 Hardie Weinberg's principle in discussion and comments made by Hardie, who was kind of mocking how trivial it was. But of course, an untrivial step was not kind of proving, you know, this identity A plus B equals C, but turning the kind of question of kind of quasi-biological question of this formal geneticism to mathematical questions. And the moment it happens, the rest kind of trivial. And this is typical for application of mathematics to biology. I wouldn't, it's not quite so for physics, but definitely so for biology that non-trivial step is guessing the right mathematics. Once you guess it, I mean, it's solved. So you cannot ask what is an important problem in biology, mathematical problem. If it is known, it's already solved. And the question is just to how to formalize what we see. And the impression we have that there are many things which can be formalized as good as we know can be formalized from classical genetics or, say, statistical mechanics, entropy, where it has been done, but it's not done. And it's not sophisticated kind of mathematics, sophisticated in the sense that it needs kind of a well-balanced structure, like for kind of formal field theory, because there are incredibly balanced mathematical structures. But essentially, because there was a way to express what you want to say. And the kind of classic example of Boltzmann, who was presenting his vision of gadgets, statistical mechanics, and mathematicians were critical of him because their language of that time conceptually was not adequate to say what he was saying. And they were translating to the awkward ninth century language was nonsense. But not because he was saying nonsense, because the level of understanding of mathematics was very primitive. Ninth century, it was multiplication table mathematics, as Hardy says. And it remained essentially multiplication table mathematics until the mid of the 20th century. And then there was a change due to coming from category 30, Grottingdijk, and then non-statute analysis and general mathematics changes. Ways, but it doesn't mean we know answers. It just only teaches us being humble. If we cannot formulate something properly mathematical, it means we are stupid. Not because the nature kind of doesn't say the right things. And so in BSA, if you were considering Boltzmann, I think there is quite an interesting mathematical question, which is unsolved. It's not even hard to formulate it and just know how much we know. What is Boltzmann equation? This is kind of a much easier problem than the one we will discuss in biology. But it's already unsolved. You see, it is almost purely, almost mathematical. So what it is? And some, a plunge mechanism, we write some kind of integral. Pum, pum, pum, pum, pum, pum, pum, pum, pum, pum, pum. Instead of this equation, it's nonsense. It's just nonsense, nice and century way to express what Boltzmann was trying to say in this language. But in 20th century language is something else, and you don't know what it is. So intuitively, very simply, as Boltzmann says, we have randomly moving particles. So they collide, and they collide according to certain law. We give this law. And then we make conjecture that they were on the collision. They are as independent as they were before the collision. Which is, apparently, that was satisfied in the Eiffel German. So it was, it might be unsensical. However, this can be encoded and written in the formulas. But of course, the way I said it is already said. And everything is there, except we don't have now mathematical precise language to express it, except writing stupid formulas. And then people working on these formulas, and they believe this is subject matter. It's not. Because the essential feature in these vague statements is functorial. What I say, unspecific applies to everything. And therefore, it means a functor between two categories. It's not equation. And then we specialize it to some very special case and think you're a Greek mathematics, which is nonsensical. Or it applies to physics. It's also nonsensical. Of course, it's rigorous anyway. I mean, it's formulated. But it's kind of wrong mathematics. Whether it applies, all this conversion, certain people prove that something converges here, there. If they have any relevance to physics, it's questionable. Because they unconsciously assume that real numbers are real. Real numbers are not real. They are just a word, a real. They are not in real world. There is no real numbers. Word is governed by something. Because people believe in the great minds that Maxwell actually saying explicitly, saying that physics mathematics is about real numbers. It's not true. I mean, there is no reason to believe it is true. We know very much nowadays. It's certainly not true. At least there is no reason to believe it being true. It's probably this way. There are all reasons to expect it's not. And therefore, all these kind of ways for computing certain cases, this logic is OK. But understanding anything is kind of not. And so one simple question, this very simple question, what is the correct factorial definition of Boltzmann equation between which categories is a function? And this is true about very many physical equations, physical mechanics. There are functions in certain categories that we should not specify. And this creates this tremendous mess where the meaning is forming as well. Meaningless. In a way, if you know notations, the hardest part to understand notations. The essence is in this sentence. Like I said, if you say, Boltzmann sentence, that entropy is the number, the log of the number of states. And I think today's translation is, entropy is element of growth in the group of such and such category, which happens to be a number by the law of knowledge numbers. And that's a number. It's not how it translated. But you don't translate it by the formula. This is a formula by Boltzmann. It's not definition. It's a very convenient computational formula without which this notion will be not useful. This is why you can compute it. But certainly, it's by no means a definition. The same, this particular formula is for specific Boltzmann equation, also a specific kind of entropy. So what I said, this idea of Boltzmann is very general. Log of the number of states, then you have to interpret what it means. And the reason is, in some cases, I'm saying it's just log of the element of some growth in the group. And then you can see and get a response to what he says. Much better than this formula. Because when people say, why this corresponds to that, and then they pretend they explain. They don't explain. Because you cannot explain, because it's not right. It's not entropy. It's a special computational formula for a special case of that. And it has deep meaning, actually. It's not exactly what Boltzmann wrote. Because in his formula was Boltzmann constant. And therefore, the whole point of this constant was deletion of a microscopic and microscopic world. And this was not a real part of the formula, not the stupid pias. Which is, well, usually in physics, you write integrals, and integrate dx. But that's minor variation. But again, the point is, it's not that. Also, there is some k. And this k, like in the quantum mechanics, h doesn't go to 0. h is specific number. And this, by the way, will be even more pronounced and will go to biology. Things are not asymptotic, interesting phenomena happen in a particular range of parameters. So this is just indicate your instance of a mathematics which you kind of almost understand. If you work hard, I mean, this can be settled. You can make pure, good formalism, Boltzmann, and other equations in statistical mechanics. Which has not been done, as he said, spoke to many people that apparently it has not been done. Everybody can believe this way. If you talk to your thesis, of course, this is how they think. And then they write these formulas because they don't know what else to do. And it's kind of, so my own experience with that was when I was quite young. And I was kind of looking at some recent literature about Boltzmann equation. And there were hundreds of papers. Just obviously, there was this function. People were substituting particular measures. And this was the meaning. They are writing absolutely the same argument, paper after paper, hundreds of papers. Boltzmann equation for different kind of situations. And in fact, I mean, computing without having, of course, simple definition, it would be just one line, sort of hundreds of hundred-page papers. But this has not been done. So Mathematics we need at this point is not kind of beautiful, elegant, symmetric Mathematics, but just to understand, to translate the simple things to simple language, like non-standard analysis. You have infinitesimals and you have to be sure you can speak about them freely. Or you have this natural object. You say, naturality means factoriality. And again, you have fantastically simple language and things become rigorous. OK, this is kind of preambler. And now we will turn to some method at hand. And so if we see what happens here, we have to eliminate. So we want to think about biological system, particularly symmetry in biology. And just start with crystals. These are not quite physical object. And this is a little thing, just for your information. And this is proteins. So just below, you see some, I think it's myoglobin. Just more or less how it looks. Of course, it doesn't look that nice, yeah? There is no picture of that. So it's made out of each ball is amino acid, its size is about a nanometer, or half a nanometer. So a small thing in the nanoballs. So the best you can see is x-ray crystallography, which in fact, highly sophisticated mathematical thing. You can see distribution, electron density in space. So we have a certain error. And then you make these pictures. And by the way, making these pictures, it is not trivial task to going through this. Because if you look at the protein database and you see coordinates, or some point that out of these coordinates making the picture, it's highly not trivial. And I will come to that. It's one of the problems about proteins. How to describe protein? You know everything, but then you want to see it in a way you can say something interesting. And upstairs, you have different crystals of proteins. How damn thing may crystallize. So this is just for your information. You see, when you have this kind of information, it's used to say, yeah, I just, if you look at this, you don't suppose to remember this, but just to have an idea what objects are angstrom. It's one-tenths for nanometer and nanometer. Yeah, let's explain what it is. And this, of course, the scales are crucial in biology. It's everything happened on a certain scale. And physics, of course, it also so. But this constant looks more flexible. And here, I think we live in some range of constant. You change them a little bit, and you are dead. I mean, there is no life. This is another remarkable thing. We can exist only in a very narrow kind of bracket of parameters, at least most of us. There are bacteria, or more. There are bacteria, archaea, or more. Yeah, now there's a few words about extracellulography. So I have a crystal. So how you can understand the complicated particle? You have a particle. By itself, it's just, it's invisible. There's no way to see it. However, if they arrange systematically, periodically, then this periodic structure carries all information about this particle. If you average various characteristics on this crystal, this averages carry information about each individual particle. Again, this is a mathematically interesting statement. So what you can see by looking at many particles simultaneously, how much you can know about them individually. How well they should be organized. If they're not, if they're kind of chaotic, just in the gas, it's not so clear what you can learn. This, by the way, another mathematically interesting question which we have to formulate, how much you can learn about individuals from the bulk information of a bulk matter. What are you do something with this and what you can say about atoms? You hit it, you see how it hits. For example, as you know, it's about water. It has this remarkable property that water is liquid at very low temperature. So very low because so H2O at the normal pressure. So up to 100 degrees, it's liquid. And molecular weight of this is 14, right? I'm sorry, 18. If you take something like CO2, which is 12 plus 32, which is by far heavy, it's in the normal room temperature, it's a gas. How could it be? If you take almost all kind of materials comparable to this, to the water, there will be gases at normal temperature, right? Just take, I don't know, nitrogen. Oxygen itself. O2, yeah, it's certainly heavier than H2O. How could it be? Why water is liquid? And so it's very peculiar substance. And nobody, by the way, truly understands it. I mean, we have a lot of questions on this too, I would