 And so the point which I want to make is that whenever you look in science, speculative science, it is always there is mathematics behind it, but it may be sometimes invisible because we don't, this mathematics doesn't exist yet. And if, so when you don't see it, there may 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 try to pursue which are not conclusive. I couldn't go far because the mathematics which I think is needed very far from what we know, but a couple of examples I can go slightly further. And I will be doing slightly different order from what is suggested in my resume of the lectures because of the subject which I want. So these are subject which are kind of, I had made some stuff related to Mendelian laws. And now there will be something about entropy and a little bit again technical aspect here, but not so much. It is Sturki, Sturki one. I live in space, which I of course with, I shall say a couple of words about that. And something will be less well shaped and I mean I have two, 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 and I just wrote 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 mathematical and here I have pretty good, well, you know, pretty good justification. This is, I shall formulate an experiment 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 just, I don't know the answer and generally speak, go around it. So the point is that kind of thing in mathematics, oops, whether internally or externally when it applies to something, there are two kind of faces to that, which is completely opposite. And just even with the 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 with those article based on quotation taken from various people, and so about reality, there is very good quotation due to Wolfgang Pauli who says that, as you know what's working quantum mechanics, and he was saying that exactly the hardest thing to understand what reality is, yeah. You don't know what reality is. You may create physical theories but how they relate it to reality or to reality is questionable and it's, and this is also true from many other aspects, yeah. So this kind of, this story related to entropy, this, I'm sorry, this story related to genetics, entropy, and this and here essentially we actually discussed it in the MOOC. It's quasi physical context and this story is kind of biological and this kind of 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 still a science. And there is a discussion and 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 with all this division, all this 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 there is scientific psychology which has completely different origin and just, which are really scientific but it's less kind of in a shadow, it has no, no. And it's pure, quite, quite, quite, quite, quite scientific. And so, but indeed what usually psychologists do, but science, psychologists, how can you, for a certain point, if you have nothing to do with science and mathematics, they are completely irrelevant. But if you look at scientific bicycles and this essentially is a college 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, well, the question of terminology. And so, this is about, so yeah, look at mathematical example just like, I love two examples saying that so mathematics has two aspects as a sort of that, one of that is miracles. And I'm 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 priori 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. I'm saying, nothing is bigger than best, right? It's best in 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, you know, sort of mistake of the languages, kind of. It's a power of the language being, these are ambiguous and so tricky, yeah. But we must be very careful of that. On the other hand, this is, we want to use it, yeah. So, but coming to mathematics, so what are miracles and what kind of logic? So this part is not, of course, accepted, not common, just from outside mathematics that mathematics is about impossible things. They start from impossible and then you have to understand them. And two examples of any, all, and I maybe even make three examples, yeah. They're purely mathematical. People sometimes smile at them, but, and because the same attitude, I look at this mathematics, the same I would like to look at this kind of phenomena. It starts with very naive things and they, in a way, they're not at all naive. They're much more profound towards the thing, naive. And historically, I don't know, maybe. As this example, I was saying many times, yeah, yeah. Two plus two equals four. Second, it will be the Pythagorean theorem, yeah. 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. Well, actually some theorem emerged, yeah. I would just, only in the 20th century, we fully understood implication of that. So what, what, so this I postponed, yeah. It is the trickiest one, yeah. So what's so special about two plus two equals four? It's very different, I'm saying. It's very, it is similar. Actually, it's already, yeah. And here is actually this related to this, yeah. Yeah, this was Archimedes theorem. And then there is, this is follows and this Pythagorean theorem. And of course, if you have something like three plus three plus three plus six, nothing interesting here, but at least you don't know anything growing out of there. All right, all right. So start with Pythagorean theorem, right. And Pythagorean theorem, if you look at the school and you give a proof and you think you finished. And this is extremely, I think, a confusing way being presented because proving the theorem is a trivial aspect of that and certainly the way it's written, it's completely, it's completely misleading because as you know, the basis of the theorem is, the essence of the theorem is Helbertian geometry. Choke breaks down. That you can define with the sum of squares, Helbertian 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 a 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 probability distribution. This is what, if you think about that, not in high school terms or not in the 19th century mathematics, but in terms of the second half of the 20th century. This is one of the way I want to emphasize. So the way you do it, you relax, you think, as Yves said, Grothin Dick would think about this. Not as somebody like Harvey would think. I'll come to Harvey in a second, this way I'll manage his name. From the modern perspective, to certain point people in science were thinking in terms of multiplication table. Well, it's not so bad, and just great men 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 things within mathematics and outside mathematics, and you come immediately, she goes somewhere. We don't know exactly where you come. So, now, doing all this theorem, why so great? Actually, this was explained to me once by Michael Atye, that it is so remarkable about the theorem and what has to correspond 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, of course. There are 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, I have to push the buttons or what the hell. And there is a map from complex projecting space to the n-dimensional simplex, so I put n here. It's that power, actually n you have to be, I hate to write it down, this is where the n is. n is not indicator of the power, of course. And there is such a map, which is a ghost via the sphere, you start