 Я могу вам предпочтать, что я буду говорить, потому что я начну с какой-то квотации, чтобы выяснить, что я буду делать. Потому что, как я буду представить, то, что мы не называем их «мathematics», потому что мы говорим, что мы не о «мathematics». Но потом, у меня есть авторитет, который говорит, что именно то, что я буду делать, это «мathematics». И то, что я буду делать, это не так. И кто это авторитет? Конечно, да. И, как обычно, если вы квотировали на панкаре, то, что панкаре говорит о «мathematics»? Так что, что это «мathematics»? Это «панкаре». И он говорит о «мathematics» в английском. Он говорит о «мathematics» в арте, чтобы дать одно имя для различных вещей. И, поэтому, это не играть с «мathematics», что «мathematics» делать, но дать это имя. И, чтобы дать это имя, вы знаете, то, что они применяют. И, в принципе, есть еще один, но Грофендик, и он тоже был «мастер» в этом. Он был дать имя для вещей, идентифицировать, что он, в большинстве, смотрел объекты, или «взрывы», и дать им правильные имя. И то же самое. И панкаре было в mind, в большинстве физических вещах, но если у вас есть время, то он тоже был идентифицировать какие-то интересные структуры, скажем, в психологии, в этой особенностей математической структуре он не имел имя, но его идентифицировали и он все еще не был обзором от «математических» даже, как он был сделан более чем 100 лет назад. Эта идея была, конечно, в зависимости от генетики, как они могут реконструктовать или старые классические генетики, как они реконструктуют шею хромосомов от хромосомов от обзоромных данных, которые никогда не микроскопиковые. Мы смотрим в цветы, в глаза, в флайзе, в дразофиле, смотрим, как цвета меняется. Они могли сказать, что генетики на структуре и как они позициями. И это математический проблем, они решили. Но, конечно, формализм генетики не был разработан, но эта вещь, в которой я думаю, она была ремакована, очень ремакована. Петерн, логическая структура в мёртвом биологии, или в психологии, не был разработан. Те же виды, которые работают, более чем те же принципы. Так, но... А, это все хорошо. И генетики в жизни работают очень хорошо. И так можно видеть, да? И, специально, я буду говорить про вероятность. Рекординг в процессе. Да. И так я буду говорить, про... Так, так, про вероятность. И потом я говорю, что я не очень любительный человек, манк математических, который был эволюционный биологист, и сделал много. И, так, если вы хотите понимать вероятность, вы должны понимать что-то. Так, что он сказал? Он сказал, что вы не можете понимать биологию, что ничего не имеет смысла в биологии, кроме того, если вы посмотрите на эволюционный. Так, что это для математического перспектива? Так, почему эволюционный дает унижение биологии? Математически, почему? Что это за те же виды? Потому что это сделает, что он коннект. Организма не коннекта. В этой степени они становятся коннектами. Во-первых, структура более интересна, чем структура. Это структура, в том числе в эволюционном перспективе. И потом у него есть унижение биологии. И, действительно, то, как мы думаем о жизни, это так. Это манк математического перспектива. Но очень много идеи в физике и в биологии манк математических, это так. Но это манк математически, и как мы манк математически развивались с этим. В большинстве манк математических делали не манк математически, в моих темпах 20 лет назад. Я научил, что 90% работа на паточках, на аудиодеференциальной эквестиции, это сделано по которым в большинстве журналистов есть динамичная и химическая реакция. У них есть очень высокая теория стабильности, и у меня есть больше идеи и субстанции. И я объясняю почему. Но это не то, что я говорю сейчас. Я хочу сказать немного о это. Так, давайте пройдем. Так, мы говорим о стабильности теории. Так, вот такой момент, что стабильность пришла от реальной реальной феномены. Первая феномена была играть на дайсе. И гемблинг. И это было исторически так, в первом человеке, который делал математическое теорию, это Галилео, который никогда не писал. Тогда это Кардана, который был опубликован позже, и потом, я думаю, первая публикация была корреспондентом с Паскалом Ферма. Галилео делал 70 лет и всё, конечно. И просто любое его уровня интеллижения, чтобы понять, как дайсе идет, и это и так. И так, но это не прихистофия. Но еще один параметр, то есть шоу, в которой математика опубликована, то есть, кто делает это? И здесь есть такой момент, который, опять-таки, questionable, дебатевый, то есть, что я говорю. Но для всех, вы знаете, это так. Почему мы должны делать математики? Мы должны идти на математику. И есть ли у нас есть лимитация? Мы сами imagine something. И так, что я думаю, это вероятно, что мы, конечно, мы, во-первых, знаем, что мы, все люди, очень-очень так же, variation between, you know, people, much smaller, between, say, chimpanzee in Africa. In these chimpanzee we have more varieties than we do, because all humanity went through several bottlenecks. And this must be understood. But many, many things about ourselves, by the way, you cannot explain professionally about human behavior, politics, whatever, unless you have background to which compare with. And this comparing is, of course, the biology of most of our predecessors of primates. And therefore, there is all reason to believe, our brains, all our brains work more or less the same. And they were programmed, in fact, much earlier, before we became primates. So there are basic programs running them and they're all the same. And that's it. So we cannot do something beyond them. When we say P1P problem, quite possible. There is very simple solution of either of them. But inside of our brain there is no way to find this. Because, of course, using computers or whatever doesn't help. There is fundamental limitation. Of course, power with computation. It allows partly overcome it. But if because you play again exponential, no computer will help you. Maybe quantum computers may changes, but therefore we are fundamentally limited in what we can imagine. And this, of course, major bottleneck for our activity in mathematics. We have not enough imagination. And so when you look at real things beyond something, not because nature is smart, but because we are stupid. This is the whole principle should be understood. Some people say, well, evolution is smarter than you are. No. It's dumb, stupid, fantastically stupid. Nothing is more stupid than evolution. Except for humans. Or animals. We are stupid because we are little part of this evolution. If you look how evolution works, the most primitive algorithm you can imagine. Leave or die. It doesn't work the right way. Take and run, it works the right way. And this is how we are made. So we are made by the most stupid process of our brains. The most primitive machine you can imagine. However it works for some reason we don't understand. And this is mathematical problem. So our brain works despite all this built in stupidity. So for the reason given to justification first great authority upon correct and then kind of this kind of a negation of any authority. Now we go to the anthropological method of this. And now we look at things. So look at things. Let's look at this picture and we return to them. The first picture will be the last we were discussing. And this will be about homology and homological probability. This is a crystal and this is a kind of kind of a organic crystal. Where ingredients are highly non-symmetric and still it displays symmetry. And it's quite remarkable thing how the hell happens. And the problem of course when you look at the kind of applied mathematics you already have somebody made models for that. And you work with these models. So you don't work with real things. These models are done by somebody. And this somebody may be great scientist or maybe second great scientist. And real mathematicians just take it for granted. So we don't know what is the right model for crystal. We don't know these are cellular structures. It's just how main brains are formed very similar to solve films. For example there are solve films. It's a remarkable structure, you know it's solve films. And then there is mathematical theory of geometric measure theory. Who said it that this is the model? So I don't know who said it. But it's not the adequate model. There are much more structure there and possibly mathematically it would be much more advantages to look at the things that it is and use all resources suggested by this structure. This is what I want to do. But on that hand if we continue that with these pictures for example this is about to be slightly mystifying on purpose. So this is picture of Morse function but I think about this protein folding by percolation. So again if you look at protein folding you can think about percolation, Morse function you can make this picture then you have to again because we give the same name to different things. So and I will show you at some moment how it works and the spectrum also part of this picture. And the last one self-avoidning framework. So probability theory just the part I will mostly talking here was came back to Flory who was a chemist and who done much lots of practical things and he invented two mathematical theory one of them was percolation theory which at his time was called Geolation theory and also the self-avoidning framework and he just not invented them like that particularly the self-avoidning framework he understood how molecules behave and he make computation and make predictions and prediction confirmed by experiment. So it is by order high level of achievement in mathematical theorem and percolation now and he also along this line he made percolation theory and also it was experimentally verified and it's called Geolation for example you know that if you boil boil an egg and this liquid stuff become rigid at some moment. So what happens? And there is got Flory theory and says under what condition this phase transition rigid one and he made specific computation of course like any scientist just not theory but computation and verification and computation is tough stuff you know when Hilbert wrote his equation Einstein equation derived by variational principle which actually only him and Einstein knew about the time it was kind of hidden for many years by Hermann Waill for some reason and Hilbert was kind of lamenting that Einstein was so good with computation he could predict this variation error made by Mercury and Einstein telling me this I was working 2.5 years in a half a life after that and I invited so many people who helped me with computation this was the hardest part of making this theory he said all equation was more or less obvious to me and then it