The last day of the conference people have been exhausted and it's been partying. Okay.  Yeah, at lunch. So now you're officially a member again of Texas? Yeah. Since yesterday. When you go back? In August. Okay. They start teaching in August too. Now that's early. So, what do you think? Wait five minutes? Whatever you like. I mean, I'm flexible.  Yeah. Is this still a problem? Yeah. Okay. Okay. So welcome to the last morning of the conference and to the last lecture of Michael Los. Thank you, Francesco. I wrote a few things down which we talked about last time. So the first model here is this thermostat model where you have a function of m variables. That's your initial condition. And you have a mass equation of the following type. You have this m particles interacting with each other. That's according to the Katz model. Totally standard. That's this piece. Lambda is just a constant. Okay. And then you have an interaction of these particles with a thermostat, right? Which is an infinite. It's an infinite reservoir, thermal reservoir. And the idea there is you draw randomly a particle of this infinite reservoir, which is in a thermal equilibrium. So, as a physicist, you would say you sample this reservoir, which is distributed according to a Maxwellian distribution with temperature beta. Beta is 1 over kT, so that's given. And then you scatter this particle with your particle of the system. And then this particle of the system changes somehow the probability distribution. But the particle which you drew out of the infinite reservoir you discard. Okay. So the infinite reservoir never changes. It just stays in equilibrium for the rest of its infinite life. Okay. So that's the idea. And then, what you can prove, this was a theorem by Bonnet, the Weideer, Natan and myself, is that when you take the relative entropy, so the gamma m is just the Gaussian function. I call a Gaussian function gamma m. And because I don't want to constantly write Gauss, right? And by the way, here, I've chosen beta to be 2 pi in order to simplify life. I don't have to carry around this normalization factor. Okay? It makes life easier. Okay. So now you look, I call it the thermal entropy, because what is it? It's relative to the thermal state. And that turns out to be the equilibrium state. In fact, we can show that the thermal entropy decays exponentially fast to zero, and mu rho is given by this expression here. And you notice it only depends on this mu, which is the interaction with the thermal state. Of course, that's to be expected, because the Katz model really doesn't have a very good entropy production. We have learned this from Wilhelm's theorem and for this counter example of Amit. Okay? Or this example of Amit. Enough. Okay. So this is the infinite thermal state. Now you say yourself, okay, let me put myself in a finite reservoir. So I have my systems coordinate V1 to Vm, and then I have the reservoir coordinate, that's Wm plus 1 to Wn plus m. Okay? So what do you do?  This time, of course, it's a master equation which acts on functions of two variables, V and W. And my initial condition is, again, the same initial conditions that I had here. The F0 is the same, but the reservoir I put in the thermal state. Okay? So that's what my initial condition is. And now you'll let this thing run. And lots of interesting things happen. The particles themselves among each other, they collide according to Katz. The reservoir particles also collide with each other. And then you have this interaction of the systems particles with the reservoir particles. That's this interaction here. And that's, again, just a collision. And note this collision rate has been chosen in such a way that when I take a particle of the system, that the rate of collision with any particle in the reservoir is constant, independent of n. The collision rate of a particle in the reservoir with any particle in the system is, however, a rate which is m divided by n times mu, which is very, very small when you think of n as very, very large. So we think of n as much, much larger than n. Okay? Good. So now there was a theorem of Ponetto, Tussonian, and Vaidyanathan and myself, which says the following. And that's what you would expect in a certain sense, namely that when you take the time evolution of this system here. This is this function. Now you multiply by a Gaussian, which is thermal. And now you take the time evolution of that system, which in some sense has the same initial condition as this one, right? Then the difference, this distance, which is called the Gabetta-Toskani-Wenberg distance, is bounded above by m divided by n, and this constant is roughly of order one. And this is uniformly in time. So in other words, when you choose n very large, this system here, which is very complicated, is described by that system, which is fairly simple. Uniformly in time, right? So these two things keep close together for ever and ever and ever. Notice, there's a funny thing about this. What's the thermal state here? The equilibrium state. The equilibrium state is a state, of course, which has to be rotation invariant. After all, what you do? You have here averages of rotation, averages of rotation, and averages of rotations. So when you start with this state, all you have to do is you average this state of all rotations in Rn plus m, in this huge space. What you get is a function, which in general is not a Gaussian. Okay? Nevertheless, this inequality is still true because Gaussians in very large dimensions and spherically symmetric functions somehow are very, very close. You can estimate this. So the spherical average of this function, and when you replace this by a Gaussian, these things are very, very close in this distance. Okay? So that's somehow the description of comparison. Now here's the question. So we know that in the thermal state, in this system here, the entropy decays exponentially fast. So far we have never had any clue what could be the entropy for this system. So what do I do? You take your solution of this large mass equation, which is terribly complicated. Now, you agree. You are not interested in the particles which are in the reservoir. That just stays there. They move around. So what you do is you take the marginal of your distribution with respect to w. That gives you a function.  It's of course not the same solution as this one. It's not. But let me call it f of vt, otherwise I'm running out of letters, okay? So what would you expect? You would expect that the entropy should also decay. Well, it shouldn't decay to zero. Why not? It's not the Gaussian. You see, this entropy is always with respect to the Gaussian. But still, what you would expect is it should decay relatively fast and to become relatively fast small, or small, relatively fast, I should say, right? Okay? That would be reasonable. Why? Because you say this system seems to be in this metric well-approximated by that system. This system has fast entropy decay. Yes, this entropy should also decay, maybe not to zero, but it should get small fast, okay? So, and here's a theorem. And this theorem has not been published yet, but it's about to be written up here. Bonetto, Geisinger, Reed and myself. So Geisinger and Reed are two German graduate students. They visited Georgia Tech and during that time we cooked up the following theorem. S, thermal of F dot and T is less than equals. Okay? So remember, what do I do? I take the solution, I integrate against D w and I plug it in here and I divide by gamma m, which is the Gaussian in m variables. Okay? So remember, the idea is that should become small fast. Here's the theorem. This is less than equals m divided by n plus m. Well, that's already good news, right? Because when n is very, very large compared to m, this is tiny. What's left? Plus n divided by n plus m. Well, that's really bad news, but what you have is e to the minus mu rho times, I have just to get the exponent right, times the entropy of your initial condition. That's the result. So you see, it's kind of interesting, right? The mu rho is precisely this number here. So what is this telling you? It tells you when you take the, and by the way, I mean, this is a real honest constant. This is really a one here and that's it. I mean, that's the computation, okay? So now you see, when you let n do infinity, what happens? This goes to zero. This goes to one. This goes to this exponent here, which is one. So that's exactly, you replicate precisely that result. Now, we know, Vaidya Natanranjini Vaidya Tan is a very smart lady. She really figured out that this was the best possible power here. So therefore, and that's non-trivial, huh? She finished. And therefore, you know that this kind of being proved is right. So in a way, it's kind of surprising that you can get for such a complicated system, sharp rates, right? That's kind of surprising. And now for the remaining time, I would like to explain to you how you might go improve something like that, okay? And things can get a little bit complicated. So what I'm going to do in my talk is I give you first steps how to approach this. And when the formulas get large, I will use slides to explain the rest of that to you, okay? All right. So now let's compute. So you do something outright crazy. What you're going to do is you take this L. Remember, you have to compute E to the LT. And the way you do it is you just expand it in terms of a power series, right? Brute force. This is what physicists do, right? And then you think afterwards what you can do with it. So what is it? So the E, so let me set up the L once more. Notice when I here, here I always subtract the identity, right? And I can actually, when I wipe out the identity here, when I wipe it out, then of course I have to correct things again. And by the way, the N here in this theorem, the N just has to be bigger than M. It doesn't have to be large. It's still true, okay? So then what you have to do is you have to subtract a certain constant lambda. And what is this lambda? It's, well, lambda S divided by M minus 1. How many terms do you have in here? M choose 2, right? So that's just lambda S over 2. What is it? M plus lambda R, N over 2 plus. And how many terms do you have? Well, here you have N, which kills this, so you have mu times N. So this is my lambda, right? And then it can go and expand the power, this thing in a power series. And let me be very generous, because you see that's the nice thing about the video. I can now write the formula, afterwards it can wipe things out and fill things in, because it's recorded, right? That's kind of nice. So I don't have to repeat the formula constantly. Sum K equals 0 to infinity, lambda T to the K divided by K factorial. And then you have a sum over lambda alpha 1. I explain what these things are. Alpha 1, alpha K. And then you have R alpha 1. Okay? So what did I do? I looked at my L. Now let me sort of expose it a little bit better. Forget about this. I'll wipe this out, but we don't need this item. So here's my L. So what you do is, I have an E to the minus lambda T, which I forgot. And then I have a lambda T to the K. What do I do? I pull out a lambda in front of the L. So I have lambda S divided by M minus 1 times lambda. Sorry. Yeah, that's right. Then I have a lambda R divided by N minus 1 times lambda. And I have a mu divided by N lambda, right? Let's just pull this out. Now what's nice about these numbers? When I sum these numbers, right, here, here and here, I get 1. These are convex weights. Okay?  