with the corresponding sphere of dimension n plus one. Maybe I put the right dimensions to, I'm sorry, to the sphere of dimension to n plus one in C2n plus C2, I'm sorry, in Cn 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 equation space. There is another, and this is 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, yeah? X here, this is a quotient under this map. It factors through the simplex. Sphere goes here via Hopf map if I go 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 the, you forget the phase to the probability measure, so you. And this is, of course, extremely remarkable, powerful thing just relating convexity in complex geometry. And this is inside, but this is inside with the theorem of, so Pythagorean theorem is the first instance of that. And it contains all ingredients of the proof. So if you think about this in these terms, not like a 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. And immediately some world of mathematics come to you. So before we had Pythagorean theorem, then that and then there was two plus three equals 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 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 you didn't quite understand that. You're always figuring why minus by minus gives plus. It was a really big discussion. This geometric interpretation is relatively recent. It's post-allerian. And the fact that you can stop 90 degrees and have a square root of minus one. And this is 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? This is my interpretation. So what's miraculous about that? You can say, this formula is so used to that. But if you think in terms of what it tells you before you write an explicit form, it tells you in a preferred three-dimensional shape, right? And this is how it generalized. If you have a segment here and you project it to three walls, so you have this piece of stick here, and you have 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 the stick. And this is a miracle. And it has squares, but it would be. And this is existence of symmetries within real numbers. And so it's unclear what is more fundamental. I was saying there were some respectful people saying something important about numbers. And the definition statement was made by 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. Maxwell himself implicitly contributed to the symmetry by introducing this Lorentzian group, which was inside of his equations. But already this orthogonal group, the fact it is there. It is, in a way, rather unexplicable. Yes, of course, we can say, ah, you understand it, but then, of course, there are layer after layer. 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 miracle is there. And the point is that if you think about this in combinatorial terms, 2 plus 2 equal 4 is not just this identity. It's the composition of four elements set into subsets of cardinality 2. 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, 3 plus 3 equals 6, here it will be something about 10, I guess. In how many ways you can decompose it. And so essential thing that 3 is less than 4. Here's 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. Because it means if I have four elements set, I can canonically associate to it three elements set. Namely, the set of this composition. Which means there is a homomorphism, a permutation group of four elements, the permutation group of three elements, which was used in algebra since, I think, Ferrer, I have approved it. I'm confused. Who proved that equation of degree four reduced to equation of degree three. Because this group is not simple. It's sort of the, well, if you divide by center, by evolution, right? This is non-simple group. It's the only one. All other groups here 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 while group of compactly group, it tells you that the group, or rather universal covering of S of four, splits into the sum of two and so three. You want to take universal covering, so otherwise you have problems with spins, yeah? And from that, it follows that, or rather of the corresponding property for the algebra, follows that certain operator split which implies that the existence of the self-duality equation. 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 in two variables, which splits into the square of the Kasserimun operator. This splits and give rise to the first order equation which is gauge equation. And then there is Donaldson theory and cyberquitian theory, and the whole world mathematics opens because this happens to absolutely impossible happens, yeah? This equation is not supposed to split. If you have general polynomial, the chance of being square of a smaller polynomial is nil. However, it happens in this case. It happens to be squared, right? Because of this identity, which comes from this identity because this group is outgrows of the permutation group by construction of the groups. And then, well, just, I don't have to advertise much for that. Now I want to use symbol logic when it comes back to science, but I have to leave this thing up, yeah? So there may be a kind of miracle you think and then you want to understand them to put them and 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 this perspective. Now if we forget about this. Now, this is kind of most pronounced because it's exactly had this flavor. So what's some... Why is this miracle, yeah? And this is one of the reasons why kind of Mendelian logic was not accepted for quite a while. So the history of this is kind of amusing. There were two points in history or three points in development, right? So one was, yeah, yeah, I hate... I have to write French names with mistakes. So I just said, it was M1, and then it was Mendel, it's M2, and then there was a third moment I had no many people involved. So Morgan was the most essential, I could put here Morgan M3, okay? In development of genetics. So Mpérti first observed he was looking at people who had six fingers, you know, sometimes people go with six fingers 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 that, which was erroneous but it was certainly in the right direction. And then Mendel was doing something for people and also this proportion appears and he may develop what I'm going to explain in a second. And then there was a next level, yes, how Mendelian dynamism 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, Sturgevant makes kind of remarkable discovery, again remarkable logically, who discovered new way to think about geometry from mathematical vision perspective. Right? It is a linkage of genes and their role. And so this is, the fact that these two numbers appear is rather incredible. You can see the certain features in the observance that there are proportion which appears one to three, sometimes. And it's, of course, nothing in biology is like that. It's probably 1.3 versus, let's say, 2.7. But for biologists it means 1 to 3. And this is quite a kind of right way of thinking, yeah? So there are these integers appearing and interestingly now that this similar thing we observed by, say, by Darwin who overlooked it. And that's, again, you just say, it's quite amusing, amusing. 