 Okay. Thank you very much for the invitation. So I'm going to speak about some work I did in collaboration in the last ten years with Andreas Grankiqov and some people of our research group, some student post-doc and collaborators. So I will review some result about almost remain manifold, that are special kind of two-dimensional generalized remain manifold, so this structure appear naturally in optimal control, and you will see appearing in this talk, so some optimal control, some theory of hyperbolic operators, so a little bit about diffusion, we study the heat equation, and a little bit of quantum mechanics, because we are going to study a bit also Schrodinger equation in this structure, so I am going to first define what is two-dimensional almost remain manifold, his basic properties, then I will define the Laplacian, so in such a way to be able to write the heat and Schrodinger equation, and at the end I will give some very recent result about the Schrodinger evolution for some alternative quantization that one can make on this kind of structure. Ok, so I will explain two-dimension too, so this is rich enough, some result that I will present are true also, hold also in higher dimension, but here for simplicity we stay in this setting. Ok, so I want to define some generalization of two-dimensional remain manifold, so let recall what is a two-dimensional manifold, everybody know this, so we have a two-dimensional differentiable manifold, and on each point in the tangent space, so I give a scalar product that I use to define the length of vectors, and the angle between vectors, so this is a remain manifold, and so what I do with this remain metric, I measure length of curves, so if I take an absolutely continuous curve, I can measure his length simply by making the integral of the velocity, and once that I am able to measure length of curves, I can define the distance between points by computing the infimum of the length of all curves that are connecting the two points, and this distance, indeed it is a true distance, and give to our manifold the structure of metric space, and this structure it is compatible with the original topology of the manifold itself. Ok, so in remaining geometry the problem of finding the distance between two points can be locally written as an optimal control problem, so how to do this? You have to introduce local orthonormal frame, so in dimension 2 you need a pair of vector fields that are orthonormal, so that their length is equal to one, and they have a 90 degree one respect to the other one, then if you want to find the curve that has the shortest length between two points, you take a curve connecting the two points, you decompose the velocity in terms of this orthonormal frame, and then what you do is minimizing now the length of the curve, if you put this formula inside this one, you get simply the square root of u1 square plus u2 square, so if you want to find the curve that realize the distance between two points, you have to solve an optimal control problem. Two remarks, so of course in this construction these two vector fields, so that realize the orthonormal frame are always linearly independent, so take in mind this, because this is what we will not have anymore in almost remaining geometry, and this construction, so the fact that we can write this optimal control problem can be globalized, it's global, only if the manifold is parallelizable, so you have a global orthonormal frame in the manifold, so only in the case which m is parallelizable, and in the case in which you are working with compact orientable surfaces, so this appears only, you can have this only on the torus, otherwise you have always to change the chart to be able to do this. So now let's define what is a two dimensional almost remaining manifold, so what we do is the following, so this is the generalized linear structure that one obtain by declaring that a pair of vector field x1, x2 that can become collinear, so they will not require anymore that they are always linearly independent, is an orthonormal frame, but we are assuming a condition that is what is called usually the Ormander condition or the Libraket generated condition, that even if the two vector field x1, x2 are not always linearly independent by computing a bracket between this vector field, the bracket or the bracket of the bracket, at certain point you get the full dimension of the space and we will see why this condition, it is important. So where the two vector field are linearly independent, so this would be our orthonormal frame and so the structure defined by them, it is a standard remaining structure. On the other side on the set where the two vector field x1 and x2 are parallel, so they become parallel, then all remaining quantity will not be defined anymore, indeed they explode and as we will see everything go to infinity, so they remain metric, the element of area and also the curvature explode on this set. We are going to see this in example now. Ok, so what we do with this structure, the idea is again to define the distance between points to measure length of distance of points, so one can define the distance between two points exactly with the same formula that we have used in the remaining setting, so one write the vector velocity in terms of the orthonormal frame and one minimize this cost and we have that thanks to these conditions, also to the Librake generating condition, so one can prove that this distance it is always finite and continuous, so this is essentially the show theorem that you