happened to be right and in addition we don't have this problem so I just want to say that it was not just writing equation which again because these guys who are involved in the nature they have intuition somebody right and then they go to computation they don't care what we call Riga because I return to that at the moment but white as by the way itself avoiding random walk what has to do with this picture and again if I want to explain this mathematical perspective and so I'm not I'll be talking about loss of subject which I really know nothing about right like random walk percolation, proteins, whatever and except for the very end when I know a little bit and so and my attitude towards all this objects or the things is not really understand them as scientists but understand but just as a source of inspiration and makes a mathematical fantasies of them and this of course I feel a little bit bad anytime I give this kind of lecture and I just quote a friend of mine because I'm not certain he want to be mentioned so I don't say who and he said that there are actually physicists, great problem like he was mentioning quantum gravity solved within say 10, 20 years but he said it will never be solved unless some other problem is solved what is another problem without which quantum gravity cannot be solved is the problem of soil depletion in 20 years the world will be starved because soil will disappear and the major problem he claims is soil science to understand how to save soil on earth it is the most depleted resource we have it's not water, it's not air it's not temperature, it's soil it's extremely complicated, mathematically incredibly complicated structure and what's the problem with that so you need people mathematically minded to do that but to do that you have to spend 5 or 6 years learning soil and there is no structure if you have young man comes and 5 years learn mathematician he has no money, he will disappear there is no structure to do that it cannot be done by a man it must be a community mathematicians, biologists physicists, chemists working on that it is however so how we still solve this problem how we go around it it is a whole other problem and then we say there are smaller creatures which much smaller heads than ourselves and they solve this problem and you know who they are our big head is no problem they are called ostriches we put your head into the sand everything is fine and now we do our mathematics and so what we do that's sad truth but this it is and still we enjoy our life and yes we can so now we go to the next so what is role of mathematics again maybe I just want to make some remark that mathematicians when we speak about different things in life and how they were treated by scientists, we speak about the rigor and say something rigorous, something rigorous and this for me because there are these dangerous errors which can crop in and so mathematics in this from at this point plays the role of immune system when there are errors bacteria or something institutional going into your body it kills them and just create pure environment in particular one of the job of immune system at least part of this so called complementary system is a particular collection protein again I'm saying that I know very little about that but yes the research I think don't think I am an expert on that but the research I think which is called synaptic pruning so when you develop this connection in your brain and this connection determines your brain and it's evolved with your age and there is synaptic pruning just destroying unneeded synapses you know most of the development of the brain consists of breaking something in, not building in but breaking in actually it's also very interesting evolutionary if you look evolution of organization of the motor system of the brain and you see my synapses they will go from frogs to to lizard much of the synapses disappear become more organized and fewer synapses those chaotic organization of synapses become more structural, hierarchically organized and there are much smaller of them and this synaptic pruning absolutely necessary we would develop without that but sometimes immune system overdone that and this what happened then and then it becomes as psychophrenic it develops and so by the way there is some similarities again for friend what I am saying, what is common between mathematicians and psychophrenics only these two people a chain of ten consecutive consecutive arguments in life one or two is enough chain breaks down but in mathematics ten still you are there which is of course miraculous where it works which we certainly don't understand now coming back to probability and so we know about this historically so with probability as a way as I said you need for biology to be understood have a connected unity by evolution the same I want to say about probability to understand probability because probability the way I see it's not one domain there are main domains and they almost don't talk to each other but they have some common sources and then we can understand them looking at the source look at the history and the first there was a prehistory of course there was a very old time I don't speak about that but starting with law of large numbers in probability it's a thing it's a remarkable simple theorem and underlies physical thinking about that and it was in modern times there are two people who contributed who used it to make kind of fundamental revolution it was three maybe yeah Maxwell and Mendel more or less contemporaries in 1960 I think the Maxwell paper was about 60 and Mendel 63 and then Todd Balsman who developed it further and then I'll say in a couple of words what each of them done and then so what Mendel the point of Mendel is but looking at statistics he was not looking at the flies but looking at the beans looking at how they developed you could say that inheritance was discrete and moreover it consists of two parts at least for with particular bean and also for human right and this tremendous kind of you can imagine what would happen if Balsman knew of Mendel because Balsman was obsessed with the idea that the world is discrete there is atomic and then this discreteness was discovered earlier by Mendel in biology and Balsman was very much kind of excited by logic of Darwin history was kind of this formalization, mathematicalization of biology and if he knew about Mendel he would really jump over the roof but biologists who contemporary look at Mendel it's not interesting in mathematics and then so this the work of Balsman was kind of culminated with Einstein's Volkhovsky who analyzing experimented by Ingell House you see this name Ingell House do you know who he is and why it's such a I don't know if he looks brown here you know a fantastic story about Ingell House he discovered the following thing he studied this and wrote the papers model tiny little particle of dust look in the water, look in the microscope they make this kind of crazy movement do you have a chop oh thank you there is this kind of little particle and it looks like that and how it could be and then Einstein and Monkhovsky figured it out and it can be said and so what is this little particle set in perpetual motion with the impact of invisible blows and the movement on sap etc who said that this was actually understood it was discovered by Ingell House in the brown colors in white brown because brown in motion because apparently Ingell House was getting brown something which was called brown in motion I don't know that another explanation feces make injection with somebody brown who invented that feces are very physically oriented we see brown color actually what person called brown actually he started brown in motion but he certainly understood much less of this than Ingell House Ingell House was a great scientist of course he never heard of that but he discovered something this object not a joke one of the greatest discoveries of the time and nobody knows about him and nobody knows about brown because brown is simple word Ingell House who can remember such a word such a name but he but it really sounds instructive to which I come back but who said that this movement mounts up was Einstein Smoluchovsky somebody else of course a translation a translation from what language from the Greek Flatten I guess it was Tito Kletsov and it was said about 2000 years ago and so this mathematical idea had to wait for 2000 years and there was this Einstein work and Smoluchovsky who described that there was been an integral and so interestingly enough by the ways that I don't know about Smoluchovsky but I read about I haven't read Einstein paper Einstein modeled this by brown in motion it is particle and it is heated by molecules and then it's what you see because when they accumulate enough you have brown in motion but more subtle experiment carefully it's not brown in motion so Einstein was wrong it was not brown in motion but outcome of the formulas was right and it was good enough to measure the size of atoms to measure Avogadro number which was done by Perilin here about 10 years after in 20s was a great physicist one of the greatest French physicists of last century so this is about history something about Mendel because as mathematicians we probably heard of Boyceman's theory of gases and the statistical mechanics and what Maxwell done and but one from mathematical point of view is that starting essentially from the low large numbers from nothing you arrive in symmetry it can become related to the symmetry of the world and this will come to this again it's rather amazing thing inside of logic of probability that there is nothing in probability which tells you a priori has orthogonal or unitary symmetry but it does you know sometimes there are definition the probability of the measure and determinants whatever we laugh at the nonsense done by non-mathematicians but inside of mathematics we do the same because we laugh people love to give definition in 99% it's made in 99s after it is nonsensical there are great definition like Grothendieck was the guy who was giving great definitions almost everybody else was I don't know who else was able to do that before after him he was really genius of giving the right names or definitions things and and so and so what about Mendel so let me say a little story and just how simple and remarkable this is that that imaginable experiment which after Mendel it was rediscovered 30 years or 40 years later and actually it was rediscovered you know what made Mendel came up paper was essentially publicly forgotten and then it was rediscovered so what was the reason for that was