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 of 1 to 3. Approximately. But amazingly always kind of closer to 1 to 3 than something else. And biologists of certain, of the time who looked at this, except for me, who was not biologist, a man who was not biologist either, by the way. But, say, Darwin himself looked at this and just couldn't make sense of that, actually rejecting it as something irrelevant. And interestingly enough, say, in Russia there was a school of Lysenko also rejected it as algebra and not, and not, and not biology. On the other hand, selection principle, this evolutionary champion by Darwin has absolutely nothing biological to it. It's pure mathematics. But because it's so simple, I mean, because exactly it has no content, biological, it's so easily acceptable. It's just kind of trivial, regardless of whether right or wrong. And this is something incredible which needs explanation. And so, it was actually Mapirici and Mendel has right ideas, but Mendel was much close to the truth. So, what is coming behind it? I just want to say the consequence of that 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 a specific example. You have two fields of flowers of different size, and here everybody white, and here everybody red. In generation after generation, here they're only white, only red, and they're the same species, but they're separated by a mountain wall. Some moment it disappears, and there's lots of insects pollinating both ways. And so, 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, p times q, different from the original one, of white and red. And then people, like, darting around him, say, ah, there was natural selection, some of them beaten, for example. You have a person who was shifted toward white or it was shifted toward red, whichever. It changed, there was a selection, it will continue, you take next generation, and nothing happens. It remains the same. This process stabilizes on the first step, and then continues being stable. It's not exactly that, because there are all the fluctuations there, of course. But this is essentially what happens. And this was prediction made by, by explain, 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, until 1908, so, Darwin was rediscovered in about, something, 1908 or something, yeah? People rediscovered his work, but they, not his work, they repeated his experiments and came to see the conclusion, and then they just started referring to Darwin. This also, in terms of Mendel, and also interesting, why they were referring to Mendel. I mean, they discovered them by themselves, and pretended, kind of, thought they didn't know about Mendel. Darwin's paper was kind of published in a quite readable journal, but at the point, there were three groups of people who discovered that, and they started arguing who was the first. And then immediately, of course, they looked at the predecessors and found Mendel. They said, this was how Mendel told. This is the only reason we know about Mendel, yeah? Because people were fighting for who was the first. And then they were just doing that and then read Mendel and realized, which 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 nature is not so easy, but this certainly was counterintuitive. Why? You have some nature transformation and why its square will be equal to itself, yeah? So, somebody is important. And then there is a probably apacryphic, maybe not story about Harje, that was playing cricket with him, by always a 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 he was kind of, I just can't repeat it, basically he was saying, he was kind of mathematics 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 top mathematics of that. In my view, he just, he wrote, people of that time, as we get it like, probably now it is like Harje, they were so smart, they immediately see the answer and they 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 material dynamics involved and it is extremely paradoxical it exists. It's indeed very strange because the map we shall describe will be essentially a polynomial map and you have a polynomial and you substitute polynomial, polynomial the degree goes up. So to have again the same p it's kind of rather impossible, right? It has a preposition of polynomials to have a polynomial of the same degree is impossible, you must say, because the degree must go up. However, this is what will happen in this example and this means that it sounds rather exceptional and rather remarkable must be happening and if you even before you just, again, yes, this, whatever Mandel was writing was algebra. So his were 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 a real complex whatever numbers and you have a, you have polynomial maps such that when you imagine also just now we are in the context of groups so imagine they just act the group generated by this map extransitive here and they all can make final dimensional space so the degree doesn't go up. So another way to say it you have a Lie group and you have, I'm sorry, you have a map say from Rn times Rn to itself which is a group law which is polynomial. Is it possible to have and so what would 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, huh, of course not. As usual this, it's too much but no. And the answer is no it's very simple, it's exactly all nearly importantly groups. So all nearly importantly groups this conception comes it's linked to this phenomenon discovered by Mendel and certainly nobody thought about that in those terms because nobody would think in those terms of groups, huh? But I'm saying this phenomenon is the same which comes to life in the first of the third dimensional Heisenberg group which again are essential as you know in quantum mechanics. So the mathematical structure again just once you have such little miracle it just, you know, it's this exactly what I'm saying but look, for him it was so simple the proof was so simple it's like 2 plus 2 equals 4 it's simple but not the question of simplicity but the question of impossibility and 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 today start today this thing concerning concerning this map of Mendel so what was sorry, I'll start with I'll a little bit start with that and then we go to the next point so what was idea of Mendel where this 1 to 3 come from again it's a pretty similar idea but I think I'm not certain exactly because one please hard to read he wrote his text as a shot and rather kind of a genetic and it's not quite clear to say what he meant but apparently he meant something else and the same speed and the Mendel is how we read it him today was as follows there are other genes and in that particular whatever they are but they represent features in that certain features 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 mixtures so there is a gene responsible for that it happens sometimes again it doesn't happen often but even happens rarely it is remarkable it must be understood that it shouldn't be so you accept from general physical principle you accept mixtures of all things any purity is there and this discontinuity this exactly what 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 so Leibniz was moving him in the development of infinitesimals and it was a very powerful very good idea and this idea has kind of amusing history but well in western culture and many things are continuous but some are not and but Mendel has kind of different idea he discovered discreteness inheritance his point was that if there are 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 the symbols but that's kind of we perceive