 Okay, thanks a lot. So today I will continue with my program and in particular I will come back to heat equations and I decided also to give you some background on heat equations on many folds in general because this is a very rich object there. So to say analysis geometry and probability theory meet together and probably if you are a probabilist it's the first time you realize that you should learn geometry and analysis, analysis to realize probably it's good to learn also probability theory and I will explain a bit where this comes from. So going back to a very basic setting take a Riemannian manifold and look at the heat equation. So on the right hand side I have the Laplacian induced by the metric here and usually there are two different communities. The people coming from analysis and the people coming from probability theory and usually you can distinguish the two communities if you look here whether there is a factor one half or not, yes a probabilist's right, one half Laplacian analysts ignore the one half as you can see today I'm an analyst and probably also. Sorry, well this is if you take Hodge-DeRam then you have another sign convention but this is a third community I would say. Okay so one would like to understand solutions to such equations and one basic question you can ask, well if you have such a solution what can you say about the gradient of u or if u is positive gradient of log u? How can you understand these things? There is a related question which comes under the name of Harnak inequalities so if you have a solution you would like to compare the solution at the point x at time s to the solution at y maybe at some different time or at the same time, yeah? And actually these two questions are more or less it's more or less the same question because you can look at gradient estimates as infinitesimal versions of Harnak inequalities and Harnak inequalities you get from gradient estimates just by integrating along curves or geodesics so it's basically the same question. Well let's come to the even more fundamental situation where you have a stationary solution which means a function which is harmonic there it's usually good not to insist that u is defined globally on all of your manifold there may be obstructions for that so typically take some domain and try to find a function on u which is harmonic and here I wrote one classical theorem which shows you why you need to learn geometry here. You may say well coming back to this equation okay this is PDE equation of course the operator is defined in geometric terms why should I worry about geometry but you can see it here so this is an estimate for a harmonic function positive harmonic function on some domain giving you estimating grad log u on the left hand side if you look at the right hand side you see this is independent of u so what do you have here? Some universal constant depending only on the dimension of n then you certainly have a term which measures the distance to the boundary because if you think that u is coming from a solution to the Dirichlet problem then the boundary values can be quite arbitrary so you cannot control the gradient u close to the boundary and so you have to take this into account and the right scaling is one over distance to the boundary and here you see you need some other ingredient which gives you a bound of Ricci on this domain from below and that's where geometry comes in when you try to understand such type of equations well as I said this is a classic result some time ago we proved such kind of estimates just using probabilistic methods Brownian motion and we did more or less provided proofs for most of Liou type estimates which are around and usually the method is quite simple and gives you a strong result so for instance of course it's not so interesting re-proving things but we wanted to understand how is the structure of such problems and how you can approach it in probabilistic terms and actually in our proof we get an explicit value for the constant in the classical proof it's just there is some constant and it was a work with Bruce Driver Bruce is very much interested in infinite dimension getting estimates there and if you know a bit the background is a famous book of Dan Struck on analysis on pass space and he writes that Yao asked him whether he can prove something Yao himself cannot prove and so he said well I tried Liou estimate and prove it with probabilistic methods and in this book he has a whole chapter which is entitled by commitment of defeat something like that so he did not manage it and so for some time it was not clear how to do it but well it can be done okay so what I said before if you have an estimate like this here you just integrate it along geodesics then you get for instance Harnock inequality take geodesic ball of radius R then you may look at the maximum of the function on the geodesic ball of half the radius you can compare it to the minimum by some explicit constant which only depends on the dimension lower bound of Ritchie on the ball and the radius maybe you need to measure distance to the boundary well now coming back to the parabolic case I would like to explain you in probabilistic terms there is an exact formula for the gradient of U because this will be important later on in terms of a Brownian motion and this is my notation you remember x small x is the starting point at time zero and this is the position of Brownian motion at time t and well to remind you how we defined this well we said it should be the stochastic flow to this operator so for instance for functions on M you should have this defining property well I need some construction to explain you the formula and this is the following fix a point on M and then define a linear transformation from the tension space at this point to the tension space by solving an ODE where you have several ingredients here on the right hand side you have something which I call Ritchie parallel and what is this I made