have seen during the lecture of Andrei Grachov that he did not call the show theorem but he explained the theorem, it is simply the fact that with this vector field they are Librake generated and so you have that for every pair of points you can always find a trajectory admissible that is going from one point to the other one, so this is a good distance and one can prove that it is continuous and that give to the manifold a structure of metric space that exactly as in the remaining case this is compatible with the original topology of the manifold. Notice, however, with respect to the remaining context that this formula in the remaining setting so this was a definition of U1 and U2 that I use to minimize of the two controls and every curve that is absolutely continuous could be written in the following format at least in the point where it is differentiable. In almost remaining structure not every curve could be written in this way, so when I write this formula here this means that when I make this minimization I make minimization only along curve that I can write in this way so this is a constraint on the set of curve in which I am going to minimize so this is different with respect to the remaining settings so this formula here we have really a constraint on the velocity and we are going to see in details now on the example. So before passing to example let's see now this definition was just local I just gave on some local charts so now we have to globalize it so this is standard we do as in remaining geometry simply by writing compatibility condition between charts to connect one to the other one or by using some language of fiber bundle so this definition, this globalization it is a bit more... sorry, this globalization is a little bit more elegant but I don't want to enter in details about that because what we need in this talk is essentially only the local structure so you can really think to pair of fields defined locally in one chart. Ok, so let me mention the fact that when we are defining this structure by only one pair of vector fields globally defined on the main if then we call these structures free so these are the most in a sense interesting and natural structure and if for instance we take a sphere or something has the topology of a sphere and you put on this sphere a pair of vector fields so you should remember that vector field on this sphere has always some zero somewhere so if I declare that two vector fields on a sphere are an orthonormal frame the kind of metric that I get it is one of these almost Riemannian because I will never be able to get a Riemannian metric on this sphere in which an orthonormal frame is given by only one pair of vector field so indeed these structures are very natural take any two dimensional manifold put on them two vector fields satisfying the romantic condition and then the metric structure that you get it is an almost Riemannian one so it is not a Riemannian one so to get a Riemannian one either you are on a parallelizable manifold or you need to change the charts to avoid to create this singularity so in this sense this kind of structure are extremely natural so let's see a couple of examples to understand this well so let's take the plane so this is what is called the Grushian plane is the most famous example of two dimensional almost Riemannian structure and take on the plane these two vector fields so one is one zero rectify is the blue one so it is always in any point equal and the second vector field it is zero x1 so it is something that is vertical like this but this vector field becomes smaller and smaller when you are on this axis where it is zero and you declare that this is an orthonormal frame so you see that now if I consider curves on this space you have that so if I consider curves on this space so the orthonormal frame in a point here is like this but in this point the orthonormal frame is something that is just one vector so you have because the other one it is zero so if you consider curves in this space so on this side you can have any curve any absolutely continuous curve like this would be okay but here for a curve to have finite length you need to when you cross this set to cross it horizontally if you have a curve that is for instance like this so this curve will not have finite length because when you decompose in one of the two controls we go to infinity so this is why we have a constraint on the velocity so I now look how appear the corresponding Riemannian quantity so what is the Riemannian metric that renders these two vector field orthonormal it is clearly something like this so because when you apply when you eat this vector on the right, on the left you have to get one so this is why you have one here and when you eat this on the right, on the left you have to get one so here you need one over X1 square so you see that the Riemannian metric is exploding and also the Riemannian area is exploding so this is one over X1 the X1, the X2 and also the curvature it is exploding that will be important later on that the area is exploding and this is not integrable so it is one over X1 so I cannot integrate this on a bounded set here and the curvature it is exploding and it is exploding and it is negative it is going to be very negative so this also will be important later on another very natural example that appear often it is what is sometimes called the quantum sphere so it is just a two dimensional sphere in which we claim that two orthogonal rotations are an orthonormal frame so I say in orthonormal frame for