Ok, so, maybe we can start. Ok, so in the last lecture, in the last two lectures, actually we saw an equivalence result at the level of calculus. I mean two different ways to define a gradient and their equivalence. Now, in this lecture, and I think in the first hour, we will be speaking about another important fundamental equivalence result, which is equivalence heat flows, particularly view as gradient flows. Ok, so, and of course I will spend some time explaining what we mean by gradient flow in different situations. A more precise statement is that the L2 gradient flow of the trigger energy, ok, so this is the standard object that we already considered in the first lectures to prove the equivalence theorem at the level of sobole spaces, coincides W2 gradient flow of the entropy. Ok, this second part sounds a bit more mysterious because I have to explain what I mean by gradient flow of the entropy in a metric space, which is the space of probability measures. And of course, to compare these two gradient flows, you should stay in a common domain for the two and the common domain is the class of non-negative functions with integral 1, right? It is clear that if you identify these functions with measures viewing F as a probability density, then you can look at the evolution problem in L2 or the evolution problem in the space of probability measures. And the final result is that the two are the same under some additional assumption. Ok, and to explain this, I will really start with the basic result, which actually motivated also the boom of optimal transportation, which is a result in 1998 by Jordan Kinderlerer. So, they've been the first to realize that there is this connection in Rn. So, I will start heuristically explaining why there is this connection in Rn for the standard data question. And then, of course, we will try to make things more precise and more abstract in general metric spaces. So, from now on, let's say we are in Rn. Ok, so in Rn, of course, we have all the differentiable calculus, the standard differentiable calculus that we can use. And the leading idea is to make p over n as a kind of, first as a differentiable manifold. The next step is to turn it into a kind of Riemannian manifold. I'm using quotes here because none of these things is rigorous. So, it will not be a differentiable manifold in the standard sense of differentiable geometry. So, not really local coordinates and not even a Riemannian manifold. However, this heuristic picture will be very important to understand what is going on. So, let's start from the differentiable manifold structure. The differentiable manifold structure is again based on the continuity equation, which you have seen appears many times. So, this is the basic continuity equation. And then the idea is that we can consider p over n as a differentiable manifold considering that there is this coupling between infinitesimal variations of the measure and the velocity. Ok, so if you stay at a point mu, so I want to describe the tangent space, I want to describe at least formally the tangent space to mu at the tangent space to p over n at the point mu, then I consider this or this as a tangent vector and they are identified through the continuity equation. Ok, so in this sense this provides to me a kind of differentiable structure and then I can define a metric. So, again, given two tangent vectors, I call delta mu delta prime mu these two tangent vectors, I define the metric to be, I define the metric to be the integral of v dot v prime in the mu, where v and v prime are the corresponding vectors associated by the continuity equation. So, you see this is a non-constant and non-flat metric because it depends on mu. It's not the standard del 2 metric. Actually, to make a connection also with PDEs, it appears like in the theory of optimal maps that the best thing to do is to take gradients. The best thing to do in this picture is to take gradients and if you are taking gradient, let's say if mu is as a density with respect to the back measure, the connection between velocity and variation of the variation tangent vectors is given by a PDE minus divergence of rho nablafi equal to delta mu. Ok, so you are really solving an elliptic PDE, well, not uniformly elliptic because rho will go to zero somewhere, but let's say locally uniformly elliptic if rho is smooth PDE. And so the connection between the velocity is given by solving a PDE. Ok, and then let's try to see that if we use this differentiable structure and this remaining structure formally, the heat equation is precisely the gradient flow of the entropy. So let's say, why? Of the entropy, of course, I mean the entropy with respect to the back measure. We are in our end. Ok, so first of all, let us compute the differential of the entropy. So I want to compute the differential of the map rho integral rho along delta mu, where mu is rho times the back measure. Ok, so this means that I have to take a solution to the continuity equation. So the variation of mu is coupled to mu by this equation and I have to differentiate this along this solution. So I get that this is equal to 1 plus the logarithm of rho times delta mu. And now I use the fact that delta mu so here I am really differentiating in the classical sense the entropy function. Now I use this identity. Sorry, here I get nabla or log rho times v rho dx. And you see