This is Jean-Balbier from ICTP. So you've been with us since 2019 in the quantitative life sciences section and you're doing research in many things, but among other things, statistical physics with connections to statistical inference, computer science and machine learning. And today you will talk a little bit about phase transitions from physics to computer science. So, good floor to you. Thank you, Zach. Hello, everyone. So it's a great pleasure to be here today for this basic notion seminar. So, as Zach mentioned, I'm a statistical physicist, but working really on various fields which may apparently be unrelated to physics. And what I want to do today is to show you that actually very important, one of the crucial notion of statistical physics which is phase transitions are actually of crucial importance also away from physics. And the aim of the talk is to give you an overview of the very large variety of systems in which this type of threshold phenomena where you have an abrupt change of the system behavior, some complex system behavior, when some control parameter like the temperature or other more abstract parameters as we will see change. And I want to show that this type of phenomenon appears essentially everywhere in nature but also in computer algorithms and these kind of things. So let us start with the basics and single molecule of water. So this is a very tiny object. It's a tenth of a millionth of a millimeter. But if I consider very many of them, and when I say very many, I really emphasize on the very of the order of 10 millions, billions, billions of them. So this is the so-called Avogadro number which is the typical number of molecules you can find at the drop of water, for example. Notice that this type of numbers are much bigger than the numbers appearing in astrophysics or high energy physics. If you take a lot of them like this, depending on some parameter here, the temperature, this may lead to very, very different results. So here you have, of course, ice. This is a kind of iceberg. Then you have the liquid state and you have the gas state. But what is important to keep in mind here is that these three very different systems are made of the very same components which are molecules of water. So at the microscopic level, they look quite similar but at the macroscopic level, of course, these systems are very different. They have different mechanical properties, optical properties and so on and so forth. So really the whole is more than the sum of its parts or as summarized by Philippe Anderson who has a Nobel Prize in 77, more is different. And what he means by different here is really different at the fundamental level. So let me continue to quote Anderson. The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of simple extrapolation of a few particles. Instead, at each level of complexity, entirely new properties appear and the understanding of the new behaviors requires research which I think is as fundamental in nature as in any other. So here is the kind of pictures that we like in physics. This is called a phase diagram. So it essentially tells you in which macroscopic global state can be found the water, the system of interest here, which are molecules of water as a function of control parameters. So here I put the pressure, which is in bar and the temperature in Kelvin. And you see different regions which are separated by lines and these lines correspond to phase transition lines where you change from one macroscopic state to another one. So for example, the ice, so the solid state is here where molecules are very compactified and they are very ordered. You have a kind of crystalline structure. Mars is here in the phase diagram. So most of the, if not only the water and Mars is in the solid form, the planet Earth is here in the middle of the liquid phase. That's why the planet is blue and not so much liquid water but we're actually close to the phase transition line and therefore we also find ice on Earth. And then you have the gas state where essentially particles are freely moving around and Venus is here. So notice that the density of the gas, for example, is much lower than the one of the liquid than the one of the ice. This is what we call a phase diagram. Let me just give you a bit of physics. The intuition, I don't know the intuition but the kind of tools that we like as physicists to understand this kind of pictures. It is called the thermodynamic potential. So essentially there exists a function that I would call the free energy that I will describe a bit later which is a function that when you plot it has a function of some order parameter. So the order parameter is what? It's a quantity that essentially describe the macroscopic state in which the system is found. So for example, at a high density, it means that particles are compact all together. You should be in a kind of solid state, okay? While at lower densities, you should be in a kind of liquid state and much lower densities, this is the gas state, okay? And there exists this function that if you plot it for a given temperature, let's say 0.2 Celsius, it will look like this. So it will have a minimum at a density of zero which in this case is by definition the one of liquid. And if you continue, if you decrease a bit the temperature, you see that at zero degree Celsius, you have a second minimum that appears and that will become the global minimum if you continue to reduce the temperature and this corresponds to the solid state. So essentially from this picture, let's say there exists this function that you can compute in some way. You can read off directly in which macroscopic global state, the system, the complex system you are studying will be found, okay? And the minimum of this function corresponds to what we call the equilibrium state in physics. So what is this function, this free energy? So this free energy is a way to understand the perpetual tension between order and disorder. So let me say that I'm an experimental physicist. I'm in some lab and I can fix for my experiments some control parameters. This is why I call them control parameters. I can control them. For example, the temperature or the pressure or in a natural environment, this is fixed by nature, okay? I will then define a number of order parameters which is a set of quantities that I can measure for my system and that essentially define that describe my system. So for example, this can be the density, the average density of molecules in the water which allows to tell if you are in the solid liquid or gas state or the magnetization which is a kind of measure of how atoms are aligned. If they have a preferred direction or any, so the magnetization is a global quantity, you look at all atoms and you look if there is a common alignment or any other macroscopic global property of the system that you are interested about. So these are the order parameters, what you can measure. So this is the way you describe the system. Then there are two very important quantities that enter into the game that really control the physics of the system. The first one is of course the energy and the E of O where O is the set of order parameters is essentially the cost for the system to be in the macroscopic state O that is described by this order parameters. So usually physical systems like to lower their energy and this leads generally towards more order. So for example, ice has a lower energy than the liquid state and ice is much more structured. You have a crystalline structure which is much more ordered. The second quantity that only enters the game where you have complex systems like the one I will be interested in today is the entropy, which is essentially the logarithm of the number of microscopic configurations of the system that correspond to a given macroscopic state O. So maybe you have many possible configurations of the atom that correspond to the macroscopic state gas or liquid or ice and the entropy essentially counts the number of such configurations and you need to take the logarithm because there are so many, actually there are exponentially many in the number of atoms or simple entities that form your complex system that if you don't take the log this number is just too huge to be compared with the energy so you need to take the log. And higher entropy leads generally to more disordered system. So the gas, the particles have more degrees of freedom to move around so the entropy is higher. There are more configurations corresponding to gas than the ice which is highly structured. And so the free energy is essentially the difference between the energy and the entropy where the entropy, there is a minus sign here and the entropy is weighted by the temperature which is therefore a way to tune the tension between order and disorder. So we come back to this picture. I have my free energy here. Now we understand a bit what it means. And so indeed if we look at a high temperature above zero degree Celsius, the minimum, so what I should observe the equilibrium state which is the state minimizing the free energy is the liquid state. And indeed at high temperature the term that matters the most in the free energy when the temperature is high must be the entropy because I have a bigger weight here. And therefore I need to go to a state with bigger entropy and the liquid state as indeed the higher entropy than the solid state. So this is the term that wins, the disordered term. If I reduce the temperature what will matter the most then is the energy term because this term will become small. And so if I want to minimize my free energy to find the equilibrium state, the one I would observe experimentally, physically, I need to minimize the energy and the solid state, the ice has a lower energy and therefore my system is found in the solid state. So now what I want to move a bit away from this classical physics models and to show you that these notions of phase transitions and the language of statistical physics and all that can be applied to a variety of other nice systems that actually appear everywhere in essentially all contexts that you can think about as long as you have enough complexity. Again, a complex system is what? It's something that is made of a very huge number of interacting entities. And what I claim and what I will show you is that in such complex systems you essentially always have the presence of this space range. So let us start with an example of imitation effects like in financial markets or votes. So I will consider a simple model of votes where essentially you have two binary, you have two candidates A and B and each person, each voter, each individual is indexed by I that goes from one to N is modeled by a variable that I call Si of T. This Si of T is essentially the choice at time T of individual I, okay? So if the individual I would be given the choice to vote at time T, it would be plus one if its choice is A and minus one if it would be, okay? So we'll consider a very simple model where essentially voters are only influenced by a global trend that is at each time we have a kind of survey of what's the trend in the population of voters and we'll model that in the following way. We'll say that the probability that at a given time the individual I votes for A which is the same as the probability that the variable Si of T is equal to one will be proportional to the exponential of the trend divided by what I call a temperature. I will come back to what it means but what is the trend first? The trend is defined to be the difference between the fraction of individuals in the population that are voting at a given time for candidate A minus the fraction of individuals voting for B, okay? That I can also just rewrite as the sum of these variables and so you see when there are as many people voting for A and B, this sum will cancel, will be zero and essentially in this case, the probability for a given individual to vote is totally random. So the probability to vote for A, so here there is a type of probability of individual to vote for A, will be 0.5. While if the trend, this quantity is bigger than zero then it means that each individual will be biased towards voting for A when it's negative, the trend means that the trend goes towards B and people will tend to vote more often for B. And what is this temperature here? I can, you see that when the temperature goes to infinity this