say. However, the structure of liquid water is a fantastically complicated and there is no good way to describe it. Again, this is described in metaphoric language. And then the micro-computational models. But there is no mathematics able to say, so what happens? Even if we knew it, we don't know how to say it. But that's immediately, you see, so peculiarly about water. And this, so-called peculiarity, is essential for life, by the way, if not for that. And this, of course, is a mathematical phenomena. Some about Schrodinger equation for these molecules. Molecules have polarized in a particular way. And it's a result of probably, probably nobody can prove it, of course. It follows from classical Schrodinger equation. And for some numerology, it changes a little bit. Parameters of atoms wouldn't be there. And we won't be here either. OK, so this is about crystalline. So in a way, what x-ray crystallography is, which I myself don't understand 100%, it gives you Fourier transform of this. So I have a density distribution in the crystal. Electron density distribution in the crystal. And you take Fourier transform of that and take absolute value of that. So it's three-dimensional Fourier transform, but not it. But it is only its amplitude. You cannot measure the phase. And exactly why you have all frequencies I've not quite understand. You must have thought carefully, because some kind of physics, which I haven't thought carefully. In any way, a reconstruction of the image, say the shape of a molecule, from its this image in the crystal, x-ray image, is highly non-trivial mathematical question. It was really big success in the early 50s when people managed to do it. And it was, in particular, say, for DNA. People usually remember Crick and Watson, but they don't trivial work in a way, much harder work. But I think it was done by Wilkins and Rosalind, if I can give you a name. They are not so much mentioned. They made them up at the bulk of the job, and then it was just a kind of children's game. But Crick and Watson reconstructed DNA, because they already, basic elements of that discovered by this mathematical analysis of very poor, at that time, x-ray, x-ray images of DNA. What is his name? Nadia Rosalind? Yes, right. Exactly. It's controversial who done the work, but that's usually forgotten the day. This was really highly non-trivial, non-trivial stuff. But again, so what I have over the mathematical question I want to bring in and just to see, this is easy kind of stuff. How crystal may come about? So what are crystals? What are mathematically? What are crystals? And in the simplest kind of perception, and this is your kind of scene, the crystallography accounts, it's a pattern invariant under some group, which contain group of this. And it's like a bigger group, crystallography group, where this is a subgroup of analytics. And well, this was actually, interestingly enough, problem raised by Hilbert. So, namely, that if you have a group gamma, discrete group gamma acting on the collision space or any other dimension, and there is discrete group of isometers, which is called, so I think the question is compact, then this gamma contain the slashes from which it follows. And this was actually the formulation of Hilbert, from which it follows there are only fine domain typological types of such actions, about 200 in dimension three. And so interestingly what Hilbert makes remark, this is kind of typical of many of his problems, he said, well, that's a remarkably difficult problem. But if you look at the hyperbolic case, it's very easy to classify the groups because you know there are infinite domain typological types and the question is closed. Yeah, now this was solved by Biberbach and this kindergarten problem, yeah, I mean just. Already everything was ready by that time, it's just exercise, yeah. Show that this group is sitting there, I mean, because they were already kind of tools doing that, I mean, and it's quite easy. I mean, it's just several line proofs. And this, of course, it still goes on and on and on and it will go forever, yeah. But anyway, but our question is it is, extremely naive picture of a crystal, yeah. It doesn't, it just tells you something, but it's just the first word of that, what will be the second. One of them is how in principle crystal can be formed. So you have this kind of strange shape, so if you have a molecule highly symmetric one and they can arrange themselves in a crystal, you may be not so much surprised. But if you have this, like myoglobin molecule, which was one of the first to be crystallized, like here, yeah, this, how you can make crystal, how such thing may organize themselves and something so well organized, shape themselves. It looks highly improbable, right? You have this potato type shape, you put them into basket and shake them and you have a crystal. Have you ever seen that? Does it happen? Why the hell it happens? And the point, one of the points is, is that they all are the same. All this kind of protein particles are roughly the same. If they would be somewhat different, probably you wouldn't have crystals. For example, say five kinds of particles, it would be different kind of world of structures they would organize. The point is that you may have this, all particles being the same. Actually, the sameness is another characteristic, not obvious feature of biology. It's where symmetry begins before crystals appears because you have on one hand, quite complicated object. It's highly organized, it's not at all, it's random shape of particular type. Again, in nature you don't see that, in non-elive nature. If you look, it's snowflakes. Snow flakes also kind of have particular nice shapes. Look at them, they're very symmetric. But there is not too identical snowflake in the world, for all we know, right? If you count the number of possible shapes, it goes into, by far exceeds the number of possible number of snowflakes, well, probably, right? And they form by some mechanism in symmetry come from intrinsic symmetry of the atoms. But here symmetry come from a completely different reason. There is no absolute in the symmetry, this is a big shape. And when it makes a crystal, it's a lot of water there, so it's kind of already watery, and so let's make it. However, it is mathematically is not impossible, right? So just now I want to say that mathematically is not impossible, and just let's, just a few remarks of how symmetry comes about. From nothing, yeah? Well, not from nothing, but from the things which things are all the same. The fact they're all the same again, yes, it has some mathematics, mathematical easy behind it. But the first level is quite simple. If you have one dimensional pattern, and you probably all heard this DNA helix, it makes a helix, and you may wonder why helix, but on the other hand, it cannot be anything by helix, yeah? Because any one dimensional structure made of more or less identical blocks will be by necessity be helical. So you make the blocks, this is invisible. Of course, there is nothing helical there, but why it should be helical? The corresponding mathematical statement is that if we have a cyclic group Z, isomatically acting on the euclidean space, the typical orbit, yeah, follow this helix. Of course, you may have parallel translate, or you may have rotation, but this are kind of two degenerate cases. Typical thing, we move and rotate, and any isometry is like that. And it's quite clear here, because if you imagine that this contact take place, which minimizes the energy, which is simplification, but quite reasonable one, so there's always preferred points of contact. And then this contact will begin the same. So you attach one on the top of another, and then one will be slightly turned with respect to another. So it means you have exactly this isometry, and so you have this helical symmetry. So anything one dimensional has helical symmetry. What's remarkable of DNA is not helical symmetry, which is kind of trivial consequence of general principle of symmetry of the space, but it's one dimensionality. And it's kind of suppressed because it's assumed, right? That's one of the problem also, that DNA is one dimension. And that's a fundamental kind of property of life. That life starts with dimension one. And nobody can imagine starting with dimension two, also main brains I will discuss about them. The 3D system will organize crystals that are typical for living organisms. They're very rare in crystals inside of your protein crystals. But one D structure is everywhere, and 2G also. It's symmetric, 1G and 2G structures. And once one dimensional, it's again. Just think about that, it will take you very far. The white can be two, three, four white things become very different when dimension grows. You cannot make anything with high dimensions. But that's one simple thing. So the helical symmetry for that reason is persistent in biology. But there are one dimensional structure. They throw molecules, they touch it out, they just laugh to become helical, I mean just. If they're non-helical, then for a good reason, not to be helical. Everything is helical. But then, still however, how it may have a crystal 3G. But there were another interesting point, a conceptual, not mathematical, that the major way to study proteins which live in your cells, to do something to them, they don't do themselves typically, they don't crystallize. In your cells protein, I think maybe in your eyes crystalline form, if crystalline is a crystal, Nadia, crystalline in your eye, is it crystal or not? As a protein, crystalline in your eye is some kind of particular protein from which your crystalline is made. Is it crystal or not? I'm not sure. Yeah, I'm not sure either. But in any case, maybe with a few exceptions, there is no crystal in nature, or protein crystals. However, there's a major way to understand proteins and in biology, one of the major means of making crystals. And that's kind of an interesting point that you understand nature, not by looking at what it does, but what it doesn't do. Otherwise, you would be nowhere. Otherwise, you would be like an ape. And you see what you see, that's it. But the whole thing you do and see what is not seen, and do what is not normally done. That's a conflict between common sense and science. It's just exactly opposite to common sense. Interestingly enough, again, its point of view is very obvious, coming from the 20th century. Say, the 19th century, somebody like Kelvin Thompson, who was a great physicist and mathematician, was emphasizing that mathematics is nothing but the extension of common sense. And from our point of view, of course, there's nothing to do. Exactly when mathematics starts, common sense dies, because common sense is just nonsense. But it was not mentality of the 19th century. It's very interesting how much it has changed, right here. Now, our common sense is very, very unreliable, and exactly because there was revolution in physics. You know, quantum mechanics came completely showing you all your common sense is just nonsense. Everything you say just naively is wrong. Period, and therefore you don't trust anymore. In a very sophisticated level of physics, not to say about common life, only certainly everybody knows nonsense anyway. So, but that's now how, but still with this, just if you use a little bit common sense mathematics, you can realize why it's not impossible still to have crystals, but it's not impossible. But it's quite interesting mathematical question, which has not been, in my view, properly studied. So imagine, so you have these particles and they have some interaction. No, they won't, they have a tendency to take minima or at least critical points. Why they will be symmetric? And that is not so, even if they are completely, completely of wrong shape. Still, you know, they seem to be group operating. They're all the same. Imagine you want to put them on some lattice and then the picture, have a very simple picture again, which has not been, in my view, properly studied. You have infinite dimensional space. You have a group gamma acting on the space. You have a function invariant under the action. And then critical points are slough to be fixed under this group, right? It's amazing, but if you have some group acting and you have some minimum point and it's very likely to be a fixed point of the section, moreover, if it happens, it will be stable, right? If not the general, say, maximum point and there is a fixed point, there is a group action, it will be stable. This is a little kind of very simple mathematics. It shows it happens. Therefore, these crystals are not impossible. They're not so improbable. Once they're there, of course, some parameter must be matched just a little bit. And then you will have stable fixed points, stable critical points, stable under and will be symmetric. However, again, in my view, there was theory of, Morse theory, equivalent, but it never approached this question. It was