know just if you can read it on again on the net articles about Fisher and what he wrote, anyway they say guessing this proportion was not at all obvious because it was marginally true, you had to because there was lots of

noise in the data, however you have it, and now why this would imply, why would this imply this what is called Hardy-Wagner rule, though it was of course non-dimensional, that

so you have this kind of picture, you have two units determining the gene, right, W and R, and their pairs, yeah, so this the distribution of their pairs, so population

given by distribution of their pairs and then when you take a random matching, so mix everybody together, it may change but it stabilize in the second step, why the re-stabilization, right, and let me explain to say go ahead of the explanation, so what is the mechanism of that

which was in my view overlooked by Hardy, of course you can write these formulas, you have polynomial, there are, it goes schematic, yeah, there are three different types, one, two, three, but these are equal, so it appears with coefficient two, and Hardy was writing formulas which I

just, you know, don't remember from high school, but quadratic polynomials, which kind of all this kind of nonsense, you don't need it and secondly see it, yeah, you don't have to know this multiplication table, it's completely, you know, kind of irrelevant but also irrelevant, you understand much better if you forget about that, and here I'll write these formulas and what

he misses is, now let me explain what he misses, what is the interesting map and what kind of mathematics it brings in, and you can ponder again about this mathematics, which immediately is also some of them full of unsolved questions, yeah, so yes you

imagine don't quite know much about what was happening, but you know it must be a quadratic type of map, we have entity here, entity here, they mix together, so we multiply some quantities,

so the same sort of quadratic map, the square which must, must, must be, must be identity, so let me show you such a map and then we shall, later I shall explain why it happens here, and the map is quite simple, we define this map on matrices, where entries are

numbers, by the way what is a matrix, I challenge anybody here to me to give a definition of matrix, you know people say matrix, mathematicians, but there is no such object, mathematics of course,

what is the matrix, anybody can tell me what a matrix is, mathematically I mean, not just for mathematically, what is the matrix, square table, you know what, written on the blackboard by chalk or, this is definition, what this is matrix, and push here numbers, yeah,

like you know, it's interestingly enough, yeah, it is not at all, you know, it's not simple, we can't, you cannot say it because we don't know it, it's not because you just, well you know, well you just know how to say it, you don't know what it is, okay,

let's work again, it's like exactly, you know, greater, better than nothing, we just have confusion of the language, we're confused by images, we don't know what matrix is, but still we use this because it's so convenient, yeah, so we have row and columns, so we take a row, we take a column, we take all elements in the row, take their sum,

take all elements in the column, take their sum, multiply them, you put them here, and this is polynomial map on matrices, what I, in order to, I want to normalize it, so it will be not quite polynomial, it will be rational, I divide by the sum of all elements,

because these will be probability distributions, and all you can see the matrices with sum equals one, and I divide by the sum to have always probability, and otherwise it wouldn't work, right, and so, so on the space of these matrices, so say

pq, p elements here, q there, I have the self-mapping, and square of this mapping equals two, that's the fact, which is

one of the way to express this, this, how do you write the theorem, and this how it works, yeah, so we had kind of here for others, here we have mothers, and we don't even assume symmetry, right, they mix their genes, this will what will be new

distribution, after mixing their genes, we repeat it twice, nothing happens, of course, this again you can do it kind of without thinking, yeah, you try to imagine the simplest possible kind of combinational thing being multiplied, and this will be of course the simplest possible thing, yeah, you can see all possible pairs of objects are matrices, you have,

and this kind of operation kind of foster you by pure logic, and in a way, if you think this what logic is, it is some underlying symmetry of the problem, built-in non-trivial symmetry in this problem, and then, and so, but again, this is a,

so Harvey was doing this for symmetric matrices two by two, so subject, I think he was not, his notation was like that, no, it was p, q,