it good or bad or what it was bad, it was jealousy of scientists so 2 or 3 group of scientists simultaneously rediscovered Mendel and so they felt very bad somebody else done it they looked at the literature just to make some guys bad look bad and found Mendel paper which everything was already done I don't remember their famous name they were great scientists also but this was rediscovered actually in mathematics I think it happens also see somebody does it maybe somebody else already have done that and how he gives names there are many terminology which reflect that whom you refer to if you discover something and already done it you say called ex-ponkare something and then you feel better it's not the guy who discovered it they covered like ponkare complexes exactly of this nature why they called ponkare complexes and so again this is a remarkable thing and because little mathematics would immediately give a flow of other things and the phenomenon is following so I have a mountain range and I have flowers of two kinds A and B say blue and red actually it might be blue and white for some reason which corresponds to some reasons would be better white white meaning nothing blue meaning something flourish and everything night and they don't direct but the same kind of flowers and then there is some little event and then this thing disappears whatever it means and then they become mixed up and then there is intercrossing and then they become mixed and then you see that after the first intercrossing look at the population and you hear 10% is blue and the rest is white what happened on the next round of that so you lost 90% 10% so what will be second what will be proportion and the Armenian point of view says the strongest survive then it will be 1% right but Mandel look at computation says nothing happens it remains 10% it stabilizes on the first step and this number of two I'm not cheating 2 sets of chromosome and this flower has 2 sets of chromosome and this is 2 white it is x squared x for x and this is 2 almost true and this and well especially kind of Darwinian we really annoyed with that and just didn't understand it and then there is story by Hardy who spoke to some explained this around some a plus b equals c wrote an article half a page in nature making excuses using this multiplication kind of mathematics to explain that and there was somebody else Weinberg who was a medical doctor who wrote 60 pages also explained that and this is called Hardy Weinberg principle and it's 10 times more than what he thought in science than any paper of Hardy with little wood whatever the most famous paper by Hardy Hardy would be crazy negative absolutely mad about that because he hated any kind of application he wanted to appear mathematician in this little applied thing but what funny Hardy I think didn't understand actually mathematics behind it and he didn't formulate correctly and what is mathematical theorem plus b equals c or something of course will make computation become some kind of formula one line computation but what is mathematical statement behind it and mathematical statement is very simple but c is mathematical statement it's not totally obvious so the fact that happens is really rather amazing and mathematical statement also how simple it comes to intuitive it says you have a matrix but I don't know mathematical object is matrix you can't mathematically define matrix because matrix some table on the blackboard not mathematics but it's okay most of mathematics is not mathematical not explainable mathematics and so you have this aig and what you do you take sum aig by i and multiply by sum aig by j and you normalize for example to say sum equals one right so you take your sum made here sum made there multiply and you make the sum we have new matrix and you multiply by scale to have sum one with zero you cannot do it so it's projective and then it says this transformation square of this it's important square of this equal to itself it's called elementary but this is still mathematical statement this is the whole theory of the Jordan algebra which is kind of a generalized phenomenon and there is Bernstein algebra and the whole mathematics and just around that but Harjes wrote a formula for some special case because he had two by two matrix which was symmetric so it was a, b, c, d and he wrote this in terms of a, b, c, d you just saw what of course and it was it's rather amazing it's only possible because you normalize otherwise it would be impossible we cannot have polynomial squared repeat polynomial degree growth but this rational map it's not polynomial and rational map may have property that square of rational map equal to itself and this is quite interesting phenomenon and so this is what kind of interesting rational map there is theory of rational map that we can develop huge field thinking this terms but Harjes missed it he was really kind of master of multiplication table mathematics he was not mathematician modern times modern times certainly I use modern perspective to say in these terms and it was not in the time of Harjes and ok, so this is about but now about Boltzmann so we come next and this I just want to I'm afraid I don't so how much I want to know what happens this time and because they say he was he was doing not an rigorous in particular it's Ramella and I know because he was appealing to Poincaré current theorem about ergodicity and whatever and then mathematician extracted from their concept of ergodicity and and just this speak about ergodicity which saying more physical term that a time average equals to the space average and so what Boltzmann didn't know ergodic theorem ergodic theorem was proven by von Neumann and then by Birger of course from a thesis point of view it's not a theorem it's obvious once being said of course it was for no one proof you have a unit transformation right and you start averaging because convexity of course converges and if you take annual p converges and alofini converges and then you have to look more closely almost every word also converges exercise for a student it's not amazingly enough it's useful it may be useful for some mathematical purposes but it's kind of for me it's quasi theorem it has really no substances just epsilon delta exercise it's kind of exactly this multiplication table kind of mathematics and what has to do with with the system of these particles absolutely big mystery nobody ever people say in mathematics ergodic theorem for balls Birger, which by the way people who claim to prove it never prove it but what has to do with because you look at this space and as Boltzmann said absolutely right we have to wait more than the time proton would decay and apply amazingly enough, the proof of that usually informative but the theorem itself completely useless and so what Boltzmann would say ergodic theorem doesn't help but what was happening in my view that people were trying to understand Boltzmann from point of view set theory and California probability theory set theory probability of theory was actually a mathematical Kolmogorov he just translated what before done I think he was the first maybe it was done before him but for all at least he was one of the people who realized probability has continuous counterpart and that in particular you can speak about probability of point on the circle and he was doing experiments actually he was throwing baguette over square kind of parkette and see how it crosses the lines and so not that this mathematics has anything to do with bakery it was coming from real life and if you just formalize mathematically you have Kolmogorov theory and this is just an exercise set theory I don't never understood what the point there was a lecture actually it was not online about history of this work by Kolmogorov because it was formulated by Hilbert how to axiomatize probability and it was really kind of people tried and didn't quite work and for me it's all not adequate for most purposes it's a very kind of Morova it never rigorously was made it must be understood because it depends on Ceremetophrenical theory Ceremetophrenical theory never was exposed 100% I guess because you use sets of measures 0 and the set of them is a modern continuum how you can work with this for me it's a big mystery I mean it's not nonsensical theory it's a very strange stuff probability is not about that probability about numbers more commutorial and not about that but how Boyceman was going around that Boyceman was going next step in mathematics actually two steps he wasn't here he has intuition and if you translate it to the modern language he was introduced two ideas was one functionality of certain operation of course if it is suggested by Leibniz and then kind of not kind of used and not considered seriously for a while but this exactly what was happening with Boyceman and if you just you interpret Boyceman in these terms you have completely different setting probability theory and so what he was saying there are systems and system are not sets they interact with them and they have kind of morphism and they have some functionality and something happening and in particular there is a particular function from the category of his objects which are not dynamical system but kind of object imitating dynamical system ensembles of particles and this is the language by the way useful in the dynamical system ensembles they were not sets and they are not set the object of categories and there was next step to the hydrodynamic equations it's enscock-chapman hierarchy and then there was the next step was B, B, G, K, Y hierarchy and P, P physicists were writing that using their physical intuition but from mathematical point of view they were kind of nature of functions in certain category and this category was introduced by Boyceman this is my understanding it was never formalized what they are saying but I am pretty certain they are giving a question and if you look at some formulas integral but if you think about in functorial terms what means functorial terms it means you speak in common language common language is how we do that by the way this is I picked up from Gerfand he hated when somebody was writing formulas or making pictures everything might be described in words mathematical is kind of about words and you have to say it correctly because you don't need computation maybe at the very end you have to make computation but that's secondary and certainly pictures are also not allowed so what is this so in common it was very simple that you have this particle moving on random and then they come collide when they move in this direction and then there is