them as polynomial variables and write this AB so genes compose of that and then there is a function phenotype given by AB so it means that what we will see is not A you don't see B is your function of two of them and this again a fundamental point which completely was missed by Darwin all these people yes this is essential part of inheritance which makes everything possible without that all this evolution of course is a shame on us as it was presented before Mendel and then in this particular example of flowers this function is color 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 well 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 being symmetric which is again sometimes true sometimes not but again even sometimes we symmetric is quite remarkable right because very strong property again many things we assume kind of in this unthinkably because we use 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 preview things which don't know and relate to them and this way in common sense it was not sensical right because it will not be reduced to what we 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 so maybe it's better to change my notation from the very beginning this two component W and A and say this color equals red unless we have double white so that's the rule if you assume it then so why this assumption is before you do any mathematics purely even purely qualitatively it is quite significant because if you have white parents flowers all their descendants will be white but if you have red parents their descendants maybe either white or red yeah? I switch from A and B color from A and B I switch from A and B to white B and W that's easier to remember white A and B is a direct notation and this specific notation for this example because I never remember who is white and who is red in terms of A and B this corresponds to A and this to B just change my notation just not to forget who is who but this annotation they are not colors and then they represent colors are function in two variables colors what you see for all else yeah whatever they call but so there are two this is a function in this way it takes and these are abstract variables and just use these letters not to confuse them and if you have red if you have red parents then they may have descendants both white and red however if you keep selecting red if you keep taking red and red and red eventually you arrive population with pure red in all generations and this of course what we use people in artificial selection for about 7000 years this estimate when people start domesticating plants and animals or 10,000 years something like that when domestication started we don't know exactly but say of order between 10 and 7000 years say 10,000 years if you start selecting on one particular feature eventually all descendants will have this feature if it's simple enough feature like this red so this is a preliminary part of the experiment so if you want to make this experiment this how it's done by Mendel and then of course by Morgan you select this pure organism it has always they have descendants at the same time and then you start mixing them and then when you start mixing them you observe this one-third ratio that and then the explanation again is quite simple so we have white and red and randomly start mixing them you may have either white white red red white and red red so this will be white this is red all this will be red and so there is this ratio 1 to 3 so you assume that everything that happened is actually probable all things are independent so again this is exactly what kind of was the point of how he was arguing of course his computation was obvious to him and to anybody kind of multiplication table in the second though I see it is not that innocuous it looks but the point that you can represent everything by this kind of maximally symmetric situation right to say that some variables equidistributed so you have a known quantities and fantastically the right assumption is to say they are equal unless otherwise stated if you have some numbers coming from some process of the same nature first conjecture is they are equal which is mathematical absurd genetic numbers are not equal but the right conjecture they are equal they are independent which means 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 there is some reason for doing that of course maybe not common sense but mathematical reason and once you make the assumptions you arrive at this one to the three ratio and then it's observing it actually happens and how plausible it is and then there is interesting controversy which I haven't followed quite but it would be interesting of course to look but carefully there were fishers who was a kind of major figure in statistics and genetics mathematical genetics of the turn of the beginning of the 20th century who on one hand he was great fan of fan of Mendel but he believed that Mendel falsified his data that it didn't follow from the real world the approximation of course this noise and Mendel kind of selected good examples and 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 the human later on indicates some one remarkable discovery he made purely mathematical related to the entropy on the other hand recently I read articles 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 the reason I don't know just if you can read it on the net articles about fish and what he wrote anyway gazing this proportion it was not at all obvious it was marginally true because there was lots of noise in the data however you have it and now why this would imply this what is called a Hardy-Weidberg rule though it was of course known to Mendel that so you have this kind of picture you have two units determining the gene right W and R and and the pairs there is distribution of their pairs so population given by distribution of their pairs and and then when you take a random mating so it makes everybody together it may change but it stabilizes in the second step why the re-stabilization and let me explain go ahead so what is the mechanism of that which was in my view overlooked by Hardy is of course you can write these formulas they have a polynomial because it's symmetric there are three different types one, two, three but these are equal so it appears to be coefficient two and Hardy was writing formulas which I just don't remember from high school but quadratic polynomials which kind of hold this kind of nonsense you don't need it in a second to see it you don't have to know this multiplication table it's completely kind of irrelevant but also irrelevant you understand much better if you forget about that and here I'll write this formulas and what he misses is let me explain what he misses what is the interesting map in what kind of mathematics it brings in and you can point again about this mathematics it's immediately some of them full of unsolved questions so yes imagine you 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 of which must must be identity so let me show you such a map and then we shall later actually explain why it happens here and the mapping is quite simple what is a matrix where entries are numbers by the way what is a matrix I challenge anybody here to give a definition of a matrix now people say matrix mathematicians but there is no such object mathematics of course what is a matrix anybody can tell me what a matrix is