You have M choose 2 times this. M choose 2 times that.  And now I divide by this total sum. These are convex weights. Okay? And these are these weights in here. And what do I do? I have to sum here over all possible pairs, which show up in the collisions. Okay? Right? I mean, there could be a collision which corresponds to these guys, collisions which corresponds to those and those. And they come up in all possible combinations. And that's what my sum is. Okay? So it's a little bit, well, a little bit. It's just terrible at the moment, right? Okay? Good. And now what you do is you apply it to a function. And then you have to ask yourself what these R alphas are doing. Well, what are they doing? So let me make a certain simple transformation first. It's always convenient to write things in the following way. I define this H0 in this fashion. Or H, sorry, just H. Why am I doing this? Look, when I go to these variables, that means that my function f of V, yeah, let's make H0. Let me write it this way. f0 of V is then just a Gaussian times m of V times H0. Okay? Now let's look at our initial condition. Our initial condition was f0 of V and W is this function times the Gaussian n times H0 of V. Agree? That's what it is. Right? Because my initial condition here was given in this word. So now what do you realize? You realize that this product here is a Gaussian m plus n. So this is a radial function. A radial function, the cuts model L, when you apply L to a radial function, well, you don't see it. It's invariant. It doesn't move, right? Because after what is cuts, cuts just averages over rotation. Rotational invariant functions don't move. Okay? So what does this mean? It means that when I stick in here my f0 of V and W, I can replace this f0 by H0 times of V. Agree? Times this Gaussian, but this Gaussian I can push through and pull it out here. Agree? That's what you get. Now, what am I supposed to do now? I'm supposed to do... So let me make sure that I get this thing straight.  Well, it's the integral of, what did I say? f of V, W and T, D, W. Okay? But there is one Gaussian, the gamma m, which is not affected by W. It just comes out. You can pull it out here. E, sorry. And then you have the integral of D to the L, T applied to what? Well, it's applied to this function H0, which gives you a function of V and W, and then you integrate against the Gaussian. Okay? That's what you get. You see, this Gaussian, gamma n, is not affected anymore by the evolution, right? Because this whole thing came in front of the evolution. It's out here. All right. And now you see this f of Vt is written as a gamma m of V times a function of T. So I will call this naturally H of Vt. Agreed? This makes life simple. Let's look at the entropy. How does the entropy look like? The entropy is, where did I write it here? I put in f of Vt, the logarithm of f of Vt divided by gamma m. You see this gamma n drops out. So therefore, you know that the entropy you're worrying about is the entropy as thermal, which is equals to the integral of H log H gamma m. That's it, right? Good. So I think we have simplified things a little bit, right? No ratios anymore. The Gaussians come in a nice place. And in particular, we will see that this Gaussian is absolutely lovely. We will enjoy this hugely afterwards. So let me now write down what we have to work out. And also, let me introduce some silly notation. And I would like to do it this way. Look, this H0 is really only a function of V. And that makes life confusing because when you apply the e to the Lt, you get actually a function of V and W, right? So what you really have to think of this H of V as a function of V as a function in V and W. So I'm going to write this H0 in the following way, just as a reminder, H0 of V. I'll write this as H0 composed with P of V and W. And what is the P? The P is just a projection. I know it's a little bit silly, but I do this only to remind ourselves that these are really functions in the whole space and this L really acts on everything, okay? We have to keep this in mind. Okay, good. All right, so with all these preparations, how does this thing look like here now? I can just continue here. Our revolution is this one. H0 composed with P, right? This guy just acts on that one. And over here, I just put H0. So now let's go and start writing out what these things are. And again, it's getting even worse, right? This is already pretty bad. Now it's getting worse. This is the integral of d theta 1, rho theta 1, integral d theta k, rho theta k, right? And then what you have to do is you have to stick in here the rotations. And what you have is the product, and I'm going to write it this way, r, lowercase r alpha j, j runs from 1 up to k of theta k, j. I mean, can you read this? No, you can't, huh? Let me write this better. So this is of product j equals 1 to k. r alpha j of theta j applied to the function v, to the vector v and w, okay? You agree? That's what it is. And you see, it looks pretty lousy, right? But notice this is a rotation, okay? And this rotation, what does it do? It acts on all these vectors. And then afterwards, what you do, you take a projection, okay? So that's what in some sense you have to compute, and that's the stuff which you have to plug in and work something out, okay? So, now let me point out a few things, which is kind of interesting. When you look at this, this is a rotation. Now, it's a good idea to write this rotation as ak, bk, ck and dk. What is this ak? Ak is an m by m part of the matrix. So this is the m part, an n part, here's an m part, and here's an n part, okay? So this is an m by n, this is n by m, and this is n by n. But now notice, and this ak, depending on the terrible ways on these alphas and the thetas and everything, okay? Now, what do you know? You know that this is a rotation matrix, right? Because this came as a product of rotations, right? These are rotations. Remember what are these? These are rotations in the plane specified by this pair alpha j, right? These are these rotations.  So what you notice is that ak, ak transposed plus bk, bk transposed must be the identity in m dimensions. I mean everybody knows that, right? That's essentially the definition of a rotation and about the other terms actually don't care. I don't just ignore it, okay? Good. Now notice, by the way, the b is not a simple matrix, right? Now what do you do next? Why do I do this decomposition? I know I do this composition because what does the p do? The p takes this vector, right? I multiply this matrix by this vector. And what the p does is it just picks out the top components. All right? So therefore, what I can do now is, and this is the point about the p, I can replace this mass by what? By h of ak. And remember, the ak depends on everything. Let me indicate this in this way. b plus bk alpha theta w. Agreed? Which just is true. So let me just check. Now what do I do next? Next I do the following. I take this whole mass, I mean you agree so far it looks terrible. But what I do is I integrate this dwe to the minus pi w squared. Remember? I mean these are the joys of the Gaussian. All right? Now you see all these numbers here, this is now gone. Remember? dh, everything divided out. What do I do? I multiply this, I mean you see these are just constants from the point of w. All I do is I take this and multiply this by e to the minus pi w squared. And this is an integral over rn. Okay? And now we can start simplifying. Look, what is this bk? This bk is an m by n matrix. So really what I'm seeing here is just a vector which has m components. All right? But you see the w runs over rn. And now you just think a little bit. You can actually factor this matrix b as 1 minus a transposed a to the... No, sorry, a transposed to the power of 1 half times a rotation, let's call it beta. And this is a rotation. Right? I mean, sorry, it's an isometry. And what kind of an isometry is it? It maps the whole space rn into the subspace rm. So therefore, now you notice when you split your coordinates into one in this subspace into one perpendicular to it, these variables here which are left over, they don't see this expression, the Gaussian is normalized, you just integrate. So what you get by this little computation is, and you can just wipe, this is a nice thing, you get one identity minus ak, ak transposed to the power of 1 half, w e to the minus pi w squared. And here the integral is now over rm. And now I think some of you guys might actually sort of see what's going on. What does this remind you of? For the experts. Let's just do this. What's the point about the Ornstein-Ulenbeck semi-group? Right? And what you have here is precisely an Ornstein-Ulenbeck semi-group, but what is your problem? Your problem is that these are matrices. This is your problem. These are matrices. Now, what's the point about the Ornstein-Ulenbeck semi-group? There is a fantastic theorem which goes back to Edward Nelson, and let me write this Nelson's theorem here. So we don't need this anymore. So nt is this gadget here, integral h of e to the minus tv plus e1 minus e to the minus 2tw e to the minus pi w squared dw. Theorem, Nelson. And I cannot really, I cannot pin down the date. I don't know. In those days things didn't really, by the way, I mean some people had remarks about Nelson's theorem, published their remarks before Nelson published his theorem, but it's very well known. It's due to Nelson. When you take nt h in the q-norm, and what do I mean by the q-norm? It's always the q-norm with respect to the Gaussian measure. That's lesser equals hp-norm. Well, so far not too interesting, but what it says is, in particular, it says that the p minus 1 is equals e to the minus 2tq minus 1. You see, that's interesting. And that's a hard theorem, actually. Why? Because what it says is that the q here can be actually larger than p. Okay? It cannot be too large, because once this is larger than this p minus 1, this upgrade is not even bounded. But when this relation is satisfied, then you know you have this estimate that the constant 1 is sharp. Now what you do with that theorem is the following. You have a little corollary. And what you do is the following. You have s of nt h is less than or equals e to the minus 2t, the integral h log h e to the minus pi v squared dv. I just wrote out the entropy, right? To remind you again that this is the entropy s of h. I write it this way, right? So this entropy, so when you stick in the Nelson's kernel, you get this estimate. And then you get some remainder. And