a rule how they go out and so the Boyceman equation describes probability of the flow of particle in a particular direction and this is how I described and there may be different rules how they interact and they have different equations and the simplest of course you have this reflection or they may have some shape or they may rotate it may be different and at some time I was looking at that it was a long time ago almost 50 years ago I guess 45 years ago and people who are doing that there are many articles just doing this function writing equation this condition this condition just using the same kind of operation what I described now it's trivial given this thing you can describe it it's functorial but Hilbert wasn't happy and he was also unhappy about this second step that it was not rigorous you write a functor in heaven but it's not rigorous in a sense that so Boyceman's operation thinking about this is infinitesimal things move in this infinitesimal moment when the collide but they real thing and they want to go to the limit and say that something happen on the limit it's still unknown there is a theorem by Lanford but it concerns the scale of time which is unphysical and this is the second model you derive one from another you have to check if it's correct or not but this is still very complicated to verify equation very hard to solve unless it's very very rarefied gas and then it's reasonably well satisfied and then there is the next step and this is Hans Koch Chapman Hierarchy so from this equation so what is the problem here because if they are independent you don't know if there is no correlation after collision after this assumption nothing happens except this so there is no correlation and again this is again an interesting point that all many physical uses of probability made on the fact on the assumption that there is no correlation these things are independent and no correlation after the shock after the shock there is no correlation so it's the same level of randomness and a priori it's not so you have to prove it under certain conditions they are saying but this is related to the time scale when half of the free motion before collision is tiny it's not a risk but again physicists look at this differently and they derive first the equation which is verifiable and they have hydrodynamic equations and like early equation, Navier-Stokes equation grad equations and there was a scheme of that suggested by Enskogel Chapman and this by Enskogel sad story, I look at the internet about him he was a school teacher he couldn't get position in academia and he done his fundamental work and Chapman who was established he realized it and trying to he was in Sweden Chapman was Englishman who done other work on different things he was great he actually could come to academia that kind of it's happened also in mathematics I don't know exactly what was the problem but and so and then Hilbert was involved and Hilbert wasn't happy that this only works if you assume smoothness of solution so this equation in the space of momenta and this equation in the spatial space like liquids reduced dimension but you have more and more complicated equations and this reduction works only if you assume smoothness but again from certain point of view it's a material because it's a function from one classical equation to another classical equation it's really natural and this is doing it by the intuition what is instructive Hilbert himself wrote alternative hierarchy using his mathematical formalism it was complete nonsense because it was physically wrong it was given see now I'm saying from somebody explain to me who knows that myself I never look at this way okay so this if you ask me why so I can't answer it but maybe I can give you a reference I mentioned some kind of I mentioned something references here and the point is that what thing happens of infinite dimension remaining manifold and you just expand something at one point like exponential map at one point and Hilbert kind of was forgetting curvature term right or this manifold curved manifold and this encyclical enchantment has physical intuition and govern by the principle that entropy must grow but if you write Hilbert equation entropy may go in the wrong direction and so so again here this when you know extremely helpful it is funtorial nature is funtorial it's not smart but not being funtorial stupid a implies b, b implies c that's it you know this step and this step and this kind of this very primitive logic everything else comes along okay so and so this let me make interruption I mean it's a moment I can give 10 minutes break so we can run away if you kind of want to ask question definitely what do you mean by funtoriality you have a funtoriality between the set of equation describing the local interaction yeah if you look you can say it in the term dynamical system which I think is wrong here one dynamical system this equation describes particle moving and here is another dynamical system differential equation time is continuous in the space of density of momentum it's functor from one to another so you modify condition of your system and according to modify there so you modify it's not to individual system to the whole class of Louisville system and this class is funtorial I never thought exactly what the class is and what the right thing but obviously how do you do it by naturality you don't care about convergence and this means it's funtorial however describing this category it will be different because there are numbers involved it is funtorial but if you think a little bit about you know how to do that because you follow the general logic of something of this funtoriality of course you modify condition according to the modify outcome this is funtoriality and no matter how you call it but again it was never formalized it's unclear from thesis that don't care about that mathematicians it would be interesting if you do that and think if you go there you can discover a new world or maybe discover nothing of course but this exactly what applies to all of those for example these people who look at this whom I know, who explained me about ex-cock checkman hierarchy they discover the whole world when they look at the meaning of what they are doing geometrically it was just some infinite dimensional manifold that's Riemannian geometry of infinite dimensional manifold and they looking in that for this infinite dimensional manifold they found new equations and they were applied and they apparently confirmed what happens in some chemical reaction right so they claimed but you never know I wouldn't these people who done claimed it but I haven't checked it myself I gave some reference to their work at some point I saw you there so and so this is this is about what you think has followed it's kind of but you have to do it you have to go through all that and it's very difficult to find correct you can easily make the wrong category it's not obvious at this moment mathematics is an art after that it becomes kind of science so to speak but to make the first step you have to use intuition now functionality helps you it tells you what you look for but then big tells you it must be not natural philosophical sense it must be natural mathematical sense and so and some categories kind of apparent here and some are not so and maybe not exactly categories it would be most interesting if there are other naturality which is not covered by functionality it would be most amazing if you discovered here what 5 minutes so I just say one word so I mentioned once a question and you know a little bit had dynamic equations and then there's BB, G, K, Y it's 5 people who discovered independently which is rather trivial stuff in my view that now you have many boys when about two particle now look at the description with K particle how it involves bigger number of particles so we have infinite chains of equations but you can neglect other terms or make some epsilon and so right where is equation derived from that we call industry writing these equations I don't know if they have any applications again but this is again mathematically would be pleasant to have a way describe it in words equation come without efforts, allow like you say what to compute and then you put into compute and compute and write formulas when the formulas may be very messy of course but now look at the next level which is more closer to what we speak and this is physical chemistry of polymers and they're folding gelation, percolation, etc and this is this will come let's say some words about percolation so it was the first paper on percolation if I'm not mistaken appeared in something I may be mistaken I would say something 1880 which I learned from a lecture by Stas Mirnov and it was an American mathematical monthly and there was some percolation model described and said well done but work you have to do something better and then of course was kind of forgotten and so by the way in any case I recommend very much if you want to learn about percolation, listen to a lecture by Stas Mirnov I found by far more informative than everything else it's more kind of kind of congenial mathematics we know and we use then more traditionally they by Castan or Grimith or more probabilistic oriented and then it was kind of it never existed and then in 1941 it was by Florie it's Florie and Stockemeyer who was done in somewhat independently and later by Florie and he was this great chemist but they speak about gelation and they are models like that so I have this active ends and then they may interact they may stick together may become stronger with certain probability or concentration become bigger and at some moment there is a this thing become rigid you become a jelly and this is critical and this is of course the same as percolation and in percolation the typical situation we have so let me formulate percolation just if you try it in the simplest terms you can say and which is of course the first model to look at not at all which mathematics is much concerned with you have have minimum information the simplest meaning also the most general you have a space X metric space Euclidean for example or I prefer maybe compact and remind in many fold which and which is involved and you take on random the N points say it's D dimensional space and there I would say number of points will be N to the power D, I think it's convenient right and then you take epsilon or constant times constant times D by N and around each pole you take a ball of this size so you take these points so they kind of they kind of you formally cover the space so a typical distance between the