that here we should not simplify, we should not simplify, I mean we could compute the gradient of the logarithm but it is better to leave it in this way. And you see that according to this scalar product so I have computed the action of the differential on delta mu and it is given by this scalar product. This tells me that the gradient to pass from the differential to the gradient I need the metric, the gradient in the vastest sense of the entropy is precisely given by this vector. So here I am using not only the differential but also the Riemannian structure to write this down. And then once I realize that the gradient of the entropy is the gradient of the logarithm of the density what is the gradient flow now according to this structure. So if you have any functional f of mu how can you write the gradient flow of f in this Riemannian structure? Well, you should write that dt mu plus divergence of vt mu t is equal to zero this is the usual continuity equation and then you should say that the velocity is for any time opposite to the vastest ingredient of your energy function. So this is a general way if you have not only the entropy but any functional measures if you write the gradient flow you should combine continuity equation which is due to the differential structure with this equation which uses really the metric. OK, now in the case of the entropy I mean if mu t is equal to zero in select value measure here you get precisely that the velocity is equal to minus, is equal to gradient I think I made a mistake with the sign somewhere here there was no minus sign perfect. And so here we get minus the gradient of the logarithm so in the particular case if f is equal to the entropy here you get minus gradient of the logarithm of rho and then of course if you insert now this information here you get precisely the heat equation you get dt rho t equal to divergence on nabla log rho t times rho t which is maybe a funny way the heat equation is an linear way to write the heat equation but it is precisely the way to understand that it is the gradient flow of the entropy with respect to this Riemannian metric. OK. So this is really the formal point of view and just also to make the connection with the rigorous theory I just say briefly that the induced Riemannian this is not so formal a posteriori can be made rigorous because the induced whenever you define a metric you can define the induced Riemannian metric the induced Riemannian distance just by minimizing the option and the induced Riemannian distance on p over n is exactly the vastest end distance that I mean we introduced that we do introduced using optimal transportation so this is a completely Riemannian and differentiable way to describe the optimal transportation problem OK, now my goal my goal is to to see why to illustrate why the same ideas well not really the same because here we use a lot of differentiable structure that can be applied in a general context and why they are useful OK, so first of all I should explain the sentence a gradient flow of the entropy x with respect the vastest end distance OK, now we are back to a metric setting and also p over n from the rigorous point of view how can we give a meaning to this statement well, here there is a very nice idea of the georgia idea of course we see that always the same ideas coming to play again here the idea is to use a sharp rate of energy dissipation so let me first explain this idea in a Hilbert space so in a Hilbert space I have a functional phi and I have the standard formulation of the gradient flow and I want to find a way to formulate the gradient flow equation which is a system in this case infinite dimension if h is infinite dimension I would like to describe this using only the metric structure and not the differentiable structure how can I do this well, the idea again is to look to minus the derivative of phi of y of t along a generic trajectory y well, by chain rule this is equal to gradient phi of y y prime then you can always bound this from above with nabla phi of y prime ok, so let's say sorry, let me put minus here and so here I have a quality let me emphasize the case of a quality here we have a quality if and only if y prime is parallel to minus gradient of phi to y right and then eventually I can apply the young inequality between I mean to positive numbers and I write it so you recognize in some sense the scheme that we already used yesterday but we apply the scheme in a different context I apply the young inequality and then again here I have a quality if and only if the modulus of the velocity is equal to the modulus of the gradient ok, so this is true for any curve, why? so for any curve minus the energy dissipation is bounded from above but this sum of squares let's say that we let's say what happens if we impose that along our curve the energy dissipation rate minus the energy dissipation rate is maximal if we impose that it is maximal I mean it is given by this quantity then we are saying at the same time that we have a quality here and a quality here and so we are encoding the fact that the modulus of the two vectors are equal and that they are parallel and so we are saying that this equation is true ok, so in this way we encode into a single inequality which is an energy dissipation inequality a sharp energy dissipation inequality the whole system ok, so now let me write down