 Okay, thank you very much. So today is the last talk in the series of talks and I'm going to discuss entropy formulas mainly under geometric flows. So this is inspired, of course, by work, which is very close to Perelman's entropy formula. And this is still a field where a lot of open questions are around. And so I already introduced this equation. Suppose you have a family of metrics evolving under Ricci flow as we discussed before and you continue the heat equation along this flow. So the Laplacian here depends at time t on the metric gt is taken with respect to the metric gt. And you would like to understand this heat equation. And so here in this equation, if you remember, I always had the trouble that time should run backward so I can make a change of variable in both equations to let time run backwards. This doesn't change much but the advantage is then I have always plus here here. So I get backward Ricci flow and backward heat equation. Yeah, I can always change to this by passing from t to capital T minus t back again. Well, so in this situation, we know now time is running forward. I have this x tx and the additional time is t. This is my space time Brownian motion starting at time zero at the point x. Yeah, so yesterday I had two parameter families of semi groups or starting at s and then going to time t. So for simplicity here, let's everything start at time s and then this process here is exactly what I explained to be a gt Brownian motion. So it's a Brownian motion but the corresponding Laplacian who is governing the local behavior of the Brownian motion is moving in time. At time t, I experience the metric gt and the Laplacian corresponding to this metric. Well, in most of the work in this field or if people did things like formulas I have in mind, the reference measure is always the Lebesgue measure. So the natural measure is you take the Lebesgue measure induced by the metric gt. So you have family of Lebesgue measures and so on. And I already said this works quite well if you are on a compact manifold because then you are finite measures and you can do integration by parts and all these kind of things. It doesn't work very well if you are on a non-compact manifold because then at each step you have to justify can I do integration by parts here and so on. So the new feature I would like to introduce here is why not working with the heat kernel measure. So the measure, this is the measure at time t given as the probability that my gt Brownian motion starting at x will be in some given set. This is a nice probability measure. And now I take this as reference measure and my positive solution to the heat equation as density. So I look at this measure mu t which comes as a density u with respect to empty or if you like it is the underlying Wiener measure we used to construct our gt Brownian motion. And here I have this ux tt and remember this is a martingale or a local martingale but I'm assuming it is a true martingale. So I'm changing the Wiener measure martingale given which I get by substituting the process in to this solution to the heat equation. And then of course the integral of u with respect to mt is nothing but the expectation here. And since this is a martingale it does not depend on t. It stays constant along the evolution. Okay, what I would like to calculate is the following thing. I would like to calculate the entropy of this measure and see how it develops as a function of time. So more or in plain terms just take u log u and integrate this with respect to mt over the manifold. This is a probability measure where you can just rewrite it. You take u log u, you substitute the process at time t and take expectation. So this is an interesting object and we would like to understand how this entropy, well I call it entropy but if you are really interested in entropy you should take minus but I do not care about sign convention and I also changing the direction of time so this also takes plus to minus and so on. At the end what is always important is that you get out something which is monotone as time evolves. Well, so one can work this out very easily how to do this. Well here you have just u log u which is a nice function of my diffusion. So use it as formula, calculate this, take expectation and you see the following two formulas. The first one which gives the derivative of the entropy is an expectation of grad u norm squared over u and if you are a various fissure entropy and so on so this should look familiar to you. For us the important thing is this is non-negative. So the entropy with all the sign convention I introduced will increase as time goes on and what is very nice here if you look at the second entropy you also have a sign here. So here is u which is positive and here you have norm squared or Hilbert Schmidt norm of the Hessian of log u. Yeah, so you also have convexity of the entropy. To calculate this here is a quite easy exercise but surprisingly it already has some interesting consequences. So for instance we used this to get something out about ancient solution to the heat equation. Well, if you remember on the first slide I said we look at this backward heat equation if you let time now run backwards you get ancient solutions. So if a solution is defined, a backward solution is defined for all t it means you get here ancient solution which start at time minus infinity and involve from there and one would like to know something about such solutions. And well in our situation what I definitely can do I know this E prime has is monotone. So I look at its limit. I will get some number which is non-negative by definition can be plus infinity but nevertheless this number is well defined and to give you a simple example of such an ancient solution for the heat equation just on the real line and you check easily this is such a solution. If you calculate the entropy of this guy you see it's just t. So it's linearly growing and so this parameter theta is just one which is the derivative of t. And then it's very easy to get for instance results like that to take a positive solution to