points will be 1 over N and you take balls so they may overlap with some constant and then you ask what are the geometry of the set and and this point which will be essential for them saying just geometry is what the hell is geometry there are so many invariants you cannot speak about them but once you have this parameter C, radius so you have not one set's family of sets and then you look at the topology of the weak invariant but when there is parameter there are numerical invariants when things change how topology changes and so we return to this picture which I had with Morse function you have this Morse function and this big space and some function may be energy or just parameter of that and we varied topology changes and these numbers are invariant numerical invariant how we make invariant of spaces and this is one basic mechanism which is parallel to how boysman describes physical system right and to which we shall return later on and and then yes I want to understand that one of the phenomena that we want to know what happens when n goes to infinity as function of C so in the limit and then this kind of phase transition that something happens in particularly discrete moments with respect to C this is more correspond to the picture of gelation but the same percolation and then you make some specific metric spaces and what you mean by balls or it may be not balls but some canonical subset there maybe if you have you for example you have Euclidean plane you have preferred lattice and you subsets to change edges of the cubical lattice on something else you have a lot of specificity which emphasize immediately the usual presentation percolation theory for reason I have to understand because except which makes in my view justified is only one thing and this is a theorem of Smirnov so there is a theorem there which is like a thing it's a real thing it's quite kind of a remarkable thing I just want to explain what it is which is not obvious which is simple and which says well maybe it's not nonsense what the people are doing there because otherwise everything else look why this model in some particular situation something remarkable happens which was predicted by first conjecture by Lang-Lenz with quotas and percolation and particular formula was right written by somebody called cardio cardio I forgot cardio and so what it is let me remind you if you don't know something curious so here is a random presentation something else and then look at some maybe I take a different picture maybe this is too elegant so to make the idea more clear so look at this shape and this shape may grow everything in the limit so you can make scale making this both smaller which is the same and you see what is probability that there is a path going from here to here in your subset so you make particular subset if you take small probability very rare distributed there will be little island spread everywhere or there may be bigger things and so what I am interested in the probability of that and below this will grow below some probability below some density of some radius whichever parameter this will be zero it cannot go from here to here there will be no path here after some moment the path appears and this critical moment so maybe before going to this minute let me give some kind of very trivial stuff which is look completely trivial there is some technical point which needs needs proof yes for proving it correctly and this is a subtle to understand what I am saying is non trivial first you look at this what they have what people are doing with smart people but let me give an instance of that so one of the common models have this lattice and so what each edge you preserve or eliminate with some probability p so this characteristic number say you preserve each edge probability p it disappears probability one over p so and so they have this graph depending on p now we have this huge rectangular or square and what you want to know to which probability there is a path going from here to here positive probability or path does not exist probability goes to zero again we have infinitely large square so there is limit and no surprisingly this is one half and this is theorem by Kaston saying critical probability one half so it means if probability you check there is one half there is no path if more than one there is a path and what happens at one half a subtle matter it is finite probability value of which is a very tricky number I am not certain it is computed for the square it may be computed but full understanding is conjectural so why is one half and one half because punctuality because this is kind of exactly it's not multiplication type mathematics however simple it says that if I have a set going from here to here you cannot go here that's clear but this nothing is efficient if there is no path from here then there is a path it may be extremely tricky something doesn't exist something else exists and this punctuality punctuality is quite remarkable in profound theorem yes I just once with my experience I was writing something elementary introduction topology kind of in the spirit of puncture and you can do every all homology theory tomoisomorphism novical browser theorem by just waving your hands except for two things you had to without thinking really just from general principle except at some moment we have to use the third theorem by night in the homotopy group of spheres and punctuality punctuality is not kind of that easy you have to do something simple but not canonical and on third theorem on the contrary some is canonical but on the algebraic level beyond my geometric understanding of the word it's not geometric picture could never make a geometric proof but and here for punctuality and it works in all dimension and that's the proof because this self-dual lattice you move it's dual to itself in general for any lattice you take dual triangulation and so these two percolation are self-dual and it was the problem of course it's obvious here but then you say aha the point is that when one pass appearing of this pass has several different interpretation this critical probability and you have to say they are all equal that on this level it's obvious and this generalizes to all dimensions if you have n-dimensional cube and there is a paper I refer to a recent paper by Carlier with some quotas and you have this any product of manifolds x cross y and you have triangle product of triangulations and you make the same kind of middle-dimensional say x cross x probably would be better so not to confuse x cross x and you have triangulation here and there product and you throw away with probability middle-dimensional faces then appear in cycles x cycles and y cycles will be one half but same reason but then the whole theory becomes very very very subtle and in general it is high-dimensional percolation very high-dimensional percolation is sorry something is sound it's high-dimensional percolation is not properly developed actually faces working on that because I refer to feralish and others even understanding random surfaces becomes much more complicated and it's kind of clear meaning to become complicated on the other hand just to have some feeling before you come to kind of really deep stuff because there are so many kind of obvious but difficult theorem and difficult well kind of mathematics I don't understand kind of computation things and so let me just explain one elementary stuff to have a feeling for percolation so traditional searching for percolation and it was introduced by Hammersley and somebody else who has no name for mathematician was in 1957 and unfortunately I couldn't find this paper because it's not in the open so in England some philosophical society it's not in the open dosto you cannot read it but I found the next paper by Hammersley so I don't know what they were saying in this first article and I'm pretty certain they were not too aware of Flory who was done more or less the same before them 15 years ago and of course not the same but they also were concerned with some real problem about really percolation problem of gazes through something filter so think about say trees just I just want to say how much it depends on a point of view say binary tree and you throw away with some probability in edge throw this, this, this with probability p what is the critical probability that there will still pass with probability going to infinity so if below this it doesn't exist after that it does and then it is obvious it's one half and one half because binary tree and if you think about this term you throw it here and here it's a little bit a mess but now think in percolation terms and then become obvious or in genetic terms you just see what happens layer after layer you go big and big and when you add new edges every time at a time and then you see what you have is the population growing either you divide or you die or you have only one and so if on the average your expectation of the number of your children is bigger than one you grow exponentially if less than one you exponentially die because you have here binary tree it will be one third right so it's obvious again but again you have to take percolation what is interesting in all this computation probability you have to write organized so that independent variables so you can make obvious computations percolation also the writing qualities and when you really come to interesting stuff like in the case of Smirnov theorem you have to use some combinatorics you have to see some secret symmetry making things equal and some probability equal or big arrangement and this part is tricky and certainly annoying annoying that might be something simpler than that up to a point now if you apply this say percolation in this space in the plane then you conclude that if probability is big enough you do go you will stop because here you don't realize all possibilities we multiply we don't have exponential possibility we only have growing boundary doesn't grow exponentially grows only linearly therefore there is critical probability which may be very high and still you don't propagate but why still there is probability you propagate so why this big probability you propagate this argument doesn't work the simplest argument non computational one use punker duality because if there are few paths then there also probability going around and having this will be also low so imagine no matter what you do this long path don't appear then this thing don't appear therefore it can escape right right so but there is another of course more computational you just repeat computation I make but now you point on the line