I think it was r, p and r, something like that, he was doing it for this matrix, and taking a writing formula in this case, explicitly in terms of p, q, r, and you have some kind of formula, but this is all always true, so what is kind of

interesting about this map, yeah, so once one property of this, as I said, it is a rational map, because you multiply in fact by some of the elements, so degree goes up, but because you normalize, degree remains, so defined in fact in the projective space,

and it's, so you multiply a product of two projective spaces to projective space of higher dimensions, and instance of that is slightly, slightly, and this map was discovered by Veronese, and I forgot, and there was another name attached to it, Segre, I guess,

and so let's just to have some feeling about this map, it's quite remarkable map in many respects, so if you just look at this, yeah, if you get a world of possibilities, a world of mathematics, if you look at this in geometric eyes, not like multiplication table, this by the way again, you can prove it by computation, I mean it's kind of easy, but you don't have to compute again, to prove it, you can prove it, it's seen from general principles, right, from nothing, if

you, this is still not the ultimate formulation to which I come, but I want to, I want to describe it in slightly better in one special case, and immediately see, so what, what, what associated to this,

and just look at it slightly again, just my point is, you don't try to literally understand what is there, just follow the spirit of that, right, and so from something

of strings, you get something quadratic, and very simple instance of that, if you have a linear form, say on the Euclidean space, linear function, take a square, you have quadratic function, and it means the space of linear function, which is Rn, goes to the space of quadratic function,

which is Euclidean space of dimension, if I'm not mistaken like that, maybe slightly more, so you have such a polynomial map, you normalize it, in a second I just give you one example, you don't remember the numbers, from R3 to R5, and you get, if you

stick it to the unit sphere, you have a map from the sphere to the five-dimensional space, where it is contained in fact in four sphere, and the map is symmetric, so actually give you

factors of a projective space, so it gives you a very nice map from the sphere to five space, and in general, so it's highly symmetrical, we run as a variety, it's five, and what kind of you think about this map and just try to understand it, and I just want to say,

so again, the map is very simple, you just take kind of, co-vector linear form normalized to norm one, take a square, become quadratic form,

so it's space of quadratic form, it's a five-dimensional space, and just there will be again normalized, so sitting in this sphere, so how it looks like, so it's two spheres, space or in general, it's highly symmetric, so it's invariant under the action of the orthogonal group, so it's completely everywhere the same, there is a isometric group, so it's orbit

of a symmetric power of the of the standard representation, right, so you take orthogonal group acts on linear forms, take symmetric power, so it acts in quadratic polynomials, and so there is a particular orbit which is projective space, but

we think about this sphere map there, but of course it's an image of projective, projective, projective space, so it's fully symmetric, you may ask are there other kind of object of that type, can you measure another surface in R5 with this symmetry, of course surface itself must have comparable symmetry, the only surfaces having

contingent full symmetry are tori, so you may ask if he can have torus sitting there with a comparable symmetry, so here the symmetry is fully orthogonal, and just we just relax it

maximally, but still we want to say it, we say aha, it will be symmetric, but only in the weakest possible sense, which is still relevant here, so we have surface that's locally measured, so it might be everywhere at the same point in five space.

And you have another such piece of a surface, and so we say they're kind of equivalent. If there is a fine transformation moving this point to this point, which preserve tangents up to second order. You always can make tangent space goes to tangent space, you always can do it. There also one second derivative goes to second derivative.

Can you have this kind of symmetry? And it will be not degenerate, namely not sitting in a small space. So here, of course, you can do it because it's fully symmetric. But this is minimal, you may expect for the two torus. And these are called free maps. So it is a question is, is there a free map of the two torus in five space?

In simple terms, it means that if you take partial derivative of these maps, so there are five derivatives, two first derivative and three second derivatives. There are vectors here. These vectors should be independent. So this map f, if you take f, f1, f2, first derivative and f11, f11, f12, f22,

meaning derivative with respect to this coordinate, there are five maps. Five vectors here must be independent everywhere. If such a map existed, again there is a kind of story related to that

because there is a paper, famous paper by Moser, and just as one of the most cited papers in mathematics, when he generalizes Nash and police function theorem, and as example he says, take such map,

and when we read this article, we couldn't construct this map. We studied this map. And then I met Moser and asked him, do you know about this map? He said, no, no. I don't know. I never thought about that yet. And still, of course, it's unknown. Such map exists, yeah? So, and from point of view analysis material,

from point of view geometry, just everything has this paper, and Moser came infinitely simply into this equation. It's a really very hard question. We don't know if such map exists or not, right? And again, just my point is you start with something like genetics. Can you go a little bit further?