this in a metric space while I will go instead over a point wise formulation an integral formulation so I say that y of t, x of t is a gradient flow you know we respect to d, we respect to a distance d of course if, well first of all well I should say that x belongs to the domain of phi, is locally absolutely continuous and then I'm writing this in an integral form so I say that the energy for any positive time t the energy at time t plus the energy dissipation as I said I write the energy dissipation in this form and here you see I am using the descending slope because I want a gradient flow to stop at minimum points dr is less than the energy at time zero let's say the energy at time zero and as you see this is a totally metric formulation of the problem because here I am using the metric derivative here I am using the slope posteriori this is equivalent if you are in a smooth setting to the ODE formulation ok, so in this way we can give a precise meaning to the notion of gradient flow of the entropy so we just take p equal to the entropy well ok, here I should maybe call y as a same distance ok so now let's see what is the connection between let me give you the idea of why L2 gradient flows vastest gradient flows well, we need an assumption under the assumption the entropy the slope of the entropy is lower semi continuous there is a technical assumption which we will see is satisfied whenever you have a curvature condition this assumption is satisfied ok, so if here I will not detail this part the lower semi continuity of the entropy of the entropy is the ingredient which leads you to the equality between fissure information and slope of the entropy yesterday I observed that in general one has only an inequality also because this might be trivial in some situations and the lower semi continuity of the entropy is the one which guarantees that there is actually equality here and so also that this object is not trivial and under this condition ok, let's say, what can we do we know precisely the energy dissipation rate in this case so we know that minus d over dt of the entropy ftm remember, we computed this very uncalculus ok, in our case we know exactly what it is so we need only to show that these two objects are related to the energy dissipation and basically you see we already one piece which is this one so the slope of the entropy we already stated that it is equal to the fissure and so I need only to know to conclude the metric derivative is less than squared is less than again fissure information ok, but this again was kuvada lemma if you remember this was precisely the kuvada lemma which we proved yesterday so put in together this and this you have that L2 gradient flows R v assistant gradient flows in particular you have an existence result for vasestan gradient flows ok, to show that they coincide one needs uniqueness theorem at the vasestan level because we already know that L2 gradient flows are unique just by the theory of maximal monotone operator but we have embedded this gradient flows in a larger class or gradient flows, a priori larger if you are able to show a larger class is also as uniqueness then of course the two flows should coincide ok and it is precisely here that curvature conditions start really to play an important role so now I really start to introduce what I already presented in the first lecture the CDK infinity assumption analytic implications ok, so we say that CDK infinity holds if for any mu 0 mu 1 in the domain of the entropy so if you take any two measures with finite entropy then there exists a geodesic such that you have the convexity inequality actually a k convexity inequality in one and here there is a modulus of continuity minus k over 2 times the square distance ok, maybe now in some statements to simplify I will put k equals 0 but this is the general definition so let me just emphasize an important technical point here we are just saying existence this is some geodesic what happens if we we put the condition that this is true for any geodesic if we a priori we impose a stronger definition well then you still have a strong consistency because in the Riemannian case geodesics are unique but a priori the problem is that you lose stability because you know when you take a limit in those spaces to show that for the limit space you have the convexity inequality what you typically do is to try to approximate a geodesic in your limit space by geodesics in the approximating spaces and a priori there might be in the limit space geodesics which are not reachable by approximation so the only apparently for the moment the only way to get a stable condition in existence quantifier ok so what are analytic implications of CDK infinity well basically all these things now will come from the typical monotonicity properties of convex of differential equations of convex functions so the most important ones are that duality formula for the slope duality formula for the slope so in general the slope is a supremum a lean soup but you can use convexity to show that this lean soup is a supremum is the supremum and here there is now a correction due to K minus K negative part of K over 2 divided by the vastest end distance between you and me and here you can take also positive part ok and for instance in the case K equals 0 you see I simply replace the lean soup by a soup lean soup which defines