a diagram where you can exactly see this is a linear application so take a vector in the tension space let Brownian motion run up to a time t starting at x then you are at this random point x t transport your vector along the path of your Brownian motion then you are sitting in the tension space to this random point so here you apply Ritchie Ritchie at this random position where you are then you get something over here and then you transport it back right this gives you this mapping I call Ritchie parallel okay and with this I solve the equation linear equation which I can solve paths by paths a covariant equation along the Brownian motion and I get this family of QTs you see if Ritchie is zero if the manifold is Ritchie flat Qt will just be the identity otherwise it's some deformation of the parallel transport by Ritchie yeah so that been introduced long time ago by people I think Paul Maliova was one of the first he called it transport or transfer amortisse where people talk about the damped parallel transport nowadays so if you do analysis on paths or loop space this is one of the basic objects you should not worry about well of course here by convention Ritchie I can read as which is defined as a bilinear form but or I can define it as linear mapping from the tension space to the tension space by using the metric converting a differential form back to a tangent vector well coming back to my heat equation we already know how to solve this equation while I'm making assumptions that Brownian motion is not exploding but I could do it much more generally so the solution namely the function U at point X at time T I get as follows I take the point X let my Brownian motion run up to time T substitute this in my initial function and then I average over all paths on the manifold so this is a very concrete and simple way to understand this equation and we already have seen how it works you write down a certain process you check this is a martingale then you take expectations the formula and it's not difficult to show that if you look at the differential of the solution to the heat equation that well this is actually the same as the heat semi-group on one forms applied to DF and you have a representation of this guy which is similar to what we have before look what we are doing now it's a PT on one forms applied to DF you take DF you evaluate this at a random point X T then you transform it back to the starting point of your Brownian motion and then you multiply it by this QT or QT I defined on TXM but these are differential forms so I take the adjoint of the QT I defined so here you see there is Ricci sitting inside this is parallel transport and this is the initial condition well how to check this same thing as above you look at this process here and check this is a martingale then you take expectations for S equals zero you have just the left hand side and for S equals T you have T minus T then gives you DF taking expectation you get this formula so this is a typical representation of the heat flow on one forms right and of course you get from formulas like this immediately estimates namely well if you estimate the right hand side you have to estimate this QT and for this you need exactly lower bounds of Ricci so if you have lower bounds of Ricci you estimate the stamped parallel transport and you get immediately estimates like this well I come back to this later on but looking at such formulas you see on the right hand side you have the differential of F sometimes you would like to have a formula where no derivatives of F are involved so for instance if you heard the talk yesterday of François Laitrapied there was such a kind of formula where you want to look at the gradient of PTF but on the right hand side you don't have the derivatives of F and I show you one type of these formulas so here it is take the solution of the heat equation and look at the differential or the gradient of U in some direction V given as a tangent vector then you have this formula where I have to explain what the different terms are the first thing we already know this is the initial condition Brownian motion starting at X running up to time t which is actually the term we needed to represent U in stochastic terms then we are multiplying it by some stochastic integral and what is this? well here I have some upper limit of my integral I could integrate up to time t because t is the time I'm looking at my solution but I can already stop I can take the minimum of the first time when Brownian motion starting at X will exit some neighborhood of X I take some relatively compact neighborhood about this point and I stop the process as soon as it goes out of this neighborhood so this is nice because usually to make this well defined I need lower bounds of Ricci but on a compact set Ricci is always bounded so I don't need any assumptions to write down my formula well I have to continue what is set this is just a flat Brownian motion in the tangent space and here is my Q I defined in terms of Ricci and L this can be any adapted process of finite energy so finite energy means if you take the derivative then in some sense it should be L2 or for experts it would mean it's a process taking values in the Cameroon-Martin space and this process should have the following properties so it should not be too weird it should not have a martingale part I wrote here absolutely continuous pass it should start at zero at the point V so remember V is the direction I'm differentiating it and it should be zero no later than either I'm at time T or I'm exiting this small neighborhood otherwise I can choose my L as I like right and I can also choose my D here as I like but of course if I take D very small I have to assure that at the time the process is exiting the L must be zero so I must drive it to zero very quickly which gives me here a large L dot so it depends on the time of the problem I'm considering if I have a huge manifold I can just wait up to