going along the lines. It was much more, a very simple question. It is more complicated to understand because the group is not compact and there are a few groups acting here. This group gamma and this group observe, this group not canonical in the Euclidean space, up to linear transformation. So there is a parameter in the group. So it's not one group gamma, but in our three, six parametric family, right? Because the basis of the group, you can change. And if you look at the modular space, it will be this radical, so sophisticated symmetric space, five-dimensional locally symmetric space. It's just this space of all metrics, modular gamma. So it's quite remarkable, kind of five-dimensional manifold, we singularis orbifold, locally symmetric space, which underlies the picture, it's crystals, biological crystals, nobody look at it. I mean, it's amazingly, right? You can make a purely mathematical theory up to some point, which hasn't been done. And again, it's not my intention. It will take, of course, certain time to develop it and a long time to explain it. But it's clear, it's in there. However, it's only, again, still kind of trivial kind of thing because you see it, yeah. And now it's just mathematics. Once at this point, it just, well, not quite trivial, but still in mathematics. On the other hand, there are much more about crystals. That's exactly the advantage you get making point outside of mathematics. You start seeing what you don't see from inside. And what you see, crystals are not just God given to you. The crystal grow. What is crystal growth mathematically? Again, physicists have this or that model, but we don't have a definition mathematically. What does it mean to have a crystal grow? And so what is abstract mathematical description of crystal growing? And the crystals are not anything, they are finite and they have defects, et cetera, et cetera. There is no mathematical language to describe. Again, the easy language for people in applied math, but they emphasize examples, there is no general principles. Okay, this I just want to, this is all my lecture with propaganda of questions we don't know how to solve. Because each of them will be certain theory and this theory will be quite interesting. Okay, what are these? Yeah, these are some talk about symmetries. So I want to know to turn to something else. Yeah, nine parameters, not six, yeah, of course. Linear group is nine parameters. So typical equation is, yeah, specific mathematical question in what I'm saying would be, so I mentioned I have a mixture of two types of particles. Still crystals are possible. So how it depends on their proportion and their relative size, probability also. Probability, meaning in the parameter space of them that they form a sufficiently robust crystal. So you have kind of huge, huge variety of question, but the following interesting phenomenon, which is not related to crystal, but may have some role here. If you take some powder of some kind, and it just says powder, and then they can add the powder and the little powder, but imagine the particles are very different size. If you mix them together, it looks like a dough, like almost fluid. All right. And I don't think there is a mathematical description of that, it's a very long phenomenon. You mix two powers of different size, and it becomes one kind of liquid with respect to another. And the whole thing looks like rather viscous, but fluid, which is systematically used in people in all industry when they put the water into there and just to make things fluid. Sometime it's more fluid, you add more powder, and it becomes more fluid. So, and the same may be true about proteins with many particles. If you have particle of different size, it may completely have completely different mathematics and crystal will be of completely different type. A priori from what I said is still maybe crystal, it still would have some symmetry, but it may be symmetry of some way different type. Okay. Well, just very naively, of course, crystal of anything. It's naive, but not terribly naive. It is local minimum, at least local, and sometimes global minimum of certain energy function of interaction between these particles. Right? Yeah. No, no, of course, they're active, they don't have to be crystallized to be active. But when they fold and you can form a crystal, you learn immediately very many things like that. First, you learn them all more or less of the same shape. So you know from the existence of crystals, you conclude that all protein molecules of given type are roughly of the same shape. And it's extremely strong conclusion by the way, yeah. Because you don't see this particle, this particle that's a few nanometers, you can't be see. Microscopy, the crystals were observed a long time ago. I think crystal of hemoglobin, or myoglobin, we're known for almost 150 years, yeah. So just looking at that, you know, whatever substance it is, it might consider identical particles. Right? Because if they're all different, I think crystals are impossible. If they really had a very heterogeneous collection of particles, there would be no crystal. But the activity has nothing to do with crystallization. The both the activity and crystallization depends on the fact that it is preferred shape and it's taken by most of the particles. That's true, but this, so this is fact. And this is a highly non-trivial fact about life, that microscopic, as much as macroscopic particles very often go in series being almost identical. Like all human beings more or less identical, right? Amazingly, I have billions of them, all the same. How could be, right? It would never so for financial objects. Yeah, I have stones, they have much more variety than human being, given so many parameters, of course. In this parameter space, we highly localized. And already that starts from proteins and two properties, both is biological function depends on that, and that's ability, our ability to understand them. But crystallizing, of course, very tricky matter. And this is a way a mathematician may be useful because crystallizing protein is a highly, highly non-trivial. It's art, which means people don't understand how it works, but art means I don't know what the hell I'm doing. Now about membranes. So these were kind of three dimensional structures. They, mathematically, we kind of approach them. But before we turn to protein structure, I want to look at membrane where, in fact, mathematically, you can say more than father's one, huh? Yes? Well, yes, if you have more or less identical particles of any kind, even viruses, you have quite big particles. In principle, you can crystallize them. So you can make a range. In principle, meaning, you usually add some substance in a particular temperature. But then, there exists, at least metastably, some local minimum of configuration of them which you'll periodic. And this is, of course, not known for all, but there are proteins which you can crystallize for good reason. And they have no definite shape. But lots of things can be crystallized, including DNA segments. Big piece of DNA will have random shapes and you cannot crystallize them. But again, I'm saying that I don't truly understand it. Because first, I don't know enough crystallography or biology and mathematically, I think it's unknown. This I also don't know, but this I know nobody knows. But the rest, possibly, some people know better. But I'm not that certain that this is dramatic well, structural knowledge of the nature of recrystallization. It's very, just how intropically, it happens. And it is not at all clear. And again, there is no mathematical study of that. So from our point of view, if there has not been studied by mathematicians of post-Grothendic time, things don't exist. So things exist for mathematicians if they were presented by post-Grothendic style of mathematics. Even if mathematics outside of this field, I think that doesn't exist. It's just pre-mathematic. It's just precursor of true mathematics. And most of the mathematics of this kind, and most of the science of this type, it's precursor to real things. Of course, when we go, this post-Grothendic will be also an old-fashioned, but anything which doesn't fit there looks rather pathetic. OK, and I'm reliable, of course. But now I want to speak about membranes. And so let me show you a picture. This is our erythrocytia. This is the erythrocytia blood red cell. And just for looking at this, I mean, just think about this. And learning it raises huge amount of mathematical problem in all radiation structures. So I saw what many years ago I attended a meeting about erythrocytes, what happens to them in their own condition. This shape is highly unstable. You put them into a glass, into a glass, and just whatever in there, it immediately changes form, become completely chaotic, irregular form. Which, again, never happens in nature, but was studied quite a bit. So the first question, why this form? So this beacon cave, this would be what emphasized by Balusius, why beacon cave. But the question for mathematicians, why it has axiosymmetry. See, it's not spherical symmetry. It is axiosymmetry. Now we have much more acquainted with spherical symmetry. And we all know of cell bubbles, and they cell bubbles are spherical. And then we say, huh, we even understand why they are spherical because they solve the isoparametric problem. And this, I have a little discussion with that. Historical, it's kind of an interesting problem. So when it was first, that's quite old. But somehow it was not influenced by erythrocytes. And for erythrocytes, we still don't know so what happens. There was a similar problem, which for some reason was more famous. It's called build more problem, which was recently solved about some other variation of problem. But the very extreme shape was also having only one symmetry. It was a torus. But for erythrocytes, for some reason, mathematicians never thought about it. Now biologists, at least it's by engineers whom I communicated at some moment, they say, oh, we can prove. It will be this shape. Of course, my understanding, what they can prove, assuming it's circular-symmetric, they can say that they have this form. Again, you have to explain in what sense. What variation of problem being solved. And why? And it's very similar to build more problem. But in my view, unlike build more problem, it is mathematically incomparably more profound and kind of opens perspectives where build more just seems to me just little thing compared to that. And this looks to me really pertains to very fundamental issues in mathematics if you understand what happens here. So that's very naively so what happens here. The problem you can formulate is as follows. That this shape, so by the way, this also applies to solve bubble. You can say, we prove solve bubble, indeed me has minimal area to give the volume. And we've proven this and there are proofs, which by the way, none of them through satisfactory. All proofs are kind of not quite nice. All on trivial and there is no 100% understanding of that. And we hardly touch the problem because we see this is not the surface of a solve bubble. It's not like a ball or something. It's not that it doesn't want to stretch or whatever. Why it would minimize area at all? Why it wants to minimize area of the boundary? We have to return to this actually, the solve bubble very much similar to the other side. The mathematics, physics, chemistry, whatever, behind it's more or less the same and it's not this naive plot problem. This plot problem is just from point of view of solve bubble, solve bubble, if you could laugh, you would laugh at mathematicians who said they understood it. Because they never touch the surface of the depths of the problem, which isn't there. Because how it's made, how it's out of soap, it's nothing, how this thing can exist, right? You know, we may be that big, how can it be? I mean, it's absolutely, again, this is a very naive attitude, that how you solve this G-door problem, of course, 2000, whatever, 3000, 4000 years ago in G-door, it was okay. Mathematics is still on the same level of understanding and now we have a retracite. So what are the problem retracite solves? But again, this will be superficial discussion, though it will really go well beyond this naive isoparametry. It is, this is what people say in bioengineering and maybe it's right, maybe wrong. This is, of course, there is some kind of heuristic reasoning why it should be so, but I'm not certain that there's good experimental evidence for that. It, per given volume it bounds and given area, so two parameters are given, it minimizes integral of the squared curvature of the surface, which is almost like wheel-more problem. I think that is slightly different. So, and then the basic, like for usually isoparametric problem, the fundamental thing is that solution is spherically symmetric. If you have linear problems, then typically symmetric data implies symmetric