this backward heat equation then the solution is constant if and only if this invariant is zero. And from there you can for instance derive results if the entropy grows sub linearly in the sense that entropy at time t divided by t as t goes to infinity tends to zero then this theta is zero and u is constant. And these are very easy consequences but surprisingly even in the case of static manifold some of these results have already have been new in this setting. So you can see this is something interesting maybe one could look at. But if you remember there is my equation. So here it is I start with the equation here but in Parallelman's work it would be interesting to get a theory of the conjugate heat equation. So it doesn't seem to work anymore. No, okay. Okay, so this is the equation which is basic in the theory of Ricci flow. So g is evolving according to Ricci flow but the u follows the so-called conjugate heat equation. So this is a backward heat equation. You have time derivative of u plus Laplacian. And here on the right hand side you have scalar curvature induced by the metric multiplied by u. Yes. And well of course because of this additional term r of u, u x t, t is no longer a martingale but I have to change the measure according to Gersonov's formula. So I multiply this thing here by exponential of minus the integral from zero to t where I integrate scalar curvature along my diffusion from time zero to t. Then this guy here u multiplied with this Feynman-Katz factor is a local martingale or a martingale so if I take expectations it's independent of t. Just as before the new thing is this exponential factor which comes from the r here which I'm putting in the measure here. So this will now serve as my reference measure. No longer just the Wiener measure before but the Wiener measure with this density. And as before I take this measure and change it by my solution of the conjugate heat equation to get a new family of measures and I want to calculate the entropy of these measures. So entropy now just would be u log u as before. Expectation in the expectation there is this exponential factor involving the scalar curvature, right? Well, if we work out the derivative we get a formula which is quite similar to the F functional of Perelman but the main difference is so we have this r grad log u normal squared but the main difference is I'm no longer integrating with respect to the Lebesgue measure I'm now integrating with respect to a probability measure. So this is the derivative of this entropy. If I work, well, I will change my definition of the entropy by subtracting this guy here. I come to the point why I'm doing this a bit later. So this will be a new functional which I also call entropy. It's the old entropy minus this term here. If I do that, then I see some thing interesting. Well, I get one new term Laplacian of u here but if I calculate the second derivative I have a reach minus Hessian log u norm squared. So I get a sign for the second derivative here and so this entropy is a functional or the derivative of this entropy is a functional along my geometric flow which is monotone. And monotonicity is strict unless reach minus Hess log u is zero. And this means, well, here I give the upshot of this. You take the derivative of this entropy called this F. This is non-degrading in time and as I said monotonicity is strict unless reach plus Hessian of F is zero. And this is what people call a steady soliton. So a manifold which has the property that if you look at reach, yeah, and there is a function such that reach plus Hessians or potential F reach plus Hessian of this potential gives zero is called a steady reach soliton. And let me say a few words why such guys are interesting in the theory. So in general, we are saying a Riemannian manifold is a gradient reach soliton. Well, if, so now a fixed manifold, mg, if there is a function such that reach plus Hessian of this function, so this is Bacchri, Emmery, Ritchie or curvature tensor equals some multiple of the metric, yeah. So rho is a constant which can be negative zero or positive and the function F is called the potential of the Ritchie soliton. And according to the sign of this constant, if rho is zero, we talk about steady solitons. If rho is positive, we talk about shrinking solitons. If rho is negative, we talk about expanding solitons. Yeah, and why are these Ritchie solitons important? Because in the theory of Ritchie flow, they serve as periodic solution. So just first small remark, if F is zero, you have Ritchie equals a multiple of the metric. This is just the definition of an Einstein manifold, yeah. And you know the classification of Einstein manifolds is already very complicated. So you can expect that classification of gradient Ritchie solitons is even more complicated. Okay, as I said, Ritchie solitons are special solutions to the Ritchie flow. So for instance, to a small exercise, suppose you start with an Einstein manifold where Ritchie is a multiple of the metric and let this manifold evolve under Ritchie flow. What will happen? Well, not much will happen. The metric will stay the same. It will just contract to a single point, yeah. So nothing changes except that we get a rescaling of the metric, yeah. If we do the same thing with starting with a gradient Ritchie soliton and we let it evolve under Ritchie flow, we have more or less the same picture. So we can define, well, look at gradient F which is a vector field. Divide this by one minus two, row T. You get a time dependent vector field here. So take the corresponding flow to this vector field. It gives you a family of diffeomorphisms, five T. And the evolution of my metric, which is the gradient soliton is nothing but pulling back this metric under the family of diffeomorphism and scaling it. So in Ritchie soliton under Ritchie flow only changes by scaling and diffeomorphism, yeah. And these solitons show up in the singularity analysis of the Ritchie flow. And for instance, it has been a problem for a long time. Or it's been observed if you do the Ritchie flow and let it run. And if the