 OK, thank you very much. So for today, the new feature will be, I will not only consider static manifolds, but I will deform the manifolds or more precisely, I'm going to deform the metric. So the situation will be the following. I take a fixed manifold, but instead of taking a given metric, I look at the family of metrics. And I will be some interval in R plus, usually from 0 to capital T or something like that. And what will this family GT be? So these are Riemannian metrics. And they evolve following some kind of geometric flow, for instance. So this is usually a C1 family of Riemannian metrics. The evolution is completely deterministic. And as I already can see here, the most eminent example is the so-called Ricci flow, which has been introduced by Richard Hamilton. And what is the idea behind that? Well, the idea is, of course, on any manifold, you can choose a Riemannian metric. But there is no canonical way how to do this. And so you may ask what is a good metric for a given manifold, which should reflect the topology of the manifold in a proper way. And the idea is, well, let the manifold decide itself what is a good metric. So first, put any metric on M, and then try to make this metric to look better, to improve the metric, to make the manifold more round, and so on. And the idea is the following. You deform the metric by this so-called Ricci flow equation. Yes, so there are a lot of things to explain. On the left-hand side, well, you have the metric, the derivative with respect to time. So this is a two-tensor. On the right-hand side, I have minus 2 Ricci with respect to the metric Gt. This is also a two-tensor. And the initial condition is my given metric I start with. Now you may ask, why this equation, why 2 here, why minus, and so on. And I'm coming to this immediately. The idea behind this equation is, in some sense, Ricci flow should be seen as a heat equation, but on the space of Riemannian metrics. So what does that mean? I'm not going to make this completely precise, but just to show you an indication for that. On a Riemannian metric, locally you can consider harmonic coordinates. So in a local chart, which is harmonic, you develop Ricci, so the ij component of Ricci in this chart. And you will see you get an expansion, which starts minus 1 half, Laplacian, but the Laplacian applied to the ij component of the metric plus lower order terms, so terms involving only first derivatives and so on. If you go with this information to the Ricci flow equation, you see on the right-hand side, well, we have here 2. This goes away by the 1 half, and we have minus, and here we have minus, so we get plus over here, plus Laplacian of gij, plus some lower order terms. So one should not think the right-hand side as kind of Laplacian on the Riemannian metrics in the sense that the Riemannian metric is a section of the bilinear forms and so on. Because if you work with Levy-Givita, then the Laplacian would be 0. But it gives you some indication how this equation should work. And the idea is the following. Well, you have this initial metric, which may be quite complicated. And then heat flow means you try to smoothen it out. Like you are, let's suppose, you are in the Himalaya, and you are confused by the very bizarre environment. And what would you do is you could try to heat up the Himalaya. Then the ice will melt down here, and the landscape will be more smooth. If you wait long enough, maybe global warming is doing the job for you. But the idea is then if you let this equation run, you will get a nice metric. And maybe as t goes to infinity, if this equation can be solved, you may see some very canonical metric on the space. Of course, there is some price to pay. This is a highly nonlinear evolution. And usually you will have singularities in finite times. So if you think about the ice melting down, it can happen that at some point, part of the mountain is just collapsing here. And so the difficulty is to understand such singularities, how they build up, how they can be characterized, and in particular how to get control on such things. Well, to give you a first idea, if you follow this equation, then of course, Ricci evolves, because Gt evolves. And you may look at the scalar curvature, which you just get by tracing Ricci. And you look what equation the scalar curvature is following. Then you see a typical reaction diffusion equation. So on the right-hand side, you have two parts. Here is Laplacian of R. R is the scalar curvature. So if you ignore the second term, this would be a standard heat equation. So if you are on a compact manifold and you would not have the second term, the evolution for the scalar curvature would just be that scalar curvature is equally distributed out over the manifold. So the flow would uniformly distribute the curvature. The second term here has an opposite effect. It tries to concentrate curvature. And so you have here competing things. One is smoothening out and one is concentrating. And you have to look which of the two tendencies will beat us to understand this equation. OK. Well, I should say this is one example for such a family of metrics. Of course, we could look at other ones, the canonical way of deforming metrics is given by mean curvature flow. So