solutions, which is not quite so, even for linear equation, but kind of that's the principle, right? Especially we have time developing linear equation. Of course, it's developed on time, it goes, goes, goes on symmetry, only gain in symmetry, not only symmetry preserved, but only gain in symmetry and at least this, the way one, one way to, it's being manifested is the second law of thermodynamics. When people say all these words for me, at least about chaos, that I, about second law, it's nonsense, it's just symmetry which increases because you have kind of parabolic flow and parabolic flow tends to make things more and more symmetric. Like that, how universe, because it is dissipation on certain scale, which means it's kind of parabolic, it is dissipation and symmetry grows and this means what they call chaos, but in fact become more and more homogeneous. Why is it called chaotic, I don't know, it becomes more and more symmetric, right? And then of course, the corresponding mathematical theorem has never been stated and certainly not proven, but that's how you want to understand second law. And again, let's say, we have to continue the Boyceman discussion and to bring it to mathematical level of today. But here it's not so and again the fundamental point, one kind of was emphasized by Turing, that certain slightly non-linear equation, even when they're symmetric, partially symmetric, may bring non-symmetric solutions. For example, if you look at the ocean and there is space and everything is symmetric and there is a wind and it also has this symmetry, however there are waves and waves break symmetry. Waves, right, they're not symmetric. They have symmetry, but the symmetry is expected to discrete subgroups, not with respect to the full group. In another famous example, when you have a certain kind of liquid, I'm not certain how it exactly works. And you start warming it from here and it goes up and then become divided into hexagonal pattern and there are kind of the whole thing and it becomes symmetric here with respect to this hexagonal group, latches and two, but feel not fully symmetric. When you have some non-symmetric thing, non-linear thing, the symmetry tends to break, though not completely, right? If you break, for example, you have this pattern on zebra, what kind, which is in agreement with shooting equations, it has some little symmetry, but not the spatial symmetry, right? But here's a different equation, they're elliptic equations, they're non-linear elliptic equations and typically having symmetric solutions means that in some sense the corresponding function is convex with respect to some structure compatible with the symmetry. But it's not kind of, even for isoparametric problem, you cannot prove it a formulating this way. It's not convex, a conceivable simple way. It is, in some sense it is, but again, by the way, there is no thematic to say in what? So it all proves, she secretly used something like convexity, but there is no good way to formulate it. We don't know how to say it. This by the way, also true in a more modern, modern scale, what Perleman proven, the Wachey proven, Pankarek injection, other thing with Hamilton flow, it exactly the certain flow and because it's flow, it symmetrizes things. And this, what the major, the main point to prove that it has enough time to symmetrize before singularity developed. Yeah, it's in again, secretly, the some convexity built into the function of curvature, which is certainly not convex in any sense. And the one of the easiest to understand in what sense and where, again, in that and this we shall come to this in a second, which is suggested by that. Now, so how we can bring this to make a next step? What would be kind of a mathematical extension of that? So here we maximize, minimize given volume, given area, we minimize, we minimize total curvature properly measured. And volume, of course, is not belonging, not belonging to the surface, it's just extra part. So even without volume, this kind of indistinct question. And so what you may ask, so what, how much symmetry this kind of problem may survive, yeah? So what will be general question of this type? How to formulate this question in general terms. And then, so you look in mathematics and ask, where is it happens? When do you observe that certain sub varieties, say in Euclidean space or somewhere else, not just minimize area, but minimize some integrals of their curvatures, right? And looking at this example, you may guess what, how to then generalize this question. So this again, this is a purely mathematical question. I repeat, you can see the surface is not three. We is given area and given volume, they bound. You want to minimize the integrated square curvature. Again, I shall let you explain why this plausible conjecture. To say they do this life, and if they're spherical and symmetric, I think you can prove they have to be concave form. This I think what people prove. I couldn't find this in references, but I spoke with people a very long time ago. So I even forgetting, forgotten. It was kind of interesting meeting with the post-doctors and by engineers and it was very interesting listening to their conversation during lunch, yeah? They were discussing cases of their patients, which were quite, quite, quite interesting. Maybe I'll tell you later. And then, so how to do it? And then you know what it is. They're complex sub-manifolds of scalar manifolds. For example, in Cn, yeah? So if you have Cn, complex space, and you take anything defined, they have a surface defined by an elliptic function, then not only it will be minimal, but it will minimize quite a few, it will minimize quite a few integrals, all its curvatures. So what does it minimize? How to, so first, white minimize, white minimize area. This I think is a major justification of concepts of minimal surfaces, exactly this example. This is a, these objects are God given to you. And the fact that they all minimize volume is quite, quite remarkable, though extremely simple. And remember when I first learned it, it was really shocked to me, and because it's not the first thing you have to see online or any kind of textbook on minimal varieties. This is why you care for them, yeah? So the simplest case would be something two dimensional. So it's a complex one dimensional, like C sitting in C2. For example, maybe graph of a homomorphic function, right, such a graph in C2. This will be minimal surface. Moral with absolutely minimizing, if you're given boundary, it has minimal area, absolutely minimal, for all possible, for all possible surfaces. And the proof is extremely simple because in the space C2, say C2 in this is, in general, you have this differential form, this exterior form. And this form has a property. So this is very, very simple. Yeah, there's a way to describe it, it looks kind of tricky, but it is C2, it's projected in two ways to see. On each of them you have area form, here area form. This is one area form coming here, here is another area form, you take this one. Right, so area of a little piece of a surface in this space is some of this, plus this, but of course you keep track of signs. If you change it, you change the sign. And of course, because these maps have points of degree, they're conformal, it's kind of obvious that if you have inside of C2 anything complex, and complex meaning it goes here and here by conformal map, then this form just equal to the area. So if you integrate this form with little piece of a complex things, it will be exactly area. But if you take anything which is not complex, it will be strictly less. Right, in some directions it's zero. If you take this long diagonal between two planes, any purely real, on the real path, it will be as zero. So it goes from zero to the maximum value, maximum value in complex direction. And this kind of observation I think maybe by Federer, amazingly it was not done 100 years prior to that. Maybe it was, but I'm saying Federer, but they haven't checked the history. I will not surprise it was known to somebody like Munch, well no, if Munch new complex analysis of course. And therefore the rest is quite clear because now this integral, what is the integral? What is this, you integrate this thing, it's your form against area. And because it's closed form integral doesn't depend on the surface. So it's always the same, right? So if you take any other surface, if you have some, for any other surface, integral maybe because integral is always the same. So if area goes down, if you move it, I'm sorry, at every point the form may become only smaller. So integral can only become only smaller. The only way to keep it the same area must become bigger. Point wise it becomes smaller. Therefore, in order to compensate and the integral gives the same number, area must become bigger, right? If you integrate smaller function and get the same result, meaning domain of integration must be bigger, right? Because kind of product, what is the integral product of area, roughly of this, by the value. The product constant, one term goes down, another must go up, so area goes up. That's the integral, right? If the moment you believe in this algebraic property is called wetting in inequality, that, that. So, and but then, how you can go to high coverage for algebraic case? And then there's very simple thing. If you take that, you can produce new objects, you just take all tangent planes to this. And this goes to high dimensional complex space. Then you can take tangent plane again, again, and again. And you're gonna get complex objects and some killer manifolds, and they're all minimal. But now the areas, if you, for example, if you take a curve, take all tangent, that will be the kind of, you're making tangent essentially are taking derivatives. So integrate area of derivative. So integrate second derivative. You do it two times with third derivative. So at each level, you integrate high and high expression of high, high derivatives. And they're all, and they're all minimal. And then you do this for smooth scale. This was motivation starting from this, from this example of a retricide. You just see how, what you have with mathematics kind of vaguely similar. And then you come to the following variational problem. You take some manifolds in Euclidean space, and you remind the manifold. Take it all tangent direction. So it's both k dimensional, say an n dimensional. So we take this, it will be grass manian of k plane on your manifold x, this big manifold x. And this was some manifold y. And then this y lifts here to this grass manian. They call tangent planes. Then keep doing it many times. These spaces have more or less canonical matrix. Each grass manian is homogeneous space, has a metric. And if y is remanian, it has parallel transport. So there is metric in the horizontal direction. There is some ambiguity. There is one parameter you have to adjust. So on each stage, there will be one parameter, which is probably very essential. So it will be, when you go on two levels, there will be two parameters. So it will be a combination of certain quantities, which are all integrals of higher derivatives. And then you ask, so what the variational problem is? So what is the solution to these problems? So abstract, and this will be the first level of that, will be this problem here. This original problem of aditricite when you go one step up. And in particular, you ask if, so if you go to the level k, you lose kind of k degrees of symmetry no more than that. So as much symmetry as you expect from counting parameters remains. And of course, we don't know that. And we don't know what will be the theory here. It will be very different from usual geometric measure theory because smooth structure involved in a more serious way. In the usual geometric measure theory, the point is you forget about smooth structure you work with very singular objects. And the power of theory still can go through. Here smoothness remains. On the other hand, when it's algebraic geometry, you may have singularities. It's OK. Things go very smoothly even with singularities. So what happens here? So this is a, this looks to me quite interesting, quite interesting issue. However, this again is peanuts compared to the real problem. This is, of course, difficult. Question is kind of something much harder, probably if it can be done in geometric measure theory, which is already hard enough. But again, compared to the real thing, it's still kind of extremely naive. Because you have to look back to this aditricite and so what makes it kind of all so bubble for this purpose? And just look at this, how it's made, how it manages to minimize this quantity. And then it's kind of quite, quite, of course, amusing. Because it is like that. So this membrane of this is not really kind of like that. But it's built out of perpendicular normals. Road-like molecules, which are, let me see if I have this picture here, called by Lipitz, which don't stick