this metric is one rotation along this axis and another rotation is along this axis and I say so this is an orthonormal frame so of course there are some degeneracies here because the vector fields are zero so the first one is zero here and here the second one is zero here and on the other side and if you go to look carefully you will see that the two vector fields become parallel on one meridian here where everything is exploding so this is why when you put coordinates on the Earth you cannot use two pairs of parallel but you use one parallel so the parallel and meridians because like this you get that your orthonormal frame is singular only on two points on the north pole, on the south pole while here you get something that is that is a singular on a full one-dimensional circle so this is why when you give coordinates in the Earth you don't use two sets of parallel but one set of parallel and one of meridian but this is not very important so this example appears in some problem of so it's called the quantum sphere because it appears in some problem of control quantum mechanical system with a tree level so this is the so-called stirrup process ok so the name almost remain geometry was given I think by us it was an idea of Andrei ten years ago when we started to work on this but indeed this kind of structure were present from long time so I think first time that appeared it was in the end of of sixties in a paper by Bawendi then was studied by Grushin so this is why the plane is called the Grushin plane then in the eighties by Franky Lanconelli then in the context of subrimane geometry so these are an example of rank varying subrimane geometry so they were studied by Gromov and Belayish in the nineties and more recently they appeared in some application in control of quantum mechanical system and also in a space in a space mechanics ok so now let's study some basic feature of this of this geometry so it is an exercise not totally obvious but not very difficult to prove that you can always find system of coordinates in an orthonormal frame in such a way you can rectify one of the vector fields and the second vector field you can write as zero on the first component and in the second component you put f it is not so difficult to do this so there is a simple proof in the book that we just write with and Barillari and this way of writing the orthonormal frame will be very very useful because simplify a bit the formulas and so the only condition that we have it is that one derivative with respect to x1 of this function in zero should be a certain point different from zero so this is the Ormander condition so the Liebrack generating condition so when I compute the bracket between this I get the derivative with respect to x1 of this if I continue I get other derivative and to be Liebrack generated I need at a certain point one of this derivative are different from zero so this is the setting the general setting that I will use in this talk and in this coordinates the singular set so the set of points where I am not Riemannian is the set of zero of this function f so where this vector field it is t0 where it is not zero it is independent from the other one then we can compute all Riemannian quantity here is very easy essentially already this computation so the Riemannian metric is 1 and here we have 1 over f2 so you see that it diverges when f goes to zero the Riemannian area is 1 over f so also diverge when f goes to zero and you have also a relatively nice expression of the curvature and also the curvature is essentially diverging when f goes to zero but the way in which diverge could be indeed very complicated and so these are the formula specified for the Grushin plane as we already saw ok so let me now mention some properties of the singular set because we don't know much but of course the singular set so the set of zero of a function could be something very bad but however it is possible to prove that it has always a big measure zero in which this function f is is equal to zero is obliged to have a big measure zero due to the fact that we have at least a certain point one of this derivative it is different from zero so this is not difficult to prove and beside this is not easy to get other condition of regularity on the singular set but in general one need a little bit more but this so if one put now some generic condition the singular set become very regular so when I put some generic condition I mean that I put some conditions that are verified by most of the system in particular if we are in the case in which the manifold is compact so when I say generic I say that I take one system that belong to an open and dense subset of the set of all this system with respect to this infinity topology and so if you make some generic assumption you get immediately that this singular set become a one dimensional embedded sub manifold and you get that you have only three possible type of points so remaining points of course where you this is these are almost everywhere where the two vector field are linearly independent so then you get the grusheen point where you have situation similar to the grusheen plane where with one bracket so here this delta this black triangle is the span of the two vector field so the grusheen point is where this is one dimensional so you are on the singular set but you use one bracket you get the full dimension of the space and so these are essentially you have situation like this so your singular set it is so you are on your manifold for instance it is compact so here it