and they only can multiply within this line if they come together well it doesn't count it's still one they don't add up but this if you make little computation it still works I haven't made this computation I just don't know how they never say they write formulas one way to make computation and you look at the worst possible case when only this can do but on the other hand when you start dying and it went long enough you have many holes and then there is extra coefficient and so I haven't seen for example what the critical rate of growth which this argument applied right it has nothing to do with punker duality pure computation, trivial computation and must be somewhere done but again these people they kind of don't ask from our point of view and certainly it's not for me to compute so I don't know but this of course once have propagation dimension 2 you have no dimensions similarly you can do something for high high dimensional percolation but then you get some little problems of course it's not so obvious what it is okay but now comes this conformal invariance and all this is just some of them punker duality it's non-trivial stuff but kind of we know that under computational stuff which I don't know so the degree of of sophistication there which I never studied but here there is combatorics and this really beautiful thing and so what is so conjecture for example if you have domain like that and for the critical probability when the first pass appears this actual probability is a number it's invariant of the domain it's conformal invariant and since conformal invariant you can bring it to particular shape and the shape convenient where the theory was proven by Smirnov is a regular triangle and you look at this pass you know and you think these four points can be brought to this picture right? and this probability if it's triangle with union sizes for the critical probability this number will be the length of the triangle if it was triangular latches latches peeled out of triangles and this is theorem it's very simple, very beautiful theorem and the proof is so he explained this and he found new proof in the lecture it's very beautiful combinatorial proof there is a feeling there must be much more general structure behind it kind of elementary which is elementary combinatorics plus with this general principle of probability theory which I didn't explain and so let me indicate what is one of the technical problems again which I learned from this lecture that good thing about this lecture he explained what the problems are not just showing or already proven which is certainly not instructive if you are not inside so you look at the following thing you have this parameter p right? so you take or some eliminates or you preserve some edge and some say lattice with probability p and then what is the probability that you have infinite path going from infinity of course if you kill already positive percentage positive percentage of edges with positive probability you will be stuck always there will be circle very small exponential decay and would exist but what's remarkable that if as I explained if it is close to zero then you will be stuck probability of going to infinity look like that it goes here and then goes to infinity like that so we know up to some point if you will be there up to some point it will be increasing function but what's not trivial continues but this is the theorem of these guys and this apparently unknown in dimension 3 this functions continues and the key point is point raduality here and this is difficult point probability indeed this is not obvious from kind of hand waving and this is kind of hard because when you think about that you assume and you don't think about that and then kind of he explains this why the theory is not trivial this is not trivial point and it's unknown in dimension 3 according to his lecture about I think 2 years ago maybe now it's known but now what percolation has to do with protein folding what a problem there and in itself avoiding random walk so another kind of think I've discovered by Florian he's more pronounced for which he actually received Nobel Prize for understanding the role of self avoiding in the behavior of molecules you really kind of explain and predict quite a few phenomena this is the following self avoiding walk so again you look square lattice or whichever you look at the path going on random but just self avoiding it wasn't mean on random and of course much more nature looking in the liquid you can see the chains of points where these distances must be one look at all this configuration take measure on there and you want to understand how typical sequence will look like if itself avoiding meaning that this distance is one or other distance is bigger than one so it's some domain all distance one is some high dimensional manifold and then some domain in this manifold this is how it looks like it's a tremendous mess so the point is the space so this is blue is where it is diagonal where it stick together and white is remaining space and if you want to understand you take random point there this random point give you a path how it looks like first it's unknown the question has any answer it may not look in any particular case this way another half this way no physics physics is very uncommon and you always believe this concentration of statistics average and typical coincide however you know it is not true in biology Wiener takes all opposite principle you have lottery average lottery and expected gain kind of opposite expected gain for every individual zero expected gain the average is total sum completely opposite picture and this by the way is a source of mistakes when physicists speak about evolution and speak about randomness and this is exactly the point when you speak about random supply you always make assumption according to your background and there is a kind of this remarkable story about Fred Hoyle who was spending some time arguing against traditional version of evolution and Fred Hoyle is a made one of the greatest discoveries most ingenious discoveries in physics I think in the 20th century and so and he was it was kind of interesting story and he never received Nobel prize he received his student and the reason for Hoyle was he was overqualified and there were some people by accident received this who were overqualified I believe Einstein and Jirak I believe in Hoyle I think three people were overqualified they were far more fantastic discoveries than anybody else and so what was the point of Hoyle you don't know so he made the following conclusion the following what he was logic life exists on earth fact conclusion there exists isotope some particular isotope of nitrogen I forgot which exist one trillions of a second which was unknown so he made this prediction on the basis of existence of life and say what was his logic and people were laughing at him of course but some people at Fermilab I believe he made experiment and found this isotope and this thing was really kind of fantastic so how did it work what is the relation so what is what isotope of nitrogen or anything else has to do with life on earth exist one trillions of a second anyway we never see it I think this fantastic stuff I just cannot resist kind of telling this because Hoyle understood where all heavy elements came from on earth where they came from because if you look at the process in the star in the sun it burns hydrogen then there is helium helium and sometime helium goes a little bit to lithium and that's it and what about carbon the key of course is carbon life is based on carbon life of course maybe some other life but the only life we know we can imagine is carbon based life because no other element has enough enough flexibility to make complicated compounds and here we make a conjecture of prediction that this come under the explosion of supernova how the super supernova because this moment temperature goes to billions of centigrade and various nuclear reaction take place and then they explode and go around the universe and sometime they come to the planets so we are kind of a stardust in very precise remnants of the burning stars but in order to have to go to carbon from hydrogen this process there are certain rules of nuclear chemistry you need, it doesn't work you make this computation it doesn't work and then he realize you need intermediate and this intermediate was some particular isotope of nitrogen and he just made the discovery and this I think he was very very good but he actually he was not he was over qualified because he was very nasty person he was fighting with everybody, people didn't like him and so he just gave this to the student the student was completely devastated because he was just he was just I wrote some paper afterwards anyway the student was really most unhappy unhappy Nobel laureate because he was really ashamed and he was a good scientist of course he didn't invent the theory he just made some more precise computations but what what I was doing he was looking at some particular proteins and there are proteins which protein which make your shape your DNA and they rate of their mutation is known is almost the same for all organisms and you take little mutation and you die right? it's not fully understand why but this is the most conserved protein and he said it's impossible to have evolution with this they couldn't come by chance, probability is not enough and and then he was fighting and he was idea of course that there are more he was not kind of believing in the god but saying there might be something else in the universe from where this chemical came on earth it couldn't be informed but the point is that him like most physicists when we make computations they make some assumptions and one of the assumptions but if you don't see any correlation the regional correlation things are independent and using that you can make reasonable prediction of the rate of mutation of proteins and some of them in experiments and there are experiments when they are artificial mutation and you go from protein A to protein A prime it changes function in certain direction by artificial selection so you modify it many times many times and select which are most close to what you want and you can evaluate we know how the rate of this changes and where you have to go and you can say how many cycles you have to make and there is a particular experiment somebody explained to me who was performing that and there is a computation with physics and it takes about 1000 cycles which will take you months and months to know and then you may experiment to do it in 5 and nobody