the negative slope the descending slope ok this is a simple convexity argument and out of this this implies that the slope is lower semi continuous lower semi continuous why because I have written the slope supremum of functionals indexed by nu each one of these functionals is lower semi continuous with respect to mu so all lower semi continuous with respect to mu keeping nu fixed this is easy because the entropy is lower semi continuous this is continuous and so remember this assumption played a role in the agreement slope of the entropy so we know that in our case this is really the feature information another important property which I think we will use a little bit later on is the upper gradient property maybe I write this here so remember I already introduced the upper gradient property in yesterday we can say that the negative part the descending slope is an upper gradient for the entropy so explicitly this means that you can estimate the difference of the entropy between two measures you can estimate by the integral of the slope dT because our definition of slope of the upper gradient in a metric space well this is a non trivial property because it will be easy to check for leap sheets functions but of course this is a convex function can be very discontinuous so the verification of this property is not trivial let me just give you a quick a sketch of the proof the idea is that you can use this property to prove a kind of one sided local and now I will explain what I mean by one sided and local leap sheets estimate let me simplify infact I already erased the additional part let me state this for k equals 0 for k equals 0 this is the formula for the ok so in particular what you get is that this bound the entropy mu minus the entropy at nu by this formula with the distance between mu and nu times the slope of the entropy at one of the two points precisely at mu and so you see that you might consider one sided because you don't have the modulus here local because you don't have a constant infront of the distance leap sheets estimate and it's a surprising fact that however out of this one sided and local leap sheets estimate you really again you can get that it is absolutely continuous the entropy along any absolutely continuous score inequality with the modulus this is a non trivial fact it's a calculus again it's a calculus lem in one variable ok, I think now we have all the ingredients to to obtain this theorem which is precisely the uniqueness theorem that we needed at the vast extent level this is a result obtained by Nicola G 3 years ago which says that under the CDK infinity condition as a consequence we know that the two classes of gradient flows coincide ok actually this is an interesting case of uniqueness without contractivity and I will explain this after giving the proof so in the libertian theory of gradient flows on maximal monoton operators that we saw in the first day we had at the same time uniqueness came out of contractivity property right here this is a case where we are able to get uniqueness even if we don't have in general contractivity ok, so how the proof works is a convexity argument ok, so the proof uses the following ingredients well, first of all this this you can check by exercise this map is jointly convex convex in the standard sense not in the geodesic sense this is jointly convex in PM plus PM I mean convex in the sense of standard convex structure of PM as an affine subspace of the spatial measures so this is convex this is the first ingredient and this implies in particular by differentiation this implies in particular that the metric derivative of a convex combination so if you take a convex combination of two curves and you do the metric derivative of this convex combination of squares this is convex in the squares because convexity here is at the level of the square distance this is the first property X and the second convexity property which I think is well known is that this just comes from the strict convexity of the entropy function this is strictly convex and I think I don't need anything else I need also this is also not trivial but the slope also is convex and finally I will use also the upper gradient property so I need this strict convexity property and for one of them I have strict convexity plus the upper gradient property so let's take two solutions from mu bar and I write the energy dissipation inequality which is the george's definition so the entropy on mu i t say for any positive tan t this is less than the entropy at the initial time now I do the convex combination of the two so maybe I will call this mu t and I know that at the level of convex combination everything improves because I have convexity and actually strictly improves as soon as for some t these two measures are different so I get that the entropy on mu t for sufficiently large t as soon as I find the t for which they are different the minus of the entropy of mu t square the t is strictly less than the entropy of mu bar and here I use the strict convexity of the entropy well how I reach the contradiction here the realistic idea is that in some sense this energy dissipation rate is maximum so in some sense I have found a solution whose energy dissipation rate is too large a curve for which the energy dissipation rate is too large how I can make this rigorous now I applied the upper gradient