is not important so you have your singular set your singular set it is something like this for instance you have two components in this way and so all these points are Riemanian so all these points are grusheen except for some isolated points where you can be where you could need one more bracket to get the full dimension of the space and these are called the tangency points and the distribution so the delta so the span of x1, x2 on grusheen point it is like this so it is transversal to the singular set and on the tangency point this become indeed tangent to the points and this happen only at isolated point so this is the type of point that you can expect when you are considering non patological cases and of course if you are in the compact case then this singular set it is just a set of intersecting circles because should be manifold should be on a compact manifold should be compact so they are clearly some set of circles like this so we will use this later on so let go on to study some properties of this geometry so this already said that I have not said but almost mentioned so the fact that in this geometry you can find set that has finite diameter and infinite area so this could look strange but it is obvious that open set like this so if you would like to compute its area you should integrate the area form so one over x1 dx1 dx2 so this is not integrable so you get infinite and then you get a set of infinite area but the diameter is finite because we have this controllability condition so we have since it is bounded you can go from any pair of points of this set with curve of finite length so at the end the diameter it is finite so this is a feature interesting that will come out later when we are going to study diffusion here so how to compute geodesic in this structure so of course if you want to compute geodesic in such geometry you cannot use the standard geodesic equation coming from remaining geometry coming from Euler Lagrange equation because they are singular because they become everything explode on the singular set however this problem is completely well defined in terms of as a problem of optimal control and so to compute geodesic just to apply the Pontragi maximum principle that was explained by Agarčov in these days so this is a very simple case because you can prove that you don't have any abnormal minimizer that is not trivial that is not just sitting at the point of the singularity and so at the end geodesic will be just a projection of the space of the Hamiltonian solution of this Hamiltonian in which you see your two vector fields and if you want to find geodesic have just to solve this Hamiltonian system and as we can see now so geodesic of course geodesic are always smooth so there are no singularity on geodesics because there is as a Hamiltonian system regular so they are always smooth and they can also cross the singular set with no singularity so these are for instance the geodesic starting from a riemanian point on the Grushin plane so this is the singular set so where the two vector fields are parallel I start from this point minus one zero and I draw geodesics you see geodesic are perfectly regular they can cross the singular set the only thing that you can remark here is that when they cross they cross horizontally because this is a necessary condition for having finite length so geodesic always cross horizontally like this so no singularity at all for geodesics because they are solutions of an Hamiltonian system they are smooth as optimal control problem there is no singularity is completely regular ok but there is another interesting fact so you know that in riemanian geometry so geodesic in riemanian geometry are not always they can cross certain points so they can create some conjugate locus geodesic for instance on the two dimensional sphere if you start from the north pole at certain point geodesic will meet the south pole they start to overlap to intersect but this in riemanian geometry do not happen if the curvature is negative so if the curvature is negative the curvature in a sense measure how much geodesic are converging or diverging so if the curvature is negative the geodesic are diverging so they will not make they will not start to envelope and in this geometry situation it is different because in the Grushin plane the curvature is always negative so it is minus 2 over x square so this was x1 square but even if it is always negative you see after some time so this is time 5.7 so this picture was 2.7 so take a little bit more time after this picture for geodesics and you see here appearing geodesics start to envelope and you see appearing some conjugate locus and some cut locus and this is curious because it happens even if the curvature is always negative so this is due to the fact of course that once that you cross the singular set all Remanian theorem relating curvature to the presence of conjugate point stop to be true because everything explode so we are not anymore in the hypothesis of Remanian geometry and indeed there are some different phenomena that make that after the singular set geodesics that before were diverging start to converge and then make some singular set so this is an interesting phenomenon that was remarked 10 years ago so let me mention another phenomenon important for us is that this is still the Grushin plane the geodesics starting from the point minus 1,0 so this is a point just before that there is some conjugate locus here the conjugate locus will start but this is not important so what happen it is that the