mathematically I mean not just for mathematically what is a matrix square table I mean what written on the blackboard this is definition what this is matrix and put here numbers it's interesting enough it's not at all it's not simple you cannot say it because we don't know it it's not because you just know how to say it you don't know what it is it's like exactly greater better than nothing it's a confusion of the language we're confused by images we don't know what matrix but still we use this because it's so convenient 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 multiply them we put them here and this is polynomial map on matrices I want to normalize it so it will be not quite polynomial I divide by the sum of all elements because this will be probability distributions and I only 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 on the space of this matrices so say pq p elements here q there I have the self mapping and square of this mapping equals that's the fact which is one of the way to express this hard dividing theorem and this how it works so we had kind of here fathers here we have mothers and we don't even assume symmetry right they mix their genes what will be new distribution after mixing their genes repeat it twice nothing happens of course this again you can do it kind of without thinking you try to imagine the simplest possible kind of combinational thing being multiplied and this will be of course the simplest possible thing all possible pairs of objects are matrices you have and this kind of operation kind of foster you by pure logic if you think this is what logic is it is some underlying symmetry of the problem built in non-trivial symmetry in the problem and so but again this is a so hard he was doing this for symmetric matrices 2 by 2 so subject I think he was not his notation was like that no it was p 2 I think it was r p and r subject like that he was doing it for this matrix and taking a writing formula in this case explicitly in terms of p q and r and you have some kind of formula but this is always true so what is kind of interesting about this map so once one property of this as I said it is a rational map because multiplied by some of the elements so degree goes up but because you normalize, degree remains so defined in fact in the projective space so you multiply a product of two projective spaces two projective space of higher dimensions and instance of that is slightly and this map was discovered by Veronese I forgot and there was another name attached to it and so let's just have some feeling about this map it's quite remarkable map in many respects so if you just look at this 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 from nothing this is still not the ultimate formulation to which I come but I want to describe it in slightly better in one special case and immediately see what is associated to this and just look at it again 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 it square and you have quadratic function and it means the space of linear function which is Rn goes to the space of quadratic function which is the 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 restrict it to the union sphere you have a map from the sphere to the 5-dimensional space where it is contained in fact in force sphere and the map is symmetric so it actually gives you a factor of the projective space a map from the sphere to 5-spaces and in general so it's highly symmetric because we run as a variety it's 5 and what kind of properties it has again you think about this map and just try to understand it and and I just want to say so again the map is very simple you just take covector linear form one take is square become quadratic form so it's space of quadratic form it's a 5-dimensional space and there will be again normalize so sitting in this sphere so how it looks like so it's 2 sphere in 5 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 an isometric group so it's orbit of representation so you take orthogonal group X on linear forms it's for x and quadratic polynomials so there is a particular orbit which is projective space but we think about this sphere map there but of course it's the 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 full symmetry are torii so you may ask if you can have torus sitting there with a comparable symmetry so here this symmetry is fully orthogonal and just we just relax it maximally but still you want to say it we say aha it will be symmetric but only in a kind of weakest possible sense which is still relevant here so we have surface just locally imagine so it might be everywhere at the same point in 5 space and you have another such piece of a surface and so we say that kind of equivalent if there is a flying transformation moving this point to this point which preserve tangents up to second order you always can make tangents space goes to tangents space you always can do it but also on 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 fully symmetric but this is minimal you may expect for the two torus and these are called free maps so this is the question is is there free map of the two torus in 5 space in simple terms it means that if you take partial derivative of this maps so there are 5 derivatives 2 first derivative and 3 second derivatives there are vectors here this vector 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 5 maps 5 vectors here must be independent if such a map exists again there is a kind of a story related to that because there is a paper famous paper by Moza and just this one with the most cited papers in mathematics when he generalizes National Police Function Theorem and as example he says take such map and and when we read this article we couldn't construct this map it was a large brick we studied and then I met Moza and asked him do you know about this map I don't know I never thought about that and still of course it's unknown from point of your analysis of material from point of the geometry just everything has this paper infinitely simple in this question it's a really very hard question we don't know if such map exists or not right and again my point is it sounds like genetics can you go a little bit further just a simple question and you arrive in very profound mathematics in very difficult mathematics another point about this Ferenese maps Ferenese varieties but they are remarkable from another perspective and again because I realized relatively recently so they map this so they projective spaces inside of Euclidean space of dimension I think maybe this maybe something else if 5 equal 2 plus 3 no no no something not quite exactly this number maybe plus minus 1 or something the dimension of homogeneous polynomial of degree 2 and there was another problem for quite a while and it was due to Borsuch I believe Borsuch proved that if you have any subset in Euclidean space of dimension 1 you can divide it into 4 pieces of dimension less than 1 say in the plane you can divide circle you cannot divide of dimension less than 1 but you can divide it into 3 pieces of dimension less than 1 and any set in the plane divides into 3 pieces of and then in 3 space I don't remember where the Borsuch was somebody else proven you can divide every set of dimension 1 and 4 pieces and there was a conjecture for high dimensions and then it was proven by Kali 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 