this remainder looks like this. Integral h log integral h. And of course the integral is always with respect to the Gauss measure. Now you notice when the h is normalized, this is zero. You don't worry about it, okay? Okay? So that's what you have. The proof is disgustingly easy once you accept this theorem. Why? Because you see when you take the L1 norm, sorry, I forgot to put an h here, when you take the L1, the integral of that, you just get the integral of h. So it's actually, you have equality when q is equals p is equals one. So then what you do is, you subtract the L1 norms from both sides, you divide by q minus one, q minus one, and you take the limit q to one, and you forget, don't forget this relation here, out pops this inequality. I'm not gonna get into this, all right? So this is the hypercontractive estimate. It's actually a theorem out of quantum field theory, believe it or not. That's what instigated this whole hypercontractivity stuff, right? Because you see, you can generalize these theorems, do infinite dimensions basically without giving up on the constants, right? One stays at one, right? You can multiply it as many times as you like. Okay. So now how do we use this? What I would like to do now is I would like to keep this formula and show you how one can prove our theorem at least in the case m equals one and n, of course, bigger than one. Right? It keeps life simple. So in this case, in which situation are we? Well, in this case, we're in the situation where the 8k is a one by one matrix. Right? Good? And now you see also that when you take the entropy of that guy, now you see all these weights are convex combinations. You see, the sum of this against that is one. The sum of all these guys is also one. These integrals also integrate to one. So by convexity of the entropy, I can simply go and stick in the entropy here of that gadget. Now in one dimension, what is my situation? In one dimension, I know by Nelson, remember the h is now because it's only a function of one variable, it's normalized in the Gauss measure. So by Nelson's theorem, by the way, it doesn't matter whether this pre-fact or here, I choose it to be positive or not. It's absolutely clear that these numbers are less than one because I have this relation. So they're between zero and one. So I can apply Nelson's theorem and I get that this whole mess in the case when m is equals one is less or equals d theta k rho of theta k and then I have a k of alpha and theta times the entropy of my initial condition. Agreed? That's what you have. And now at this moment, there's absolutely no reason to rejoice, right? Because you have absolutely no clue for the moment what the sum could possibly be. Now, here's a little idea. It turns out you can actually compute the sum explicitly. Okay? And so how do you do that? Let me just... Yeah. That's right. Okay. So we have to compute the sum. Well, here's the idea. You choose h0 of v to bv squared. Okay? So now you have two things. The first one is the formula. Maybe I'll write this formula down somewhere and stash it away. So this was just integral e to the minus lambda t sum lambda t to the k over k factorial. k goes from 0 to infinity. Then you have the sum over all pairs. Then you have the integral over the rows. And finally, you have the h of ak. I'm going to write it this way. v plus 1 minus ak k t 1 half w e to the minus... No, that's just what it is. No, I can't actually do that. So this is this formula. So now what are we going to do? We're going to choose the h0 to bv squared and plug it in our formula. So then what you get? When you choose the h to bv squared, so this is h0, sorry, h0 is v squared, what you get? Well, you get the square of that gadget. So in our case, when these things are one-dimensional, what you get? You get the ak squared because that's this guy. v squared. The integral against the Gaussian is 1. And here, what you get? You get the w squared 1 minus a squared. And then you get the w squared against the Gaussian. It's a one-dimensional integral. That's 1 over 2 pi. So now you get this sum. And you see something. You see that this sum here, together with this ak squared, is precisely what you want. So if you have a chance to compute this thing in a different way, then you're home free, right? And the idea is actually quite simple. What you do is the following. d v squared e to the minus pi w squared d w. That's precisely this sum. This whole mess is precisely that. And now you differentiate. And what you get is the integral e to the l t, and you have an l in front applied to the v squared e to the minus pi. And now I think I propose to you as an exercise to compute what this is. When you do this computation more or less carefully, what you find out is that this, let's call this function u of t. It depends, of course, on v, right? u of zero. And of v is what? Well, when the t is equal to zero, it just gets v squared integrated against the Gaussian w. That's v squared. When you differentiate this out, it's a completely elementary computation. You'll learn that d u d t satisfies a differential equation. And this differential equation looks like the following mu rho one over two pi plus one over n v squared minus mu rho n plus one over n of... That's what you get. I just recommend you to do this computation. It's extremely simple. Why? Because all you have to do is you have to apply the L to this v squared.  And once you do this computation, out pops this formula. Now, this is a linear ordinary differential equation so that everybody knows how to solve it and what do we get? We get that u is equals to one over two pi n right this way, n plus one one minus e to the minus mu rho n over n plus one over n. That's perfectly reasonable because that's your rate of decay, right? t plus... And then you get one over n plus one plus n over n plus one e to the minus mu rho n plus one over n t v squared. That's what you get. So therefore you know that this piece here must be this sum. Why? Because this is precisely what you get here. And you notice this is precisely what I claimed in this theorem. Okay? So this works pretty well. And so now you say, well, now I can do this one of course. It's absolutely obvious. Now you do it for n equals two and you run into a wall. You see, I gave you a little presentation of Brass-Kambleep theorems and so you're pretty sure now that this Brass-Kambleep theorems must be used somewhere in these arguments, right? And that's the reason that's getting a little bit complicated so I decided to put up some slides about that. Okay? So remember, this is the guy up here which corresponds to Nelson's estimate, right? I mean, to which we have to apply Nelson's estimate. But you see, this is very complicated, right? The ak is just an m by n matrix. Now what you're going to do is you're going to do what is called make an orthogonal single ovality composition. In other words, you see, I might have some zero eigenvalues in the single ovality, some singular values which are zero, I keep them, okay? So the u is orthogonal, and gamma is a matrix which has non-negative elements in the diagonal. Here they are. Okay? And remember, all these are functions of alpha and theta of all these patterns. Good. Here's your theorem. When you apply Nelson's estimate, this one here, well, I can't see it. Here. Repeatedly. You have to repeat it. Why? Now you see, when you iterate this estimate, how do you do it? Well, you see this function has the ak, has a u, gamma, v transposed. Now the v transposed just hangs in there with the v. If you replace variables when you compute the entropy in terms of v, this is just an orthogonal transformation. Everything is rotation invariant. This v drops out. What does not drop out is the u, and the u stick together with the h. Okay? So how does it look? I get this h, u here, and I take marginals. Why do I have to take marginals? Because you see, when I apply, for example, Nelson's theorem to a function of two variables where I have different eigenvalues here, then what do I get? I cannot assume that the function, the marginals, are normalized. I cannot assume that. So what I have to do is I have to put the eigenvalues in terms, and that gives you this huge mess. So what is it? It's the sum over all subsets of the sets 1 to m. When the index is in the complement, you put the gamma i squared, one of these eigenvalues in the single-valid composition, and when it's in the set, you put the 1 minus gamma j squared. Here, you have to take the marginal in the variables which correspond to sigma. You write this, and this is a complicated function. You multiply by the log, you integrate it against the u, but you have to integrate it now in the remaining variables, which is an r sigma complement. So this estimate you can get. This is just a repeated application of Nelson's theorem, but you have to be a little bit careful. Here, I've written out what the h is, h of u kb. And the h sigma here is written out as this integral. You notice something. Here, just as an important part. Look, what is this? This is actually equals to that. You see what the difference is. Here I integrate over r to sigma complement. Here I integrate over all of rm. Now, what do I do? This is the trick. You see, I can simply complement these variables here to variables in all of rm. It's just a 1, right? When you integrate. And then I can just remove the sigma here, right? Because after all, these functions only depend on the sigma complement variables. And then finally, I can move the u over and I can onto this function here and that's what I get. This is this complicated expression, okay? I mean, I understand that you're not going to follow me really, but you want to get sort of the gist of it, okay? All right, next. How do you... So this was Nelson's Theorem. And now what is it? This is the right-hand side of Nelson's Theorem. Right as sum over all of rm, this is the integrative of all the rows. It's a huge mess, right? And what I want to do is that this whole mess is less or equals the entropy. That's what I really would like to have. Maybe with a factor. Okay? That's somehow the goal, you agree? So how do I do that? Well, here comes brass complete, right? I just written it down again. You have your Hilbert spaces, K of them, they have dimensions T i. You have these maps going from rm to the h i's. And you know that this B i, B i transposes the identity on these Hilbert spaces. Now, what you also need are these numbers C i that when you look at this matrix you add up to the identity, right? And what does brass complete tell you? This huge mess here is less or equals that mess. This is a nice mess. This is a terrible mess. Good. So now here's a consequence. I remember I told you about the Legendre transform of the entropy. If you apply the Legendre transform of the entropy, you get to this theorem here.  