you embed your manifold as some sub-manifold into Rn. And then you look at mean curvature flow and you pull back the induced metric from Rn to your manifold. And you get the family of metrics. And that's what we call an evolving manifold. And of course, there are many other things, but mainly I will stick to the case of Ricci flow. There are actually two kinds of Ricci flow equation. The one I introduced was with the minus here. This, as we have seen, gives us heat equation because if you look here, the minus goes away. The other one is with a plus sign, which is a backward heat equation. And both are equally important to understand. And both equations give you a deformation of the metric. Well, now, if you remember what we did yesterday in this crash course, I tried to explain what is flow to a second-order differential operator. And in particular, we constructed flow to the Laplace-Beltrami operator. And the defining equation was this here. I wrote it here in differential notation. And for a Brownian motion on a static manifold, we said, well, it should be the stochastic flow to the given Laplace-Beltrami operator. Here we have lots of Laplacians because for each t, we have another one. And so it's natural at time t, which Laplacian should we take while the one we have at time t. So this is the Laplace-Beltrami with respect to the metric gt. And so this is some natural thing to consider and such a flow corresponding to this evolving family of Laplacians. I will call a gt Brownian motion on M. Well, to give you a rough idea how to think about this object, this is my notation. I start at time s at the point small x. And this is the position where I'm at time t. This is no longer homogeneous because the operator depends explicitly on t. I cannot just shift time. I really get or should look or say as I start at 0 and run up to time t, but this is not the same as starting at s and run up to time s plus t. Well, of course, this is still an elliptic diffusion. So I will have a nice transition kernels. And they have a density t. So I look at the probability starting at x at time small s of the position at time t, x t, to be in some given set. This is a measure which has a density. But I have to decide density with respect to which reference measure. Well, because usually the reference measure you take is the Lebesgue measure induced by the metric. But now we have lots of metrics here. So at time t, here I'm at time t and looking at the position at time t. So it's natural to take as reference measure the measure I have at time t. So the one corresponding to the metric gt. And well, then we get a nice kernel here. And we can look at the corresponding evolution or the corresponding heat equation. You see what you get looks quite familiar, what you expect. So the time derivative of this kernel equals the Laplacian, but now depending on time. But you have a new term here, which is given as derivative of the metric trace of that. And so if you take the special case that gt is given by the Ricci flow, g dot would be plus minus Ricci. So trace of Ricci is scalar curvature. And so you get the following evolution. For the forward Ricci flow, the heat equation is, well, as you would expect here, I just let the Laplacian depend on t. But I get a new term here, which is scalar curvature multiplied by pt. And I have plus or minus depending on whether I'm looking at the forward Ricci flow or the backward Ricci flow. So this is already an equation, which is no longer so easy to handle. But I'm coming back to this. So what one could do now is, well, natural question is, it's been over decades, a lot of works are done to understand heat equation on Riemannian manifold, how it depends on geometry, and so on. So a natural question would be, now, look at the heat equation. But under, for instance, under the Ricci flow. So you let the metric evolve according to Ricci flow, forward Ricci flow, and study the heat equation and the situation. Well, this is a typical forward heat equation. There is a lot of interest also in the so-called conjugate heat equation. I take, as well, forward Ricci flow here. But now I look at time derivative of u plus the Laplacian. So if I put it on the other side, I would have minus. This is a backward heat equation. And well, so the question would be, what can we say about such system? But I want to explain first why this is interesting. And actually, this is one of the crucial steps in Perlman's work to understand, for instance, this conjugate heat equation. Well, let me give you a very brief introduction to the basic setting of this theory. If you want to study Ricci flow, and I remember having heard lots of talks already quite some time ago. And what the people always said, why is Ricci flow so difficult? Because it's not a gradient flow. And this you could hear for quite a long time. And it's true, but it's not completely true. And the new idea really came from Perlman in this direction. What he did is look at the following functional. This is not a functional just so we have a given manifold. And we look at all the Riemannian matrix on this manifold. And I define a functional which assigns some value to a metric, but not just to a metric, metric times functions. So my functional