together. They stick together not because they're sticky. They stick together because for purely probabilistic reason. No, it's already proposed to viruses. So I don't have this picture here. And so you have this kind of molecules, and they are road-like molecules, and they make this, they make the membranes. Why they make these membranes? Essentially, I mean, of course, I'm against metaphoric, because this is the most probable state for them. They don't interact themselves. There is not force between two of them, keeping them together. And in fact, you can see it in the aditricite. Actually, aditricite, all cells made this way. But in most cells, besides these, there are lots and lots of proteins sticking inside. In aditricite, there are very few proteins. They are most pure, pure, this by Lipitz layer. And you can see, there is a microphotograph seeing how it moves. These things, it's two-dimensional liquid. They completely move freely, one with the pectoralis. They keep this shape, but within themselves, it's two-dimensional liquid. There is no forces, no stickiness between them. But what keeps them together is probability. It's most likely configuration in for certain measure. These are the measure in the space of such molecules. So these configurations are most probable. And the same, by the way, reason for cell bubbles, which we make in space. They take this shape because it's most probable shape, which you can come using. Of course, you must be careful what is the measure, and how this measure comes, where the measure comes from. And this measure is not only in the space of these molecules, but in the water. If not for water, it wouldn't work. The water is called hydrophobic. And they have certain affinity for water, positive or negative. So hydrophobicity, whatever. So water molecules are highly polarized. They are oxygen. There are these two hydrogens. And it has high polarization. And so there are these electric interactions between different molecules. And these kind of molecules and any other molecules, in particular those. And these are not polarized. So they don't like to have water, especially at the ends. Especially, and these are by lipids. So here is a lipid ends, which don't want to be close to water. And so, yes, if you write down the energy, kind of, boyfriend function for that way, one term is probability. Of course, another is interaction. This tends to be minimized when this sits inside. So they want to be hidden. And this is a shape when this center becomes hidden. So, of course, energy is involved, but not so much energy of interaction between the molecules, but the interaction with water molecules. And then, from this probabilistic reason, they exist. And then this solution, the reaction problem, enters here. However, it's still not that easy. And it's unclear what should be kind of proper mathematical theory. So what, of course, interesting is that, so this differential equation, which eventually defines either sol-bubble, erythrocytes, whatever, come from this statistical ensemble. So if you have a statistical ensemble of two kinds, water and these particles, there is energy attached to it. And of course, there is always entropy meaning the phase volume when they leave. And from this, you write something, some kind of function in this space, which avoids some function, which kind of minimal which corresponds to that. However, there is always parameter in the temperature. And again, this is how kind of non-mathematical thinking gives a new perspective. You hit it. You should not do mathematics with minimal surfaces. You never ask, right, mathematically, what is happening if you hit a plateau equation. But here it is. You have to hit it. Everything disappears, right? They all decay, live decay, and they go, all right. You cool it down, and they still come, so they come back. And we don't have simple mathematical model for that. Absolutely. And this is quite remarkable because you may think, because this process of heating, whatever, is kind of a, of course, kind of phase transition, but this is just words. What is, how much of the structure, for example, so bubble as spherical, and this has some symmetry. Can you see the symmetry in the high temperature? What happens when you hit it, it becomes really randomly moving particles. Can we see symmetry there? In what sense? You know that after cooling, it will become symmetric because if you have a long parameter, it's an elliptic function. So everything already there, all knowledge exists if you know high temperature. How you can read information would happen at low temperature. Again, this is a metaphoric verse, yeah? That's exactly the issue. How to make a mathematical model? We know it is there. I mean, there's no question about it. And I wonder is a universal thing for many dramatic differential equations. So my kind of dream that all meaningful equation, I'm not any stupid equation, but really good, like Hodger equation, you know. Minimum surface equation, whatever, they all of this nature. The only low temperature limits of subject much more interesting, and this interesting lives in a different world. But in this world, all the symmetry, all the properties are there if you know how to state them properly. And if you know them, you know much more about the original objects. They're just very degenerate representative of some fundamental statistical law in some phase space. So that's about membranes. So we had one dimensional, we had crystal, we had membranes. And I want to talk about proteins. No, maybe I bought three dimensional and say two words about viruses. But again, we have another kind of variational problem which we again have no mathematical formulation of. So let's look at the virus. It's not very apparent if you look at this virus. This is, I think, an electronic photograph. It's rather realistic. But you may be heard that many, most viruses are highly symmetric. All small viruses have icosahedral symmetry. So they have this icosahedral group acting on them, which was certainly they discovered. It usually is described to titan, the epithelium body. Icosahedral was described by titan. But of course, viruses are much longer here than titan. And most small viruses are icosahedral. There are bigger viruses which have these helical symmetries. Recently, I was alone, people discovered viruses without symmetry. And there was a little shock. There are huge viruses without symmetry. Now again, I said why. But if you first look at this, why the hell a virus must have icosahedral symmetry? No, we don't, people, don't have it. And no other organism of moderate size have this kind of symmetry. Of course, there are five-fold symmetry for the c stars. Actually, I'm wondering if anybody can give rational explanation why number five. Two, bilateral symmetry is kind of clear because of evolution. But when it comes to high symmetry, it's tricky. But here's icosahedral. So it's a group of 6 to 120 elements. And how it can be? Why the symmetry? And this is the cubical virus, for all I know. So why icosahedral? What, again, they do solve some variational problem, but very of a very different kind. And that's, again, this is a little bit speculative, but interestingly enough, this was predicted before they actually was determined by Watson, the same Watson who was speaking Watson. On the basis of data, they had crystallographic data. And some speculation, which I'm going to explain, they arrived at the idea of icosahedral symmetry for viruses. And the reasoning, speculative reasoning, it would be quite interesting to make it mathematical. I mean, it's kind of mathematical, but speculative, again, we don't have enough means to say it exactly. So part of the reason is the same as for usual crystals. So we have virus made out of some proteins, a particle. And this particle inside there is DNA. But if you look at the virus code, it's just made out of proteins. And many other things goes into there, but basically they're proteins. And they make this code of a virus, and they assemble. And having symmetry is no more surprising than having symmetry of a crystal. It's slightly more surprising, because you have to go around and exactly make fit. So these are one parameter you have to adjust. But it's not impossible. It's not completely. See, if you have 100 parameters to adjust, it would be absolutely improbable to develop such system evolution. No evolution is too much. You have a fine fitting of 100 parameters. But here it's about 100. The group has 60. The group has 60. Think, so how you can do that? It's not one-cycle group, but rather tricky, simple group. And then the reasoning, I think, suggests by Rikin. What's in these follows? What is the problem? What is the variation of the problem virus solves? The point is, it does solve some variation of the problem. It minimizes something. Of course, he wants to minimize the size of its code. But what it must contain inside? You see, it's not a volume like a cell. A cell contains a molecule. It want to have a maximum volume inside if it cares. But what virus contains? What it contains inside is its DNA RNA, depending on what kind of virus it is. The whole purpose of this code, to protect and prepare this DNA for an adventure, it must enter somebody's cell. In fact, the purpose of the code contains as much DNA as possible in order for this DNA to be more efficient in doing what it's supposed to do. And this viral code is coded by this DNA. So we have this very mathematically very interesting kind of coupling. We have a surface which must, in gross, contain something. And this something encodes the structure of the surface. It encodes. So the relation is extremely physically mathematically very hard to say what it is. And then it's kind of intuitively clear that symmetric patterns are optimal. Because symmetric pattern can be done of identical particle of proteins. And it's much easier to encode one particle than many, even if this particle must be of special kind. Because in material, you have the same size of genes encoding any protein, whether it's this shape or that shape. So therefore, relatively short gene can encode proteins such that it will mesh itself such that they make the face of icosahedron. So it has this internal symmetry, like triangle. And then they make icosahedron because of the symmetry. And this is the optimal way to do it. It should be noted, by the way, that bases in DNA much heavier than amino acids. So what's usually, you think information is kind of physically lighter simply than what you make, right? You have instruction to make some object. And it's written in thin paper, and then we make big objects. In biology, it's opposite. You have this big stuff, information. And then after this, you make something 10 times things smaller, right? Each piece, so the amount of information, the weight in DNA, to make one of protein, I think DNA is 10 times heavier than protein encodes, at least 10 times. Maybe a moment that I've forgotten there. But it's much bigger. It's kind of paradoxical somewhat. But so it is. So it's a huge thing inside. This information is very expensive. It costs you a lot. Yeah, it's very heavy. Each extra information to encode second, such molecule would be an enormous, enormous enlargement. And therefore, why do you do that? The question is how to formulate mathematically. It's certainly purely mathematical phenomenon, expressed in this language. But you don't know how to, well, you say it is simple, of course, you can make specific models that are tied, but we bore. Clearly, it's something very general. So that's the question of what it is. So it's, again, a very, very amusing issue. And then, of course, there will be other mathematical questions. I haven't thought about that. Imagine once you have a general mathematical question, more or less the way I stated here, you can push it very far and look at other examples. And then a symmetry group emerge from that level. Can you imagine such model explaining symmetry of the physical world? By the way, biology, we see that this reason of some of the symmetries come. You make symmetric thing because it's kind of the easiest thing to do, in a way. And the easiest, and here it's very specific sense, because information, they have to encode it, terrorizes. Easiest is not emotional, but it's really some process. It's a very technical sense, probably the simplest. I'm curious if it's possible to give such a model. Actually, people were trying to do that. I remember somebody was trying to do this for explaining gravity. Gravity has its nature. It's a kind of information type phenomenon. Or people don't accept it, of course, but there is such a model now of gravity. But anyway, this logically can be suggestive. So it's another example of mathematics coming from Belgium. About two years ago, I was trying to write an article about that and try to formulate these questions and just to make at least one step for describing them. But then they