Yes, ask simple question, and you arrive at very profound mathematics, and very difficult mathematics. Another point about the Spheronese maps, the Spheronese varieties, but they are remarkable from another perspective. And again, the way it was idealized relatively recently, so they map this,

so the projective spaces inside of Euclidean space of dimension, I think, maybe this, maybe something else, yeah? If five equal two plus three, you know, no, no, something, not quite exactly this number, yeah? Maybe plus minus one or something, yeah? The dimensional homogeneous polynomial of degree two in n variables.

And there was another problem for quite a while, and it was due to Borsuk who had proven, I believe it was Borsuk who had proven, any subset in Euclidean space with diameter one, you can divide it into four pieces of diameter less than one.

Say, in the plane, say you can divide, say circle, you cannot divide in two pieces of diameter less than one, you'll be exactly same with one, but you can divide it into three pieces, yeah? Of diameter less than one. And any set in the plane divides into, it's easier,

into three pieces of, and then in three space, I don't remember whether Borsuk or somebody else proven, you can divide every set of diameter one in four pieces, and there was a conjecture for high dimensions, and then it was disproven by Calli, and these are exactly counterexamples.

This set, if you look at them, it's obvious more or less, you need roughly exponentially many pieces. We divide them into smaller pieces, they have quite remarkable geometry, very, very simple, but very complicated set of opposite points. So it depends how position the points of diameter one, and these are also counterexamples.

So, I mean, these quite remarkable objects in many respects, they really contradict the naive mathematical intuition, and in this kind of Mendelian conclusion that something may stabilize in the second round of reproduction. You make selection, you do something, and then nothing happens afterwards, everything stabilizes.

You see, it's really quite remarkable. Now, I said how to go without computations, what next level of mathematics. So what about matrices, yeah? I had to explain what are matrices. Again, I'm saying there is no unique concept of a matrix, that's the whole point.

In different contexts, they have different things. Now in linear algebra, I want to define, what do you mean column and rows, yeah? It's just a range, a bunch of numbers. But how do we think in modern terms? Modern meaning late 19th century,

in set theoretic terms. So more modern terms will be categorical terms, to which we turn. So we look at the set of all these objects, you know it's a vector space. But it's a vector space, but also, and this is what you're doing with that, we're taking some of these elements. So, it's a vector space X,

together with a linear function, from X, distinguished linear function to you, yeah? So it's a linear space with distinguished co-vector. Okay. And that will be for us, rows, columns, whatever.

So I call them space, and I make this, this kind of scalar product with one vector. It's kind of not scalar product, but just linear function. For this you can take tensor products, because you can just multiply these functions.

So it's closed under tensor products. So these tensor products are matrices. Right? Because again, now they have this tensor product decomposition, and you have it. Now, what is this operation in these terms? So when you have this tensor product,

what you can do, you can go to X. You just take vector this. So, so, so how, how you do that?

You take X times Y. Right? So if you have element here, you just take this, because you get, this will be number, right? Right? This is the value of your linear function. So you can project X, Y to X,

and you can project it to, to here. So in general, tensor product, you can project it to its components. The reason I think is projection. That's one problem in quantum mechanics. You cannot go to subsystem. You have system, you have two spaces representing your different quantum.

You bring them together, become tensor product, you cannot go back. We need to say they're entangled. But of course, mathematical says there is no canonical projection back. There is no canonical map. But if they come with this kind of distinguishing, you can do it. Now, so what is our operation?

I took this projection. Now I take this summation and this multiplied them back. And then it's the formula because it's kind of obvious. Yeah, it says that, it actually says how you put brackets, yeah? So if you have to, so you first project it,

and then take the tensor product back. Now, but because on the second round, your object already is a product, so it's meaning it being product, right? When you repeat again, it will be the same product. Because if it was from the very beginning, it was X time Y.

And now we apply this operation. It will be X times Y, and this will be X time Y. And if you take the tensor product, you get X times Y again, yeah?

Because you normalize, I'm sorry, when you, this make projection, you normalize with this sum, right? So this will disappear because it makes normalization. So this term disappears, so it remains the same. So square will be itself. There is nothing into computers. It's fully tautological.