essentially means that this trend does not matter anymore. It means that individuals are not influenced anymore by the global trend. While when it goes to zero, this exponential, this shape becomes extremely peaked and it means that people are strongly biased, much strongly biased by the trend. So I can think of this temperature as the average independence of individuals with respect to the dominating trend, sorry. So it's a kind of level of rebellion if you want. So let's simulate this system and see what happens. So here is a small code where I essentially simulate this dynamic of people voting at different times and that are influenced by this global trend. So let me consider first a very small system with five individuals. Here what happens is that over time we see that, so minus one, when this curve touch minus one it means that at this time all individuals would agree on voting for a candidate B. When it's plus one all agree on candidate A and in between it means that there is a different fraction for A and B. And you see kind of fluctuations like this and there is no consensus that appears essentially. It's quite erratic. I will increase a bit the number of individuals. I simulate again. And you see again the same kind of patterns with a kind of change of behavior of the population like this at times goes on, but it looks still quite random. Now I increase a bit more, 20, and we start to see something interesting. You see that the time window on which individuals agree starts to increase, okay? So you see a kind of patterns that appears. Now I go to 100, so it's a kind of large system. And what happens? What happens is the following. You have what we call a polarization effect. So this is a kind of breaking of the symmetry between the candidate A and B or the state minus one and plus one if you want. So essentially the trend created by the population becomes strong enough so that if some individual starts to try to vote something else essentially they will be, the population will have a stronger effect and will win against that. And essentially the whole population is stuck there in the state, which is the minus one state in this case. And this is what we call a spontaneous symmetry breaking in physics. So you have a breaking between the plus one and minus one symmetry. And you see that if I simulate the same model but on much longer times, what happens is that you see that the system will first polarize. So the population will agree essentially on which candidate to vote for for a very long time, but suddenly for some reason because of subtle correlations and probabilistic effects you have a drop and all the population then changes. So they change choice and then they vote for the other candidate. Okay? This kind of phenomenon is very generic in complex systems. And this illustrates this nice sentence, which is again, more is different. For 10 individuals you have this kind of erratic behavior while for a hundred you have this polarization effect that creates a spontaneous breaking, spontaneous symmetry breaking. But keep in mind that these two systems they follow the same rule. The only thing that changes is the number of individuals the number of variables that enter your system. And this is the type of phenomenon that leads for example to cracks in financial markets. Okay? Actually the model that we just simulated which is a baby model of imitation of course oversimplified, but still it contains a lot of interesting phenomenon. It's actually very closely related to an important model from physics. I think most of you will have heard about it. So-called Ising model of ferromagnetism. Ferromagnetism is the fact that you have magnets that essentially atoms in magnets tend to align in the same direction. They all carry a kind of little magnet themselves. And when all these little magnets align together you have a global magnetization and therefore you get a magnet. And this is what we call spins in physics which are just up or down variables which interact through this green interactions here. This interaction what they do is that they try to align spins. So you can think of that as energy contributions that are lowered when the spins align together. Okay? And the temperature is a kind of measure here of the fluctuations of these spins. Okay? How strongly they are affected by this alignment by this energy term that try to align them. But high temperature, what happens is that essentially most of the spin are in random positions. So you don't have a global magnetization. Half of them are in up, half of them are in down and they fluctuate like this. And this is equivalent to our simulations where I take a high temperature which means that our voters are very weakly affected by the global trend. And in this case, you see that there is no global consensus. The average vote which is the equivalent of the magnetization in this system. Again, the magnetization is the global alignment of this arrows, it's the sum of all these arrows is around zero and fluctuate around zero. But if I decrease the temperature which in our voter models means that people are more akin to follow the global trend. At some point you have a phase transition towards the ferromagnetic state which means that almost all spins will align together and you create a global magnetization. We can again read that from our free energy that I plot for different temperatures here as a function of the magnetization which is our order parameter. And you see that when the magnetization, when the temperature is large, it has a minimum in zero. So there is no global magnetization. We are in this region of the phase diagram or equivalent in this type of states here. Well, if I decrease the temperature at some point there are two minima that appears here. So it's here, I'm at the critical temperature and I see a global magnetization, a global alignment of the spins that starts to emerge which is the equivalent of this polarization phenomenon, this imitation model that we consider. And the same type of transitions also happen in a collective animal behavior. So let me show you some nice pictures. So here is a picture of huge colonies of insects what we call so active matter in physics where each individuals can make decisions by themselves but then you study huge colonies of them. This is a swarming state which is a kind of disordered state where individuals essentially do not follow specific direction actually. They look that they follow specific direction. This is more a polarized state, it's an ordered state. Disordered would be more like flies packed together with a center of mass that don't move much. And here is two other states of this active matter systems, these animals, here are fishes found in the so-called miling state where they kind of rotate around the center. And you have a polarized states here of birds which are huge colonies of starlings. I would like to show you a nice video. So here is a real video of these birds that form this kind of very complex colonies. And what is nice, what is important to notice is you see that this kind of system are kind of critical. What does it mean? They're really close to a phase transition in the sense that they are nor liquid nor solid. So they're kind of liquid because you have this very, you have kind of independent parts but at the same time you have a global coherency which is like a solid as well. So these systems are really critical. They are at the edge of a phase transition and this is not for nothing. These are, this comes from biological reasons of course. Okay. So this type of systems can be very simply modelled by this so-called, for example, V-shek model where each individual, what it does is that at each time it looks at its closest neighbors. What it does is that it approximately computes the average angles of all its neighbors and it tries to follow this direction. Of course the computation is not perfect and therefore we model that by adding some noise. So in equations, this is very simple. The angle in which the individual eye will move at time t plus delta t. So at the next step is essentially the average direction of its neighbors plus some noise which models the imperfection of the computation by this bird. So this term is a kind of energy term that will tend to create order. This term will contribute to align all the birds together to create a kind of magnetization for the birds, global alignment. While this term, which is the noise term can be found as an entropic term that tends towards more disorder in the system. Okay. So maybe I can show you a small simulation, this model. So this simulation was not done by me. So here they define an order parameter which is essentially the global alignment of all the velocities. So you sum over all the individuals in the population their directions essentially. And this is called the velocity correlation which I like to call the magnetization but this is the same. And so it is essentially zero if the motion of the birds are totally random and one when they're all perfectly aligned. And what they will do is that they will tune as time goes on the strength of the noise which reduces here. And you see that as the strength of the noise reduces this global alignment increases this correlation between the alignment and you indeed see that there is more and more structure in the movement of these birds. The beginning it was kind of totally random. And at the end we see a kind of global behavior where they follow each other which I think is really nice. All right. So let's now go towards maybe systems in which I think most of you wouldn't imagine that type of notions would appear but actually they do. And in particular I will start with problems in computer science and discrete mathematics. In graph theory and combinatorial optimization and I will tell you what it means. So the fundamental question behind that is a kind of basic question but at the same time extremely deep which is the following. Why some problems are easy to solve while others are not at all? So this is something very clear. But is there a fundamental reason for that and you must have a guess the fundamental reason between different hardness for some problems is related in some way to phase transitions and this is what I want to illustrate now. So let me introduce the father of what we call combinatorial optimization and graph theory which was a problem that people were wondering about in Konigsberg which is a city in Russia which is now called Kaliningrad in the 18th century. So the name has changed, it is here on the map. And so the game was the following. So this is the map of the city at this time. So there were an island and then there are two banks here and another island here. And there are one, two, three, four, five, six. There are seven edges connecting these different parts of the city. And the question that people were trying to answer is can we find a path that crosses all bridge a single time? And really what is important here is a single time. So people were just trying like this different solution. Okay, let us try, let us play a bit. Okay, so we missed one. Unfortunately, we cannot reach that bridge from here. So we didn't find a path. Let's try another one. We still unfortunate. We didn't find a solution. Okay, so people were trying thousands and thousands of combination and never found one. And actually the reason is that there is not and this has been understood by Leonhard Euler who was a mathematician. And what he did to answer this question is essentially develop what is now a very active field of discrete mathematics which is graph theory. So we took the map, define the scheme. So he simplified the map and then he put on each banks here on each side of the river and on the islands, a dot. Okay. And what he did is that he connected these dots if there is actually an edge on the original map that connects these parts of the city. And what you obtain is the first graph ever which is the following. And here, so for example, this node correspond to this one and you see that you have an edge connecting to this one and an edge connecting to this one. Actually you have two edges connecting to this one which are these two edges and so on and so forth. And Euler told us after some work if more than two nodes have a node number of edges there's no solution. You can not find a path that crosses all bridge at a single