realized it would take me several years. I had to write some article fast. And so I wrote only third of my introduction. And this is what I was talking to you about entropy, the path which you can go and more or less make steps inside of mathematics. But all others I found extremely slow process to doing that. You see, in a way, it's harder. In a way, it's harder. In a way, it's easier than doing a mathematic per se. Actually, I was thinking about the following examples. So what is hard and what is easy in mathematics or not? So one fact is that nowadays we have reasonably good programs which even can prove rather difficult theorems. On that hand, there is no hint of a program which can formulate this kind of program, can formulate problem of this type, translate something simple into mathematics. So let me give a very simple example, much easier than what I'm saying when we perfectly know what to do. Like with Hardy, when he was formulating this problem of Mendelian dynamics, Mendelian genetics into mathematics, he didn't have to think. His mathematical instinct immediately told him what would be the mathematical shape of the question. And he even didn't notice it, that he was doing something. He translated this naive thing, he had population, random mating. There is this formula. They multiply. So obvious for him, not even obvious. He didn't have to say obvious. It was like breathing. He don't say it's obvious how he breathes. Yeah, he just breathes. And then even this mathematical computation he had to make, it was multiplication table, but he had to make conscious effort to make it. On that hand, if you try to formalize it, it's opposite. You can easily make any kind of simplifying a formula on a computer. But this first step, absolutely, we don't know how it works. No, it's automatic in our brain. It's not, we don't know what this mechanism is. It's a very simple mechanism, but we are not aware of it because it's never conscious. So let me give an example, a very, very, very simple example of that, which is, I think, is amusing this kind of psychology in a way. And you start with the following question you give to children. And this used to be in Russia, given to high school children on the Russian Olympia. It was an easy question. And most of you are aware of that. You have six people, group of six people. And then there are three of them, such as either any three of them mutually acquainted or not too acquainted. Already, I made the picture graphically. And then, of course, immediately formulated. So I gave a full graph on six vertices. And it's too colored. Then contains triangle, which is monohomatic. And proving this is exercise. Any computer, you can easily write computer program proving it with more general theorem. It's actually a special case of a theorem of Ramsey, which applies to simplices of any size. It says we have a simple of huge size and divided into five to many columns. It always contains monohomatic paths, or better like the large, even if you color not edges, but also faces. And this kind of passing from here to general theorem is obvious. Proving this general theorem, any computer program can do. But making this step from people being acquainted or not to graph, it's obvious if you're a mathematician. And it's just, how we do that? Why it's obvious? It's obvious because we know what graphs are. If you don't know, it would be not that obvious. Probably you could invent the concepts of graphs, though you wouldn't have the words. But that's, I think, psychologically a basic mechanism which in our brains, we don't know what it is. And this, of course, here, we face the similar question, but it's not so simple. We have this kind of imprecise statement. And there is no doubt in our minds that adequate to this thing in graphs. And why we are so certain? We can't regress to justify it, whatever. This step is mathematical, but mathematical a completely different style, completely unknown to us. So certain, so logicals must be mathematics. But of course, it's not mathematics, you know. OK, so, but now, I want to come to proteins. And the proteins are, of course, famous problem. Because it's a famous folding problem. But I want to say this again, it's a technical problem. And full of, it just kind of can be precisely formulated. And so people love it exactly because it's precisely formulated, but it's exactly what makes it not probably a wrong problem. And again, a good instance of that is what Hilbert was writing in his problems. All his problems generally stated are beautiful problems. But whenever he tried to make it precise, it either trivial or wrong or not interesting. It's incredible how it works, just. All his problems, if you just, what he meant, it was really great. And when he wanted to be precise, it was just nonsense. Almost, I think half of his question, yeah. Half still OK. But the original one, but originally what he stated, half of what he said precisely was just absurd. Either it was obvious, or it was misplaced, or it was missing the point, et cetera. For example, yeah, just a famous example. He wanted to say, develop theory of the function equation. So solving this definition. This is a good question. But instead, he wanted to be precise. He said, let's make an algorithm for solving the function equation. The moment he said it, there was a solution. Let's count the example. Completely mathematical useless. Completely kind of ingenious, but having no mathematics to it. And that's it. But the first question goes and goes and goes. Yeah, we have the algebraic arithmetic geometry. Fantastic field. That attack actually closed what he was doing along the lines he was doing in different contexts. But the question was, only another question, to prove in it that you cannot make function many variables, superposition of function a few variables. And again, he wanted to be precise, say a continuous function. It certainly was a count example. But for the found, completely in material. And the question in general is, what the hell makes function of many variables? What are function of many variables? Why specific function, the roots of the equation, so hard to represent? These are vague questions. However, they stay. And they may lead to mathematics, which has not been known. This mathematics is still ahead of us. And this, again, is only another question as mentioned about the discrete groups. By the way, actually, I heard. I don't know what it's true or not. But by impression that very many people's papers were held between correct. He was writing very rigorously, unlike Poincare. But unlike Poincare, he was making lots of mistakes. So one is his solution of warning problem was wrong. And then his solution of Dirichlet problem was wrong. And in the case of Dirichlet, the correct solution was found by Poincare. And yes, prior to him. But for some reason, his solution was accepted and was corrected, et cetera. But I'm curious, by the way, how much mistake he was making. People who have tried to be rigorous and precise usually start making mistakes. When you keep vague, like Poincare, what he was saying is right, but sometimes vague. He was making mistakes, of course. But usually, objection to him, he was not sufficiently precise, but he was saying the right thing. And he was trying to keep precise to make a mistake. Oh, well, it's natural, because it's great ideas, not time to make them. Not because he was stupid. I mean, just was a mistake to be precise. When you do too fast. I don't want to be insulting to Hilbert, but there is a very good joke by Pauli, who was saying, I am not objecting to your thinking slowly, but I object to your publishing faster than you think. So he couldn't have no time to think through everything he wanted to do. I think so. Because he's everything in his direction. All the ideas were fantastically good. When he was trying to make out details, of course, it takes time. And he was not doing good. So this folding problem for proteins is, I think, has some kind of fame attached to it. But it is, I think, not the most interesting problem. So it is very technical. And it says, in general, you have this molecular chain. And proteins with molecular chains. There's some interaction between molecules. And then, eventually, it comes to make the spatial configuration. And you can try to formalize it and say, how you know interactions. And just, you want to solve problems with these equations. It's tremendously computationally difficult how to bypass this difficulty. But when the moment you start doing that, you're making mistake after mistake after mistakes. Because we don't know interaction. It's not like that. They're not all fault. And there are lots of misfeeds and tattered ties, much more complicated and more interesting situation. So once, in general, in biology, it's like you come to the zoo. And but it's not quite zoo. So this is the whole point. So I'm climbing the proteins. Here. So the difference between zoo, animal, and proteins, is two-fold. First, of course, animals, you see by your eyes. And proteins, you don't see, but you read them. You just read about them. And so it's by your mind, eye, you see them. And it's unclear, so we gain and we want what you lose. Because when you see animals, originally, your visual system tricks you in certain perception, which may be right or may be wrong. So you immediately be manipulated by your visual system. So you don't know what you see. Anything you see by your eyes, you don't know what you see. Especially with animals. You see shapes, you're not there. And no matter how much you see, it's very little mathematical there. You have to make some effort to forget in a way what you saw to become mathematical. And this mathematical way of thinking about animals, I think is rather recent, yeah. I think Galileo already had it, but I don't know any indication before Galileo. He invented this. He realized this kind of proportion of different size section of the bone, the size of the bone. And then Buffon also thought about that. But it was kind of still relatively simple, the kind of mathematical thinking about animals. However, if you look at the proteins, you immediately know that this is mathematics. This is mathematical objects of a certain type. In a way, as diverse as animals and this perception of real things, like you go to the forest or you see different kind of trees. On the other hand, you see that they're made out of mathematical unions. But we don't understand them. However, as poorly as you understand animals. By the way, again, this idea is interesting enough that in all the times, I think, it's like people like Aristotle were very much confused why free thrown stone would move. It was clear to them why animal would move. Animal want to eat it moves, it was clear. But why stone would move would be unclear. And understanding that we don't understand animals, it's rather recent. At least, as the first, I read it in some way before. He says, oh, the hell, this horrible, complicated machine. How could it be? We understand nothing about how it works. And this, not understanding, was very recent. With proteins, it's good. You just immediately don't understand anything. So good, yeah? It's mathematics. And you don't understand it. And so you want to understand it, yeah? So what about this folding? So what is the logic of this cell? It's really kind of very perverted, very converted logic. So the easy DNA, it keeps information. It's the keeper of information. And it does nothing, essentially, except just information, but this information makes no sense of information, by the way. What the hell is information? It's not Shannon information. It's information, metaphoric information, which has no mathematical definition. It's one of the questions. Is there meaningful mathematical definition of this information? It's systematically used in biology. And it looks kind of metaphorically good concept, but we don't know what it is. Now, this information is read by some protein. There are some proteins which read this information. And they are kind of infinitely more complicated than this DNA. Like, indeed, we have letters written on the paper. And a reader, a person who goes and read, I mean, come on. These letters are no big deal. People say DNA is most important, anyway, because they are slaves, anyway. People, they are slaves of this DNA. However, they are proteins, they are complicated machines. They read it. And among this instruction is doing many, many things. But the most kind of interesting is they can do themselves, and they can do DNA. So among this instruction, how to reproduce information again, and how to make the builds. And of course, such thing could only evolve a step by step, because you have to start somewhere and nobody has not only idea how it could happen, but why it could happen. Just everything we know about life outside of living matter says this is impossible. So that's this machine of producing transforming information into real objects which move, protein move, do things, and eventually ourselves. I mean, it's absolutely inconceivable this could happen. And some people believe it only happened because there are so many universes. Because in one universe, it would be not enough. It's very unlikely. So it's probability of the soul, though. You need something like 10 to the 100 universes. Then they feel more comfortable. Then maybe one of them is like that. And that would be true. I mean, who knows? It's maybe true. One day, maybe we shall learn. It's at least theoretically. It's not impossible. We shall be closer understanding that. But anyway, it is there. So in each moment in this protein, this information turns into action. And this is exactly the moment of folding. And this is a kind of folding, apart from this physical meaning of that. Of course, it's kind of physical process. But besides being physical, it's also conceptual, and I don't know how to say it, information theoretic process. So this information, when it folds, become completely different. Now, all inside become more or less irrelevant. Then they keep it. And now, the own protein knows what happens on this surface. And what it does, well, not all proteins, of course, most protein for, but not all of them, and a tremendous amount of information being closed. And this, I think, is a crucial information being closed. Again, information in what sense, which means you can change this hugely, the sequence that we have absolutely the same functional protein. And this happens, you know, in the course of evolution. You have two homologous proteins doing absolutely the same and having almost no common feature on their sequences. In sequence amino acids, there are 20 plus epsilon. 