Ritchie flow gets trapped by such a Ritchie soliton then nothing is changing anymore, yeah, just by scaling or diffeomorphisms and so on. And so it's an important step in this whole program to understand what kind of Ritchie solitons you can have and to classify them in certain situations. For instance, in dimensions three, yeah, the original program has been. Okay, so what we have so far is we constructed a functional which is monotone along the flow and is strictly monotone except I'm sitting on a steady Ritchie soliton. And steady solitons are not enough. One also needs to understand or in particular, one needs to understand shrinking solitons, yeah. And the idea of Perlman was to construct a functional just as his famous F functional which is also monotone along the flow. Monotonicity is strict unless you are on a steady shrinking soliton, yeah. And I'm sure how he did the calculation has been, he knew what he wanted to have and then did the calculation backwards. And by this way, he came up with this at first sight quite weird looking functional, yeah. So this is now unlike the F functional which was defined on the space of metrics and the space of function depending still on an additional ingredient which is just a positive number, yeah. And you look at this functional, well, in the classical worker, you should assume that M is compact otherwise you already run in trouble that this guy may not be defined. So here you have again gradient F norm squared plus scalar curvature. Then you do something, you add here F, you subtract the dimension, you multiply this, the first two terms by this number tau. And then you write not the volume measure but e to the minus F as we had before but divided by two pi tau to the n half, yeah. And everyone looking at this formula first, my goodness, I've seen that this looks like Gaussian heat current, what is the idea behind that? And I tried to explain this a bit, yeah. But what Perlman did while he writes down this functional looks at gradient flow, so how he has to deform the metric, the function and this parameters that he gets the most increase of the functional and it turns out these are the induced equations. So G should evolve under reach flow, F follows some backward heat equation but there are still some additional terms on the right hand side while the scalar curvature is not surprising gradient F norm squared. It's also okay but here we have dimension over two tau which, well, we don't know yet what to do with that and this tau is just time moving backwards, yeah. And if you take a solution to these three equations or a GF tau evolving according to systems of equation, plug it in to this W functional, look at the time of the derivative, wow, what you get on the right hand side is now Regi plus Hessian minus a positive multiple of the metric, yeah, so we see this functional is non-degrading in time and monotonicity is strict unless we are on a shrinking Regi soliton, yeah. And as I said, I'm quite sure he when developing this functional, he knew what he wanted to have and then probably he calculated backwards to find out how to define the W functional. Well, this looks a bit messy but let's do the same trick we already did before instead dealing with F. Last we introduced a new variable U which was E to the minus F, here I normalize it by this four pi tau to the N half. Then I get something which looks a bit better in as far as the heat equation is concerned. I have the usual conjugate heat equation, this is okay, this is Regi flow and time is moving backward, yes. And just rewriting W instead of the functional of the original F, now with U I get this expression for the W function. Well, and the nice thing is we get something monotonic along the flow. And the term here which governs the slope of derivative of this functional is given by Regi minus has log U equals G over two T. Well, how to think about this functional? And if you look at the term we had, it reminds you about the Gaussian kernel and the entropy of the Gaussian kernel. So just to do a small exercise, look at the standard Gaussian measure on our N and try to calculate the entropy of the corresponding backward heat equation. So just take this here as density and instead going forward in time, you write tau which is capital T minus T and then calculate the entropy of this thing, right? Then, so that's the entropy of the standard backward heat equation on our N. Then you find this expression. Yes, and now you do the following. You have now two such solutions or two such use. So one is on your manifold which is evolving under Regi flow solving this conjugate backward heat equation. And the other one is on our N solving the same backward heat equation. Of course, R is equal to zero, yeah? And you want to compare the two things, yeah? And so you do the following. You look at the relative entropy. You look at the entropy of U which is on M and you subtract the entropy of the standard solution on our N, yeah? And relative entropy means you take the difference of the two, yeah? And then, well, so this one we just calculated we get this expression and you want to study how this depends as a function of T, yeah? So you want to see how much faster the entropy of the U grows compared to the entropy of the standard heat kernel. And if you take the derivative of this you get an expression like this here and taking the derivative of H multiplied by tau, you get this weird W function. I will come back to this in a moment but there is related work of Laini who did quite similar things on a fixed Riemannian manifold. Namely, the idea is the following. Take a static Riemannian manifold, yeah? Look at positive solutions of the heat equation, yeah? Calculate its entropy, yeah? And subtract the entropy of the standard Gaussian kernel. So this will be the difference between the Boltzmann entropy of the measure U of all the X, yeah? To the Boltzmann entropy of the standard Gaussian measure on our N, yeah? And then try to see how this relative entropy evolves as a function of time. And if you calculate the derivative in this setting, you