And that's, it's that theorem. By the way, this is a theorem which I learned from Eric Carlin and Cordero Arauskin. I mean this whole sort of circle of ideas. So now what we're going to do, we're going to apply that theorem to this mess. And you see you have to abstract a little bit. So here's a sum. For all practical purposes is also sum. You agree? I mean, what you can do is you can approximate such integrals always by discrete sums. I wrote D row, I hope you don't mind. It goes a little bit faster. So you have this stuff. Then you have this remaining thing which we got out of Nelson's theorem. And now, you see, this has precisely the structure which we want. Namely, why? Because the h of v is here. What is f i of b i of v? Well, this turns out to be this logarithm. Right? So what do I do? I make this correspondence. Sorry, this was wrong. Here should be the logarithm of h. So this is going to be my correspondence with the logarithm here. This integral is one. Why? Because this thing, I can simply, this is a marginal. I can complement this to an integral and it goes out because it's a rotation. The Hilbert spaces correspond to these sub-spaces where are they? Yeah. In which these guys project, the b i's correspond to these operators here. That's where they are. And now you have this distinctive pleasure to figure out whether something like this here is true. Notice this is the b Let's go back to Brazcom Blib. Right? I have to check this relation. This relation is trivial to check. This relation can be checked. So that's this one. I have to sum over all sigmas. I have to integrate over all rows. Okay? That's what I have to do. And what it is, it turns out that this mess is equals the identity times a number. And what is this number? Right? Now, why is this a miracle? Because now, let me just argue how you can do that. If you take this integral here, then you see these guys here have nothing to do with the sum on sigmas. So you can write it this way. And now you notice that when you, for example, look at this matrix here. Well, this matrix is a diagonal matrix which has only zeros and ones. So let's pick a place where there's a one here. So that means, let's take the place one, one. Okay? This matrix of them. Well, either it's zero or it's a one there. If it's a one there, that means I have to sum over all sigmas which do not contain one. And then you can actually sum that stuff and what you get is precisely gamma k squared. Now, what is this?  And now you realize again that this a k, a k transposed where does it came from? It came from this product of rotations. That stuff you can compute. And it turns out this is precisely this number c k m. Okay? So what does this mean? This means that in this huge mess here when you put here the entropy you can replace this whole sum by e to the minus lambda t sum lambda t to the k over k factorial and then you have to c m k. That's what your estimate is times the entropy of your initial condition. Well, every child knows how to do it since kindergarten what this is precisely the theorem which I told you. Okay? So I know this is a little bit brutal and you can imagine it took us a while to get there. But I really urge I mean, I think what I really like to convey is you see, when you have a complicated system it doesn't mean that you can compute something pretty explicitly. It's really you get rates and you know what's going on. Right? Now what you can of course complain about is you can say wait a moment this is not really a good theorem, why? Because what you should be doing is you should take the entropy relative to the equilibrium state. Now that opens up a whole totally new kind of worms. I think it should be able, one should be able to do it but you have to invent a whole slew of new hypercontractive estimates of atomic groups. And then you have to also produce a version of brass cum bleep on the sphere. We have some experience with that. But this would be a totally new project and at the moment I'm too exhausted I won't be able to do it. All right? Good. So I think you got a little bit of the message how this works and what I'm going to give you now for the remaining few minutes I'm going to give you a few results here. Some references, okay? So the fundamental paper about all of this is of course the one due to Mark Katz from 1956 where he also mentioned what we call Katz conjecture which then was proved by Elise Jean-Ress. The computation of the gap was done in this paper here. I wrote this down because it's just some lecture notes very short and it's sort of in a non-fancy way. Just the basics. And that's sometimes very nice to read at the beginning. Now this whole thing was elaborated in the paper in Akta a long paper where we also compute the gap or at least prove the Katz conjecture for momentum preserving collisions. A nice paper is David Maslens who really computes much more about the Katz master equation. And many of the things I talked about was of course in this beautiful paper of Cedric and the title is interesting Church and Yannis conjecture sometimes true and always almost true. Well in a way he's right. In us in this paper is where he actually shows this lower bound on the entropy production and Amid was the one then Amid Enaf who provided the upper bound. Now you see what I in some sense told you is kind of a preparation to read some other papers in here. So these are the really fundamental papers in this business. Namely by Stefan and Claymore Moore. So this is a nice paper at the beginning you get a first outline of what this whole thing is and if you then got up this is the appetizer and then when you're really hungry you can go to Inventionist for the next 147 pages it's worth doing it actually. It's a very nice paper. This is very fundamental. I would say this is really seminal work. So now about the thermostats we started out with Ranjini in this paper in Journal of Statistical Physics where we proved this exponential decay of the entropy. The connection between the infinite thermostat and the finite reservoir was actually proved in this paper. This is the student, the graduate student of mine. And if you're interested in this Gabetta Dosgani Wenbeck metric here this is the right paper to look at. It's actually a very interesting paper because we often look things, what is it, L2, LP no, there are many other interesting metrics and they really provide a slew of those. Braskam Lieb Here is the classic paper of Braskam and Elliott Lieb with all the constants and Young's inequality. It's converse in its generalization of more than three functions. That started the whole story. And then Elliott produced a huge paper of Gaussian kernels of only Gaussian maximizers. Now that's actually worth reading again, why? Because for him e to the Ixy is a Gaussian kernel. So he gets all the sharp constants in the house of Young inequality and all these kinds of just all comes out of this paper. This is seminal work. And then the stuff which I told you I learned also lots of it from Keith Ball and that this is beautiful because he uses Braskam Lieb inequalities to prove theorems about sections of cubes and related problems. It has a huge impact on convex geometry interestingly. And you can also look at this paper these two are very very nice papers. He writes extremely well. That's really worth reading. And then the one so this by the way I have to say Keith did it in the rank one case where these matrices be I or rank one matrices. Now Frank Barthe in another terrific paper in Inventions he actually proved all this kind of stuff for the rank, the arbitrary rank case. And then Eric, Elliot and I we discovered this correlation inequalities on the sphere and then we realized that heat kernels can do a lot of good to prove all these inequalities and we did this in this paper also for the rank one case in Braskam Lieb and that was later done in the general rank case by Bennett, Carberry, Christen, Terry Tau. I also would like to mention Eric Carlin and the paper of Eric Carlin Dario-Coderra-Rauskin, namely sub-additive entropy in relation to Braskam Lieb type inequalities. This is where many of these things have told you show up. And for historic reasons the very first papers were by Loomis and Whitney this was at Braskam Lieb type inequality but of course much much more elementary right. The one which I showed you what you use for the Sobler's inequality and there is an analogous entropy inequality which was proved by hand. And you see this goes way back. Hypercontractive estimates Edward Nelson had this famous paper, The Free Markov Field that's a paper about quantum field theory where he actually has this inequality and the guy actually sort of realized that this inequality which I told you this one here is actually closely related you can prove it with the sharp logarithmic Sobler's inequality. So in other words, Nelson's theorem implies the sharp logarithmic Sobler's inequality and the converse and of course about logarithmic Sobler's inequality there's lots of stuff I just put this here for historic reasons of course these papers of Lenny Groves. And with that I would like to thank you for your attention in particular would like to thank Francesco for organizing such terrific conference and I would also like to thank the supporting staff for this fantastic help in running this place. Thank you.


You can't see the DNA of any plants or animals from the outside. Neither do most people see the law. Only the lawyers see the law. We have a duality of law where we have constraints and then we have enabling legal institutions and we can use the latter to get around the former. And lawyers know how to do that. That's one of the most powerful ways in which wealth is created today. The legal system, which is governed in principle by the idea that everybody is equal under the law, can nonetheless be used by very powerful strategic actors to create private wealth for themselves. Until 1880, most of the source of wealth was in rural land, which was protected by legal devices. What came in instead? Legal modules, private law, contract law, corporate law used to flip objects, ideas, promises into wealth-generating assets. Asset holders have found lawyers who have coded their assets in law and have amounted resources that they can use then again to hire better lawyers and code more assets in law. Enormous amounts of wealth have been created with capital. And if you critique capitalism, people typically point to the Soviet Union and say, there is nothing else that could possibly work. And yet there's something fundamentally wrong with capitalism. It is unjust. We have seen many upheavals of capitalism lately. There are structures built in that make the system both relatively fragile and at the same time create major advantages for some at the very top. We created this mechanism in the 19th century in an age of capital scarcity, but it's not clear that we want to maintain that and still allow investors to make a lot of money as if the world had not changed around us. As we rethink how to deal with problems such as inequality, such as climate change, understanding how the law really works is of critical importance.