time. So this became, this has a name now, this problem of course in reference to Euler. This is can we find an Eulerian path? And so the question is the answer is very easy. Just look at your graph, answer the following. Is there more than two nodes that have a node number of edges? Here the answer is yes. We have one here for example, one here and one here. And therefore there is no Eulerian path. If the answer would be no there would exist an Eulerian path. So essentially we have a problem here that is easy to solve. We have a simple algorithm which is just to check that to answer the problem, okay? By the way now the map of the city had changed a bit. They essentially destroyed these two edges. So therefore now there exists an Eulerian path. So in 2020 the solution is actually trivial. All right, let me now discuss another problem that looks very closely connected but that is fundamentally different. She's the so-called Hamiltonian path problem which is the following. Can we find a path that encounters each node a single time? So before for the Eulerian path problem I wanted to path through each edge, each edge. Now I want to path through each node a single time, okay? And actually to the best of our knowledge the best solution is just to try all paths until finding one that works. So there is no smart way to find a solution to this problem. But the problem is actually that there are exponentially many solutions, different path, not solutions, different path to test until hopefully finding one solution if there exists one. But let's just remember all together again how fast exponentials are increasing. Exponential 10 is 20,000. Exponential 30, 10 to the 13, so 10,000 billions. So it's really a huge number. So forget it for bigger numbers. Actually it's easy to show that even if n would be of the order of 50, so n is the number of nodes in the graph for which you want to find an Hamiltonian path. It would take you more than the age of the universe on the modern supercomputer to test all path. Essentially you can just forget about it. This problem is not solvable efficiently. At least we don't know how to do. And therefore answering the question is there an Hamiltonian path is a very hard problem. So there are easy problems and hard problems. And hard in the sense here that we don't know efficient algorithms to solve that. And there is a name for that in computer science. These are problems that belongs to so-called P-class where P stands for polynomial. It means that there is a polynomial time algorithm. So an algorithm that makes a polynomial number of operations, polynomial in n, the number of nodes in your graph to solve the problem. NP means that there is no such polynomial algorithm. The only thing we know at best is an exponential time algorithm. So a very costly algorithm that usually you cannot run. And actually this problem because belongs to a special class of such complicated problems that we call NP complete class where complete here means that if you can solve efficiently this type of problem, again, we don't know how to but if you would be able to, you could solve an extremely large class of other complicated problems. Essentially almost all hard problems for a computer. So if you solve one, you solve them all. And you solve them all. And proving that these two classes are not the same in the sense that there actually is a class of problem for which there is no polynomial time algorithm. Because we don't know there is not, we just know that we didn't find any. Proving this inequality that these two types of problems are really fundamentally different. It's one of the most deep questions in computer science. If you can solve it, you get $1 million. And that's not a joke. This is one of the millennium prices. All right, so now I want to ask the following question. Are all NP complete problems, so essentially hard problems, are there really all the time hard? What do I mean here? These problems are hard in general. This is what we've seen. But sometimes they're actually very easy. Here is a graph. And if I ask you to find an Hamiltonian path or this specific graph, the answer is absolutely trivial because of the structure of the graph. Just start from a node and follow the graph. And you just found an Hamiltonian path. So this is very easy. So something must happen in between these two extreme regimes. You have a regime where the problem is really hard. We don't know any algorithm able to do something smart. And there is another regime where the problem is absolutely trivial. Okay, you give it to a two year old kid and he solves it. So something must happen to bridge this kind of two complexity regimes. And of course you have guests, probably the connection, what separates this kind of two regimes are phase transitions. So to illustrate that, let me introduce another such NP complete problem which is very nice. Which is the so-called coloring problem. So it's very easily stated, but at the same time very hard. That's why it's such a nice problem. Imagine you have a map. And I ask you, can you find a coloring which means a way to color the different countries so that two countries that share a border do not have the same color? Okay, there is actually a theorem which is the so-called four color theorem that tells you that four colors are enough to properly color any map. Properly color means again that no neighbors have the same color. So this has been conjectured a long time ago and it took more than a hundred year to be proved. And the algorithm to do that efficiently took additional 20 years to be obtained. So how to turn that into a graph problem coming from graph theory. So just add, just associate a node to each country. Then put an edge between any two nodes that share a border on the original map and you just got a graph. And now the four color theorem in this language reads the following. Four colors are enough to properly color any planar graph. So planar graph means that you can essentially draw it on a surface so that two edges will never cross. There are no crossing edges, okay? Right. So now we want to study how we go from simple instances of this coloring problem to hard instances which are really difficult to color. So to do that, we need to define a kind of ensemble of graphs, which will be random graphs. So how to generate a random graph? You just take a certain number of nodes, let's say 100 nodes, then you fix a number of edges, in this case, 218. And you just connect nodes randomly with this fixed number of edges, okay? So here is a representation of such a random graph. And the problem that we will fix ourselves here is can we color this random graph? It's just three colors. Of course, it would have access to infinities to as many colors as you want. The problem would be trivial. You just color them with a different one. Here we constrain ourselves to have only three colors. Notice that even if we would have four, it would be non-trivial because here this is not a planar graph. You see that things intersect a lot. So this graph, you cannot represent it on a plane, okay? Without intersection. So now we have this ensemble of random graphs. Let me define a control parameter that will allow us to tune the complexity of the problem. So our control parameter, like the temperature in physics, is here the density of edges, which means essentially the number of edges divided by the number of nodes, okay? That you can fix as the average number of borders between countries in the example of the map, okay? So when this thing is high, it means that you have very many small countries that have a lot of neighbors. When this is small, it means that you have essentially very few large countries and it's much easier. So a small C, the graph typically looks like this. You have many nodes that some are just separate of all the others. You have few connections like this. So it's very easy to find a coloring. While for large C, it's very hard to find colorings because you have many connections and therefore the nodes are highly constrained. If I put a color here, it means that I cannot choose the same color for all these guys, which is a hard constraint. All right, so now let me take the best algorithm that we know to solve this problem and run it on many instances or many different random graphs for a fixed average connectivity, which is my control parameter. And I do the statistics. I compute what is the fraction of random graphs that I can color with my algorithm as a function of this C. And what you will see if you try the experiment is that until some threshold, essentially your algorithm performs extremely well. It always finds solution. And suddenly you have a drop in performances like this and above this threshold, which is extremely sharp and that gets sharper as the number of nodes increase. We go from 50 to 100, 100 is the red curve. Above this threshold, essentially your algorithm is totally stuck, it won't find any coloring. And the reason is that actually deep in this region with high probability, there are no ways to color the graph. But here you have a region where still you find some, but you see that the computational time required to do so explodes seems to grow extremely fast like this. And this is nothing else than the appearance of a phase transition in a combinatorial optimization problem. The algorithm trying to solve this problem is experiencing a phase transition and therefore the problem in this region becomes extremely hard. While in this region, the problem is extremely easy and in this region, the problem is essentially impossible to solve. You don't have colorings for your graph. And this defines the phase diagram, exactly what I've shown for water at the beginning, except that we're talking about the behavior of an algorithm and you do have different phases for this coloring problem, okay? And this can be put in analogy with physics. Again, the easy regime can be thought as the liquid phase where essentially particles can move freely around. So a lot of degrees of freedom, it's easy to move things and to find colorings. While the hard and impossible regime, the graph is highly constrained. It's very rigid, which can be linked to the glass phase in physics. And actually it can be more than linked. The tools coming from the study of glassy systems in physics, in statistical physics, are really the ones that are appropriate to describe all this region of the phase diagram. Let me now discuss not a last another set of problems which I find fascinating and where again, phase transition will appears which are problems of signal hidden in noise. So this is called inference. And in inference, the problem is always the same. You have some hidden signals, some piece of information that has been corrupted by noise and your task is to recover the signal, okay? So let me discuss the father of inference problem which is the so-called communication problem which gave birth to what we call information theory in 1948. So the problem is the following. You have an emitter and a receiver that wants to exchange information that you can think as bits like zero and once. You can always represent information zero and once in the numerical world. So our dear Bob is trying to say to Alice that he feels kind of strange because he's blue, which is indeed strange. But unfortunately Alice is not hearing him properly. So what she asked very naturally is to repeat. Can you repeat the message please? And indeed, Shannon, who is the father of information theory who's an engineer from MIT, understood why it has to repeat is actually trying to communicate at a rate, at a speed if you want, that is too high. So let me explain what I mean here. So the communication problem, you have again this emitter and receiver that wants to exchange a piece of information. But there is no way, there exists no perfect communication channel. There is always noise, interferences. There is always some source of randomness that will corrupt the information that you want to exchange. And this can be modeled, for example, in the simplest case, but what we call in coding theory and communications, the binary erasure channel or BCP. What does it mean? This channel, which is a probabilistic model of information corruption, it takes, for example, as input a zero and with probability one minus P, it will output the bit without any error. But with probability P, which