There is no computation or no discriminants or nothing. All these are artifacts of four notations of 19th century, right? Actually, you know, Mendel had a pretty good education. He was studying physics with Doppler, or you know, for Doppler effect. And then there was another thesis less known,

but who was trying to develop combinatorial description, combinatorial models in physics. I forgot here all kind of books about combinatorial, I forgot his name. Combinatorial models in physics, and from there, partly probably Mendel was inspired to make this thing in there. So this is what we have here, and this is completely kind of simple.

And so just no computation involved. I mean, just pure, pure tautology how it should be, right? Simple thing might be simple. Yeah, they don't. Computation, sometimes it happens, of course.

There's some unexpectedly out of complicated computation. But it's so rare, I mean, it's a double miracle. So we don't usually expect it. Now I want to make little turn. Maybe I want to make five minutes break because now I'll turn to entropy. And so let's make five minutes break.

Now who wants to go? And here I have some, very few of them, papers, one of the thicker one where I speculate on something which I cannot prove. We'll come back in five minutes.

Okay, so I say just a few words, continuous, so on a technical level I shall go into this in the following lecture. So the essential map was that I have a row here,

I have column here, and I multiply them, and I have this kind of matrix. And this is a map from product to linear spaces. Today, this is the standard map corresponding to tensor product, which has quite, quite, quite amusing properties. And there are several, several layers of that.

And so to which it will converge, not today, but I just want to say is the following, when you start again, relax and generalize, maximally you arrive at the situation of the following type. When you haven't, say, community of algebra, topological algebra,

for example, maybe in this example it will be algebras of truncated polynomials. So you can find dimension algebra, maybe more general algebras, and you do the following thing. On one hand you have algebras and you have some number of endomorphisms.

And it will be quite simple. Here you just restrict your polynomial to some subspaces, and then extend them, extend them by zero whenever possible. Then you take multiplicative endomorphisms, which will be certain product of this phi j.

When you multiply them, you get multiplicative endomorphisms. And then, thirdly, you take the linear combination, certain linear combinations. So that gives you a class of transformation.

By now they are not endomorphisms of these algebras. So in relevant examples, there will be this rational transformations. There will be rational transformations of polynomial transformations in the space of truncated polynomials.

But this is quite interesting also for such algebras like algebra of function with convolution. For example, in generalization with Hardy-Wein principle, that in very many cases such a transformation, many cases which are explicitly describable,

usually describable in terms how this endomorphism act on the spectrum of this algebra. If this dynamics is simple, especially often, these dynamics are simple. And simple I mean that they have a fixed point, either attractive or where a repelling part can be described,

where the simplest and most famous instance of that is the normal law. The fact that when you convolve, you convolve a random variable with itself many times and normalize, you converge to normal distribution is instance of that. So and Mendelian dynamics is another instance of that.

And there are other example in genetics, what's so-called formal genetic because real genetic, of course, nothing is so simple transformation much more complicated. But still the core of that is and this happens in again, if you look at your model situation more general than the one

which I described, namely, you have not one gene, but you have many genes, it's one point. And secondly, when it is not deployed organism, when there are only two copies of each chromosome and organ, but several, you arrive in this more general picture, when you have this endomorphism of tranqate polynomial algebras,

this kind of maps and then you can prove that there are these fixed point properties similar to the law of the normal, the central limit theorem, whatever it's called. So this is one line of thinking, but there is another one much more in the same respect,

which I just was mentioning. And this you associated to the name of Morgan's instrument. The mathematics is much more kind of less well shaped and for more interesting.

When you remember that from these statistics, in this statistic you encode or you construct also geometry of chromosome. So we know today by molecular biology that genes in DNA comes in strings.

They are polymer, they are one dimensional molecules and this is the most essential feature of life. It's one dimensionality. This where mathematics enters for some reason, life is one dimensional, right? It's encode by something one dimensional because chemically it's impossible

to make something heteropolymers multi-dimensional. But before it was discovered experimentally in molecular biology, it was shown by Sturgevant by Mendelian type of experiments.