Okay, welcome back everyone to our second lecture on faith, BC 111. We're going to take some time just to answer some of the questions and then we will move forward. So there's a question here from Krisha. If you know God has given a clear direction, but our faith gets disturbed as we get scared of unknown and end up taking a different direction, does it mean God will be upset with us? How to fix it and come back in faith? All right, so sometimes this happens because maybe we get afraid or we're scared, we may not go in the way God wants us to go. Remember that God is always our Father. He's always our Father. And we may make mistakes, but God is bigger than our mistakes. So it's not like, that's why he's God. We may make mistakes and God is always bigger than our mistakes. As long as we go back to him and say, God, I'm sorry, I should have gone that way, but I was scared. I didn't want to go. I went this way. Please get me back on track. He will get us back. So God is bigger than our mistakes. He can fix it. God is greater than time. He can accelerate things. He can make up for lost time. Right? So God can do that. In Joel chapter 2, I think it's verse 26 or 27, he says, I will restore the years which the locus has eaten. So God can restore time. Our times are in his hands. Psalm 31 verse 15. Our times are in his hands. So he controls that. So the answer to your question, Krisha, is all we need to do is to come back to God. Say, Lord, I know I made a mistake or I didn't do what I was supposed to do, whatever that situation is, we acknowledge that before God. And now God, please get me back on track and God will do that. Sometimes getting back on track make take a little bit of time because it just depends on the situation. But God will restore. God will bring things back. And God is greater than our situation. Okay. Hope that answered your question. All right. Okay. I can't hear. I think we need a mic for the class. Can we exercise our faith practically in a practical way in? Okay. So can we exercise our faith practically? So you mean like as while you are in Bible college? Is that what you're asking or in this world? Oh, can we exercise faith practically in this world all the time? Right? So in every situation, in everything we do in life, we exercise faith practically. So you and I, we, so the question that was asked by one of our students here in class was can we exercise our faith practically? Right? So I'm just giving an answer to that. So the thing is in every situation, you know, we must choose to act in faith.  Right? At that moment, two things. One is exercise your faith. And then secondly, you do what you need to do practical.  You're taking care of your health.  But at the same time, exercise faith. You say, in Jesus' name, I speak to this sickness. I command you to leave. My body is a temple of God. So you use your faith. So even in a small, in a thing like that, you're facing with your face with sickness, you exercise your faith, but you do what you have to do also practically because God has given us understanding to take care of our physical bodies.  You use your faith in every situation. Example, if it comes to money, you use your faith for money. Right? Or for example, okay, right now, right now, we are, okay, there was a problem with our central location. So we had to find a place. So we finally got a place. But now I'm using my faith. I said, God, we're starting an eight o'clock service. We're going to do a 1030 service. I want to see the whole full for both services. So I'm using my faith for that. Right? So practically, right now, how are we going to see the whole full and both services? One is, of course, we had to do our work. That is, we have to go do all those things. The practical side is there. But in the spiritual side, I use my faith. So Isaiah chapter 60 is a big chapter. So I was meditating it because God said, Arise, so I have to base my faith on the Word of God. So I base my faith on Isaiah 60. He said, the Lord will arise on you. His glory will be seen on you. Then people will come. So that's how I'm building my faith. This I'm doing during my prayer time. In my prayer time, I use that. So God, I believe I'm going to see the whole full of both services, eight o'clock service. I want to see the whole full thing. And then I'm believing God. So God, I want to reach one million people in the city of Bangalore. I want to see thousands of churches raised up all across India. So for all those things which are yet to be done, practically, right? I'm using my faith. So like that for so many things that, whether there are small things or that there are big things, we have to use our faith. So in every, in area, while I was a college student, and I can share many stories, but while I was a college student, when I needed money to pay for my tuition, use my faith. When I needed to come out of a difficult situation, use my faith. When I needed to travel on mission trips, I didn't have money to pay for those things. I used my faith. So during my college years, that's how I used to use my faith. You know, for all these practical things I needed from God, so use my faith. And then, you know, so different seasons of life, whatever situation you're facing, you've got to use your faith for those situations. Okay. So there are lots of, lots of practical situations where we will have to use our faith. Thanks. Yes, go ahead. Another question. Okay. Very interesting question. So there's a question here from one of our students in class. He says, can we change God's will by faith? Interesting. Now, generally speaking, the answer is no, meaning when we're talking about God's plan for the human race, God's plan for things that he wants to do, those are things we cannot change. So we have a lesson, I think later on, called the boundaries or the perimeters of faith, that means what faith cannot do. So faith cannot change God's will. Faith cannot change somebody else's will. That means I cannot use my faith to control another person, you know, because that person's will is, that's independent of my faith. So there are boundaries to faith. We will talk about it in that lesson. So in general, the answer is no, God, you know, we cannot we cannot change God's will for the human race. We can't do that. But what we do see is that we can change God's mind about a matter. For example, suppose a person is living a very reckless life, and we find this in the Bible, especially in God's dealing with Israel, God said, I'm going to destroy them. Moses says, God, don't please, don't please forgive. That's okay, Moses, because you're praying, I will forgive them. So in that sense, right, even though the people deserve judgment, because somebody was praying or interceding, standing before God, God said, because you're praying, because Moses said, okay, I will not, I'll give another chance. So in that sense, we can do that's what prayer and intercession will do when we pray for people, intercede for people. But the overall purpose of God, we can't change the plan of God for the human race. That's the will of God that will be established. But in these matters, the mind of God on certain things, we've got calls us into partnership and we can do that, especially through intercession and prayer. Okay, all right, let's see now. Another question here, Surya, I am having so much of faith that prayer for life. What should I have to keep focused, whether on faith or on prayer, for God's reply? Okay, I think I understand the question. The question is, I'm having a lot of faith, but maybe, so he's saying faith is stronger than life of prayer. What should I pay attention to? Should I focus on faith or on prayer to get God's answer? Here's how I would respond to that question. See, all these are important things. I mean, faith in God's prayer and intercession is also very important. So you're doing a full course on prayer and intercession. So all these are important. Now, when we're dealing with the situation, we must exercise faith. Faith is necessary. Prayer without faith is useless because Jesus said, in many places, I'll quote example, Matthew 21, verse 22, whatever you ask in prayer, believing you will receive. So prayer always has to be in faith. Prayer without faith is empty, is useless. So we've got to have faith in God and when you pray, you pray believing prayer. You pray with faith in God. It's not just making lots of words for God. It has to come out of a heart of faith. But having said that, also understand that in some matters, prayer and intercession are very important, especially when it has to do with other people. So if I'm having faith for my own life situation, so on, faith is important and usually just by faith you can deal with it because Jesus said, you speak to the mountain, you tell the mountain to move, it'll move, nothing will be impossible to you. So if it's a matter of your own life, your own life situation, so you just exercise faith, you speak words or faith things will happen. But if it is dealing with somebody else, maybe you're praying for the salvation of a family member, maybe you're praying for another person to grow spiritually or to walk in the ways of God. That's where prayer and intercession is very important. And you have to pray in faith, of course, but prayer and intercession is dealing with what is affecting that person. So in a situation like that, intercession, prayer becomes very important. Of course, all of it has to be done in faith. So or when you're dealing with God to work in your city, work in your nation, prayer and intercession becomes very important. So my answer to the Suresh, if it's concerning a personal matter, you're dealing with a life situation that's in your personal life, you just use your faith, like Jesus spoke to the storm, the winds, the waves, he didn't even pray for that wind or the storm, he just spoke to it, he spoke and ministered to people that way. But he also had a strong life of prayer, so he would go away into the mountains and he would spend hours in prayer. And when they are praying for people, when they're interceding, when they're dealing with other people, that's where our prayer and intercession comes in. Okay, all right. So, all right, I think we've answered the question. Any other questions in class? Yes, long question. Okay, try to make it short and ask your question. It's okay. Is there is a pastor or praying person? Sorry, there is a pastor or any praying person? There's a prayerful person, okay.  They have faith for receive healing, but the praying person have no faith. It's healing is happening or not happening.  