20 plus three amino acids, right? 20 plus three amino acids. Different amino acids. No, yes, Kugel told me that you're right. And another two. No, no, no. I think there is a form of metianine similar to salient metianine. Yes, but this metianine, it is not an amino acid. It is separate amino acid. It is just metianine with salient something, salient group attached, but it is metianine modified. It is not new amino acid. OK, that's the question. Maybe I understand, but this too, it is just modified other amino acids. It's not new amino acids. OK, it's not true in you. OK, but I just, I could misunderstand what he said, but he says, and there is some other. I think, I'm not certain he said metianine. This came to my mind because herself, of course. Exactly, that's why it was a serious addition of amino acids. But anyway, that's, again, usually we say 20, but everything you say in biology is literally true. It's 20 plus epsilon. It's unclear, you see, even the question, the whole question, what does it mean that I only 20? If you call it true amino acids or modification with other amino acids. This, I think, is very, I feel very good with this. I mean, because this is a real life, and things are not precise, but it doesn't mean they're wrong. I mean, again, precision is deadly in this context. Idea of precision has its limitation yet. And, but the moment it falls, so it loses tremendous amount of information. And this process is transformation of information into action, and it's very hard to say what it is. So when you make all this modeling of shearing machine, there are two aspects of them. Information theory can actually something be happening. And they're kind of different, and they're very confusing, of course, in the description of shearing machines, and all these, which are, of course, not quite mathematics. It's very poor quality mathematics. And one question is if we're trying to describe cell, you can create more interesting, more beautiful mathematics. Everybody who looked in the shooting machine is so ugly. It's exactly what Hardy said, no room for ugly mathematics. However, this mathematics is without us. It is nothing, I think, ugly you can imagine in shooting machines, idiocy, mathematically. But on that hand, you need it in some sense. And the question is, if there is something more beautiful? Even more ugly, I think, is that von Neumann invented this self-reproducing thermometer. And this extremely ugly thing, you just describe them and describe them in some kind of a tremendous bore. On the other hand, it's nice to know it's possible. But what's remarkable, again, about that, and that's indicator of the imperfection, doesn't know how to formulate the theorem. What it means, you have self-replicating thermometer. I mean, you know, you can describe this process and say how you take this machine, imagine this machine, and it does its own copies. But what is the statement of that? Before you make this description, that's very amusing that even in the as if formulated precisely theorem of Hewitt's universal machine, if you look carefully, they're cheating on how they're formulated. You first make a proof, and then adapt your formulation to the proof, to the construction. It's not your mathematics when you have clear-cut question and then you answer. It's not like that. And this partly is different kind of mathematics. It has some motivation. But I'm saying that the way it's self-organized, the way it's replicated, and the way it transforms information looks much more beautiful for some reason, logically, than done either by a Feynman model or by a Huring machine. And we have no way to describe it. So we don't have general picture and put them, compare them, and to say, this has this property and this is property. But for example, there is no simple mathematical model of replication of GNA. I mean, of course, you can describe everything, you know. But saying it is simple, but exactly what are features of that, we cannot say it. We feel it, but it may be, of course, an illusion. It may be, of course, an illusion. It looks so very simple and beautiful, but mathematical. I mean, not kind of from some kind of emotional point of view, beautiful life. So this is one thing about proteins. And so just again, what I want to say is a couple of words about proteins. So how do we know by the way they fold? So what is this folding and what we can do? This is two words about ribosomes. If you didn't know, yeah, that's probably the most mysterious thing in the universe. If you understand the least of anything, the ribosome could be involved. Of course, every second, millions or trillions of them works on ourselves. And we cannot imagine they could come. It's a machine which makes proteins. And the most complicated thing apparently in the universe, by far more complicated than stars, we understand the explosion of stars, billion years, light years, they were much better than we understand ribosomes. It's incredible. And they're all in us, just making these proteins. And this considered exactly the thing which could not happen. This one universe was not enough. Our universe is about a diameter, 100 billion. Light years are just tiny compared to improbability of such machine to come to life as a ribosome. Absolutely no idea how it could happen. No model, no energy, so you can add one impossibility for another. And they, however, they work incredibly well. And so they make this protein. And so the logic of that is separation of, yes, on the physical level of this polypeptide chain. There are chemical bonds, which are rather weak bonds but still they're chemical bonds that are quite weak. In a sense, we have just even green, I think, green photon of light would break it. That's another interesting thing that is a balance, more or less, between chemical bonds and light of the sun. So if the sun would be a little bit cooler or a little bit hotter, life, as we know it, would be impossible, right? Because if photons would be a little bit stronger, they would kill, destroy all chemicals responsible for life. If they would be weaker, they would not create this chemical, right, because they involved in their creation. And as we know, if there is some problem with this ozone layer, then we have stronger photons coming here, and everything will be dead, yeah? All vegetation, everything will be dead in a matter of days, yeah? Very fast. So it's again incredible that we have this layer. How could we create it? Because life didn't come with this layer. It was created by biology, biological created oxygen. Yeah, it was created by the activity of green. We don't know of green what, right? Because again, conjectually, it was not green plants because plants came up afterwards. So it was not even cyanobacteria. So we don't know who were the first organisms to create oxygen-producing organisms. They were completely different in nature, and worked very slowly, and then with more and more oxygen, and then ozone layer, IP, and then more sophisticated creatures could come in, which would be sensitive to. And probably this first one, we were not on the surface. Because if the ultraviolet were here, no life would be possible on the surface of Earth. They'd be hidden somewhere, right? So everything we see now in life is certainly what's very different from how it began. So just general remarks. So these are simple amino acids, how they make. Again, for every shape of amino acid, there is a general scheme, but immediately there are counter examples. They're all the same except without cycle, but one of them has a cycle. They're cyclic amino acids. And each of them, of course, has individuality, right? If you want to look at proteins, you have to know personally every amino acid. And each has lots of interesting features. And it's like if you have a language and you only have 100 words, each word has its personality. They're not just words. And there is a good reason. I think there is a good reason for that. The building blocks, they're very individual, very structured. And there might be some good logical reason why you don't have very similar homogenous blocks. On the other hand, for building DNA, you have very homogenous blocks for making proteins. Amino acids on one hand are similar, and on the other hand, it's extremely different. In many ways, they're different. So the 20 of them, and if you look by characteristics features, three features, and they are all different groups. Almost not two amino acids would be in one group. Probably. And again, it's very hard. It's definitely, it's definitely, it's definitely mathematicians, this is how they're being glued together. It's a process involving water. Everything in biology goes along with water. They come together and one molecule of water goes out. And when they break, water goes in and they go out. And it's very energetically, almost zero energy level, I think, in the most direction. So anybody here, but there is this hook, yeah? So it's kind of, you can imagine, there are this hooking and one chain and another like that. And once they're there, they're at the stable. But if you, but it's not energy, it's entropy which keeps them together. And I'll come to that in a second. Yeah, there's one interesting issue which you should be understood. We might be, yeah, this is just a little information about amino acids. Just to have this, again, it's specific of this, not that essential. But there is some logic in that which is, I think, is crucial, which is very hard to isolate. So if you imagine you have to make an artificial life, even similar to ours. And there will be these blocks. And they might be separated and structured from the formal organization. It must be similar, well, vaguely, to what we have here. Not the particular properties, but something. And this something here, when you look at this, it's like learning it, you feel it. But it's very, very hard to articulate in this one of the issues. This was kind of rather famous protein because it was the first for which folding was kind of postulated or established. And so, just again, just saying that all proteins have individuality, look at this. This very common protein, it is enzyme which cut RNA molecules. And it uses, it's automatically by cows. The cows are how cows function. They cannot digest the food themselves. So they eat this food and then there are bacteria living there. Actually, something I don't quite understand there. But the point, of course, is that as a result, the cell, eventually, they have to digest a much smaller than the cells of vegetables. So a vegetarian cell can contain more, more, more, just water, a little bit of sugar, all kinds of sugar there. Very little, say, of RNA material. But inside of bacteria, there's lots of RNA. Therefore, in order to digest this bacteria, and cows digest not directly this vegetable, but they digest bacteria, and so they have digest lots of RNA. It's unclear to me, so it means that proportionally, this RNA come anywhere from. From plants, that concentration of some RNA is getting up. I don't quite understand this. So why they needed more than we do? If we eat meat, I think we have a similar problem. But we don't have this RNA in our stomach and some large quantities. Right? Yeah, I