is typically not too large, it will output nothing. So it's essentially erase, it fully destroys the bit. And it is the same, it is symmetric with the bit one. So if you try to communicate through this channel, the fraction P, random fraction P of the bits would just be erased, fully lost. The question is, is there a way to robustify communication with respect to noise? Is there a way to communicate anyway despite the fact that there is noise? And actually, Alice, without realizing for the solution directly, there is a solution. Just repeat, you need to add redundancy, which is the key idea behind communications. So here, what you can do, for example, let's say you want to communicate yellow, which in binary form could be 001. Instead of just sending these bits like this and taking the risk of losing a fraction of them so that the message could not be decoded at the receiver side, just repeat it. So repeat it three times, for example. So here I just repeat my message three times. This repetition code are free. And what I call the rate is the number of information bits divided by the number of actually transmitted bits through the communication channel. So in this case, I want to transmit three bits and I send nine and therefore the rate is one third or equivalently the rate is one divided by the redundancy and here the redundancy is three. Okay, so it's very simple. And what Shannon understood is something absolutely fundamental, which is the fact that there is an absolute limit to the rate at which you communicate. So what do I mean by that? If you allow yourself a target error because there is no way to communicate perfectly, there will be always error. But what you can do is to try to minimize this error. So let's say you allow yourself a very small target error, almost zero. Shannon tells you that actually there is a rate above which you cannot communicate at maximum for this channel, for this eraser channel with probability 0.1 of erasing bits. You will never cross this communication rate. Okay, if you allow yourself a higher error, this curve is moving a bit. We don't see it on the graph. It's moving a bit on the right side. You can communicate at a slightly higher rate, but not much. And this is really a fundamental limit to communication. Above this line, it is impossible. And below this line communication is actually possible. By impossible here, I mean independently of any coding scheme. So for example, here we use the repetition code. The repetition code has a rate of 1 third, which is here, but it leads to a very high error. It's not, at the decoder side, you will get many errors. What you can do to lower the probability of error is to increase the number of repetition. This is five, et cetera, et cetera. And this is 61 repetition. So you see that we arbitrarily lowered the error, but the price we pay is that we tremendously reduced the rate of communication. So there is an absolutely fundamental trade-off between communication rate and error, probability of error, the fraction of bits that are lost at the receiver side. If you want to increase the rate, you need to pay the price of having more errors. If you want to reduce the number of errors, you pay the price of reducing the communication rate. And this is the best curve you can reach. So our dear Claude Shannon answers to our friends, Alice and Bob, whatever you do, and even if you add access to the most advanced alien technology that will ever exist in the universe, you will never be able to communicate at a rate exceeding the capacity of the noisy channel you are using. There is a fundamental phase transition that prevents you to do so. I'm really sorry. And so our friends are a bit disappointed and poor Alice is still not hearing very well apparently. All right. So let me discuss something that I find very nice, which is the strategy that nature has found to solve our problems. So what do I mean by that? We have this plot that I just showed you. So you observe that with our repetition code here, we still have, even if we repeat a lot, we still have a very large gap between the optimal curve, okay? The optimal trade-off between error and communication rate and what we actually reach with this communication scheme, this repetition code. So is there a way in some sense to close this gap? And nature has found a way to close this kind of gaps. So let me illustrate that again, coming back to water. Imagine that we have water at above zero degrees Celsius. You can do the experiment if you lower the temperature of this water slowly enough. And if the water is very pure, it will stay liquid even until minus 40 degrees Celsius. So the water is trapped in a so-called super cool state. So essentially you see that because you are in the liquid state, which is this minimum of the free energy, you see that when you decrease the temperature, the minimum is now here, but there is a barrier between those two states, which is here, but if you don't help in some way your system to jump over this barrier, you will just stay here forever, okay? You will get stuck for very long times and actually for infinite time if the size of the system is big enough. So your water remains liquid. But if you help the system a bit by punching on it, by adding a bit of energy locally, essentially you allow this ball to jump in the equilibrium state, which should be solid at the temperature, which is below zero degrees Celsius.  This water is at the temperature, which is below zero. And this crystal propagates along the system as a wave like this. And therefore at the end, thanks to this nucleation effect that starts from the seed of the nucleus here, the whole system finds the equilibrium. And actually, quite amazingly, we can use this ID to obtain error-correcting codes, so ways to add redundancy for communicating that allow to reach the Shannon capacity. And this is called spatial coupling. The idea is to create a code, a way to correlate our bits, a bit more smartly than just doing repetitions in a way that you create a kind of 1D chain like this. And the idea is that you have a kind of nucleus. So let me first say what are these boxes? These are called parity checks. These guys tells that their neighbors, so the bits to which they are connected, the sum has to be even. So essentially the emitter and receiver for communication, they agreed before communication that whatever signal will be transmitted, it will verify all these constraints. So for example, the bit number one, two, and three, the sum of them has to be an even number, okay? So now we send our bits that verify these constraints and at the receiver side, Alice, she receives this version. So some of the bits has been erased unfortunately. But you see we designed the code in the way that the first bit is connected to a single check. So even if it is erased, we know for sure that it has to be a zero because the sum has to be an even number. It cannot be a one. And now you see that now that we inferred that we reconstructed this bit, we can now reconstruct this one because I have a zero, a one. And if I want the sum of these three bits here to be an even number, this one has to be a one. It cannot be a zero. So this information, this seed helped me to reconstruct the next bit, which will help to reconstruct the next and the next and so on and so forth. And actually you get a reconstruction wave which is behind. It is exactly the same physics as this super cool water. And this is not an analogy. I mean, this is really the same phenomenon that appears. And you have a phase transition from this super cool state to the equilibrium state, which is the crystal. And in the context of this problem, crystal means reconstructing the information. So the information that was sent to this binary channel, the razor channel here was this image of a woman. And you see that you see a wave of propagation, a wave of reconstruction that starts from the first bits and that is propagated inwards the signal like this, exactly like in the water, which I find really beautiful. So thanks to this technique, we managed to fully close the gap between what we obtain algorithmically and what we can obtain optimally. And so Shannon is happy because Shannon told us there is an absolute limit, but he never actually gave us the recipe to reach this limit. And it took more than 40 years actually to get error correcting codes that are able to reach this limit. And what I just told you about, this partially coupled code is a class of codes able to do so. Actually there are only two classes of such error coding codes. Let me quote Richard Feynman who was a Nobel Prize from 65. The science of thermodynamics began with an analysis by the great engineer Sadik Arnault of the problem of how to build the best and most efficient engine. And this case constitutes one of the few famous cases in which engineering has contributed to fundamental physical theory. Another example that comes to mind is the more recent analysis of information theory by Claude Shannon. These two analysis incidentally turned out to be closely related. So Feynman of course has understood at this time that information theory and statistical physics and thermodynamics were extremely close fields of science. Let me discuss another example coming from machine learning and artificial intelligence. So in this example, I will discuss one specific field of machine learning which is called supervised learning. So the idea of supervised learning is the following.  So each image is what I call a data point and these data points are labeled which means that for all of these data points I also give you the answer, the category to which belong to this data point. So this bunch of data, of labeled data is called the training data. And what we want is to have an algorithm, a procedure that outputs from this training data a predictive model for new unseen data which were not part of the training data. What do I mean by that? Once we have our predictive model, what we want is that if I take a new example that is unlabeled, that was not part of the training data. So our predictive model never seen this image before. I want that our model is able to generalize in the sense that for this new data point it outputs the proper label, in this case a dog. So let's look at a cartoon version of this problem. So here is some labeled data. I have one cat and two dogs. So I train an algorithm that in this case we just try to find a linear, just a plane to separate the data. So let's say that this is what the algorithm output, this is our classifier. In a sense that when we are on the left side, the algorithm would tell you this is a dog. When we're on the right side, the algorithm would tell you our predictive model would tell you this is a cat. Okay, let's test that on new unseen data. Unfortunately, we have errors because here you see that this dog would really, that interface is not properly classified. Maybe it would be classified as a cat and this cat would be classified as a dog. So our algorithm is not yet very able to generalize. So now we give labels, so a bit more data to this algorithm to train it more. The algorithm is able to fine tune. This is our new predictor, our new model. Let's test that again, some new data. There are still errors. You see that this cat would be classified as a dog and this one is still at the interface. So our model is not yet perfect. So maybe by giving some more labels it will improve and indeed, now we have something that looks quite nice and if I test that again, a new example, it will properly classify it as a dog. This can be done through the so-called Perceptron Learning which is the simplest neural network that exists which is a very simple model of artificial neuron like we have in the brain. The idea is that you have a number of input neurons that will correspond to the input image. So these are the values of the different pixels of this image. So let me emphasize that this machine, this algorithm does not see this image. What it sees is really a huge vector of numbers and that's all. So it's a very hard task to make sense of this huge vector of numbers. For us, it is very easy to guess that it's a cat because we're trained for that. But for this machine, this is just numbers, a lot of numbers. So the task is not so easy. And what this machine does is that it takes these inputs and it projects that onto a so-called