understood why it's a mathematics of this because probability is not things are not independent there are correlations which are not physical whatever in the matter of things and this is extremely well seen or unseen in protein folding there is a plastic problem showing that there are some very basic mathematics which don't understand absolutely not how proteins fold and as a model question much easier but very well shaped is question why avoiding random walk and the self avoiding random walk but the way described it is really interesting question why I say this measure why I take all paths must have equal measure why not to say you may add one molecule at a time so it makes one stop at a time but then of course any elementary exercise every random walk become blocked on the planes about 70 steps will be blocked region you cannot go out any dimension now and I think this random walk protein folding are closely related and you have to understand this picture so one point of course the space embedded curve say well understood for closed curves topology is extremely complicated if this path all this complication appear geometrically again if you introduce some parameters and create this topology in more artificially so it's a very complicated topology of not theory implicitly in there in dimension 3 in dimension 2 it's another story and it's maybe eventually solved but what is the problem elementary problem and it seems extremely difficult so just on the plane say usual random walk goes if you make n step will be square root of n the question is what happens to the self avoiding walk viewed more than square root of n or less than square root of n and 4 in dimension 3 predicted some number will be more it goes further and he made very good prediction using this idea that just part of the walk exclude some volume and so it takes change probability space volume changes and he make prediction well correspond to experiment but approximate and then of course there's more but nobody can prove it neither in the plane nor in the space that you go further than the square root all you can do roughly that you cannot be become concentrated just go moving nowhere this is just what you can say you just cannot fill the space completely which is extremely trivial trivial result and very kind of not exciting so what the difficulty here so this is very and it's kind of if probability at all can work here it's meaningful to ask this question if you look at what happens to a partner how they move it's another story of course then you have different world and you have different questions you have moving molecules they interact with them and there are very many things happening and the most remarkable thing of course is protein default that's our protein essential molecule and some of this it's a chain usually not very long about say 200 amino acids and when it falls it always take a very fast, very particular form and if again you get rough computation milliseconds otherwise you would die and this is the reason a rough computation it goes into days or years it's long process and you don't know its mechanism but there are physical models now more or less very rough with reasonable prediction but there is no mathematical theory there are computer programs which can guess what will be the shape of the fault and it's important for understanding function of protein is not the last word and this has something also with percolation and also exactly with dimension 3 I guess if it were dimension 2 or dimension 4 it would be very difficult much easier problem and the folding would be impossible so apparently what happens this kind of you can fantasize and this happens in cell indifferent when you have this thing and it moves around and it has to cross itself and the moment it touches itself certain bases start interacting and it may happen that there are arrangements because they are not random sequences they are rather special but still not terribly special they heat with reasonable probability in the basement they are not only stick but also oriented and then next moment you don't go away but you slide along for example which happens in cell when you have cell and you have this DNA and there is some protein which find the place where to do it and if you do by diffusion everywhere it's very unlikely but what it does it goes somewhere becomes sticky a little bit and then go along and then becomes much faster and this was interesting kind of theory of random walks when there are really very medium has different dimensions so some random walk become dimension 3 and some one dimensional random walk but of course if it sticks it certainly jump back and forth it certainly computation becomes messy but mathematical description of this is very simple there is a general principle very simple implementation of this like equation of motion they are very simple you follow Hamiltonian that's it but then there are different Hamiltonians the same again described in the same universal language but specificity of a specific parameter become computationally maybe difficult but mathematically transport and so there may be if you start thinking those terms you may can get some intuition about what that but that's one of the unsolved problem and unclear what it is and again there is one amusing result there is also a hexagonal lattice of Smirnov with Hugo who is here not in this audience again but in this institute that it doesn't tell anything about the diameter of that it's beyond what we can do but it tells you what percentage of pass we will be self avoiding and this number is square root of 2 plus yeah it is some number which is involved that properly understood percentage is like that you see so you count how many different random paths are there you take log and divide by n and go to the limit obviously exist it's a multiplicative function and this is the number and so it's not complete nonsense right again you can say it's all the theory it's just so natural many times ago about this kind of things with Jean Bourguin and I'm saying he has probability there are so many interesting problem we just take this geometric problem say random and then they have and then he says well trivial questions and either they are obvious all the percolation he said for him it was all obvious before Smirnov or it's a completely unreachable so it's just not good however now there is progress and certainly it's not nonsense and and this is one of what you call thing yeah it's not just idea it's real stuff happening there and we have to look from outside and understand what it is okay my time is over and I certainly not through what I wanted to say today let me see maybe I say a couple of words but I wanted to say today yeah this is what you are saying yeah this is Poisson equation ah right yeah I wanted to say something about languages maybe do it we will be down in 10 minutes just so far it was about probability theory and saying where it goes and where it doesn't go and certainly saying it needs and another instance of that of completely different kind of problems are in natural languages which is similar to what happened to evolution but as I mentioned this confusion with hope and even more this happens to natural languages and so I want to make some some people who understand languages unlike myself also I need to say about for Neumann and about entropy well probably we will not have time to say it but now I want to make some quotation discuss some we can find some references this is not a if you have a copy of that it's not final version I will certainly rewrite it in some moment in a more systematic way but yeah here is some reference to something I mentioned but now I want to say something else about probability in linguistics yeah so the point is that the way the way you use probability affecting probability in physics in my view is you basically to symmetry you can make assumption of something independent something is equal because thing is highly symmetric like particles moving in space and there is huge symmetry there which I, my next lecture I explain how you can exploit it from different angle but it's not so in the language so what is probable what is probable in language and so there is some quotation by of Chomsky where is Chomsky then I explain some particular instance of this how probability comes yeah here is what so before that I was I'm taking this as just copies from which I'm in the process of writing on languages thought that you can can you use probability in languages and what Chomsky says no because every sentence almost every sentence in life which comes is said for the first time and will never be repeated which is not quite true much things are repeated but it's with long enough of course not probability to it and so it's complete nonsense however especially in the modern techniques of language, artificial language processing you use probabilistic models and they work pretty well and and then objection to Chomsky is as follows two fold first in physics also you assign probability in a particular position of atoms so many atoms and this is infinitesimal number they never enter this number they never enter any particular position if you have billions of particles and they this here there of course is exponentially small number right like 2 to the 10 minus 10 to the 25 or something probability of any state probability that if you divide in two halves in each particular particle you either heal that you'll be 2 to the power of this Bagadran number absurd probability makes no sense but what does make sense the two probabilities are equal it makes no sense and this was one of the reasons by the way some people were skeptical about the atomic theory but in languages it's different because you don't know how to equate them and how to count them and so what I want to say that probability in fact in languages is not a number but something else so you have to think as a probability 34 language as a function from the category whatever it is of languages to the category of some objects which represent probability and one of them is a tree like picture let me see if I have this picture here of some tree ah I think I forgot to make this tree wait a second ok I make a picture there so if you look what is the frequency of a word and what is the frequency of a sentence if you look at the some string string of symbols some word of sentence you see how often it appears but when it appears just where appear in what you just look at the longest strings so it may happen like that may be odd and then may be like that for example if you take strings peaks fly and then it looks like that