Is that what you said? The person who's sick has a lot of faith, but the person who is praying doesn't have faith. So the question is healing happening or will the healing happen? Yeah, so I will answer it based on the Bible, because every situation will vary. So the situation that Francis described is there's a sick person who has faith in God, but there's a person who's praying who does not have faith in God.  That's the question. I hope I understood correctly, right? Okay, so Francis, we know, so what does the Bible say? The Bible says, for example, Matthew chapter 9 verse 29, according to your faith, it will be done for you, according to your faith. So that's what God looks at. So the sick person will receive according to their faith. So the person who was sick has faith in God, believes they will see the results, that they will be healed. Now, if the person who's praying, if he doesn't have faith, okay, the prayer is actually ineffective, because remember James chapter 5 verse 15, 16 says, you know, about the elders, they have to pray in faith for the prayer of faith will heal the sick. So if they're not praying the prayer of faith, that prayer is ineffective is useless. So the prayer may be useless, but the person who's sick has faith in God, and through their faith in God, they will receive, right? That's, I'm giving a biblical answer. Now, in reality, the situations are so different, because we don't know every what's in people's hearts and what they're going through. We don't know, but I'm just giving a Bible answer. In practice, we may see all kinds of things happen, which are different. Okay. All right. Okay, shall we move forward any more questions from online students? Any questions? Are we all okay? All right. Okay, let's move forward to now, what we're going to do for a few weeks or for some time, I know a few weeks, but for some time is we're going to break things down. That means, okay, you know, I get in the chapter 10, we talked about speaking your faith and so on and so forth. Now we're going to get into different aspects of exercising faith and so on. So we're going to talk about each piece, like when I talk about confession, we're going to talk about endurance. Now, we're going to talk about how to, you know, pray and receive various parts, piece by piece, right? We're going to talk about it. So in chapter 11, we talk about how confession releases faith. You see, one of the things God, you see, God has designed us in such a way that we are able to express what is in our heart through the words of our mouth. So what is in our heart is expressed through the words we speak and what we will see in scripture, both the Old Testament and the New Testament, is that our words are very important. The words we speak are very important and God has used or God has instructed us that our words are a vehicle or a way to express or release our faith, our words, the words you speak, is a way to release your faith. So we need to understand that and we need to use it.  So what is your problem? He says, I have a tumor. Okay, how are you going to get the tumor out? You and I are not doctors.  We can't do that. How are you going to deal with the tumor? Well, Jesus said something. He said, speak to the mountain. Speak to the mountain. So you can even, you know, mountain basically represents some material thing, something in our natural realm. And he said, if you have faith, you can speak to that. So that's why you can say to the tumor, tumor in the name of Jesus. I command you to leave or I command you to dry up and disappear, whatever, you know, you give, you can speak to it. Now some people who don't understand this, they listen to you and say, gone. This person has gone, lost his mind. He's talking to the tumor, but they don't understand spiritual truth. That's in the Bible. But in the Bible, for example, Jesus, he spoke to things. And if you just turn with me to look, chapter four, and I'm sure that you are familiar with many examples in the Bible, but I'll just point us to one example. So if you turn with me to look, chapter four, please, and please look at verses 38 and 39. Look, chapter four, there is not in the notes. I'm just taking a little side journey. We'll come back to the notes. Look, chapter four, 38 and 39. It says that he arose from the synagogue and entered Simon's house, but Simon's wife's mother was sick with a high fever and they made requests of him concerning her. So this is Simon's wife's mother. She had high fever. I don't know, maybe those days they may not have had thermometers, but maybe it was 102, 103, 104, high fever. What did Jesus do? Verse 39. So he stood over her and rebuked what? The fever. Jesus is speaking to what? He's speaking to the fever. It says he rebuked the fever and it left her. That means you can imagine Jesus going, you can just imagine, Jesus going to Peter's mother-in-law. She's lying in the bed, high fever. He goes and says, fever, leave. What is this? He's talking to the fever. He's rebuking the fever. What is this? But Jesus rebuked the fever and what does it say? It says, and it left her. See, he used words and he spoke to what? He spoke to the fever. Notice he did not pray. So again, we go back to one of those questions. Should we use faith? Should we pray? In this case, it was not about prayer. It was about exercising faith. It was about speaking to the fever. He spoke to the fever. So you can speak to the sickness, whatever that sickness. Speak to it. Words, speaking words to that sickness. You speak to the tumor. You speak to whatever that condition. Speak to it. Tell it to go. Rebuked it. Meaning they say, go, leave. And she was healed. So I'm just giving one example. You look at the ministry of Jesus, for the blind or the deaf person. He said, be opened. He's speaking to the deaf. He's speaking to the ear. He says, be open or to the dumb. Be open for the person who cannot speak. Speak plainly. So God has given us this. So use your words. Use your words to do work. Spiritual. This is spiritual. So we must understand that words are important in the exercise of faith. And that's what we're going to spend time on in this lesson. Lesson number 11. Now the word confession. There are different contexts in which the word confession is used in the New Testament. We see we have to confess Jesus as Lord of our lives for salvation. We have to confess sin to receive forgiveness of sins. The word confession is the Greek word homologia. Right? And it is really made up of two words. So homologia is made up of two words. Homo means same. Homo means same. And logia means to say. So homo same. Logia means to say. And therefore it simply means homologia simply means to say the same thing. To say the same thing. Right? So the word confession means to say the same thing. So if I'm talking about confession of sin, I say the same thing. That means I say I have sinned. Right? Suppose I got angry. And I say the same thing. I said God, I got angry with somebody. I shouted at somebody. God please forgive me. I should not have done that. Confession of sin. I'm saying the same thing of my sin. God, this is what I did. Wrong thing I did. Please forgive me. That is confession of sin. Saying the same thing. Confession of Jesus as Lord. That means we're saying the same thing. Jesus Christ is Lord. Jesus Christ has risen from the dead. And he has ascended to heaven. So we are saying the same thing. That is confession of Christ as Lord. And Jesus said, if you confess me before men, I will confess you before my Father who is in heaven. That means if you say Jesus is Lord, then he will say that boy, that girl, that person is, I am his Lord. He is my son, my daughter before the Father. That means you are acknowledging Jesus. Jesus is acknowledging you. Confession.  He will say the same thing about you and me. So the word confession is a very simple word. It simply means say the same thing. Say the same thing. That's what it means to confess. Now, the Bible also teaches us that we have to confess our faith. Now, we have a full book called Speak Your Faith. You can download the PDF for free, but the book will be printed. We've just added a printing, so hopefully it'll be printed and you'll get a copy of it. But that book, Speak Your Faith, basically has scriptures from Genesis to Revelation that teaches on speaking your faith. So you can go through the whole book. But here, we're going to only talk, reference a few verses, a few verses. So we see this confession of faith throughout the Bible. It's not just in the New Testament or only in the Old Testament. It's throughout the Bible. People say the same thing. They say what they believe. That is confession of your faith. Say what you believe. And of course, you and I believe God. You and I believe the Word of God. So we must say what we believe. So look at some examples. In Joshua chapter 1, and I've not put this on, but Joshua chapter 1, God is telling Joshua, is instructing Joshua. And he says, Joshua, this book, and at that time, they just had the books of Moses, the first five books. And God is telling Joshua chapter 1. Sorry, I turned to Deuteronomy. Okay, Joshua chapter 1 verse 8. He says, this book of the law shall not leave your mouth, depart from your mouth.  Keep it in your mouth. But you meditate in a day and night that you may observe to do according to all that is written in it. But then you will make your way prosperous and then you will have good success.  Let it always be in your mouth. Don't let it leave your mouth. Meaning, in other words, to put it in simple language, always say what my word has said. This book, don't let it leave your mouth. Meaning, let the words of this book be the words of your, always say, you always speak my words and you meditate in it. You carefully do it, observe to do and then you will make your way prosperous. You will have good success. Okay, so you follow my word, you will have good success. Now, I want to, I want us to look at a very interesting passage in Deuteronomy chapter 30, verses 11 to 14, because why I say this is very interesting is because this same passage from Deuteronomy chapter 13 is quoted by the Apostle Paul in Romans chapter 10. Very interesting that the Apostle Paul in Romans chapter 10 is quoting this passage from Deuteronomy chapter 30. So we're going to read both. That means what God taught his people in the Old Testament is still being given to believers in the New Testament, this particular truth. Okay, so what did God tell his people in the Old Testament? Deuteronomy chapter 30, verses 11 to 14. Let me read it please. For this commandment which I command you today is not too mysterious for you nor is it far off. So God is saying, look, my word which I'm giving you today, it's not mysterious. It's not something you don't understand and neither is it far away from you. Verse 12, it is not in heaven that you should say who will ascend into heaven for us and bring it to us that we may hear it and do it.  