length of the sphere so if I take one point I take the sphere of a certain radius so if the sphere is intersecting the singular set what happen it is that his length usually it is infinite so you can have some sphere even of small radius but such that the length of the circle of the contour of the sphere is infinite so this is due to the fact that this curve so this is the sphere so the set of point reach at for instance here I don't remember this time probably this time something like for the end of point geodesic time for so this is not horizontal so it is not like this so the length of this it is infinite because here there is accumulating it is something go to infinity and so there is some accumulation and the length of this front it is diverging so this is important when we are going to studying later on some random walk on this structure ok, so let me just mention something that is very well known for people that is working in subrimanian geometry so this is the set of geodesic this picture and the end point of the geodesic when we start from a point in the singular set so these are the geodesic you see geodesic start to overlap very close to the starting point there is a lot of singularity that are accumulating here there are some conjugate locus accumulating in infinite number of conjugate it is very complicated this singularity but what it is important it is to remember that for every epsilon so you can always find a geodesic that has lost his optimality before epsilon so there is a non-uniformity with respect to lost of optimality when you start from from a point of the singularity this is something that never happen in in rimanian geometry because in rimanian geometry small enough if you start from a point the sphere of small enough radius are always smooth there is never a conjugate locus in this case ok so now I would like to start the second part of this talk and to discuss a bit and to discuss a bit diffusion on this kind of geometry so what I would like to do here is now to try to study an equation type heat or shoddinger like this free for a free particle so no potential just free evolution on such a structure dynamics we have more or less seen that geodesic are regular can have some conjugate points and so on but let's see what can we say for a quantum mechanical particle so satisfying shoddinger equation or for the diffusion for the heat equation so the first question that we have of course is the right operator that I have to put here so what is this delta so what is the Laplacian that I have to put here so this is already not completely obvious question because I could say I could use this Laplacian the sum of the square of the vector field defining the structure but this would not be a good choice because this would depend on the orthonormal frame so will not be invariant by change of orthonormal frame and then it will not be global if the structure is not free it's not defined by a pair of vector field and have to change the chart so this would be a bad choice so if I want something it is an invariant operator by change of orthonormal frame I need an expression of the form the divergence of the gradient so in this almost remaining geometry there is no problem at all to define the gradient so this is simply because so the gradient of a remaining geometry the gradient of of a vector field you write like this so the sum over j so you take the differential of your function and sorry sorry of a function you have to compute the differential then you put the index up with the inverse of the metric so this work very well in almost remaining geometry because it is true but here you have the inverse of the metric so in this structure if you want almost remaining geometry is precisely a geometry in which you have the inverse of of the metric it is regular but can become zero so this is a well defined object and what we have to do now is to define to define this Laplacian to define the divergence with respect to some volume so the problem is to choose the volume that you want to use in this definition of this operator I would say that we have infinity plus one choices so either we choose volume from outside and then we will have some diffusion some study that will depend on the choice of this external volume or we use the only volume that we know to be intrinsic in such a geometry that is our area form that unfortunately diverge so become infinite so we have to care about operators that will be diverging somewhere so this is the point of view that I will use so I will consider the Schrodinger the Schrodinger, the heat equation for delta that will be simply the Laplace-Beltrami operator given by the Riemannian Laplace-Beltrami operator that will diverge on the singular set so I will use this point of view that is the most intrinsic because we have to deal with this kind of singularity so let's see what is the form of this operator so it will be the divergence with respect to our area form of the gradient and so this is the expression in terms of an orthonormal frame so this is x1, x2 is an orthonormal frame and you see appearing the sum of the square of the two vector fields and then you have some first order term which we see appearing the divergence of the two vector fields and will be this the term that will diverge because the volume is diverging and so we have that this term here will possibly be going to infinite so let's see for the Grushian metric so for the Grushian metric x1 it was 1,0 so it is d over dx1 in the second component so it is this and so for the