classifier which essentially can be taught as the synaptic weights of this machine. And here you have an activation function that will take the sign of the sum of these inputs. The xi are the values of this input neuron. So the values of the pixels of this image times the weights which represent the synapses here. You can think of that as the sign of the linear projection of our input onto the vector w which is essentially the normal to this line here. So this vector w parameterizes our classifier. And this sign, so the output neuron gives you this sign and this is actually the class that you predict for this data point. And what the perceptron learning is doing is from many examples, it tries to learn these weights, the synaptic weights in a way that it properly classifies all the data from the training data. And then what you hope is that if you use it on a new example, it will output the proper label. So if you try to study this simple neural network for a simplified data set, a bit simpler that is extremely complicated that a set of images of cats and dogs for a type of data for which you can really develop a mathematical theory and understand what happens. Then you can get formulas that tell you what is the generalization error that is the performance in classifying new unseen data of our perceptron algorithm as a function of the amount of training data, okay? So the amount of training data here is our control parameter is what we can play with. And this is our order parameter which will define in which algorithmic complexity phase we are, okay? And what you observe is again a phase transition. So if the amount of data is not big enough, if you look at the optimal algorithm and by optimal here, I mean that this curve has been computed thanks to a formula. We cannot actually run the optimal algorithm because it would take an exponential amount of time to be run. But fortunately in this simple model, we can compute an exact formula. We have a theorem for that. And this is this red curve. And you see that there is a critical amount of data where you have again a phase transition. It drops suddenly and above here you have a kind of perfect learning. It means that essentially you find exactly the proper classifier between the two classes. But before this transition, you don't have perfect learning. So this can be defined as a phase like in physics. And now let me test that again to efficient algorithms that I can actually run now on my computer. These are the black and blue dots. And what you see is that this fast algorithm, the black one is able to match the optimal performances except in this region here where you have a mismatch between the optimal algorithm which would lead to perfect learning and your actual fast algorithm which is not perfect. But if you give it a bit more of data then you have another phase transition and suddenly you enter this so-called easy learning phase where now you have a simple fast algorithm able to actually reach optimal performance. And in between you have again this odd regime like what we've seen in coloring, for example. Which is the kind of glassy phase of the problem. All right, so I'm essentially done. So I hope that I conveyed the main message which was that the variety of systems in which this kind of phase transitions and statistical physics notions can be applied is absolutely huge. And may a priori look like very different and separated systems, but actually they can be described into a single framework. So I've discussed physical systems, animal behavior, computer science problems, communications, inference of signals, financial markets and then votes. But I could have talked also about quantum systems, neuroscience, DNA and protein folding which is how proteins place themselves. The spreading of diseases in populations, of course, traffic jams, neutral breaking cosmology. And let me also emphasize that behind all that there is not only the physics and algorithmic aspects but there is tons of beautiful mathematical questions which actually I'm very interested in and I will stop there. Okay, so thank you very much for this beautiful talk. Maybe we can all unmute ourselves and first give you a round of applause. Okay, so here comes the difficult part. Now, if anyone has any questions, we just want not everyone to speak at the same time since we are almost 100 participants here. So maybe if you can either raise your hand or just write in the chat that you have a question and I can try to choose an order. So any questions? Maybe the diploma students. Okay, so yeah, raise your question. So, well, sorry for the pronunciation. Not a problem at all. Thank you, Jean. Very nice, very, very nice talk. Hi, what? Very clear. Yeah, very clear and lots of nice examples. I'll start just a question from backwards. I mean, taking the algorithm's question, you wouldn't know a priori if you quit, how an algorithm will follow that curve. If we look at your final slide of the black and blue algorithms, you wouldn't know a priori how it will behave. So the answer is, in general, no. If you gain, after working a lot in the field, you gain a kind of intuition in the sense in which, if I take the machine learning example, in which scaleings or the amount of data, the problem will be easy or hard. The precise numbers, of course, no. But in general, no. There is no recipe to get, to know in advance in which phase, algorithmic complexity phase you will be, without doing, actually, quite involved computations. And this is the whole job of the statistical physicist and of the mathematician to compute this free energy or these generalizations of free energy for these more abstract systems to actually get the phase diagram properly. So to follow up, so knowing that there is a phase transition in whichever system I'm working on, does not knowing it exists somewhere does not necessarily help me solve it? No, but it at least helps you to be careful in the sense that this kind of, let me take the example of the Shannon capacity, which I think it's very enlightening about this point. It's very important to know that there are fundamental limits and when you can quantify them, because imagine that Shannon would never got his theorem telling that there is an absolute fundamental maximum rate until which you can communicate. Maybe a full generation of coding theorists working on communications would have worked a hope, I mean, without any hope at the end to develop algorithms able to beat this limit that they didn't know about by improving algorithms, but actually there is no way to beat this limit. So it's very important to know in advance if your problem can experience phase transitions or not and if you can to try to quantify it because you can then just lose time developing algorithms to get performance that there is no way to attain. Thank you. Okay, so we have another question as well from Ralph Gebauer. Yes, hello everyone and thanks Jean for this nice seminar. And there's one thing which is not very clear to me, in this part where you talked about Shannon and communication and error in the communication, then you said one can reconstruct or correct for errors thanks to these constraints, which in your example are the sum of three consecutive bits must be even or something. So, but what happens if you try to use this and want to send a message which does not satisfy this constraint? Yes, so maybe I did not emphasize that enough. So what I said is that before communicating the receiver and emitter, they agree that the only possible messages that will be communicated will verify this constraint. So if you allow yourself constraints that generate a dictionary big enough, then you can map one to one, any of these messages verifying the constraints to a message that you want to transmit. So you can always find what we call a code book which is this huge dictionary of messages that verify the constraints to communicate. Okay, so the example of this image which was reconstructed, imagine the rules were different from just the simple sum. Yes, yes, yes, of course, of course, of course, yes. Thank you. Okay, so it seems there are no more questions at the moment. If you do, of course, you can still write. So I can seize this opportunity to tell you that we will do a little questionnaire here just to see if everything works well or not. I will start this little poll that takes 20 seconds soon and please answer that. It will be helpful before you leave the meeting. And then I'm also thinking that maybe there are some diploma students, for example, who didn't dare to ask a question in front of all the hundred participants all over the world, right? So maybe the diploma students can stay a little longer and they can get another chance to ask questions to the speaker. And will we get wine and food and snacks outside now or not? Yeah, this is what happens normally, no? This is something maybe you can take up in the poll and we will try to fix that for the next meeting. Okay, so I will watch this poll now. Yes. Okay, Fernando is suggesting a group photo as well. Let's see. Okay, so Jean, there is actually a question here from Syam saying his mic is not working. So how do you relate the second-order phase transitions to various systems is his question. Okay, so I think what Syam has in mind is that there are different type of phase transitions in the sense that the most common phase transitions are what we call first-order and second-order phase transition. So what it means is, so first-order phase transition, it means that the first derivative of the free energy, which is the fundamental object that allows to describe the thermodynamic and the global properties of the system, there exists a first derivative, which is discontinuous, okay? A second-order phase transition is when you have a second derivative that is discontinuous. In the latter case, when you plot the order parameter, like anything that I plot here, the algorithm, the performance of an algorithm as a function of the amount of data, for example, second-order means that it would be continuous. It would be smoothly varying. Well, the first order means that you have a discontinuity, like most of the cases I've shown here. There is no rule in the sense that in these systems, you may have the two types of phase transitions. I emphasize more the first-order type because they're more visual and in this kind of high-dimensional problems, this is more typical in the sense. But for example, in the biological systems, when we've seen this V-shek model where the correlation, the alignment between the birds was increasing smoothly as a function of the decrease of the noise, this was more a kind of second-order phase transition. But the two can happen. This really depends on the system. Okay. Very good. So now I guess we can ask to take this group photo that was suggested. So when I ask everyone to please switch on their cameras, switch on the video, and we will see if we manage to take this picture. Yes, with all the 60 plus participants that are still here. Suddenly it becomes a bit more... Yes. Yes, and now you can see everyone who was here. I suggested it, but I'm not quite sure how to do it. Yeah. Well, suddenly we don't all fit in one screen. So what I could do is just stay still and I take a picture one at a time with my phone. I think a mixture of technologies maybe... You can do a screenshot maybe also. Yeah, but I tried this and... Before it didn't seem to work. I can try that, but... Okay. Fernando, I can do it. No worries. Okay, Sabrina can do it. I think I see that there are three groups which come up. Yeah, we can do three. I will do... Well, the screenshot actually works. So I can do that too just to have... One... So people want to turn their camera on? No, not all. Bogdan, Banju... No, they're saying that... They haven't come around. If they are so kind... Sergio, Núñez, Sochog, Yikram, Dennis, who? Well, maybe they don't want to. So we don't have to watch anyway. I mean, those that are willing, it would be nice to turn the camera on. I'm on page two. I'm going to try another screenshot. Somebody has a penguin. So that's not real. All right, so... Oh, now we have just... Yeah. People left. Yeah, people just care about the science, not this kind of stuff. Wait, wait, one more. Okay, I think that I have them. Okay. Thanks. So thank you, everyone. Okay, so I think everybody is a diploma student.