So what is Mendelian which was done by Morgan, who developed all these techniques and developed overall picture by making experiment with Drosophila. They are small kind of flies which you can breed them very fast. You develop, you make these pure breeds. You have certain feature being repeated and then you start mixing them

and you see how different features are being distributed, right? So look at these statistics and from that Sturgevant concluded that genes lie on the line. So these features you observe, you observe typically the shape of the wings,

the color of the eyes, three more characteristics and the stripes on their bodies. I think these are three first things which he has done and you can say the line from the line and that one of them in between the two. So the basic relation here is between. Line is not a metric line nor it is ordered line.

It's a between relation. So and you can say which gene lies between other genes. Though genes were quite abstract and Morgan himself was emphasizing that it's a material for what is the nature of these genes. Whether the water they are kind of molecular nature. There may be transcendental features and what he says is that between the genes,

which the gene is studied and the phenotype you observe, there is a world of embryonic development and so immensely complicated. So it's completely irrelevant what you know about genes, you know, to say how they are related. And this still almost true today.

And this is by the way, he was hated I think in Soviet Russia because it was considered an idealistic statement. Because he didn't care about the material nature of genes. Though he perfectly of course understood that there might be some hidden molecular biology but it was not his business. And Sturtivanko was a student at that time.

He was about 19 years, 20 years old. And he made this conclusion from the experiment in this what's called linkage by Morgan that there are these linear structure in genes. And these are things quite remarkable that you look at statistic of something and you can reconstruct geometry.

And the only example I know where it was considered before, maybe there are others. And as you point correct, and in one of his books, a forgotten issue of them, either it was science and method or science hypothesis. He wrote these two books.

He looks at another question. He certainly didn't know about genetics. But he speaks about geometry. And we know the world we live in has certain symmetry. And the trick in symmetry is this one. It's a simple group.

It's a very complicated object. It's an atom which is not divisible into anything. Unlike other, the first truly kind of new phenomenon in group theory. It's not compact, simple, first compact simple group.

Yeah, if you forget about this. We have taken double covering here. And how we can manage to reconstruct it. We live in this world and we turn around, but we see the same world. How could it be? How can we match images here? You turn and you see the same thing. Because what comes to your eye will be nothing.

So you have here, you moved it. And these two functions are not close. And it's again people who are analyzing images. Until recently, they just could not even conceive that because they were naturally reminded. They're always comparing images by taking their difference and looking at the difference.

But you move this image to this position, they're maximally separated. However, the same image. Only for the last 10 to 20 years, people arrived to this understanding that Poincare had 100 years before. So how it can be? And essentially, Poincare suggested the same way, the same model,

the same kind of presentation of geometric structures. So I just say it and then we will elaborate on this later on. So what is geometric structure? So without saying what structure you construct, what you see and how you want to reconstruct it, yeah?

So you have a set with geometric structure. And you don't know what structure is, but what you know, right? For example, look at the images. For images, it's kind of clear, right? Forget about dimension three. Just look at dimension two. And imagine you are shown images on the screen,

but the screen itself has no geometry. So you just screen and then a pixel and then somehow enumerated. So you have like 200 by 200 screen. Let's simplify it. Let it be 1,000 by 1,000.

So I have two to the sixth, two to the sixth entries. And then maybe I add the black and white. So we have a sequence of black and white, enumerated in a certain way, always enumerate a pixel. In each image, it's a million of dots.

Black, white, white, black. Imagine you have a million of such pictures, right? Each of them is a sequence. This is what comes to your brain, yeah? Your brain is essentially linear. There is no kind of dimensionality in it. So the problem is as follows.

Again, I repeat. You have million of strings. Each of them, well, maybe a million just not to confuse numbers. Imagine you have more of them. You have 10 to the seventh strings of the length to each 10 to the sixth, yeah? You have lots and lots of them. Each of these lengths and there is black and white. I purposely don't say zero and one.

People just say, huh, I say zero and one. I don't want to say it. I don't know what numbers are. Numbers are an immensely complicated thing, yeah? There is black and white. And once you say numbers, you immediately make all mistakes. People exactly were doing that, making numbers that tend to add them, subtract them, which is not relevant here. They're just black and white dots.

How can you say, can you say that these are pictures of the world with dimension two with Euclidean symmetry, right? That's the question. And Poincare kind of analyzes this question and he suggests something which I explain more, what he says.