Laplace-Beltrami operator here you get the square of the first vector field the square of the of the second one and then you get this diverging first order term that go as minus 1 over x1 and this will make all the stuff that I'm going to tell you interesting so another choice could be to take the standard Lebesgue volume on the Grushian plane so if you take the standard Lebesgue volume on the Grushian plane the two vector field become divergence free and so you get just the sum of the square and you have not anymore a diverging first order term and this give a completely different diffusion so this is another completely different study and what I would like to do is to study this kind of of operator ok, so my question now is the following so we have our manifold let's consider a compact a compact case for instance so we are on a manifold like this with some holes I don't know it is compact and then I have some some singular set assume for instance it is like this so we are Riemannian on the two side and we are not Riemannian on this piece we know that the geodesic go on the other side we have no problem so there is a classical trajectory go on one side to the other one and we are asking if the heat and the Schrodinger equation so the heat will flow from one side to the other one and also if solution of the Schrodinger equation so if a quantum particle that initially it is localized somewhere here you will be able to find somewhere somewhere here of course we have to give a meaning to this divergence that appear in the operator so this question it is essentially equivalent to the following so let's concentrate to just one second so it is equivalent to the following question consider omega a connected component of our manifold minus the singular set so for instance omega is this part here so omega is this part here that is the part that I am going to consider and I consider delta my Laplace Betami operator define here only on this side so everything is Riemannian here singularity is on the boundary I consider this on omega then what I am going to to ask so I am asking if this operator it is essentially self-adjoint in L2 so understanding if there is flow or not is essentially equivalent to this question, why so if I can prove that this operator is essentially self-adjoint so this means that the Cauchy problem it is well defined completely on this side so here I don't want to enter in details about the theory of self-adjoint operator but the only thing that you have to to keep in mind here is the following that if I have an operator that it is essentially self-adjoint on some space then if I consider the Cauchy problem for the heat or the Schrodinger equation then this is well defined I have existence and uniqueness and you don't need any boundary condition so this means if my operator is essentially self-adjoint I don't have to care of what happen here on the boundary and I have that my evolution will be just on this side and nothing can flow from here to here because since I am essentially self-adjoint on this side nothing can cross ok, on the other side if I prove that this operator it is not essentially self-adjoint then I have to give boundary condition to give a meaning to the corresponding heat and Schrodinger equation and I can choose a condition so these are killing conditions at the boundary reflecting conditions or I can have other conditions that connect what happens on one side and on the other side but this problem is not related to this geometry even if you consider on the real line on R you consider the Laplacian d2 over dx2 so this is on the real line d2 over dx2 it is essentially self-adjoint you have a good evolution for the heat of the Schrodinger equation but if you eliminate one point so if you consider this on R minus 0 if you eliminate one point you are not anymore self-adjoint you have to give what you have to give some boundary condition here to define the evolution for instance you should require this boundary condition or Neumann boundary condition or you can give a condition that connect your solution from one side to the other side to give the evolution in our case it could happen that the evolution is already self-adjoint complete on one side so this could mean that nothing flow and what you are going to expect it is indeed we expect a negative answer so we expect that this Laplacian it is not essentially self-adjoint because since jodesic classical jodesic go from one side to the other side we expect that since jodesic go that trajectory can we have the impression that also Brownian motion can go from one side to the other side that a quantum particle can go from one side to the other side so we expect so this at least was what I was expecting when I was studying this problem to get that we are not essentially self-adjoint and the result that was surprising that we obtained already some years ago it is indeed the opposite so the result it is a positive answer so we are indeed essentially self-adjoint to this result we obtained with Camille Laurent so we are on a compact orientable two-dimensional almost Riemannian structure so I have one hypothesis that I discuss just now so if I consider a connected component of my manifold minus the singular set like this then if I define my operator so this is the technical way of saying that it is essentially self-adjoint I start to define with domain the set of functions infinity with compact support then we get this essentially self-adjoint on L2 and so that nothing can flow from one side to the other side so this is a