simple but very complicated set of opposite points so it depends how position the points of dimension 1 and this also count example so I mean this is quite remarkable objects in many respects they really contradicts the naive mathematical intuition and the Mendelian conclusion that samsik may stabilize in the second round of reproduction you make selection, you do samsik and then nothing happens afterwards everything stabilizes it's really quite remarkable now I said how to go without computations what next level of mathematics so what about matrices 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 context they have different things now in linear algebra I want to define what they mean column and rows it's just a range a bunch of numbers but how we think in modern terms modern meaning late 9th century in set theoretic terms so modern terms will be theoretical terms to each return so we look at the set of all these objects you know it's a vector space but it's a vector space and this is what we'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 distinguish linear function to reals so it's a linear space with distinguish covector ok and that will be for us rows columns or whatever so I call them space and make this scale a product with one vector not scale a 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 the tensor product composition and you have it now what is this operation in these terms so when you have the tensor product what you can do you can go to X you just take vector so how you do that you take X Y right so if you have element here you just take this because you take this will be number right this is the value of your linear function so you can project X Y to X and you can project it to here so in general tensor product you can project it to its components the reason such thing is projection that's one problem in quantum mechanics you cannot go to subsystem you have system you have two spaces representing different quantities you bring them together become tensor product you cannot go back because we used to say they are entangled but of course in mathematics you said 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 this summation and this multiplied them back and then it's the formula because kind of obvious it says that it actually says how we put brackets so if you have two 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 it again it's 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 because you normalize yeah I'm sorry when you this make projection you're normalized by this sum so this will disappear because it makes normalizations 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 computational no discriminants or nothing all these are artifacts of four notations of 19th century actually you know Mendel had physics with Doppler or you know for Doppler effect and then there was another physicist less known but who was trying to develop combinatorial description combinatorial models in physics he was one of the I forgot he wrote kind of books about I forgot his name combinatorial models in physics and from there partly probably Mendel was inspired to make this thing so this what we have here and this is completely kind of end and just no computation involved yeah I mean just pure pure tautology how it should be simple thing might be simple computation sometimes it happens of course unexpectedly out of complicated computation but it's so rare 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 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 where I speculate on something which I cannot prove you come back in five minutes okay so I say just a few words continue so on a technical level I shall go into this in the following lecture so 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 of two linear spaces today as the standard map which has quite quite quite amusing properties there were kind of several several layers of that and so to which we'll converge not today but I just want to say is the following when you start kind of again relax and generalize this maximally you arrive at the situation of the following type when you have a community of algebra topological algebra for example maybe in this example it will be algebras of truncated polynomials so you can find dimension algebras but 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 5j when you multiply them you get multiplicative endomorphisms and then thirdly you take their linear combination certain linear combinations so that gives you a class of transformation by now they are not endomorphism of these algebras so in relevant examples there will be this rational transformations it will be rational transformations polynomial transformations in the space of truncated polynomials but this quite interesting also for such algebras like algebra of function with convolution for example and generalization of this hard divide in principle but in very many cases such transformation many cases which explicitly describable usually describable in terms how this endomorphism act on the spectrum of this algebra if this dynamics is simple it's pretty often this dynamics is simple and simple I mean that they have a fixed point either attractive or where 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 random variable itself many times and normalize you convert to normal distribution is instance of that and Mendelian dynamics is another instance of that and there are other example in genetics what 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 you 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 an organ but several you arrive in this more general picture when you have this endomorphism of polynomial truncate polynomial algebras this kind of maps and then you can prove there are these fixed point properties similar to the law of normal 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 instrument the mathematics is much more less well shaped and more interesting when you remember that from this 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 the polymer the one dimensional molecules and this is the most feature of life it's one dimensionality this where mathematics enters for some reason life is one dimensional encode by something one dimensional because chemical it's impossible to make something hetero polymers multidimensional but before it was discovered experimentally in molecular biology it was shown by Sturkiwand by Mendelian a type of experiment so what is a Mendelian which was done by Morgan who develop all these techniques and develop 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 features being repeated and then you start mixing them and you see how different features are being distributed right and so look at this statistics and from that Sturkiwand concluded the 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 thing which is done and you can say the line on the line and that one of them in between the two the basic relation here is between line is not a metric line nor it is ordered line it's a between relation and you can say we gene lies between other genes though genes were quite abstract and Morgan himself was emphasizing that it's a material what is the nature of the genes whether the water the molecular nature they may be transcendental features and what he says is that between the genes the study 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 