He says more that I'm going to say. And the same, what is. Sure, this one does, in fact. But how you would do that? This kind of purified question, yeah? Can you do that? Is it possible in principle, yeah? To say, but again, the point is I take pictures from the real world. There are not any sequences.

The sequences, I took a screen, I take arbitrary enumeration, a very idiotic one, ta-ta-ta-ta, I don't know what. But then I take real photos of this room, of this world, of the sky, of people, faces, whatever, and not just random distributions. And if they're random, they're random, yeah? If they're just random, poof, you have noise.

Nothing you do, you always have noise, yeah? But from the real world, and they say, yes, you can. You can, looking at them, say, they're coming for images, and you can actually construct the screen. You know who the screen is, and then you project them, and to see these images back. You can find what the enumeration was.

How? Exactly using this idea of Churchill, which is, Poincare made next step, actually, Poincare reasoning was two step. Churchill made only first step, because his geometry was simple. And you do, as I said, the essential thing,

it came from the real world. And so what does it mean, the real world? And the essential thing of the real world, that it kind of very, there's very little variety in this, yeah?

Things tends to repeat. Namely, if you look at the image of something, say, black and white, and you take a point, and it's black. Then you take point nearby, probability it's black is very high. Boundaries are small, and interiors are big.

So if you have any image, and you look at some point, and you look nearby, it's very high probability. By far exceeding what happens far away, it will be the same color, will be the same as the original point. Right? And this is a kind of basic fact of life, which allows us to operate in this world.

If not, we will become, you know, we wouldn't exist here. Nothing could be possible. And this is not the only feature of the world, but this is a basic feature which we exploit. Once you know that, aha, you look at this correlation function between two points, and out of this somehow, you make distance. Right? So it's function in two variables,

correlation between two points. So we have these numbers, you take number, so there's millions of them, you take, I don't know, number ten to the four, and number seven times ten to the four. There is some correlation of colors, you put it there. So I give you function in two variables in the space of these things. And you make some function that you call distance.

So now you're set of sequence, you have a distance function, and then you observe that it's symmetric. Well, approximately, of course, with respect to the symmetry group, and there is unique, essentially, representation after rotation of this on the screen. If it happened, probability of this being accidental, exponentially small.

Of course, it may not happen. And if it doesn't happen, well, nothing, so your approach was wrong. But amazingly, how often this works. And when it works, you know it works. And this exactly was done in a slightly different terms. You look at this conjectural gene, they have a known geometry.

Again, I'm saying not exactly what is done. You make this correlation function, and look at this space with this geometry, and see it's a line. Right? Just from correlation between points, and you, of course, it's not like that, not that simple. For many reasons, in a way, it's more simple, in a way, it's less simple.

Because he doesn't reconstruct geometry as metric. It's only geometry of, in between geometry. He knows who between whom. He need less. On the other hand, these kind of things are less, well, what I say would work if this thing were rather independent, whatever. But there's a first step in this logic of Poincare Insurgivant.

And, of course, if you try to do this in the real world, I guess your brain wouldn't do that. So nobody knows if there is a plausible algorithm doing that in a realistic time. So if the algorithm comes,

yes, you come to one's mind, you'll go an exponentially long time. There is a kind of shortcuts, and Poincare suggests some idea how they may go. And essentially, it's because you move. It's because you can make experiment. You can interact with the environment. And that makes mathematics more sophisticated, and it's not quite known.

But now I want to turn, the last ten minutes, to return to the simplest aspect of this Mandel and see what else, where else it brings you. So, and this, as I said, was multiplication of entries here.

You have entry ai, entry bj. I hate this notation, yeah, because they're meaningless. I and j symbol, they're not numbers, yeah? And here, this multiplies, it has cij equal ai by bj.

Again, because, as I said before, there is no such thing as a matrix, yeah? Yes, it's just a word. It's not a mathematical object. However, it's very hard not to use it, yeah? Actually, I don't know how to live without it, without, say, without, without, call them vectors, and there are all vectors, okay.