In our last video, we explored the work of the first pioneering antiquarians of the modern age. We discussed how archaeologists Arthur Poznanski and Neil Steed, along with many other astute individual researchers, unraveled a possible key to unlocking the true purpose and indeed historical significance of the site. They concluded that the site, due to academia's reluctance to tag any ancient ruin with a date of more than 4,000 years, is the oldest ruin on Earth. The archaeologists discovered an alignment with the solstices and spring equinox, which only occurred around 17,000 years ago. However, there are many other intriguing areas of interest yet to be fully understood. Along with the volumes of photographic documentations of precise measurements, independent researchers also discovered that the site's gray stonework also possesses a curious magnetic property. The question is, if these ancient people knew of this interesting characteristic, what was the purpose of using said stone? Was the stone slowly magnetized by a technology once present at the site, now lost to history? Along with the gray stone, however, the site also contains an equal amount of red sandstone, which was used to build the site, yet this red stone does not share the enigmatic magnetism of its gray counterpart. Perhaps the sandstone is somehow immune to what was responsible? Perhaps this is why the Great Pyramids were built from sandstone to avoid the masonry taking on this magnetic charge? Perhaps, but I digress. Puma Panku is what we like to call a smoking gun, a site which clearly displays masonry skills of its ancient constructor, precision-cut stone masonry, which today could only be achieved with the use of advanced stone-cutting machinery. Shrouded in mystery, the archaeological side of Puma Panku is one of the biggest headaches for mainstream archaeology to explain. So how would an ancient culture, one which was far less capable than modern man, cut, shape, and transport from many miles away, carvings out of some of the world's toughest stone accomplished with such incredible precision? A group able to transport blocks of stone, sometimes weighing far more than 50 tons to the site, effortlessly placing them in position, often using a placement technique indicative of polygonal methodology? Interestingly, after investigating possible causes for this characteristic, it was realized that the building material was not granite, as it had long been assumed, but was in fact andesite. Andesite is the most iron-rich volcanic material we are aware of. It can contain around 15% iron oxide, and can have up to 4% magnetite. Thus, the stone, it seems, could have already been displaying magnetic characteristics when placed where it now lay. Yet the question still persists. Why did this stone get selected as the building material for the temple? It was built by a civilization that brought the stone from many miles away, so the suggestion that it was the only stone available would not be a logical conclusion. It was chosen by a group who were seemingly meticulous in their application. So it is quite possible that the magnetic characteristic was somehow utilized by the builders. Thankfully, it is only a matter of time before Puma Panku's secrets are fully understood, and we finally discover who built it. It is a place we find highly compelling. There are countless unexplained ancient enigmas, which can be found atop a remote mountain side of Peru, within a site only rediscovered within known modern history, now famously known as Machu Picchu, polygonal masonry. Although it is a technique found all over the world, the standard of polygonal masonry on show within the ancient sites of Peru is undoubtedly some of the most impressive found anywhere on Earth. However, as we have discussed in the past, with varying degrees of reaction to our discourse, ancient civilizations the world over, nearly all currently unexplained ancient ruins, no matter where they are found, has some form of celestial significance built into its construction. The controversial question, however, is why? Why do we see the monitoring and indeed celebration of solstices throughout ancient culture? And indeed how, were these precise alignments accomplished? Is into a hattana yet another astonishing relic left as a nod to the advanced knowledge of its builder, people who somehow constructed the clock, and indeed Machu Picchu, the surrounding settlement? Now tied to countless legends as to its origins, the most popular, however, is that In te huatana is, quote, the place where the sun becomes tangled, end quote. It was constructed with precise alignment, perfectly angled to face the four cardinal points. It is located at the top of the mountain, atop a structure adorned with seventy steps leading to its position. It is unquestionably an ancient upart, and is considered, by all who are aware of its existence, as a wonder of ancient technology. A solar clock, somehow created to indicate when it was precisely the winter solstice. It is, coincidentally, a time which is hugely significant to modern-day Peruvians. Known to the Incas as inti-raming, the long-held local celebrations seem to coincide with the purpose of the clock, once undoubtedly created using tremendous effort and knowledge. It is all in the effort to signify this same date, one which is officially the most important celebration of the entire empire. Is this pure coincidence? Or is this celebration a surviving tradition dating back to a forgotten antiquity? Incredibly, there are, in fact, two inti-huatanas or solar clocks within ancient Peru. Crane is located in Pesach, and the, as Fore mentioned, Machu Picchu solar clock, positioned on what is now classified as a sacred mountain. On September of 2000, a beer company was making a commercial when one of their cranes hit the solar clock. They unfortunately broke off part of the point and left the relic in a terrible condition. The National Institute of Culture, however, the INC, sued the company for damages in 2005 and was awarded an undisclosed amount. Who built these incredible ancient solar clocks? Why were so many ancient civilizations obsessed with the sun and indeed its activity and precessions? Were these ancient civilizations trying to tell us something? Why were so many ancient quarries left in a state of seeming abandon? As if the ancient people, once undertaking the movement and placement of ancient megaliths so huge, we today still cannot explain how they were moving them, seemingly vanished, right in the middle of said undertakings. Are these solar clocks, like the ancient quarries which still mystify all who explore them, a surviving clue as to the fate of this past civilization? They are undoubtedly highly compelling. The cylinder dredges up mud from the seabed in the form of long cores. The types of tiny fossils found at different levels in the core shows the sea temperatures of the past. Scientists have collected enough sea cores to form a detailed history of climate during the last million years. Dr. James Hayes leads the research. The climatic record in these deep sea cores tells us that there have been eight ice ages in the last 700,000 years. It also tells us when they have occurred.  In some regions in particular, displaying overwhelming masses of evidence supporting the posit of a past highly advanced builder, these areas often littered with displays of incredible ancient feats. Yet our next place of interest possesses some of the most incredible rock-cut chambers to be found anywhere. And just like that of the Giza Plateau, or the Inca trails of Peru, Turkey, along with its ancient counterpart Lebanon, still contains a smorgasbord of ancient uparts, mystifying masonry skills, and gigantic stone trilithons, all found within what would appear to have been a major settlement of this now lost civilization. The reason why we attest to many of Earth's ruins, having once been the work of a past now lost civilization, is the number of unexplainable features nearly always discovered at these puzzling ruins. Therefore, to understand that all the knowledge utilized to build such sites, the methods for lifting such stones conveniently forgotten, is to suspect that they were instead the work of an equally forgotten civilization. It appears to be a logical hypothesis to pursue, one which we indeed have been, which we have found bared much fruit. We believe we have now amassed enough evidence to support our claim beyond any reasonable doubt, subsequently discovering a far more fitting tale of events in regards to the true origins of many of the world's largest of ancient ruins. Hattusa is a melting pot of baffling construction techniques and surviving ancient artifacts. Within permitted timelines, the site predictably has a well-explored period of inhabitation, and any explanation as to how these more recent ancestors achieved its construction conveniently eludes modern academia. Hattusa was also known to have been the capital of the Hittite Empire in the late Bronze Age, added to the UNESCO World Heritage List in 1986. The main focus of this video, however, is in relation to a rather curious anomaly at the site, a unique object which could be described as an out-of-place rock cut. Known as the Green Rock of Hattusa, this mysterious stone's origin, or perhaps more importantly, its past function, is unknown. The green stone was once so perfectly polished, it originally had a mirror finish. Yet why this particular stone is here, why they chose this green stone specifically, or who brought it to the site, is a complete mystery. There are many other impressive features found at the site, including holes bored through massive megaliths with seemingly laser-like precision. It also has the ruins of what has been confirmed to have once been a Sphinx. All of these characteristics indicate that at one time in the very distant past, Hattusa was a place of significant importance. Fearing an uncanny amount of similarities in build technique and layout to many sites of South America, yet its green stone is unique to this site only. The question is why? Why was this green stone quarried, cut, place where it lay, then polished to perfection? Was the stone merely a gift from an Egyptian pharaoh, as modern academia would have you believe? Or did the green stone of Hattusa once serve a more profound purpose? One day we will find out. It is an object which we find highly compelling. Chufut Kale is an astonishing ancient city, argued by mainstream historians as dating back only to medieval times. However, this we suspect is possibly due to it being documented during this time as having served as a fortress within the Crimean Mountains. A national monument of Crimean Caratus culture, although argued to be