to say how they relate and this is 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 material nature of genes though he perfectly understood from hidden molecular biology but it was not his business and Sturziband who was a student at that time he was about 19 years 20 years old and he made this conclusion from the experiment and this was 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 that was considered before maybe there are others and as you point correct and in one of his books I've forgotten which of them either it was science method or science hypothesis he wrote these two books he looks at another question he didn't know about genetics but he speaks about geometry and we know the world we live in has some symmetry the tricky 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 the first truly new phenomenon in group 30 it's not compact it's a simple group if you forget about this we have taken double covering 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 because what comes to your eye will be nothing you have here you moved it and these two functions are not close and it's again people who are analyzing images till recently they just could not even conceive that because they were analytically minded 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 arrive to this understanding the Poincare headed more than 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 said and then we will write on this later on so what is geometric structure without saying what structure you construct what you see and how you want to reconstruct it so you have a set with geometric structure and you don't know what structure is but what you know for example look at the images for images it's kind of clear forget about dimension 3 just look at dimension 2 imagine you are shown images on the screen but the screen itself has no geometry so you just screen and there are pixels and they are somehow enumerated so you have like 200 by 200 screen let's simplify let it be 1000 by 1000 so we have 2 to the 6 2 to the 6 entries and they may be black and white so we have a sequence of black and white they are enumerated in a certain way always enumerated pixel and each image it's a million of dots black, white, white, black imagine you have a million of such pictures each of them is a sequence this is what comes to your brain your brain is essentially a linear there is no dimensionality in it so the problem is as follows again I repeat you have a million of strings each of them maybe a million just not to confuse numbers 10 to the 7 strings of the lengths each 10 to the 6 we have lots and lots of them each of these lengths and there is black and white I purposely don't say 0 and 1 people just say say 0 and 1 I don't want to say it, I don't know what numbers are numbers are an immensely complicated thing 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 is not relevant here they are just black and white dots how can you say that this is pictures of the world with the same dimension 2 with the Euclidean symmetry that's the question and Poincare kind of analyzes this question and he suggests something which I explain more but he says he says more that I am going to say and the same what is but how you would do that this kind of purified question can you do that, is it possible in principle 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 obituary enumeration very idiotical one I don't know what but then I take real photos of this room of this world of the sky of people faces whatever distributions and if they are random they are random if they are random you have noise nothing you do you always have noise but from the real world and and I say yes you can you can looking at them say they are coming for images and you can actually construct the screen you know what the screen is and then you project them and to see these images back you can find what the enumeration was exactly using this idea of Churchill which is Poincaré made next step actually Poincaré reasoning was two steps Churchill made only the first step because his drums were 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 it is very little variety in this 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 right boundary is a small and interior is a big so if you have any image and you look at some point 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 the kind of basic fact of life which allows us to operate in this world if not we wouldn't exist here nothing would be possible and this is not the only feature of the world but this is the 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 so it's function and two variables correlation between two points so we have these numbers you take number so this million of them you take I don't know number 10 to the 4 and number 7 times 10 to the 4 there is some correlation of colors you put it there so I give you function and two variables in the space of these things we call distance so now your set of sequences have a distance function and then you observe that it's symmetric approximately of course with respect to the symmetry group and there is unique essentially representation 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 then it will work but amazingly how often it works and when it works you know it works and this exactly what we have done in a slightly different terms you look at this conjectural gene they have a known geometry again I'm saying not exactly what it's done you make this correlation function and look at this space with this geometry and see it's aligned right just from correlation between points of course it's not like that 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 a geometry in between geometry he knows who between whom he needs less on the other hand these kind of things are less what I say would work if this thing were rather independent whatever but that's the first step 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 yes you come to one's mind you'll go 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 and it's 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 NTBJ I hate this notation here because they're meaningless I and J symbols they're not numbers and here this multiplies you have CIJ equal AI by BJ as I said before there is no such thing as a matrix it's just a word it's not a mathematical object however it's very hard not to use it actually I don't know how to live without it without column vectors and a row of vectors okay but matrix I saw handy for some reason here because you're right 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 product of column by rows and so in the space of all matrices of order say PQ space of PQ of matrix of rank one namely those which are product of column rows and this is an instance of one of those Segre-Erbanez-Varanez variety but what they are from from kind of statistical point of view if you a little bit know elementary probability and this entries are positive numbers with normalize sum equals one so what you have so these are now simplices of dimension P and Q and so we have product of these two simplices which maps to the simplices of dimension PQ so it's a rather tricky object sitting inside this simplices have the curved surface in a very integrated way and so what is what and it's kind of 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 extremal set for so-called Shannon inequality