it always appear in the same sentence you see that one peak fly and there are very few sentences just I'm sorry I'm wrong picture on the concrete like that and then it will be like that it will be few but very long and then they don't branch but if we have simple words we are usable with branches immediately and look more like that therefore the methodics of this tree is more essential than the particular number of probabilities attached and by the way trees especially with weight attached to them makes a category very similar to category of numbers you can compose trees you can add trees almost like algebra I don't have time to explain that and this may be however it's not tree they are more complicated graphs and that's quite interesting phenomenon in languages and for formalizing it's not obvious because this is a vinagrad schema challenge so the package doesn't fit in my bag because it's too large and because it's too small and you have to figure out what is true on the base of statistics and you see it's just agree, look at the Google however if you replace doesn't fit by fit and explains it's all it will be the same pictures it will tell you nothing because this example I took it because it's just a classical example in the study of automatic analysis of languages but if you look at something much more elementary if you look what cat eats what was more typical cat eat mice or cat eat grass and you look at the Google and then you learn that cat eat grass but not mice I made this experiment some a while ago but almost no cat ever eat grass or if kid eat chocolate and then you find out that my kid my cat never eat chocolate but your kid very often eat chocolate and there is a good reason for that So So there are these combinatorial arrangements of words they are still combinatorial so probability is there but that's another kind of problem how to find the right setting for that and of course in machine learning this subtle thing is ignored the machinery of the neural networks which works remarkably well but again it works this opinion works the worst possible system but we don't have another one they are not really something especially good about that it just was developed and there may be much better system so our brain works slightly better because the number of computation make much less achieve comparable results so for example this machine which plays chess they definitely made more computation than human brain done but they play still comparably to human slightly better but comparably to the best chess plays and the mystery how these people work how people can play chess this much more mysterious with such premissive brain computer very slow billion times slower so this is the last point I wanted to make where which is a source of ideas on probability theory Refreshing Chomsky it's not he says that you can't apply probability but you have to say you have to modify probability to apply it and make sense of languages and this is of course we will be different probability and then in my following lectures I will explain where you can go from there so we look at this and now kind of the same pictures we had from a different kind of perspective what probability will be from more and more topological and geometric and we will have no application to real world but mathematically quite amusing and based again on some mathematical facts which is no ok for today it is all free to ask questions now so this Schmirno's theorem that he said it's for the triangular lattice whereas the Pankareto entity that you mentioned was for the square so why doesn't this method for the triangular lattice go through to prove no because he just eventually they use a simple formula this interpretation of the cardia formula which is for triangles for regular triangles this is kind of a technical level make a simple form and you can see exactly the length of this edge but my interpretation would be the following that the symmetry group of triangular lattice rotating around put irreducible but square is reducible so it's more symmetry there and therefore it was most apparent in the original proof now they have simple proof they developed calculus or kind of kind of calculus on this lattice and because it has more symmetry it's closer to usual and when it's split like the square it's probably not solved this possible explanation but they do play some game with hexagon of course there is duality between Pankareto still applies but we go from triangular lattice to hexagonal lattice and this game is played in this proof yeah they and they prove non duality but some kind of duality they have three thick quantities which are related by some analytic relation and from there it comes based on some combinatorics well I just you have to just look at this paper or listen to him I just I don't know the details I just don't know officially there is no but but if you start thinking about that you see that in all dimension there must be something like that there is a universality simple principle you expect them to be true and then they somehow not obvious how to prove them how shape how it depends on the shape, some continuity some symmetry how it's involved there is one of the most amazing results in the theory so you there is no simple explanation for that it was a big problem and it was one of the most spectacular results in mathematics you can say easily I don't understand it I don't want to say what I don't understand but actually everything I was saying I don't understand but exactly my point was to convince you that you also don't understand so you may start thinking that's my justification of my non understanding you're supposed to not understand that ask me but else you don't I think we don't understand what probability is I mean in people yes you develop some branches but of the whole picture what it is what it is we don't know and wait how it should go it's impossible again I want to say you can play with names but you never go to the bottom of things unless you know where the things are you have to know where the things are and play with these things and then you come back to your names and probability mushroom that was playing with the names sometimes very sexually like in these examples but also with completely abysmal failure like this self avoiding random block all these three dimensional percolation all many other another example I want to question I can bring forth that there is a percolation of discrete kind of quantities there is one dimensional thing and even when they closed up closed up and there is this where we are measuring the space of closed curves with some effort there is a theory of random surfaces or maps from surfaces somewhere using conformal parameterization and then you stops in dimension three and so conjecture would be there is no probability theory for higher dimensional many faults or cycles or whatever very special thing but not general impossible because when you start closing it there is so much break and dependence that you cannot do anything thing becomes absolutely chaotic and uncontrollable the question is how to formulate precisely before trying to prove it so there are limits where probability theory works everything not everything you can formulate you can do at some moment you run into principle in some way unsolvable problems or not solvable in the language or style we know and this is I think is a point but this is a kind of traditional probability and another aspect that there are many alternative probabilities I what are the alternative probabilities exist in the Romanian by now nowadays there are several alternative theories and some are developed and some in the course of development probability is very peculiar part of mathematics it's not truly it only has one leg in mathematics and one outside and we don't know where the thought is why did you say that growth and dig was one of the best naming things well because this is a parallel you can bring between probability of theory and algebraic geometry that growth and dig developed this definition of scheme and then he had this definition of topoids or the other and this definition of scheme came a few years after appearance of the book by Andrei Wei in the foundation of algebraic geometry when Andrei Wei defined algebraic varieties as a set of points of some universal field it's exactly parallel to how growth and dig how Kolmogorov defined probability this universal probability space that I have all that and growth and dig are shown it's not right definition it's not a set like it's kind of a scheme it's kind of a function from category of category of algebra to category of sets he gave the right concept definition name for the algebraic variety and this is not done for probability theory because probably there are several probability theories they are different of course growth and dig is on the last word it's time to revise it every such concept must be revised every 50-70 years if you use the same definition fundamental logic 50-70 years and don't change it something is wrong it's time you have to be in order of course accumulated evidence from from mere symmetry that there is something else in algebraic geometry it's not really properly grasped but concept of words given names given to growth and dig he introduced many other names in algebraic geometry and he gave a fantastic way of giving correct definitions and sometimes which is not taken by other domain mathematics but he himself kind of unity well organized in algebraic geometry and it has tremendously positive impact while algebraic geometry was developing so fantastically well right and this is the same if you do it another domain it certainly may be not possible it's desirable may be not possible but it definitely was not this was not happening it has not been happening yet but growth and dig was a must of giving correct name we know in algebraic geometry these words and the stick they are so kind of guiding you very well but and of course probability in mathematics were other examples of that but in the recent time growth and dig's name comes to my mind first maybe that I don't know if there are similar what is algebraic curve and it was before growth and dig was not way to say it properly well no context you cannot define objects just out of context you have to create this context and he understood the right context to define algebraic curve and with probability you can talk several of them where should be defined in my the way I see it is just out of date of course technical formalism everything remains but the question if you want to go next step you have to change foundations so ok thank you