No, I'm not kept it away from you. And then verse 13, nor is it beyond the sea that you should say who will go over the sea for us and bring it to us that we may hear it and do it. So God is not saying, look, I didn't put my word far away from you in some other place that you have to go across the seas to go and get it. So what God is saying is, look, my word is not far from you. It's not so far away up in heaven or far in some across the ocean. It's not far away from you. But what he's saying verse 14, but the word is very near you. The word which God has spoken is very near you. Where is it? In your mouth and in your heart that you may do it. So God is saying, hello, my word, it's not far away. It's not in heaven so far that you have to go to heaven to get it. It is not far away across the ocean. You have to go across and get it. No, my word is very near you.  It's in your heart so that you can live by it, follow it, do it. So that is what God told his people in the Old Testament. Very interesting. Let's go to Romans chapter 10. We'll explain this to me close. Romans chapter 10. The Apostle Paul, the book of Romans is a powerful episode. So in your third year, you will be studying the book of Romans verse by verse, beautiful, beautiful study. You will understand the book of Romans. It's the, it's the more, I shouldn't say it's the best, but this book, episode of Romans is the highest from a doctrine standpoint, from a teaching standpoint. It's one of, you know, Paul, the Apostle's best writing. Of course, the epistles talk about Christian living, but Romans is talking about Christian doctrine, you know, right from sin, salvation. It teaches us so many things, powerful, powerful. You'll study it in your third year, but in Romans chapter 10, very interesting. The Apostle Paul is quoting from Deuteronomy 30 and then he's explaining to us how to practice it. So it's very interesting. Romans 10. Let's read it. We will read from verse 6. But the righteousness of faith speaks in this way. So he's talking about speaking and he's saying you have to speak like this. That means people who have received righteousness by faith, they talk like this. Okay. So you and I, have we received righteousness by faith? Yes. So we are supposed to talk like this because he's saying for the righteousness of faith speaks like this. Do not say in your heart who will ascend into heaven, that is to bring Christ down from above or who will descend into the abyss, that is to bring Christ up from the dead. Verse 8. But what does it say? That means what does the scripture say?  So he's quoting Deuteronomy 30 verse 14. The word is near you in your heart and in your mouth. For what? That is the word of faith which we preach. So which word is he referring to? See in the Old Testament, God was referring to the book of the law and he was saying that word is very close to you. But now Paul is referring to the word of faith. That means the message of faith in Christ and everything that Paul was preaching. So whatever Paul was preaching, the word of faith, the word that brings faith, the message that about Jesus Christ, what he has done for us on the cross and all of that. So this message that produces faith, that word is very close to you.  But what must you do? Next verse. So he doesn't stop there. Verse 9. That if you confess with your mouth.  For what? For you to confess. If you confess with your mouth, the Lord Jesus and believe in your heart. So the word is in your heart. For what? To believe it and believe in your heart that God has raised him from the dead. You will be saved. Look at verse 10. For with the heart one believes unto righteousness, with the mouth confession is made unto salvation. So Paul is quoting from the Old Testament and then he's explaining how it works in the life of the believer.  He says with the heart man believes, with the mouth confession is made. What is confession? Say the same thing. So word is in your mouth. You confess the word, you got to say the same. Say what the word says. Say the same thing. With the heart man believes. The word is in your heart. So what do you believe? You believe the word. And then he also tells us what believing does. With the heart one believes unto righteousness. That means believing puts you in right standing with God. You come into the right place. To be righteous means to be in right standing with God. You're in the right place before God. So when you believe in your heart, you believe the word, it puts you in a place, right place before God to receive. With the heart man believes, verse 10. With the heart one believes unto righteousness or resulting in righteousness. So when you believe with your heart, what is the result? You're in the right place. You're standing in the right place before God. Then what happens? Next verse 10. And with the mouth confession is made unto salvation. That means now when you confess, it results in you experiencing God's work in your life. That's the salvation. So two things. When you, the word is in, so let's break it down. The word of God, the word which brings faith, word of faith. It's in your heart and it's in your mouth. So if you and I want God to work in our lives, we shouldn't say, oh God is so far away in heaven. Who will go and bring him? Or oh Jesus died. Who's going to bring him up? No, don't talk like that. So Paul is saying, don't talk like that. The righteousness of faith speaks like this. This is how we're supposed to speak. Speak according to the word because the word is in your heart and it's in your mouth. With your heart you believe, you believe the word. When you believe the word, it puts you in righteousness, puts you in the right place before God.  That means when you confess with your mouth, after having believed in your heart, you're in the right place.  So example, we'll just mention a few examples and we'll take questions. Suppose a believer is sick. Suppose you're sick. Don't say, oh, who will bring Jehovah Rafa from heaven for him to heal me? I got a Jehovah Rafa, but he's in heaven. How will he come and heal me? Or maybe Jehovah Rafa didn't rise up from the dead. Who's going to raise him up? Don't speak like that because he has already given you his word.  His word says, himself took our sicknesses and bore our diseases. His word says, by his tribes we are here. So the word is near you. What must you do? You believe that word in your heart and you believe that word in your heart. It puts you in righteousness in the right place before God to receive.  You confess that word. It brings salvation. It brings God's work of healing into your life. Confession is made unto salvage, resulting in salvation. You got it? Or let's say there's a need, a financial need. Don't say, who will bring Jehovah Jara from heaven? Or is Jehovah Jara dead? How will he meet my need? No, the word is near you. It's in your heart. It's in your mouth. What is the word? My God, child, supplier, all my need, according to his riches and glory through Christ. Jesus, God is able to make all grace abound toward me. That I always having all sufficiency and all things will abound to every good work. The Lord is my shepherd. I will not be in what the word is near me. It's in my heart. It's in my heart. It's in my heart for me to believe it with the heart man believes. So you believe that word. When you believe that word, it puts you in righteousness before God. You're in the right place to receive now because you believed it. And out of a believing heart, confession is made. That means you must speak and declare, my God, child, supply all my needs. The Lord, my God, makes all grace abound toward me. The Lord is my shepherd. I shall so confession is made unto salvation. If you see the result of God's working in your life, your needs will be met. Understand? So we will pause here. Let's say any questions. Let me check the online class. Any questions? We have five minutes. All right. Any questions from online class students? Any questions? We have five more minutes for a few more minutes. Yes. Sorry, we have great faith. Okay. So Ren's question is, if we have great faith and people think that, oh, this person is very proud or things like that. So one is, see, we don't all, we don't have to speak our faith in front of people all the time. It's very important and we will continue this. You know, it's very important, first of all, to speak our faith before God because Bible says Hebrews 3, 1. Next class, we'll talk about it. Jesus is the high priest of our confession. So we need to make our confession before Jesus. Secondly, not everybody understands faith. So what you do? Keep quiet before them because they don't understand. They'll think you lost your mind. They'll think something wrong. So before them, just keep quiet. Now Jesus put it like this and it sounds hard but Jesus said, don't throw your pearls before swine, before pigs. And there was this simple interpretation is, don't, these valuable things that you have, spiritual things, don't put it in front of people who cannot understand it. They'll only laugh at us. So why even bother? So don't do that. So most important, we speak our faith before God, if a Christ, because he's a high priest and a few people who understand our faith. You know, I'll just mention this verse, 1st Timothy 612, Paul tells Timothy, hold fast your confession which you have confessed before many witnesses. That means these witnesses, of course, are the believers, people who understand. So before people who understand, then you share your faith. They will support you. They'll encourage you. But if they don't understand, don't waste your time. Don't say anything, just keep quiet. Otherwise they will think something is wrong with us. So don't do that. And if they misunderstand us, don't worry because they don't understand where we are coming from. We are coming from faith in God and His Word. They don't understand that. So don't worry about us. Okay? All right. We are going to close one minute we have. Okay, let's close. Maybe I'll just pray to the, yes, you're saying sometimes, if we, as faith increases, we also become proud. So we have to avoid that. Just don't worry about, I mean, sorry, we should watch our own hearts, our own lives, that we don't become proud. Because pride can hinder the exercise of our faith if we let pride come in. Sorry, what were you going to say? How to make sure that we don't let pride come in? Okay, let us answer that next time. Okay, just remind me, next class, we will start with your question. Okay? So when my last question didn't have time to answer, I was thinking about it. But you asked me, next class, we'll start. Okay? Okay, let's just close. Father, we thank you for this time of learning.  Thanks for being on the class. Okay, take your quick break and please get ready for your next class. Thank you.