time T then it goes to zero by itself I can choose L even deterministically and to get such a formula here so everything is explicit we have here Brownian motion we have this process we can choose here we see the influence of Ricci how Ricci comes in and this is just the initial condition of course from here it's very easy to get estimates of the gradient of U in terms of the initial condition F well this is typically called Bismuth type formula and the way how to prove it in particular if there are stopping times that have evolved is well of course it's some integration by parts on the venous base but the point is you should do integration by parts always on the level of martingales if you do this on the level of martingales you can still introduce stopping times you stop the martingales there are still martingales and at the very end you should take expectation which gets you back to the level of PDEs that's the strategy and quite often people do it the wrong way and then they end up taking derivatives of expectation of functions where some stopping times are involved and stopping times usually depend very badly in the sense of taking derivatives because if you disturb things a bit and then you may not exit at a given point you may come back to the domain and so on okay so this here more or less as background about heat equation on Riemannian many foods and remember the program we are interested doing things with respect to moving geometries and I guess it's definitely not a surprise that all I explained so far can be adapted to this situation because all my arguments rely on martingale arguments and this just means I have to adapt the setting to get things now for instance heat equation under geometric flow like under Ricci flow or later on I will come back to the work of Perlman by looking at the conjugate heat equation but this will probably be tomorrow so just to remind you we constructed this object we called Brownian motion on space time and here I do it by fixing some time capital T and let then time run backwards from there which gives me Brownian motion with respect to moving metric but at time R I really have the metric G T minus R that's the right setting to deal with such equations here than as before if you have a solution let time run backwards capital T minus R and here you substitute the process starting at X at time zero and let it run up to time R this will be a martingale taking expectations you get a formula as before expectation F of some process the only thing which is more complicated is this S here because it's no longer homogeneous but take S equals zero you have a formula exactly as before so if you take S equal zero in this equation well there is similar representation for the gradient of U the only thing I have to redefine is this damped parallel transport this QT and I have to add to reach G another term which is given by a time derivative of the metric so if I take a general evolving or a family or one parameter family of metrics I have to introduce this derivative of the metric to get the right damped parallel transport then I can go into the formula and everything works as before but you see here this is interesting because well if we suppose we are following the reach flow this will be zero here so it means that my QT is just the identity that means in my gradient formulas at the end reach will disappear and this is a well-known observation that in some sense many folds evolving under reach flow behave in many respects like reach flat many folds because reach by the dynamic point of view reach is killed by the deformation of the metric the metric is deformed in such a way that it kills exactly reach and then you can see here that this will go away and you get gradient formulas which look like formulas you have on a reach flat on flat spaces and well I come back to this immediately well I would like to say something about how to relate curvature bound to certain type of gradient estimates so again I'm going back to the case of a static Riemannian manifold to show you the idea and looking at this formula for the gradient of U I told you already you get such an estimate for the gradient so look here you take PTF which is my U where F is the initial condition you take gradient of PTF on the right hand side you have PTF of gradient norm so this is the heat flow when you start with gradient of F and if you have a lower bound of reach it was obvious from my formula that you get such an estimate and actually this is equivalent so you can show having a lower bound of reach e curvature is equivalent to such a functional inequality if you do it in L2 you get a similar equation just with L2 and you can write down a lot of equivalent conditions so you can express it also in terms of a loxobolef inequality you can write it in terms of a Poincare inequality these are all conditions equivalent to lower, to characterizing lower bound of reach e and there are still, I could continue probably until tomorrow to write you conditions and this is still a very popular domain of research so if you know a bit the work of Lot, Villani, Theo Sturm and so on they are very much interested in understanding curvature on spaces which are more general than Riemannian manifolds to define reach e, well you need a differentiable structure but if you have, if you can give equivalent conditions which may be which are more robust and make sense on more general spaces so for instance here if you have corresponding heat flow and you can define the gradient of a function then you can make out the sense of such equations and you have a chance to define reach e bounds at least in situations where there is no obvious notion for reach e there's also lot of conditions in terms of entropy the convexity of entropy if you take papers of Theo Sturm but the point is by such equations you can only characterize bounds for reach e, you cannot really characterize reach e by itself so this is still a bit far off of what we like