But matrices are so handy for some reason, yeah? Because you write on two-dimensional space, of course. Not for any deep mathematical reason. So we have this map. So what, and so we have this matrices of special kind, matrices of rank one, right? So they're a product of column by rows, and so in the space of all matrices of order, say, pq,

the subspace of pq of matrices of rank one. Namely, those which are product of column rows, and this is an instance of one of those Sagre-Arbenese-Veronese variety. But what they are from, from kind of statistical point of view,

if you're a little bit know elementary probability, and this entries are positive numbers with normalized sum equals one. So what you have, so these are now simplices of dimension p and q,

and so I have product of these two simplices, which maps to the simplices of dimension pq. So it's a rather tricky object sitting inside. This symbol has a curved surface in a very intricate way. And so what is, what, and it's kind of an equilibrium position,

what people might say in statistics, and it's related to the entropy in the following way. So now entropy enters, and so this subset is extremely set

for the so-called Shannon inequality.

Now everything I was describing in this Mendelian dynamics is actually like statistical mechanics. It exactly follows the rules, the entropic rules. Entropy goes up, you just take random processes and then you give them up, and this is like, very much like in many physical, chemical dynamics

where entropy goes up, and in particularly here, this kind of, there is entropy behind it, and so, and the map, this Mendelian map in the space of genes increases entropy and moreover brings it to its maximum value. So this equilibrium states, which I said, your mixed population, if you have only deployed organism,

you look at only one gene, after the first generation, you arrive at equilibrium, and this equilibrium characterized by having maximum entropy. Right, so now we have statistical equilibrium, immediately entropy comes to our mind, and this was one of the things you should have studied by Fisher, and so, so what is entropy?

Now I want to say, let's extend, so another five minutes, how you think in the relaxed term also about entropy. So 19th century entropy. So actually the boundary, it's not 19th century,

something I was saying 1950, where kind of a, mathematical change, it's very much due to the work of Grothendieck, when it moved, went beyond multiplication table. So before that it was mainly multiplication table, there were little indication here and there,

but basically it was multiplication table, and then it changed, yeah. And so, in particular about entropy, so how you think about entropy, not in terms of multiplication table. So in terms of multiplication table, this is the definition due to Shannon, and just following of course, Boitsman, and the Gibbs, it was the following expression,

in kind of a central contribution of Shannon, I'm joking a little bit, putting here a log on base two, yeah, instead of nature log, which we don't care. So, so p i are numbers, so in terms of numbers, they are positive numbers,

sum of p i is equals one. And you write this expression, and then you use different notation, which I hate because you never remember them, this entropy, the whole thing you, you call this p, this bunch of numbers, the order is material because it's symmetric, so entropy over p is a number,

attached to this collection, so what it is, and what it has to do with this Sagrale-Benez varieties. So, so I want to define entropy in, in categorical terms.

So, again, so, it must be something, and we know it a posteriori, something remarkable associated to this bunch of numbers. And everything remarkable must have simple,

many kind of, many properties, which come together, which are kind of rather incredible, there's so many properties, one entity may have, which I don't want to discuss, this is kind of secret motivation, but pretending, you know, there may be there, you don't know there, you want to define it. First, what about this,

a range of numbers? From a point of view of, kind of, geometry of naive, naive, in a set theory, if you have these numbers i, taken from some index set, you have a simplex in the Euclidean space, acibi. This is not a point, I just hate to write,

so I'll do it like R to the N, yeah? It's completely meaningless, meaningless notation, if you try to analyze it, it's unclear what you mean. It's set. This makes sense, yeah? We have set, and you take, find the set, you take all the real, all the varied functions of the set, and then you have the space. But what is N? N is not a set,

so you cannot say what's R to the power of N. So, but this makes sense, and this is simply there, and this particular set, enumerating it, and you can say how these are just points here. It's one point of view. But there is another way to think about that, if you take categorical point of view, you say how,

whatever in the world, not a set, it might be a category, and indeed, yes, but these points, they're not just different points, they're related to each other, and when they're related to each other, the set changes, it's not the same set, right? Here you have immediately something, you're uncomfortable, because the set is specified,

but maybe different sets, right? They're just set, they're not just one, two, three, N, or whatever. Maybe tables, maybe genes, maybe, you know, whatever, they're just sets, and how these different things are related. Even the number is variable, and the point is, that finite probability spaces, or infinite, whatever,

but finite, make a category. You have a well-defined notion of morphism between the two, and this is a super simple concept, because if you think about them, and that's kind of a good way to think, and much better than any kind of formalism, it's a bunch of stones of different weight, we should normalize total weight to be one.