fairly recent archaeology, we believe, however, that upon closer inspection, a far more intriguing, far older story for its origins begins to appear. Identifiable advanced features so often discussed here on our channel, found throughout the ancient world, remnants indicative of a far older, far greater technologically advanced group having once been responsible. A group who had access to knowledge and stone cutting technology that has not only left the same signature scars all over the planet and investigative undertaking we have previously explored regarding similarities and differentiations to identify signature methods, an ongoing investigation in an attempt to identify separate lost, highly advanced civilizations and, most importantly, how many there were. The site was clearly built by a group who were also capable of quarrying and placing enormous stones sometimes over 1,000 tons in weight atop one another. Many people are aware of the remarkable underground city known as Derinkuyu, a settlement we have covered in the past. A gigantic ruin carved straight down 80 meters into the bedrock, rumored to have been lit using a pipe system which teased natural gases out of the strata, lighting a candle-sized flame at regular intervals throughout its entire structure. We know of this amazing ruin due to its incredible rediscovery. When a local purchased a property in the area, he set about doing some renovations. However, upon knocking down a wall in the house, he was confronted with the entrance to an amazing site. This in turn attracted the media, thus Derinkuyu's public popularity, regardless of its controversial nature, was cast in stone, with rather difficult to explain origins. However, Chufut Kale, although itself a partially underground, partially above ground, yet no less incredible ancient city, also carved with incredible precision from the bedrock of Earth. The site was discovered and studied in depth initially by mainstream-funded individuals. As such, predictably, it has subsequently been little shared publicly. This regardless of its incredible nature, its prehistoric appearance, and the fact that it even appears to be older than its more famous counterpart, Derinkuyu. The site is largely ignored, and this is undoubtedly due to the institutionalized powers that be who constantly monitor and thus control the financial incentives. In requesting exploratory funding, whether within such fields as education, archaeology, or history, will simply be refused any application for a funded expedition. This reluctance to approve any in-depth public exploration of the site has long kept the lid on these ruins, and we feel for good reason. For although Chufut Kale was once masterfully carved from solid stone, created to house many hundreds of families and individuals, the erosion present on the site is also staggering. Many of the once refined chambers are slowly losing their form and returning to nature, with some caverns seemingly identical in appearance to many sites academia would simply dismiss as natural formations. Yet regardless of this dismissal and the deliberate overlooking of its grandeur by certain fields of study, we find Chufut Kale highly compelling. We have in the past covered countless ancient anomalies found amongst the many ruins of ancient Peru. Hillside fortresses, mountaintop sanctuaries, completely self-sustaining technologically advanced group whose ruins still contain countless as yet unexplained methods of construction and often incorporating inexplicably large megalithic blocks once quarried, transported, and then somehow seemingly effortlessly placed atop one another. Masters of architecture, irrigation, stonework, and horticulture, this group, although claimed to have been that of our far less capable recent ancestors, the Incas, built self-sustaining earthquake-proof settlements high among the clouds. Sites often build at altitudes far higher than 2,000 meters above sea level, with these ancient once indigenous builders, also one installing simple yet incredibly effective gaps in the pathways to such sites as Machu Picchu, allowing the inhabitants to draw the bridges to the site, cutting it off from any possible invaders. Once these bridges were removed, sites such as Machu Picchu became virtually impenetrable. We have previously covered many incredible Peruvian ruins. The Indihuatn, for example, is yet another relic we recently covered here on the channel. It is yet another example of this now lost civilization's past knowledge and extraordinary now lost capabilities. A solar clock precisely bored into being directly out of the bedrock of Earth, which precisely indicates the solstices. We discussed how certain characteristics of many ancient sites, most notably the apparent Mayans masonry, Incan, and Neolithic sites, such as the Stonehenge within the UK, all display a past obsession with solar precisions. Furthermore, the constructors of these sites all displayed an uncanny urge in particular and undoubtedly most prominently at the site of Machu Picchu to undergo a mammoth undertaking to create what now appears to have merely been a quirk of engineering, entwined within the architectural planning of Machu Picchu itself. It is often perceived as overkill, so much polygonal masonry is present virtually everywhere could be laid. Perhaps these efforts of stoning up literally every crevasse at the site, regardless of whether it would be on public display or not, may have merely been due to a purely aesthetic obsession by a once highly capable now lost civilization. And who must have perceived such, as yet unexplained tasks, as child's play? The incorporation of natural geological features into the sites is yet another curious characteristic of Machu Picchu, which many individuals who visit the location are perplexed by. It would appear that the ancient civilization responsible for this incredible site's existence, like a number of the other sites we have covered previously, incorporated the living rock of the mountains into the construction plans of their past sanctuaries. Rather than have simply carved them flat, many ruins display a collaboration of such natural stones into the buildings themselves. The Temple of the Condor is one of these incredible examples. A natural rock formation, which was formed millions of years ago, was spared destruction and was incorporated into the building of the site, subsequently becoming a place of worship. Many believe the Temple was a pilgrimage of religious worship. The Masons who manipulated the Temple of the Condor into the site skillfully shaped the rocks below the main man here, into the shape of outspread wings, of a bird, largely believed to be that of a depiction of a condor in flight. According to a number of studies of the ruin, upon the floor of the Temple is the carving of the condor's head and neck feathers, flowing up into the body, which is the natural formation we still see today. This completes the posited figure of the three-dimensional bird. The Temple of the Condor is undoubtedly one of the most spectacular examples of what these so-called pre-Incas were once capable of. Like so many other ancient sites found all over the world, share so many characteristics with ancient Peru. The question is why did the builders of all these sites go to such great efforts not to displace or even incorporate seemingly common rocks into the build of the sanctuaries? Who were the builders of Machu Picchu? Were they a world-faring civilization? We find the evidence to suggest such highly compelling. There are many ruins of ancient antiquity still in existence, which tell a vastly different tale for the history of human civilization than that of its contradictory counterpart, namely modern paradigm. One enforced, funded, and massively supported. A tale of events merely written by a victor. However, thanks to our ancient ancestors' incredible abilities, many of their ruins and relics so well constructed, often made from complex, elaborate, or gigantic materials, can tell us all what a well-funded modern academia simply cannot explain, thus is staunchly rejected and ignored in favor of the illusion of an authority over our species akin to that of an oracle. One of the most popular and predictably easily argued of ancient sites for our case of events, our conviction in the belief that there is a hidden history of man and indeed earth. One of a far greater antiquity for our species, with many now lost civilizations having once come and gone. The site is additionally one of the largest ancient anomalies of our planet. It is of course the Great Pyramids of Giza. Due to the structure's immense sizes, the evidence for several conservation efforts in the form of casing stones littering their four-sided facades. The sheer size of the stones involved in the bill, the amount of stone used, the precision of the once hermetically sealed, constellationally aligned ventilation shafts, and the many different stages of erosion present on these varying sections of the construction efforts, indeed makes the Great Pyramids a great avenue of study, and one of strong conjecture in the face of an adverse and conspiring academia. Our hypothesis, we feel, due to our compiling of the many inexplicable factors of antiquity, has now all but been virtually proven beyond doubt. For our opinion, regarding the true history of earth, is not just based upon the objection of the illogical explanation, widely given for the origin of the Pyramids of Egypt. It is also based upon many other areas of ancient history, which not only support our claim, but if one sticks to mainstream historical ideology, we'll simply find the task of explaining these ancient uparts simply impossible, they simply cannot exist. And our next artifact of interest is no different. Like King Tut's dagger, the Beth Shearim glass slab. Found in Switzerland by amateur archaeologists, it is known as the Hand of Preals. Not only is it the earliest, most elaborate example of bronze and goldwork ever found, but quote, we do not know either the meaning and the function attributed to it. Its gold decor suggests that it was an emblem of power, a distinct sign of the social elite, even of a deity. The hand is extended by a hollow form that suggests it was originally mounted on another object, perhaps as part of a scepter, end quote. Who made the Hand of Preals? Is it really 3,000 years old? If so, how can it be the earliest example of this form of metalworking yet be so elaborate in nature? The Hand of Preals is unquestionably an ancient upart, and could quite possibly be a surviving relic of a once highly advanced yet now lost civilization. We feel this hypothesis is far more logical, and as such, makes the Hand of Preals highly compelling.