now everything I was describing in this Mendelian dynamics is actually like statistical mechanics it's exactly follows the rules entropy goes up you just take random processes and then you give them up and it's like very much like in many physical and chemical dynamics where entropy goes up and particularly here 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 this population if you have only deployed organism you look at only one gene after the first generation you arrive at equilibrium and this equilibrium characterizes by having maximum entropy we never have statistical equilibrium immediately entropy comes to my mind and this was one of the things which was studied by Fischer and so what is entropy now I want to say to this extent so in another five minutes how you think and relax entropy so 19th century entropy so actually the boundary it's not 19th century something I would say 1950 where kind of a mathematical change very much due to the work of Grottending when it moved when 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 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 was definition due to Shannon and just following of course Boilsman and the Gibbs it was the following expression and kind of essential contribution of Shannon I'm joking a little bit putting here log on base to you but you don't care so p i are numbers so in terms of numbers they are positive numbers some 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 call this p this bunch of numbers the order is a material because it's symmetric so entropy entropy is a number attached to this collection so what it is and what it has to do with this sagra-albanese varieties so so I want to define entropy in categorical terms so again so it must be something and we know it but it's very remarkable associated to this bunch of numbers and everything remarkable must have simple it has many properties which come together which are 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 only know they may be there you don't know they want to define it first what about this a range of numbers from a point of view of naive set series if you have this numbers I taken from some index set you have a simplex in the Euclidean space there's another point I just hate to write as long as you do it like r to the n it's completely meaningless notation if you analyze it it's unclear what you mean it's set this makes sense we have set a finite set we take all the real value function 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 n so but this makes sense and it's simply there and this particular set and you can say how these are just point here it's one point of view but there is another way to think about that if you take a categorical point of view you say whatever in the world it must be a category and indeed 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 here you have immediately something uncomfortable because the set is specified but maybe different sets they're just set, they're not just 1, 2, 3, n whatever maybe tables, maybe jeans, maybe whatever they're just sets 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 there is 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 much better than any kind of formalism it's a bunch of stones of different weight and the morphism you bring some of them together and they add 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 there is essentially a unique error between two spaces well, not quite unique sometimes there are several but typically it is unique so what's the point to say error why not to write just a P bigger than Q and this is what usually people do in measure 30 they use this notation saying one sigma algebra bigger than another sigma algebra and there is a very funny reason which I think extremely kind of notationary significant because specifically when you speak about entropy and this will be relevant you can write entropy of F whatever it is it makes sense but you cannot say entropy of this sign because in order to say it 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 whom is respect to whom and that's I think impossible it's just wrong it's indicating of wrong relation and this is very convenient and in fact relative entropy and entropy of amorphism immediately by the way if you have entropy of an object there must be entropy of amorphism now what kind of simpler thing you can assign to a category so you 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 even any category which actually in this case will be a priori semi-group but you can extend it to a group given any category you can make a billion group or semi-group of this and this is a very simple rule if you go arrow A B C you say element to this arrow to this arrow will be alpha and beta and to this combination I'm sorry to composition you assign this sum to this category and take the semi-group then you can insist on being a group in this case it will be semi-group 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 just on the algebra you remember a little bit of geometry analysis it's topological category so you have to do it in topological sense I don't explain again on my next lecture and once this is being done you have your definition because if you do it like that you have a huge group a very very big group like an uncountable group which is not good so you have to make it topologically and when you make it topologically you obtain certain group or semi-group and then you think a little bit and realize that this group equals to a multiplicative group of real numbers greater or equal than one it's semi-group so numbers bigger than one make it a semi-group multiplicatively and the reason for that why this abstract group happens to be equal to the semi-group is just the law of large numbers you just go through all this definition you formally apply it but you continue it to go to some limit and then this limit you apply to this Bernoulli theorem and you arrive at that and you take log of this and there is again no trivial reason for taking this log it's not a teleparant, why should it take log and then you get used up to normalization and get entropy so entropy is just log of the value of this and this growth in the group and this of course advantages because the formula some of PI log PI come hyperspiratory and then you can prove it's really given very easy once you know what I said once you know these fun factorial properties and it actually immediately act not only for objects but also for morphisms so I give you entropy and then it applies in a situation slightly more general than I said and the full extent of this definition is unclear it immediately brings you to the domain where I think it becomes unclear and this what I will pursue sorry, I finish this in my following lecture then I come back 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 multiplication table type mathematics see how it is kind of funny he is making fun of what he does but again from the perspective of today he is almost 99% multiplication table kind of mathematician he was doing all his numbers he was playing his numbers all his life not his kind of more combinatorial or geometric quantities which he 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 in this example numbers are still serve you very well but then when you go to the next level the other structure is not so clear ok so for today I finished