get an integral like that. And so supposing that Ricci is a non-negative, we know that we have the differential Harnock inequality for U, which is usually written in this way but it's easily seen, it's equivalent to the condition here. And this is exactly what shows up over there, yeah? So it tells you in a nutshell the following. If you have a manifold such that Ricci is non-negative, yeah? Then look at the entropy of positive solutions to the heat equation on this manifold. If you can compare curvature, Ricci greater equal than zero, then do the same construction on our N. Look again at the positive solution of the heat equations or just take the standard Gaussian measure there. Look at the relative entropy and then you get a sign for the relative entropy. So it's saying that if you can compare curvature, you can compare gross of the entropy of solutions to the heat equation. And this is something quite interesting because in particular because if you calculate the derivative, then you come up with an expression which is exact, which has a sign according to Liou, yeah? And usually Liou, so that's this inequality or it's equivalent to the one here. It's a typical gradient estimate for you we discussed and people usually are only using that this is non-negative but it tells you more. This formula gives you a quantitative interpretation of Liou because how big this is exactly shows up if you compare the volume gross of positive solutions of the heat equation on M compared to solutions on our N. Well, so in this case H is non-degreasing as function of time while here time runs forward. So don't tell the only interesting point is that you get something more not talking. Well, so coming back, I still have five minutes or something like that. Okay, yeah, okay. So coming back to our original problem which we got from the Parallelmann's W functional, I want to do the same thing now with respect to heat kernel measure. So define the entropy as I did before with this correction term and do the same construct the same functional on our N using the standard Gaussian heat kernel, then you get this guy here and if you work out what this entropy on our N is, so you just block this in and calculate then you see also the role of this correction term. It takes out the linear gross of this thing here. That I'm ending up with the term I like to have. And if I normalize you appropriately so I can compare the two things, my basic object will be the relative entropy. So this functional on M under evolving geometry and the corresponding one on our N. I take the difference and I'm interested in the time behavior of this functional. And well, so I have this H of t which is the relative entropy and from there I define Wt just as before H times derivative and then I find the following two interesting formulas so the first one gives the derivative of this relative entropy. We have here on the right hand side quite a lot of terms but this is very nice because I will show you this expression here in red has a sign usually and this is related to a Liou type estimate for such heat equations under Ricci flow and the W functional has the properties we want. It is, well this E star remember that was Wiener measure with the exponential density. So this functional here is monotone and as I want it is monotone strictly monotone except I'm getting a draped by shrinking a soliton. And so as I said the relative entropy is non-degreasing in time and the right hand side is non-positive due to the Liou inequality for solutions to the conjugate heat equation just as we had in the theorem of Laini for static manifolds. So this here has a sign which tells you that my relative entropy also has a sign as function of time. And the W functional is non-degreasing in time and monotonicity is strict unless we are sitting on shrinking a soliton. So we have constructed now functional, this WT functional which can serve as a tool for our analysis. Maybe just one small remark, the case of dimension two of a surface. There are notions of entropy. So there is a notion of Hamilton surface entropy which is take the scalar curvature, log scalar curvature and integrate it out over the surface. And I think it was Ben Chow who observed that this here, this entropy is non-increasing along the Ricci flow. And this, the point is, on a surface things simplify because in the general setting we always had this pair of G and solution of the heat equation. But on a surface the scalar curvature itself satisfies the conjugate heat equation. So you can simplify your formulas, you don't need this U, you just take the scalar curvature and rewrite the old formulas. But I'm not going into details here but well, a few words about possible applications of these functionals. Well, in, for instance, one can use them to prove so-called non-breather theorems. So what is a breather? This is just a periodic solution for the heat equation. So suppose you have two different times and the metric at a later time is just a pullback of a metric you had a time one before, yeah? Multiplied by some constant here. And according to the value of C you call such a breather or such a periodic solution shrinking, steady or expanding and so on. And one is interested in ruling out such a breezers because they cause trouble when dealing with Ricci flow. So there are theorems like no steady or expanding breezes or every steady breather is Ricci flat, every expanding breezes, gradient solitons and so on. And such things you can approach, for instance, with the formulas I introduced and the advantage is they are well suited also to non-compact manifolds here because the reference measure is always a probability measure. Okay, I think that's a good point to stop and thank you for your attention. I have been asked whether it's possible to put the slides somewhere on the net. I shall do it. So either it's on the page of ICTP and if not, then I put it on my own webpage. Then if you wanna still check some details because working with slides usually is too fast and you first get just a rough idea of what it is about. Thank you very much.