property and this is where the CDK assumption plays an important role by the upper gradient property this is less than the entropy of mu t plus the integral between 0 and t of mu t prime here I really needed the upper gradient property ok and then eventually you simplify the extreme the entropy term here and you get that this integral is strictly smaller than this integral which is not possible and so we get a contradiction so this means that the gradient flow is unique why I am saying that this is an interesting case of uniqueness without contractivity well because if you are sufficiently nice setting you can really compute more or less explicitly all these objects and so you can make in a symptotic analysis this was done by OTTA and STURM so OTTA and STURM made many papers on finseler spaces in particular let's say that we consider Rn with norm and we let back measure so this will be my m and then what they prove is that the gradient flow the heat flow so now we know that there is a unique heat flow when I say it flow can be understood in any sense the heat flow is contractive is W2 contractive if and only if norm is ilbertian so it comes from a scalar product so in general even in this simple situation we have a uniqueness of the gradient flow but you would never have contractivity unless you are ilbertian ok and we will see in the second part in the last part of my lecture where also this assumption plays a role improving contractivity in general metric spaces so already in light of this result you start to see that this is an important assumption which of course translates into the fact that the trigger energy is quadratic or translates into the fact that the heat flow is linear so in this setting all these things are equivalent ok very well ok I think maybe this is a good place to stop for 15 minutes and then we go a little beyond the CDK infinity theory ok so in the last hour I can really reach the latest papers that I announced in the introductory lecture so now we can really speak about the Riemannian spaces with curvature bounded from below with Ricci curvature ok so I think I will not have time or maybe very little time at the end to speak about the dimensional theory so I will just assume n equal to infinity which does not mean that the spaces are really infinite dimension they could be only means that we are not putting any upper bound on the dimension so we speak only about CDK so lower bound on curvature no lower bound on the dimension and we added an R to remind that we have a stronger definition in Riemannian definition and the basic result of my paper with Gili and Savare my second paper with Gili and Savare is that the following three conditions are equivalent of looking at the theory for different canality of the infinity of the dimension or if I look at either one or higher you mean on the real line? no sorry dimension is infinity no no in fact it is a bit misleading because n in this theory is a kind of upper bound on the dimension so when you say n equal to infinity you are saying I am not putting any upper bound on the dimension so you can safely say that Rn is CD zero infinity of course this is not a very precise description because you already know that n is finite dimension but you are simply neglecting you are not considering for the moment any upper bound on the dimension in fact the more precise information but I don't know if I will have time to reach this is that this space is CD zero n anyhow one should not think that all these spaces are necessarily infinity dimension equivalent? well first of all we are enforcing the CD k infinity condition so we say that it is CD k infinity and the heat flow is linear but as I said this equivalence between 1 and 2 is really easy in consequence of previous results you could say that let's say the l2 gradient flow ok let me write in this way l2 gradient flow of the chigar energy of the chigar energy is linear or you could say as we said that the vastest time gradient flow of the chigar energy is additive we can't really say linear because we are on an affine subspace but we can say that it is additive and the equivalence between these two parts is clear because we proved that these two objects are the same in their common domain pardon? oh sorry you are right entropy while the really new condition which is equivalent to either 1 or 2 and I need to explain is that any new bar in POVX is the starting point of an EVI EVI k solution EVI k gradient flow of the entropy ok so this third characterization calls for for a new definition of gradient flow stronger than the previous ones which is the EVI the notion of EVI gradient flows and before I go through the definition of EVI I explain why in some sense we need this additional condition well we should keep in mind that our goals are always to keep consistency so we are doing a stronger maximatization we want to keep consistency and stability otherwise in such as this theory is useless ok what happens if we add that the shear energy is linear is clearly still consistent with the Riemannian case right? but stability is extremely difficult to show using only we know that by itself taken along the CDK infinity condition is stable but the combination of CDK infinity and the linearity of the shear energy is by no means elementary so at least we don't know any proof which does not use property 3 and maybe some of