mean, when you start thinking about it, you read it, and you see, they say, I find it extremely annoying. People say it in books. Immediately, they start asking all these questions. How balance is being kept? And they never answer it. Apparently, they never ask themselves these questions. It's immediately looked paradoxical. Why cows need more of this RNA? Could you take out this? So this is an enzyme. Did it drop something? Sound sounds like it dropped. The enzyme which cuts RNA molecule. And this was the first about which it was in this very kind of robust enzyme. Well, it has some built in a very strong way. Usually, the attachment with the molecules in the protein is relatively weak. But there are some of those, another strong Dyson feedball. And then it was discovered that the first you do something bad to this protein, you put it, I think, in acid or something, yeah, or heat it. I forgot exactly how you didn't rate it. And then it starts working. And it works very well. It can discuss RNA. I'm not certain what exactly the test for that. And then it was found by, I forgot to say, Afrinus and I think this guy, that Jesus was a big discovery. I'm saying it found. It's a discovery. It's a work, yes and yes, a work. That when you cool it back, bring it to the back condition, it comes back to original shape. Namely, at least it becomes as functional as it was before. See, it's function is a very remarkable thing. What it does, actually, this enzymatic activity of each protein is a kind of a miracle, because nothing of the kind you can do artificially. Enzymatic, and I say again, I don't think I have much time. This is kind of, they catalyze this extremely efficient catalyzes for certain chemical reactions. Much more efficient than anything we know. And so this was how we discovered. OK, maybe I just say one word about catalysis, and that will be the end of it. So what is catalysis, and what enzyme do? And this is, and again, an interesting thing you need in textbooks. And they say something like that. So you have to go from one state to another. No, it may be not a very good picture. You have the profile of energy. And of course, a system would like to go from here to here. But it cannot, because there is potentially a barrier. And then there is some catalyzer, which kind of cut this barrier, and then you go. And what people don't say, this is just metaphor. It's just a way to help in mind, but this tells you nothing. We don't know the hell what happens. What do you mean this barrier? You have a space, even in classical terms. First is a quantum mechanical process. Forget about that. Even classically. You have systems with hundreds of degrees of freedom. And so what do you mean this barrier? So it's very complicated, lots of unscathed. What happened there? What do you mean barrier? How barrier can be made? Barrier, where? What is the probability of making a barrier in a high dimensional space? It's very, very different in one dimension. It's very easy to make barriers. To make barriers in dimension two, you might have the whole circle of them. Yeah, in dimension C, whatever. So what enzyme does? Enzyme adds degrees of freedom. So we have a more high dimensional system. What do you mean it doesn't? What are the sources he uses for doing that? Protein, as enzyme, internally has very little energy inside. It's a very weak thing. It breaks covalent bonds very, very strong. How could it be? So none of this is actually allowed. Something is known, but essentially it's unknown. And when you read the book, people pretend that they understand. That's if I'm extremely unknowing when you read the book in biology. You read it as if people know what to talk about. They just like, you know, just they say kind of one incredible thing after another in a matter of fact way. Which is extremely, extremely unknowing. Because you know, some chick is known, some chick is unknown, and some people know it, some don't. It's a huge field, of course. Very few people understand the Jamaica activity. They have different perspective on that. Mathematically is, what are the shape of this landscape, and how the shape being changed for introducing new substances? So what are the mathematics of that? The most primitive description, I say one word, I thought a little bit. It's extremely primitive. It doesn't tell the whole story. But it is of the following kind, that one way to describe immense kind of possible specialization. They have no time to go into that of this question. Both of folding, they are more related. And in Jamaica activity. And I just thought a little bit what mathematics can be account for that. Not proving, of course, anything about specific enzyme, but at least giving a background. Of course, you have to know things also. Things happens in a variety of way. But even this naive picture, which I described, may have purely mathematical development. If you just look more realistically, high dimensional space, how to make sense of that, it's not so easy. But one thing which I'm using, which describes some of those systems as follows. That you can describe proteins in the early system like that as trees with measures on them. So tree have lengths of edges. So there's some distinguished point here. So distance to this point represents energy. And there is thickness of these edges. It's kind of entropy of them. So these are the objects. And the simplest one of them will be just measured on the line. The whole tree is a line. It will be just measured. And this classical physical mechanics is concerned with. You have a physical system. And what you look, you see how entropy distributes with respect to energy. How many states are they with given energy? And this was statistical mechanics when this tree reduces to light. And then various operations of the measures make sense for trees. For example, convolution of measures makes sense for trees. And when you bring new particle, meaning they convolve your tree with somebody else. Tree is the only background of what happens with protein. Not the whole story. But it's still already informative and it's mathematically transparent. But while I have no time to talk about that, and actually even with a head, I have pretty little to say. Except that some formalism of measures extends to the trees. And they make kind of a nice category. But say enzymes are quite, quite mysterious. But they are not understood physically, chemically, biologically, neither mathematically. We just don't know how to start talking about them in context of classical mathematics. Of course, some process are, of course, quantum. So again, just when you have a physical system, you can consider this function of energy and see how much particles, how much phase volume is in the given region of energy. And this is the basic kind of, usually you take a Laplace transform of that with canonical distribution, but these kind of technicalities. That's what statistical mechanics is about. You describe physical system by a measure on the line. And the measure says how many stuff has given energy. But when you look at a more complicated system like that, it may happen that there are several states which may go from one to another with the same energy. So you have a tree, and you have energy, and you have two branches with the same energy. And they have many different ways. And then there are extra structures there, how fast you go from one branch to another. So the system are multi-stated. There is no unique equilibrium state for this system like proteins. Proteins, or even more, when you have a system of several proteins, there is no unique, again, state here is a measure. It's state not the point. It's a measure, and again, it's questionable language. The whole of this language of statistical mechanics is very deceptive when it comes to it. It's not adequate. But this kind of tree is part of the reality of certain biological systems. So they describe this kind of trees, which says how many possibilities are per given energy. And so it partly catches the idea of connectivity of the space. Let me give a very simple example. A simple remark where it's motivated by if you have a function in the space, in the Euclidean space. There is this tree of levels. You can see these levels, and they connect the components of the levels. They have a tree, and I think it was introduced by Cronroth. So it is a topological tree. And statistical mechanics has measured theoretic projections. So if you combine the two things, you have a Cronroth tree, and it also has a measure. How much is there? So there are different connected components in the space of states. Of course, if they ever recover, they mix up. If not, they're not. So this tree is not the end of the story. But it's already may have some meaning in proteins. And there is little mathematics behind them. Even if, for example, you multiply two trees, and then you'll enter this energy. And so we have three again. So this convolution on these trees. But of course, when you introduce enzymes, it's more tricky. It's a trickier thing than convolution. So it distorts the tree. And then, well, I thought a little bit, you can say something about that. What it means to have a quasi-quasi-convolution, which corresponds to enzyme. But again, this is just mathematical, very naive things, which don't want to reflect the whole picture, as usual. It's like a description of this erythrocyte solution of variational equation. They only touch the simplest part of what happens. And then there are layer after layer after layer. But all of them may be mathematically, not all of them, but some of them may be mathematically significant. OK. So I'll stop here. I mean, I was asking questions. Now you may ask questions. This is one question about biology. So the folding, the way the folding folds, the only thing that matters for biology or everything is also matter. For how this is spreading in caves. Is the folding the only important thing there? Not all protein folds. In the folded state, it becomes active with folds. But somehow it misfolds. And it's equally important because if they misfold, you die, if you misfold too many. So cells have to keep track of the proper protein, which misfolds, for example. So misfolding is, as important, I mean, more than important. What is more important? What is more important, be it bread or potassium cyanide? What we have more effect on you? I'll answer this question. If you speak about importance, what is more important for you, bread or potassium cyanide? Potassium cyanide. Which is more important? What do you mean? Yeah, so they wouldn't eat it. Important doesn't mean good. Many important things, but there you go. Why is it different? You have two different proteins that fold exactly the same pattern. When they fold it the same, do they behave the same? Yeah, they behave more the same. It's not only fold, but they have the features. It's not only the shape of the folding. There are those molecules which will be on the surface of proteins. How strongly it will be fold, how flexible it will be. There are lots of parameters. But still, the number of parameters is much fewer than the number of ingredients. You see, therefore, you can change protein and have a parameter the same, and it will work the same to two proteins. Again, I must be very careful in saying that, because they do the same thing, but in different environments, in different organisms, this is what we observe. So you might be somewhat careful if it's the same. It is a wheel in the bicycle, and there is a wheel in the car. The shape are important. It's the same. You can't replace one by another. But what's important, they are both wheels. So I mean, this might be quite dangerous to make simplistic conclusion. But again, the sameness, by the way, is already mathematically retrieval issue. What does it mean they are the same? There's a context, and the context is so variable, unlike physics, so many, many contexts. And then it depends on context. But this, by the way, about potassium-senate is, again, mathematically, a very interesting question. You can eat a lot of stuff, and everything happens to you. You digest them, and then a tiny little stuff, and you have a problem. So in what sense you are stable? What a mathematical description of stability of dynamical system, which is stable in some range, and the Mikhaili unstable in different range, right? Now in France, actually, there is a lot of poisoning by mushrooms, yeah? And I forgot the most efficient. I don't know what is the English name for that. The most efficient kind of poison, which stop your, block your RNA, RNA synthesis. Bledný paganky. How could it be in English? How called is chemical in them, which stops, stops RNA synthesis? I mean not, I mean I eat something like that. I have very, very efficient poison. OK. It's very interesting, you know, as a way of being poisoned by this kind of mushrooms, they disrupt your production of RNA. OK, you guys have to remember.