is defined on the space of metrics of M and functions on M and gives me a value in R. And what or how is this functional defined? Well, you look here at gradient F norm squared. So be aware, what does it mean? The gradient depends on the metric. You take the metric you have here. Norm squared also depends on the metric. So the metric is then you add here scalar curvature with respect to the metric G multiplied by e to the minus F and integrate it over the manifold. In this theory, usually M is compact. Then you definitely have no trouble with defining this functional. And one should say, if you look at this work, usually there is always the assumption that M is compact because you want to use freely integration by parts. And as soon as you go away from having compact, you run in a lot of troubles. And what I'm going to do is later on, I will show you that my computations are no longer in terms of integration by parts. I will do everything by Eto's formula. And there, I don't really need that the manifold is compact. All I need is that certain local martingales at the end are true martingales because I want to take expectation. And I do not want the expectation to depend on time t. Well, you may say, OK, nice. So we can define this functional. But the point is now, well, what is the gradient flow to this functional F? Or in more naive terms, you may ask, how to deform the metric and how to deform this function F to get the most of increase or degrees of your function F? Well, this is quite a general problem. So one first assumption is you have here the volume measure but weighted by E to the minus F. And you should insist that this product here gives you a static measure. So the volume depends on t if you deform the metric. And F will also deform. But this measure should be a static measure. And then you realize the following. Well, you should deform the metric following this equation, which looks quite similar to what we had before, except that here is now a Hessian of F. But people having some background in probability theory, you know, this is Ricci plus Hessian of F. This is the Bakri Emery Ricci tensor or the Toulousean Ricci curvature. So that should not disturb us too much. F should follow not the heat equation but the backward heat equation. And here is still this scalar curvature. And so the upshot or what I want to tell is I'm not explaining now in which sense this is exactly the gradient flow. But take the reversed point of view. Suppose that G and F evolve according these two equations here. And then look what happens to your F functional. So well, calculate the F functional and look at its time derivative. Then you calculate things. And you see this term here on the right-hand side. And what is the interesting thing? The interesting thing is here you have something norm squared e to the minus F. So this is positive. So along the flow or if we deform G and F using these two equations, our F functional will be monotone. And this, if you are familiar with the theory of geometric flows, this is always some very powerful result. If you find a functional which behaves monotone along the flow, then you are always in a good position to get information out of this. And so here the Perelman F functional behaves monotone along the flow. And well, there is this annoying point. This is not just exactly Ricci flow, what we would like to have because there is this Hessian. But there is a well-known trick how to get rid of this, which is the following. Take the time-dependent vector field, which is the gradient of F. So this is a vector field. And it generates a flow, phi t. And now you pull back the metric using this flow. And at the same time, you pull back the function using this flow. Then you get two new families, G star of t and F star of t. And if you look then at the evolution of these pulled-back quantities, you see the metric now evolves exactly following Ricci flow equation. Here we still have a back what heat equation. We get one new term, but for the moment we should not worry about it. And of course, it's trivial to check that under simultaneous pull-back of metric and functions, the F functional does not change. So this does not affect the value of the F functional. In particular, we have the same monotonicity as we had before. So up to now, we have the following information. If G evolves under Ricci flow and F under this bit strange backward heat equation, then the F functional is monotonic. And even more, you see, well, this is an integral over Ricci plus hashen of F norm squared. And we see that monotonicity is even strict unless this term here Ricci plus hashen of F is 0. And what does it mean? Ricci plus hashen of F being 0, this means we are on a steady Ricci soliton, which is a generalization of Einstein manifolds. If F would be 0, then you know Ricci being 0 gives you an Einstein manifold. And more generally, Ricci plus hashen of some function being 0. These things are called Ricci solitons. And we see our monotonicity is really strict unless our flow gets trapped by such Ricci soliton. And so one of the major steps was then to understand Ricci solitons. What can we say about such things? Well, finally, to relate it a bit more to a probability, instead of F, I will pass