you know that there are also from the point of view of calculus of variation convergence there are examples of limits of Riemannian matrix which are not Riemannian for instance in homogenization so it's very easy to lose the quadratic structure when you take limits and so I mean this is a nontrivial fact the fact that the combination of the two is still stable so we still have a good theory is precisely due to this characterization and now we spend a little time explaining what is the notion of EVI again I can go back to the Hilbertian framework I am in a Hilbert space I consider it a solution the notion of EVI goes back at least was strongly inspired by some work of Beniland in the 70s what can we do also we can we can write the sub-differential here it is more convenient to write this in terms of a sub-differential inequality sub-differential inclusion in fact this theory applies to convex functionals so the EVI theory in some sense is designed for convex energies so let me start from this and I can write this differential inclusion in this way that phi of y minus phi of x t is always greater than so minus x prime is in the sub-differential so is minus x prime of t y minus x of t let's say for any t for any y ok let me assume that the curve is differentiable everywhere so let me activate this definition ok but now the point so phi convex ok now the point is that I can write however this has the time derivative of one half x t minus y square and you see that if I say that phi of y minus phi of x t is always greater than this derivative I'm using only like in the georgie definition and so I say that let's say in a metric space y dy in absolutely continuous curve y let's say in AC2 y is an EVI solution ok now I will put it into also the parameter k I mean in this inequality k was equal to zero and in general you have an additional term if the time derivative of one over two distance y square y t z plus phi of y t is always less than phi on z so you have a kind of test you have to test the derivative is like in the alexandro theory you have to test the time in this case the time derivative while in the alexandro theory you are testing the convexity properties you have to test the time derivative of the square distance from a generic point z and you want that this derivative plus the energy at time t should be less than phi time z of the test point z and I forgot sorry I forgot here I should add the k dependence is plus here you should add the k over 2 when you have a k convexity I mean if phi is k convex here you should add k over 2 so the derivative plus phi plus k over 2 d square of y is less than phi well a few words about this notion this is much stronger than the gradient flow based on energy dissipation so for instance just to illustrate to you the strength of this condition I will show you how EVI implies contractivity again which means that in this situation we were before for instance of a non-ilbertian norm there is no possibility to get an EVI solution whenever you have EVI solutions they are always contractive so this means that this notion of gradient flow is stronger than the one based on neural energy dissipation for instance let's say that you have two solutions y1 and y2 EVI let's say let me put k equal to zero maybe I just give a sketch of the proof of meeting some technical details but then of one half of y1t z is less than phi of z minus phi of y1t this is the EVI for y1 but then you have also the EVI this is a kind of argument which comes also in the laws it's a kind of kruzko scheme of t and then the idea is which can be made rigorous even if you have inequalities for almost a very tie is to take z equal to y2 here and w equal to y1 here and then you add the two inequalities by 2t so by labinets rule this is the sum of the two partial derivatives plus plus one half d over ds one half s equal to and then to estimate these two partial derivatives you use precisely the EVI information and you get that this is non-negative so this is a formal proof but it can be made rigorous and so you have contractivity for free and I mean this property has very strong implications so EVI EVI implies as we saw contractivity it implies also stability as I said it is quite easy to show that this notion of gradient flow is stable and this is much less elementary this was proved by denerian Savare this implies convexity along let's say k convexity along all geodesics and you see this is an extremely interesting fact because as I explained at the beginning of today's lecture there is no reason no abstract reason why by itself convexity along all geodesics should be stable conditions because there might be in the limited geodesics which are not reachable and so on and it is surprising the fact that you can find a stronger property which is the EVI which is stable and implies this is a priori unstable condition ok and now we will give some ideas about how we can connect conditions 1 and 2 to 3 and this is where really all the calculus that I explained in my first lecture plays a role more precisely I will explain how one can pass from one or two how we can try to derive the EVI property assuming at the same time convexity and quadratic structure of the trigger energy so the scheme of the proof is this there are two ingredients one which is elementary as we will see which is the derivative of a sustained distance along