Okay, well, what is the signature? What does it look like? It looks like the thing in the big parentheses here. That was just how Alice created them. The i-th signature is her private key times the thing that came out of the hash for the i-th signature, times the random polynomial she chose for the i-th signature. Notice something very important here. In each signature, the C sub i's are different and the y sub i's are different, right? But her private key is sitting there waiting to be plucked off the data if you're clever enough to figure out how to get at it. But I mean, it's obfuscated. It's hidden by being in this expression. Okay, so the big parentheses I just substituted in what s i was. The C I just copied. Now I simply do a little algebra. I've used the distributive law. I split it into two sums, because I have a sum there. The f of x doesn't depend on i, of course, right? So I can move that out in front. So this line to this line, just a little algebra. But let's think of what these sums look like. So let me first think about what this sum looks like. The y sub i's and the C sub i's really have essentially nothing to do with each other. Alice picked the y sub i's at random. The C sub i's came out of a hash function. The hash functions produce things that look very random. So these products look quite random. And their coefficients of the y's and the C's were sort of uniform. We're symmetric around zero. Okay, so when you take these products on average, as t gets big, this is actually gonna get quite small. What about over here? We're here I have C i of x times C i of x inverse in some sense. C i of x and C i of x inverse, those are not independent of each other, right? They both depend on i. And if you do an analysis of this, this average actually converges if the C i's are chosen randomly to something non-zero. And I'll give you an intuition on it. Remember, a ring is essentially a cyclotomic ring, a ring of cyclotomic integers, right? So if you think of x as being an nth root of unity, then x to n minus one is really x inverse is complex conjugate. If zeta is a root of unity, one over zeta is the complex conjugate of zeta, right? So in that sense, essentially, if you analyze it from that viewpoint, the quantity here, this product of these two C sub i's is a complex number times its complex conjugate. Okay? And if you do that, you get a positive real number, right? So if you add up these positive real numbers, their average will be something non-zero. That's where it's coming from. And one can work out even explicitly. This, as you take more and more random C sub i's, it roughly converges to four n over three. So the top line is what Eve uses to attack the system. Right? Eve knows this top line. And she computes it and gets, well, and actually she should multiply this quantity by three over four n to get rid of the three over four n, okay? And she'll get a polynomial with real coefficients. And she runs the coefficients to the nearest integer because f, remember, has coefficients zero, one, and minus one. So if you know f to within the coefficients of f to within, say, a tenth, you know f, right? Because its coefficients are just zero, it's ones and minus ones. And that's how Eve would recover Alice's private key. If Alice were careless and didn't do the rejection sampling step, okay. Now let's suppose she does it, she signs things using rejection sampling. I claim that in that case, this transcript reveals no information about Alice's private key. Now the first thing I want you to think about is this is a perfectly valid English sentence. This list of numbers reveals no information about this other quantity. But how do you make that a rigorous mathematical statement and then prove it, okay? So, I mean, this isn't at the same level as Diffie-Hellman's idea that you could create a crypto system that used two separate keys. That was just a mind-blowing idea even if they didn't know exactly how to do it. But this is a similar sort of thing. It's mind-blowing that a list of quantities created using this secret information would contain no information about the secret thing you used. Once you realize that might be possible, one can think about how to do it and then figure out that, yeah, there are ways to do it. But just the idea, I found really, really astonishing. So, how do we quantify this? What does it mean to say it reveals no information? Well, I'm gonna phrase this as a conditional probability. Hope people have seen a little bit of probability theory. So, my claim is that if Alice chooses the private key F zero, okay, regardless of what F zero is and regardless of which, remember the signatures are in this box, that's the rejection sampling step, that all the coefficients of the signatures have absolute value at most K minus N, okay? So, signatures have to be in this box. But if I take a potential public key, a private key, and a potential signature, what's the probability that a signature created using F has S equals S naught, given, this is the conditional, that F is F naught, okay? And I want that probability to be independent of F naught. So, that would mean that if I used F naught to create signatures and you used F one to create signatures and that person over there used F two to create signatures, our probability of getting a signature that included S naught is the same. And can you see that means that the signatures convey no information about the private key because the probability of getting any particular signature doesn't depend on the private key that was used, okay? So, our goal is to prove that this probability, well, we'll compute it exactly, really, is independent of the F naught. Question, and very loud. I cannot understand you. Bjorn, could you take the mic back? And also, I know there was another question in the middle also, we'll do that one next. It's in the back. Where is he?  The signatures, they are published always with a certain hash of a message. So, even if there's the same probability to generate a given signature for some C. Yes, I'm sweeping that under the rug. What this is actually doing, giving a security under what's known as a random oracle model. So, you're making an assumption on the hash function that it is completely disassociated from everything else in some sense. And even that's not a precise statement, but that's a good point and it can be made precise and I'm ignoring it. Yeah, Bjorn, there's also a question in the middle here somewhere, yeah. Thank you. Do you ever get the same signature with different private keys? Can you ever get the same signature with different, with different private keys? Well, the answer is yes and no. You can certainly get the same S's. You won't get the same S's and C's. Well, I guess even that could happen, but there would be different random Y's that were used to create them. But in fact, the sample spaces for all these things are so large. The probability of that happening is indistinguishable from zero. I mean, the space of possible S's and the space of possible C's usually would have like 260th elements in it. So you're almost, it's highly unlikely to ever get a collision like that. That's a good question. Okay, so we wanna prove this probability doesn't depend on Fnought, even though it's a conditional probability with an assumption that F equals Fnought. So let's do the probability calculation. The first line here is just what was on the previous slide, except I made it a little, I abbreviated it. So the probability that S is Snought, given that the F you use was Fnought, where C and Y are uniformly randomly chosen. C with coefficients minus one to one, Y with coefficients randomly from minus K to K. And this relates to that question back there. If there's actually any correlation that you know of between Y and C, then you might be able to exploit that. So we're basically assuming the hash function is really what's called cryptographically secure. Okay, so what's that probability? How do I compute that probability? Well, I take all the possible C comma Y values and I look at all the C comma Y values that make the Snought, which remember is C F plus Y, that the Snought uses Fnought. And I divide that by the total number of possible Cs and Ys, okay? So this is just how you compute this probability by listing the number of elements in the set that are winning numbers, divided by the number of elements in the set total. Number of winning tickets over number of total tickets. Well, let's analyze this a little more closely. How many C comma Ys are there? Well, once I pick my C, remember Snought and Fnought are fixed. Once I pick my C, there's only one possible Y, namely Snought minus C Fnought. And that Y is only allowed in this pair if its coordinates are at most K.


Alright everybody, what is up? Welcome back to another video. Today we're going to be looking at the number 99th best player, voted by the players in the NFL of 2019. It was my first season I'd ever watched, first season I'd ever followed. I still didn't pick a team by the end of it, but I did have some select players that I did enjoy watching and I'm hoping that they're going to all be on this list.  So anyways guys, number 99, Derek Henry, a running back for the Titans. Haven't heard of him, but I do like running backs. Let's crack into it. You just worry about yourself Taylor, okay? Number 22 for the Titans. Let's do this. He's a big back and he has a speed.  Oh, boom, straight up. Saquon Barclay. What number is Saquon Barclay? What number is Saquon Barclay going to be this year? Top 10? You're giving him a chance. Obviously, like he showed in that 90 yard touchdown line, you're giving him a chance. Oh, he got a 99 yard touchdown. Obviously, the first player that comes to mind is the Jacksonville given the one-yard line. One of us said something like, let's just make this a long drive. Let's just go one player. And we all kind of juke in the lap, because I think it's all happened a lot of the time. Marioda off to, what's his name? Henry. Henry breaks the first one, gets into open space. Boy, eh?  Too high, mate. Too high again. Get off me. Get off me. Too high again. Man, watching that play live, that would have been amazing. He just stiff-armed three guys in a row against Jacksonville. One of the best defensive teams in the league. Nice, man. Bang. See that? Look, watch this. He jumps in the air. No, sorry. Go back here. Gets the hand off, jumps in the air. He's jumped off his left foot.  The defense thinks he's going to the right, but he's going to the left. Bang. Breaks the first tackle. He's in open space. Now it's up to him.  Oh, that's a beautiful stiff arm. And another one.   That's another good one, too. Sick. Yeah, it actually reminds me of Marshall Lynch's Beast Quake, to be honest. I was going to say, is the only reason why he's on this list, that run, or was he solid throughout the year? Did he get a thousand rushing yards? Did he? Oh, check that. How'd he stay on his feet there? Oh, that's nice. Oh, that's a beautiful run, man. I wish I could see that in real time. He just bit about four people. Off the left, off the left, bang. Actually, how many did he beat? One, two, three, four, five. I think we don't even want to try. Especially in the second half of the season. I mean, that was Jim Brown's stuff right there. He showed, like, all his attributes at the running back, that I'm his vision, I'm his strength, and his speed. Vision, strength, and speed. That's what you need to be a good running back. Isn't it Barclay? Anyways, guys, that was number 99. Derek Henry, my first look at the running back from the Titans, but it's definitely got my interest, and if I do happen to catch a Titans game, which I most probably won't, then I'll be looking out for him. Number 22, Derek Henry, and I think you guys should, too. Anyways, guys, that's another video done. Just one player per video I've decided. Number 98 is next, and that is, give me a second, Tyler Lockett, a wide receiver from the Seahawks. Another skilled player, skilled position. They're all skilled, but, you know, skilled. I'll see you in the next one. Peace out.