And the morphism you allow, you bring some of them together, and they add, the thing adds up. They only become bigger, of course. And these are your morphism. It's kind of super simple, and you may think it's even stupid, because it's just,

because these are essentially unique arrows between two spaces. Well, not quite unique, sometimes there are several, yeah, but typically they're unique, right? So what's the point to say arrow? Why not try just a p bigger than q? And this is what usually people do.

In measure theory, they use this notation, saying one sigma algebra bigger than another, sigma algebra blah blah blah. And there is a very funny reason, which I think extremely kind of notationally significant, because, specifically when you speak about entropy, and this will be relevant, you can say, you can write, entropy of f,

whatever it is, it makes sense. But you can't say, entropy of this sign. Because in order to say, you have to put here both p and q. And then you say it will be relative entropy of p with respect to q, you have to remember who is respect, who is respect to whom. And that I think is impossible.

Well, it is relative, not, ah. It's just wrong, it's indicating it's of wrong relation. And this is very convenient. And in fact, relative entropy and entropy of morphism. Immediately, by the way, if you have entropy of an object, you know there must be entropy of a morphism. Now, what kind of simple thing you can assign to a category?

So I have a category, and you assign something new to this category. And there is a very simple kind of rule, you don't think about that, you say, aha, given any category, you take growth and the group of this. Which actually in this case would be a priori semigroup,

you can extend into a group. Given any category, you can make abelian group or semigroup of this. And this is a very simple rule. If you go arrow a, b, c, you say element to this arrow and to this arrow will be alpha, and beta, and to this combination, I'm sorry,

to composition you assign the sum. Right? So you can abelianize your category and take the semigroup. Then you can make, it exists as being a group, and in this case it will be semigroup. So do that. Of course, this is the topological category. So the object here, and this is kind of, as often happens,

yes, you don't use algebra, yes, on the algebra, you remember a little bit of geometry analysis. It's topological category, so I have to do it in topological sense, which requires certain effort, which I don't explain again on my next lecture. And once this being done,

you have your definition, because if you do it like that, you have a huge group, a very, very big group, yeah, like an uncountable group, which is not good. So you have to make it topologically, and when you make it topologically, you obtain a certain group, a semigroup, and then you think a little bit and realize that this group,

a semigroup, equals to a multiplicative group of real numbers greater or equal than one. It's semigroup, yeah? So numbers bigger than one, make the semigroup multiplicatively, and the reason for that, why this abstract with a new group

happens to be equal to the semigroup is just the law of large numbers. You just go through all the definition, you formally apply it, and continue, you go to some limit, and in this limit, you apply this Bernoulli theorem, and you arrive at that, and we take log of this, and there is again non-trivial reason for taking this log.

It's not that apparent, why should it take log? And then you get usually, after normalization, you get entropy. So entropy is just log of the value of this in this, in this growth in the group. And this of course, the advantage is because the formula, such sum of p i log p i, come a posteriori,

and you know, and then you can prove, it's really given very easy, once you know what I said, once you know these functorial properties, and actually it's immediately, act not only for object, but also for morphisms. So I give you entropy, give you reality of entropy. And then it applies fact in this situation, slightly more general than I said,

and full extent of this definition is unclear. When you start thinking further, it immediately brings you to the domain where things become rather unclear. And this what I will pursue, sorry I finished this in my following lecture, then I come back from, slightly from this perspective

look at the Mendelian dynamics. But again, my point is that you can see different things if you look from mathematics, I'm saying second half of the 20th century, compared to the, compared to the multiplication table type mathematics. He hardly is kind of funny, he's making fun of what he does,

but again from the perspective of today, he's almost 99% multiplication table kind of mathematician, yeah. He was doing all these numbers, he was playing with numbers throughout his life. Not his kind of more combinatorial or geometric quantities, which he just apparently was missing. And that is a kind of characteristic

so that eventually you come to numbers, eventually, maybe not, like Maxwell was suggesting, maybe not because in these two papers which I brought to you that's exactly instances when you, the number seems to be not appropriate.

That numbers only, in this example numbers serve you very well, but then when you go to the next level, to the other structure it's not so clear. Okay, so for today, I finished.