bit surprising because everything that is classical and it is crossing so let me just mention what this hypothesis means so this means that with one bracket so I'm assuming that with one bracket I get the full dimension of the space so this means that we have only Riemannian and Grushin point so we don't have these tangency points in which on this line this is turning so the set of admissible velocity and then become tangent like this so this is not present and also implies that z it is a manifold that is a finite union of non-intersecting circles so this hypothesis of course we need to give a meaning to the boundary of this set but the fact that we need only Riemannian and Grushin point this is a technical hypothesis and one of our most difficult question that we have in our research group that we are not able to answer is what happen if a theorem like this it is still true in the presence of tangency point so this is this is not clear I can discuss later if somebody is it is interested ok so corollari corollari already said so you consider a Schöninger equation the Cauchy problem for the Schöninger equation in L2 or for the heat or for the wave it is the same then if you start with some initial condition that is here this initial this your function will never find on the other side yeah this means that they discuss this now wait one second yes of course this is surprising so what happen so now then we looked for some because the proof I can give the proof later if there is time at least in the Grushin case but the proof is not very speaking so and it is not completely obvious why we have a phenomenon like this so the first imagine that we had it is essentially the following facts so if this is the Grushin line the Grushin singularity and the Grushin plane so what happen it is the volume is exploding here so in a sense random particle when arrive to this volume they are lost in this infinite volume and also quantum particle it is like to have a lake you send some water here you want to get some water here but if your lake has an infinite area you can never get anything at the end so this is a kind of explanation but it is not it is not very very satisfactory and this theorem that we gave indeed it show that many result that work well in Remanian geometry is not true anymore in this setting the famous result by Varadan that relate the asymptotic of the heat kernel so Varadan prove that minus 40 log of pT so pT is the heat kernel of the Laplace-Betrami operator for t that tends to 0 tends to the distance square so result like this is not true anymore because pT the heat kernel if you start from here will be 0 on this side because there is nothing flowing and on the other side the distance here it is finite so result like this will not hold anymore and also all semi-classical theory in quantum mechanics that essentially in a second essentially says that for h bar that go to 0 so the equation should tense in a certain sense to the corresponding classical solution so also this is not true anymore because classical particle cross and quantum particle don't cross and this phenomenon is today called the quantum confinement and there were quite a number of papers that gave other result in this more general result in this setting ok so now what I would like to do is to try to give you what time is it a little bit of time to give you a kind of interpretation of this phenomenon in terms for the heat of a random walk and then try to give something similar for the Schödinger equation but as we will see for Schödinger equation situation it is a little bit harder but let's start from from the heat where we have a kind of nice interpretation of the phenomenon in terms of of a random walk so you know that you can build the Brownian motion so solution of the heat equation as a link of random walk so what you can do is to start from a point q0 then you send randomly a particle on a sphere of radius epsilon in time delta t uniformly on the sphere then you continuous you go on to write an equation for the density of probability so let's do this so to do this I do the following so let's feed the density of probability let's say the density of particles so we say that we have to write a diffusion equation so I say that how much my heat so my density of particle is increasing at this point it is proportional to the difference between how much I have a heat here and how much I have on the sphere of radius epsilon so if the temperature let's say is the same of number of particles then nothing will happen because in equilibrium if here I have less temperature than what I have on this circle then it will be some flow so what I say it is the variation at t of my temperature my density of particle here it should be equal proportional constant of proportionality I put here equal to 1 to the difference between the average that I have on this circle and what I have in this point so the average is computed here phi of the exponential map so these are the geodesic so these are this point is the function at this point and d theta is a measure of probability I take a uniform on the length of this circle then I divide by delta t and then I use this parabolic scaling so I say that delta t is equal epsilon square so epsilon is the radius of my sphere delta t is the time and then I send delta t to zero and I get an equation of type d t phi so this become the derivative with respect to t so this become an operator that this operator is defined in this way and this part after the limit it is the generator of the random walk