absolutely continuous course ok in the EVI formulation it appears derivative of this object and it is computed and this is the most the more elementary ingredient the really non trivial one is the derivative of the entropy along geodesics ok so let me explain the little bit the two so we have my curve in the space of measures I have my test measure which I call nu remember that in the EVI formulation there is a test point to be used and I consider the optimal transport problem between mu t and z associated to this problem as you have seen in Guido's lecture there is a Kantorovich potential ok which I recall is defined in this way that W22 mu t nu is equal to the integral I hope I am following more or less the same conventions of Guido plus the integral of phi of the c transform of phi t the nu ok so for any for any t I have a maximizer in the dual formulation which I call phi t Kantorovich potential ok now using this I can really compute at least the first part the derivative of the vastest distance and here the observation is that since phi t plus phi t c my definition is less than than the c of course here my c is the distance square I know that W22 phi t plus h nu is always less is always larger because I am in the dual formulation is always larger than integral on x of phi t plus phi t d mu t plus h plus the integral of phi t c d nu so this means that at points where this distance is differentiable at any point I can compute the derivative of this function knowing the derivative of this function is the usual comparison argument which appeared also in the proof of Kantor and for instance so this means that the time derivative for almost every t the time derivative is equal to the time derivative of this but we know we know what is this derivative because mu t is solving the heat equation maybe now I will start to call mu t in t equal to rho t times m dm and now I will write it in this form I mean I am just using the usual integration by parts formula I can write this less than the limit or the limit as epsilon goes to zero of the trigger energy of rho t minus epsilon epsilon phi t minus the trigger energy of rho t divided by 2 epsilon ok so roughly speaking what I am computing here the Laplacian is not linear a prior in rho but this expression is linear in phi so the expression higher here is the differential of trigger energy at the point rho t or rho tm along the direction 50 differential should be linear ok this is based on the integration by parts formula ok so let me remember that there was an inequality between minus see if I can reconstruct this there was an inequality between minus phi delta rho dm was always larger than phi star d rho was the integration by parts formula and this is smaller actually it is by convexity by convexity I think you can check that this is smaller than this one if I remember well well if you assume that the trigger energy is quadratic is clear right but but ok I think here there is you know maybe you should not apply the integration by parts formula but you should really really revisit ah no no no this is definition of sub differential right you apply the definition of sub differential minus delta rho is in the sub differential so you see you can bound this difference by you see what I mean by definition of sub differential ok let's apply the definition of sub differential is greater than minus delta rho because this is the element in the sub differential times minus epsilon phi and then you divide by epsilon there is a 2 somewhere yeah yeah now maybe the normalization the normalization the trigger energy was normalized with the factor 1 over 2 so there is no 2 here ok ok so you get this expression I'm not using yet the assumption that the trigger energy quadratic I want really to emphasize the point where this assumption comes into play I will assume k equal 0 also still to simplify some computation well the non trivial fact is that let me go back now to this picture I was doing before I'm taking out the geodesic maybe I will call this geodesic mu s t mu s t will be the geodesic from mu t to nu so the index s will run between 0 and 1 t will be just at the time parameter in the heat flow and the second ingredient is that the derivative of the entropy along the geodesic so basically you prove this property you prove it for any fixed time the time t plays no role in this proof is just a geometric formula of the derivative of the entropy along geodesics this is larger than another limit which is the limit as epsilon goes to 0 of the chigar energy of phi t minus the chigar energy of phi t plus epsilon f t rho t divided by epsilon and so you see that well I will not have time to explain where this formula comes from it really uses the calculus that I described in my first lectures more or less you see we already computed with this calculus a difference quotient of the entropy so more or less the ideas already appeared in my lectures on how to compute estimate the difference quotient of the entropy using optimal transport here I just want to emphasize the difference between this term and this term because this is again a kind you can view this as the differential of the chigar energy but now at the point 50 along the direction and there is a minus again along the direction rho t so in general there is no reason if you are in a non-remanual framework if your norm is not libertian this