to u, which is e to the minus F. And then the g is still Ricci following Ricci flow. And u is a standard backward heat equation. And on the right-hand side, I have just multiplication with scalar curvature. So this will be, you see, this is exactly the equation I said at the beginning, study heat equation under Conjugate, or study the evolution of Conjugate heat equation. And so we have now, in terms of u, the functional F writes like that. And the derivative is monotonic, and strict monotonicity is strict unless this here is 0. Well, there's always some thing people talk about entropy in this relation. So where is the entropy here coming up? Actually, it's not directly written out in Perlman's work, but it's a quite easy calculation and a nice observation. So to the following, you have this u, which is a solution of the backward heat equation, which is a positive solution of the backward heat equation. Take this as a density for a measure, for a measure mu t. And look at the entropy of this measure. So what does that mean? Well, you look at the Boltzmann-Schellen entropy of this measure. So you can just define it. Well, I take here not minus, but plus. So while I get too many minus signs. So take u log u and integrate it with respect to the volume measure. Here, here, here we are. This is u, each one u. e to the minus the corresponding volume. e to the minus f volume measure at time t. That should be a static measure. But here I'm not. So that means that, well, so here, OK, I'm sorry, this here actually stays constant along the flow, which is a consequence of my hypothesis. And basically, this is the same as to say that this follows a backward heat equation. So if you have this equation, this will be trivial. e to the minus f volume t. Now, this should be a static measure. So just think about it. The total mass of this measure should, yeah, for instance, yeah, yeah, yeah. No, no, no, no, no, no, no. OK, thanks. So look now at the entropy of this measure here. And of course, what I said, if you would not have u log u, if you just take u there, it would not depend on t. So this depends on t. And try to understand the behavior of t. And you see some very amazing fact. If you take the derivative of this entropy, what you find is exactly Perlman's f function. So in other words, we can start with this solution to the backward heat equation under reach flow. Look at its entropy. Take the first derivative of the entropy, get the Perlman's f function. And the derivative of f, which is now the second derivative of the entropy, is monotone. So you have some convexity of the entropy function, right? OK, so this already introduces the third item of my title, which is related to entropy. Basically, all my entropies I will look at are somehow or correspond to measures, where I have some reference measure, and the density, which is given as a positive solution of some heat equation. OK, I still have 10 minutes, I think. OK, now coming back to stochastic analysis to what we did yesterday, I showed you how to define Brownian motion. Now you will say everything just depends on t. So you have metric depending on t. So do things as we did yesterday, but let all things depend on t. And maybe if you're happy, you get the right GT Brownian motion. There is one major difficulty, because if you remember yesterday, we not just constructed Brownian motion, we constructed as well parallel transport along Brownian motion. And if we would just do things as we did yesterday, letting the quantities depend on t, we would lose that the parallel transport gives you isometrics. This would no longer be true, because the metric changes, and there is some derivative of the metric, which must come in. So let me briefly explain the new idea. Instead of the tangent bundle of m, we consider the tangent bundle not over m, but over spacetime. So m times this interval in r. So sections of this bundle are now just time-dependent vector fields. And to do the same thing as yesterday, where we had on a Riemannian manifold the Levy-Givita connection, we have to give this bundle a connection. Of course, if we just look at things in the space variable, if x, so sections of the bundle, are time-dependent vector fields, then the covariant derivative of y in the direction of x will take the usual covariant derivative you have at time t. But there is one new thing. We also have now on m the time direction. And we have to explain what is the covariant derivative of a time-dependent vector field in the time direction. And there is only one reasonable way to do this, which is given by the formula here. You take the usual derivative, and here you have one term where the derivative of the metric comes in. And it turns out this connection is compatible with the metric. I think I saw this definition in a paper by Hamilton a long time ago. And you can check. This is basically the only way to get a connection on the tension bundle over spacetime, which is metric. Yeah? OK. So now this connection allows to define parallel transport, but along curves in spacetime. So here you have curve in spacetime, also has an explicit time component. And I will look here at typically at curves x, t, where rho t is some monotone transformation of t. And I'm only interested