Let's try and learn about the multiplication of two numbers with the same base of course when the supplements have different signs basically. So let's take two numbers like 98 times 104 now how we can perform this multiplication again we first need to identify the base of both numbers and we require that the base is same for both the numbers and then we proceed on writing these two numbers in a vertical fashion and then writing the supplement of it the supplement of 98 is minus 2 since it's 98 minus 100 and the supplement of 104 is plus 4 now how we can perform this multiplication note that 98 and 104 are on the left hand side minus 2 and plus 4 are on the right hand side let's quickly draw the horizontal line and now we are ready to find out the multiplication the result that we are going to get has two parts to it one part is combining the supplement and the other number the other part is the multiplication of the supplements let's first take the combination of a supplement and the opposite number so which is 98 plus 4 or rather 104 and minus 2 in both the cases we will get 102 because the base is 100 with two zeros we will add two zeros after 102 next thing is to multiply minus 2 and plus 4 and the result that we get is minus 8 so basically what we want to do is that we combine the two results and now we have to combine this number and this number here let's do that 102 with two zeros is basically 10,200 plus minus 8 which is basically minus 8 itself and the answer that we get here is 10,192 and so the multiplication of 98 and 104 is 10,192 let's take one more example so let's say we wanted to multiply 991 and 1014 let's quickly identify the base here the base of 991 will be 1000 same thing for 1014 and now we will write these two numbers in a vertical fashion 991,1014 and let's write the supplement for both the supplement for 991 is going to be 991 minus 1000 which is minus 9 let's write minus 9 here and supplement for 1014 is going to be 1014 minus 1000 which is plus 14 let's write plus 14 here we can complete one part of the result which is combining a number and the opposite supplement so 991 plus 14 which is going to be 1005 and then because the base is 1000 we need to pad three zeros in here then for the next step we want to multiply the supplements which are like this so minus 9 times 14 which is going to be minus 126 so let's put a minus sign here and we'll write 126 the final result will be this number minus 126 which is going to be 10,4874 let me quickly write the concluding result 991 times 1014 equals 10,4874 note that we have done subtraction because the signs of the supplements are indifferent


This algebraic geometry lecture will be about elliptic functions and we will be using them to show that several cubic curves in the plane are not birational to the affine line. The previous lecture proved this using an algebraic method, so this lecture is going to be different because we're going to use an analytic method. So we have to recall the definition of the Weierstrass p function, p of z. This funny squiggle is actually a letter p. It's some sort of weird calligraphic letter p from some 19th century calligraphic alphabet and as far as I know every letter of this alphabet has been lost except for the capital p which was used by Weierstrass for his function and has been copied by mathematicians ever since. Anyway the Weierstrass p function is an elliptic function. So an elliptic function is one such that f of z plus lambda is equal to f of z for all lambda in some lattice l in the complex plane. So a lattice is just something looks something like this. So if this is nought and this is a and this is b then the lattice consists of all integral linear combinations of a and b. So for example a plus b to a and so on. So this function has two periods a and b. Now the function cannot be holomorphic unless it's constant because it's if it were holomorphic it would be bounded in a fundamental domain and therefore bounded everywhere so it has to be constant by Lieuville's theorem. So f has to be meromorphic. It must have poles somewhere. So how can we find functions f with this property? Well one obvious way of finding it is to take any function g and just sum over all lambda in the lattice. Seem to be confused about whether I'm calling it l or lambda. So we sum over all elements of the lattice lambda and if we define f of z like this then it's completely obvious that f of z plus lambda will equal f of z. So what do we choose for g? Well it has to be something with a pole otherwise sorry it has to be something that converges. This means it must decrease reasonably faster at infinity. Well let's try g of z equals 1 over z. Well in that case it's easy to check that this doesn't converge. See the number of lambdas with absolute value less than some constant c grows quadratically in c and this only decreases linearly in c so that's not good enough. If we take gz equals 1 over z cubed this converges well. If we take gz equals 1 over z squared this is borderline by borderline I mean it doesn't converge but it only just fails to converge by the sort of minimum possible amount it could fail but like. What this means is that if we modify this very slightly we can actually get it to converge. So the vice-trice p function is defined like this. We take p of z to be 1 over z squared plus sum over lambda not equal to zero of 1 over z minus lambda squared minus 1 over lambda squared. And what you see we've done is we've just added a sort of constant and the constant is doubly periodic so in some doubly periodic so adding all these constants shouldn't really affect the fact that this is doubly periodic. And the nice thing about this constant is that it's the constant term of this expression here. So z is large then 1 over z minus lambda squared is approximately 1 over lambda squared plus even smaller terms. So what we're essentially doing is adding this sort of this constant sum of lambda does not equal zero of 1 over lambda squared to everything. The trouble is this does not converge so this is not convergent but we're sort of adding this non-convergent expression to this non-convergent expression and producing a convergent expression. So we've got a perfectly well-defined function. There's one slight problem that since we've twiddled everything by adding in this constant it's no longer clear that this is doubly periodic. However this is not very difficult to show because if we take the derivative of this then we're essentially summing all the translates of 1 over 1 plus c cube which is convergent up to some constant derivative of this is 1 over z cube not to a constant that should be a 3. So the derivative of this is doubly periodic and this means that this function here is doubly periodic up to a constant. So we find that rho of z plus a is equal to so not rho p of z plus some constant and this constant must be zero because this function here is an even function. So the fact this is even implies this is constant. So this slight change the definition to make this convergent fortunately doesn't affect the fact that this is still doubly periodic. You've got to be a bit careful with this argument because there are some quite similar functions where you do this trick and adding in this constant actually stops it from being periodic but anyway we've got a nice doubly periodic function. Next we have a look at the Laurent expansion of p by strice p function at z equals zero and this isn't very difficult to work out it's 1 over z squared and the constant term vanishes so we get naught times z plus something times z squared and we don't really care all that much for what this is. So we can look at the derivative well this is going to be minus 2z to the minus 3 plus again the constant term is going to vanish and we're just going to get a linear term something in z. Now we take the derivative and square it well this is going to be 4z to the minus 6 plus something times z to the minus 2 plus something times z to the 0 plus something. On the other hand we can take p of z cubed and multiply it by 4 and this will be 4z to the minus 6 plus something times z to the minus squared and so on. Now you see these terms cancel out and we're just left with terms in z to the minus squared and a constant term so what we can do is we can take rho prime of z squared and this will be equal to 4 rho of z cubed plus something times p of z I'm sorry I keep calling this rho I should be saying p so we take something times p to cancel out the term in z to the minus 2 and then we take some constant to cancel out the constant terms and then plus something that vanishes at z equals 0 so this is doubly periodic and it has no poles and vanishes at z equals 0 well if it's doubly periodic and has no poles it's bounded in a fundamental region so it must be constant and since it vanishes at 0 it must actually be 0 so this expression here must actually vanish so we find that bias stress p function satisfies the following differential equation plus some constant times rho which is traditionally called minus g2 for complicated historical reasons that satisfies this equation here okay well what on earth is the point of that and the other question is why these things called elliptic functions when they seem to have nothing to do with ellipses well answer the second question first because it's a little bit easier the reason why these things are called elliptic functions is if you work out the arc length of an ellipse it ends up with integrals like integral of dx over the square root of 4x cubed minus ax minus b so this is arc length of an ellipse so these integrals are called elliptic integrals now if you look at the denominator of this it looks very similar to this bit of the bias stress p function in fact you find the bias stress p function as an inverse if you sort of invert it it's you get the inverse function as the integral from something to p of the integral of dp over the square root of 4p cubed minus g2p minus g3 which should be a g2 so the inverse of this elliptic function p is more or less given by this elliptic integral and um someone i'm not quite remember who first noticed that these things are doubly periodic but the inverses of these functions are doubly periodic so doubly periodic functions end up being called elliptic functions as you can see the connection with ellipses is very round about and it's really rather dreadful terminology but it's probably too late to change it anyway now let's go back to algebraic geometry and look at the equation 4y squared equals x cubed minus g2x minus g3 so this is some sort of um cubic plane curve might look something like this so here we've got a some sort of affine plane curve and um if you look at the equation differential equation for the bias stress p function you see it looks very much like the equation for this cubic curve if we put y sorry put the four in the wrong place four should be there if we put y equals the derivative of the bias stress p function of z and x equals p of z then we see at the point x y lies on this curve so what's this saying it's saying that we've got a map from the complex numbers onto the set of onto the curve y squared equals 4x cubed minus g2x minus g3 which takes a point z to um p dash of z p of z well that's not quite true because this is actually not defined when z is in the lattice so this should really be c minus the lattice l and it maps this onto the curve well it's a bit annoying having to remove the lattice points so we can actually get a map from c to the projective curve where you remember we we turn this into a curve in the projective plane and just add a point at infinity and of course this maps the point zero to the point at infinity of this projective curve well this is a doubly periodic function so we actually get a map from the complex plane modulo the lattice l to this projective curve and it's not too difficult to check this is actually a homeomorphism of topological spaces now this projective curve is very nice if you take c and think of it as being r squared modulo a lattice that's just r modulo a lattice times r modulo a lattice so this is isomorphic to s1 cross s1 as a topological space in other words it's just a torus which you can picture like this so over the complex numbers this curve if you add a point at infinity is homeomorphic to a two dimensional torus and this implies that it can't be rational because any rational curve is the same as p1 up to a finite number of points and p1 over the complex numbers is just isomorphic to a sphere s2 so we've we've just got a a nice sphere there on the other hand the curve y squared equals 4x cubed minus g2x minus g3 over the complex numbers this is topologically isomorphic to a torus s1 times s1 and there is no way to turn a torus into a sphere by removing a finite number of points and adding a finite number of other points now if we had a rational map between these two spaces the rational map would be defined except to define that number of points and it would have an inverse that was well defined except to define that number of points so any two curves that are birational are the same topologically if you're allowed to add and add and subtract a finite number of points so these two curves are not birational um by the way I just just make a quick comment um so we've constructed um a doubly periodic function by taking a sum 1 over z minus lambda squared um except we're not retaking this because it doesn't quite converge but we can make it converge by um sort of fiddling around a little bit you might wonder why don't we do this with singly periodic functions well you can take the sum sum of 1 over z minus lambda and this does not converge well it's not absolutely convergent but it's so close to being convergent that it's very easy to make it convergent in fact you can make it convergent by summing it carefully in in um by carefully summing over lambda in order of absolute value and then it becomes conditionally convergent but not absolutely convergent and this is a periodic function it's not difficult to figure out which it's actually pi times the cotangent of pi z um when people are doing trigonometry they don't often mention this I think because you do trigonometry at um high school or something and at high school you're not allowed to talk about complex numbers um this actually gives rise to the product formula for the sine function because you can write sine pi of z is equal to z times a product over n not equal to zero of 1 minus c over n well you've got to be a bit careful about this product because that sum is not convergent and this product is actually not convergent either unless you're a little bit careful about the order um so what's the relation between these well if you take the logarithmic derivative of both sides then this identity here becomes this identity here um so um the analog of the via strice function is really just this function here incidentally this product formula also has an analog for the via strice function where that's that's getting to the very interesting theory of um theta functions but that's not strictly speaking part of algebraic geometry so I'll stop there for the moment