object a priori has nothing to do with this object because the rows are inverted but if you are quadratic they are the same so the chigar energy quadratic implies that they are the same and precisely because they are the same I can prove the EVI property let's see why combining these two facts we get the EVI property so since this is equal to this I can say that this is smaller than this one but then now it's a matter of convexity I mean if for a convex function again I have the monotonicity of different quotient along geodesics this is the right derivative at time zero which is less than the difference quotient at time one which is the entropy minus the entropy of m at mu t and so you see that I have got precisely the EVI property derivative of the distance squared less than this and so this proves in one direction why one and two imply the EVI property ok maybe just a few words to conclude and maybe I will leave some time for questions remember the Buckley-Hemmery condition the Buckley-Hemmery condition which I also introduced in my very first lecture is an inequality between the Laplacian and now all these objects you see are well defined in our in our framework ok then I have to tell you what is now this object instead ok so the question is can we write the Buckley-Hemmery condition in our framework is it true does it hold this is the other side of the theory in our bounds well to do this all these objects are well defined the Laplacian I have to tell you what is this now and this is defined as follows there is again not an elementary result which tells us that the formula so if the trigger energy is quadratic the formula goes to zero remember before we were looking at this derivative under the integral so we were computing the derivative of the trigger energy now we are really trying to compute this derivative pointwise there is no integral here and this limit exists in L1 and it defines bilinear form a continuous bilinear form in W12 times W12 so now we have a pointwise almost everywhere defined pointwise object let's say in L1 function and of course this object plays exactly the role of the scalar product between gradients which is now emphasize again is pointwise defined in the theory of Buckley-Hemmery this is typically denoted by gamma because for them is a totally abstract object so they use an abstract notation gamma fg plays the role of nabla g so all these objects make sense in this theory and one of our results is that theorem in rcd k infinity space the Buckley-Hemmery condition holds and informally well I will not give a technical statement let's say undersuitable non-degeneracy assumption let's say trigger quadratic this is the necessary plus the Buckley-Hemmery condition imply rcd so posteriori we can really say that these are two sides of the same theory not exactly because when I mention non-degeneracy assumption or as usual these examples of spaces for which the trigger energy is identically zero of course if the trigger energy is identically zero then all this object becomes trivial and also the Buckley-Hemmery condition is trivialy true so you can't expect to get any geometric implication on your space so you should add some kind of mild non-degeneracy condition for instance for the semi-group where regularizing condition and then you are able to restore the validity of the implication ok, this is our third paper where we basically prove that the two sides of the theory are equivalent maybe just a few words about what about the dimensional theory this is much more technical and still not completely understood I just say in a few words what is the main difference one of the main differences is that to put the dimension into play you should not work with the standard entropy the Boltzmann-Schannon entropy but to use a dimensional entropy so an entropy which depends on the dimension which is called Reini entropy which is this the integral or rho and again you start to ask convexity for this for k equal zero cd zero n means precisely that this energy is convex along geodesics for k different from zero things becomes extremely more complicated and actually it is maybe nicer to write the Reini entropy in a constant term of course which does not affect the convexity properties if you write the Reini entropy in this way minus the integral of rho rho and you multiply by n and if you scale of course this has no effect on the convexity properties because you are adding a constant you are multiplying by a constant normal limit also in a rigorous sense as n goes to infinity is really so in this way you have a really consistent theory which in the limit as n goes to infinity gives you the cdk infinity theory ok and on the course in very recent papers basically this year kn theory started to be developed so rcd roughly speaking again means that you add to this condition that the trigger energy is quadratic between quotes because there are many subtle aspects to be investigated and on this and then of course again you can ask what is the connection with the Bacry and Mary theory and there is a first very nice result by Erbar, Kuvada and Sturm and there is also a work in progress I have with Andrea Mondino and Giuseppe Savare but this is really recent theory I mean and as I said not yet completely understood ok so I think it's a good idea to stop a few minutes before so that if you have questions you are welcome