to do well coming back to the heat equation along some geometric flow I denote the solution well as I said this is not homogeneous anymore so I take the initial condition at time s and I look at the solution at time capital T so I get a two parameter family of semi-groups indexed by s, t and f is the initial condition and what I just said before looking at such gradient estimates I said evolving under reach e flow would correspond to being reach e flat in a static setting so one expects estimates where this exponential factor would be just one and so we can do the following well look at so-called super solutions to the reach e flow which corresponds well if you take the analogy to static manifolds reach e flow would correspond to reach e flat and super solutions would correspond to reach e non-negative and for reach e non-negative if we go back to the gradient estimates the k is zero because zero is a lower bound for reach e so it's not surprising that the gradient of this two parameter semi-group can be estimated by the semi-group of the gradient so here there is no exponential factor because as I said if you go to my formula I explained which tells you how reach e enters into the equation reach e is killed by or it's you have on the well it's not completely true what I'm saying if you go to this dammed parallel transport you see you get some term which is bounded below by zero and this you use for estimating and doing as before we can write down functional inequalities now characterizing super solutions to the reach e flow completely analogous to what we did before characterizing lower bounds of reach e in terms of gradient estimates you see here on the right hand side there is no exponential right but you may ask okay super solutions to the reach e flow this is probably not so interesting can we also characterize solutions to the reach e flow so can we write down estimates which characterize not just super solutions obvious these kind of estimates are equivalent to the greater equal here there are two week but can we find other inequalities which characterize exactly reach e flow and it turns out yes we can do but we have to go from the manifold to the pass space over the manifold remember if looking at the heat equation you take the initial function you evaluate it at x t and then you take the expectation so it means this is on pass space this is just a one point cylindrical function where you have a pass you evaluate it at one point and then you take the expectation okay let me explain the setting I'm going to deal with we go to pass space so pass space now on our space time manifold time as usual is running backwards and I have a pass on m and I carry a long capital T minus t so this gives me a pass in space time and brownian motion based at x capital t I define it gives me a measure on pass space measure induced by this process so x t is the brownian motion with respect to the family g capital T minus t of matrix and the important point is this gives me a measure on the pass space the pass based at at x t so at time 0 x t will be x and this will be capital T well on given such space time curve I take an evaluation at k points between 0 and t so I evaluate the x component here at time sigma 1 sigma 2 and so on sigma k this is what people call a cylindrical or one can define cylindrical functions on pass space which only depend on this or which factorize via this evaluation at k discrete points so a functional on pass space which is defined by looking at the pass x only at this k different times and so evaluating combining this sum u and mk which is nice compact support and so on will be my notion of a cylindrical function on a pass space well then I define type of gradient on a pass space this is basically the same as a Maliova derivative on a pass space written for a cylindrical function so you take derivatives of this u in the different direction parallel transport it take the sum and take this thing here composed with the evaluation at the discrete points which gives me a notion of of parallel gradient for cylindrical functions on a pass space well with this object I can now go forward and can really characterize solutions to the Ricci flow namely I get I need to have gradient estimates but no longer for one point cylindrical functions as I had before such estimates are too weak but I have to go on pass space and I take the gradient of expectation of such a cylindrical function but estimating by the parallel gradient or Maliova derivative of f expectation of this so this here should be also an e like the one here as before the only point is it's no longer on m it's on pass space and what I'm saying this is an equivalent condition to a cheat following Ricci flow and well this is interesting because as I said before people on many faults or spaces more general than many faults have been interested in characterizing Ricci in terms of functional inequalities and it's a longstanding problem in the theory of Ricci flow how to or to come up for instance with a notion of weak solutions to the Ricci flow equation because of course if you look at the program of Parallelman while he lets Ricci flow work and at certain points singularities appear then he investigates the singularities he does some surgery and then he continues with the Ricci flow and until he ends up at infinity and there he sees something interesting but as soon as you have a singularity in your equation it's no longer well defined so one would like to have a notion of weak solutions or something which makes also sense going over singularities or something like that and while here you see at least something which is very promising in this direction because you have a characterization in terms of functional inequalities which you may be able to extend or to make sense of such inequalities also in cases where you have trouble with the original Ricci flow okay if I understand correctly I have to finish now thanks a lot for your attention