and so now the question is if this operator here it is or not the Laplacian and indeed what happen beside constant this operator it is exactly Laplacian so this is well known and you can also be used this formula to define the Laplacian operator indeed operator is computing the average of the function the different of the average of the function on the circle the sphere minus in the point and one can also prove that convergence of the operator implies but this is a little bit another story but let's stay simple here so this is what happen and so now let's see what can what we have in the Grushin plane in the Grushin plane we have the following so the idea is the following I start here and I randomly jump on a sphere of radius epsilon so now I have fixed delta t and epsilon then here I do the same I go on as soon as I go I arrive closer and closer so my sphere become a bit become a bit more like this and then at certain point I get that one of this sphere so as I said epsilon delta t I get one other sphere that is like this and it will intersect that will intersect the singular set so this sphere intersect the singular set but is not made like this it is a little bit more like this in the way in which I compute the probability is on the length so I take a uniform probability on the length so I have the length so I take something that is uniform, uniform and uniform what happen when I am here it is the length now that is from being finite to being infinite and then and then I have that this two point has an infinite quantity of length that is there so when I jump from this point on this sphere I am obliged to jump on these two points and at the next step when I am here I have to jump again on my sphere and it has this shape that we saw and if I jump from here to these two points again I will jump here because there is an infinite length that is because the length is infinite so the only contribution is given by these two points so what happen is that as a random walk once that you arrive to this singularity your process it is killed you arrive there and you cannot move anymore and that is the reason why at the end you get this this self adjointness because the process cannot continue on the other side once I arrive to the singularity it is killed there and can I have three minutes more or ok ok so let me just say two words of something that is very recent concerning an explanation of this impossibility of flowing on the other side for the Schoeninger equation so for the for the Schoeninger equation this random walk interpretation was satisfactory at least for us to have a good intuition but for the Schoeninger equation situacionality is a little bit more complicated for several reasons so in quantum mechanics you find often theorem of the type that if you have a wave packet solution that is sufficiently concentrated then for h bar that go to zero so the maximum should essentially satisfy the classical equation of the motion so this is the semi-classical theory that is an enormous theory that is true only in certain under a certain hypothesis a little bit delicate but however on a remaining manifold because here I am considering a particle that is free so no potential flowing on a remaining manifold it is already not completely obvious what is the quantization procedure that you have to use so we have our classical Hamiltonian so this is the Hamiltonian that give given by Pontragi maximum principle and what I did it is just to substitute my classical Hamiltonian with h bar square divided by 2 and then the Laplace-Beltrami operator but I gave no justification of this so for the heat we know that this is the right way of doing if you want also by the reason random work that I just said the mechanic is not so obvious and indeed there is a huge literature about geometric quantization that say that in the Riemannian geometry depending on the scheme that you have to use you have to introduce together with the Laplacian some correction factor that keep contribution of curvature of the scalar curvature indeed if you go on the literature you find that this c is a constant that depends on the procedure that you use so in some paper you find that c should be equal to zero so the standard Laplace-Beltrami or 1 over 12, 1 over 8 this is a nice paper by Anderson driver about this and this is really depend on the method for geometric quantization you get something from 0 to 1 over 6 if you pass integral you get something different so there is an ambiguity there is an ambiguity that depends on the way in which you want to quantize the operator and you see that when you are in Riemannian this term k it is not extremely important in the semi-classical limit because when you send h bar to zero so this go to zero so of course Laplacian will not go to zero because there is some derivative but this contribution becomes smaller and smaller as h bar go to zero, the contribution of the curvature but in our case this go to infinity and so this contribution it is still there and then we proved that instead if you use this is my last slide if you said we use this operator, we study the self-adjointness of this operator self-adjoint if and only if c is equal to zero so this means that for any other quantization that is not the one corresponding to c equal to zero so this operator it is not essentially self-adjoint and so one should in principle be able to recover some semi-classical limit but this is something deep that we still don't beside this result we still don't know much, thank you very much for your attention