at least here in two cases where rho t is t, so where I just have as additional component t here, or where time runs backward from some capital T. I look at capital T minus t. Then I do exactly the same as yesterday. I go to the frame bundle. Now the fiber of my frame bundle are now isometries. But between rn and the tension space at point x with respect to the metric gt, this gives me again this decomposition in horizontal and vertical. And the summary is more or less here. We have a notion of horizontal lift. We can lift tensioned vectors downstairs, so on my spacetime, up to tensioned vectors on the frame bundle. A typical tension vector downstairs well is something which is a tangent vector plus a time derivative. This is the additional ingredient. And we can horizontally lift it up to some, we just lift up this tension vector. And we have the horizontal lift of time derivative. If you work this out, what is the capital DT, you will see there is a derivative of the metric in it. But for the moment, we don't have to worry about it. In terms of the horizontal vector feeds, we have again this horizontal aplassion on the frames. And one last notation take, so DT was the horizontal lift of the time derivative down on the spacetime manifold. And I look at this vector field, DT, but multiplied by rho dot of t. In our case, rho dot is just 1 or minus 1. And now, as yesterday, I solved this equation here on the frame bundle. But I introduced a new term, which is constituted by this additional vector field on the frame bundle, which is the horizontal lift of time derivative. So this gives me a process on u. I start with something on Rn, on flat space, for instance, with Brownian motion. And I solve this equation. I get a process u. I project it down here. But the projection is over spacetime. So I get a process on the manifold and my original rho t. So either t or capital T minus t. And the summary is the following. If we start with Brownian or Wiener process on Rn, we do this procedure. We get Brownian motion on m. But it's evolving while you have to be careful. At the beginning, I said gt Brownian motion. But what gt is depends on the rho t. So the metric you experience at time t will be the metric g rho t. So it can be gt or also g capital T minus t, if the time is running backwards. And in the same way as yesterday, we have parallel transport. But the nice thing is now, by construction, these parallel transports are isometries. And of course, if you would do the calculation a bit down to Earth, you would have to elaborate this term here. And then you would see you need an additional term which involves derivative of the metric and so on to achieve that parallel transport finally are isometries. OK, so I have now two cases, x dt or x t capital T minus t. If that is Brownian motion, this gives me Brownian motion on my spacetime. OK, let me just use the last minus two minutes to give you a first example. So look at the heat equation. But the Laplacian should follow some flow, where the general deformation of the metric. Then you work out these two formulas. You calculate the drift of u log u and of grad u norm squared over u. This is basically Pochner's formula. Then you see here, here is Hessian norm squared plus this term here, which is twice Ricci plus time derivative of t. So if you know that dG over dt is greater equal than minus 2 Ricci, which means that we have a super solution to the Ricci flow, this here is non-negative. So you have on the right-hand side a positive drift. From that, it's quite easy to check that a combination of grad u norm squared over u and u log u, if time runs backward, gives you a sub-martingale. So a sub-martingale is a martingale plus something monotonically increasing. And if under some integrability conditions, or if it's a true sub-martingales or boundedness or something like that, we can take expectations and compare the expectation at time 0 to the expectation at time capital T. At time capital T, you see I have written here capital T minus T. So at the right endpoint, this term goes away. And the left endpoint, if small t is 0, I have here x0, which is just a small x t. So I end up with this formula. And if I work it out, what this formula means, I get a so-called gradient entropy estimate. I can estimate grad u over u norm squared by the entropy here. And you already, this is just a translation of this thing here. And you see already if this is entropy where this u is normalized, u x t is just a constant. It does not affect the expectation. If you put the 1 over t on the other side, you see that the entropy is greater equal than t times this equation. It already gives you a gross rate for the entropy. So the entropy has to grow at a certain rate. You can play around a bit with this equation. You can introduce some delta. Or if you like, you can look at the maximum of u, or a supremum on m and the time interval. So then it's less or equal. You take here the maximums. It's mt over utx log. Then only this thing here depends on random, but the expectation is 1. So you immediately recover from the simple sub-Martingale argument Hamilton's gradient estimate for grad u over u in terms of the maximum of u. OK, thank you very much. I think it's time for lunch.