 OK, well, first of all, let me begin by thanking Hugo for the opportunity to give this course here at IHS. And also, thanks a lot to everybody for coming. So let me begin by giving you an introduction to what this is going to be about. So there are two canonical ways of constructing surfaces uniformly at random, which are homeomorphic to this sphere. So what I'm going to do is first just briefly talk about the discrete approach. And in the discrete approach to this problem, what you do is you imagine that you fixed an integer n. And then what you do is you pick a quadrangulation. So I'm going to pick m sub n uniformly at random from the set of quadrangulations of the two-dimensional sphere with n faces. So this is a finite set of objects. So there's no difficulty in making sense of this operation. And then once you have your quadrangulation, you can view it as a metric space And the way that you do this is that you just equip it with its graph distance. So using the graph distance associated with m sub n. OK, now we have a random metric space. And it was shown by Langoll and Meermann independently that if you take this random metric space and you rescale distances by the factor n to the minus 1 quarter, then if you take a limit, these random metric spaces converge. So the spaces converge to an object called the Brownian map. And I'm going to be abbreviating the Brownian map by the acronym TBM. That's a little bit shorter to write. OK, so they constructed this object. And what is the Brownian map? I'm not going to give the definition of it. But let me just remind you of some of its basic properties. So the Brownian map is, this is a random metric space. It's actually not just a random metric space. It also has a measure. So it comes equipped with a measure in a very natural way. So it's a metric measure space. It's homeomorphic to the sphere. So this was first proved by Ligall and Pollan. And its half-storff dimension is almost surely equal to 4. So this was something that was also shown by Ligall. OK, so you have this random sphere homeomorphic metric space. It has four dimensions. But it does not obviously come with an embedding into the sphere. So this is sort of an abstract object. And it's not clear a priori how to embed this into the Euclidean sphere. OK, and the reason that one might care about this is that it's often very convenient, for example, to put something like a statistical physics model on a planar map, so on a quadrangulation, and analyze statistical physics model there. Because this is sometimes actually much easier than it is to analyze a statistical physics model on a planar lattice. And if you have an embedding, if you know how to embed your limiting object into Euclidean space, then what this allows you to do is it allows you to make comparisons with statistical physics models, the corresponding statistical physics model on a Euclidean lattice. So if we had a way to embed the Brownian map into, say, the Euclidean sphere, then we can make comparisons between statistical physics models on a random geometry, for example, the Brownian map and on, well, the sphere or the complex plane. OK, so this is one motivation for wanting to understand embeddings of scaling limits of random planar maps. OK, so this is the sort of discrete side of the story. Now let me give you a high-level introduction to the second approach, which is a purely continuum theory of random surfaces. And in here, what you do is quite different. And the starting point is the following thing. So if you have S, which is a Riemann surface, and you assume that it's homeomorphic to the sphere, then the classical uniformization theorem tells you that you can always find a conformal map. So there exists a map phi, which goes from your Riemann surface to the Riemann sphere, which I'm going to denote by C hat. And this map is conformal, OK? And the very nice thing about this is that if you take now the metric, the Brownian metric associated with this surface, and you put it into coordinates, so if you put the metric for S into coordinates using this map, then it takes a very nice form. So the metric can always be expressed as the Euclidean metric, so dx squared plus dy squared, times a function, which is often written as e to the rho of z. So z here is just x plus iy. OK, and because our surface is a nice smooth surface, this function rho is a smooth function. So what this is telling you is that you can parameterize the space of surfaces, which are homeomorphic to the sphere, in terms of the space of smooth functions. So you can parameterize surfaces, which are homeomorphic to the sphere with the space of smooth functions. OK, and now if you want to use this as a starting point to develop a theory of random surfaces, what you need to do is you need to put a measure on rows. So to make a random surface, what we need to do is we need to specify a measure, a probability measure on rows, on functions rho. Once we've done that, then we've specified a probability measure on surfaces. And what we're going to be interested in in this course is a very special case. And this is what's known as Liouville quantum gravity. And I'm going to abbreviate this very frequently with the acronym LQG. This is going to correspond to the case where rho is equal to gamma times h, where here gamma is going to be some number, some positive number, and h is going to be an instance of the Gaussian free field. OK, so another way of saying this is that what is Liouville quantum gravity? So Liouville quantum gravity with parameter gamma, this is just the random Riemann surface with metric e to the gamma h of z times dx squared plus dy squared, where here h is a Gaussian free field. OK, and the reason that, yep? Is it on purpose you stress that rho has had to be a smooth function, but now your measure is on things which are not even functions? Yes, that's right. So that's the whole, right, that's going to make things more challenging now. Yeah. So that's also why I put this in quotes, because this is not to find a real Riemann surface. By the way, the full Liouville quantum gravity has the lambda exponential phi nonlinear potential. So is this equivalent to this? No, so I'm just considering the simplest possible case. Yep, that's right. So this construction, of course, is not rigorous. And the reason is that the h, the Gaussian free field here, this is not a function. Not only is it not a function, I'm sorry, not only is it not a smooth function, it's not even an L2 function. This is just a distribution valued object. So it's going to require some interpretation in order to make sense of what this actually means. And previously, in what sense was this thing understood? This has been only understood before mathematically as a random area measure. So in other words, people have constructed the volume form associated with Liouville quantum gravity, but not the metric, not the notion of distance. OK, so we have two objects. So we have the Brownian map, which is the scaling limit of our random planar maps. And this, again, is a sphere homomorphic, four-dimensional, in the Hausdorff sense, random metric measure space. And again, it does not obviously come with an embedding, so no obvious embedding into Euclidean space. And the second object that we're going to be considering is Liouville quantum gravity. And Liouville quantum gravity, at least to start off with, is just going to correspond to a random measure, which is defined on a subset, well, either defined on the subset of the plane or the whole plane itself. And so what we're going to try to do is we want to sort of unite these two things and say that they are, in fact, the same. So now let me explain what the kind of conjectures are. So what you can imagine doing is you start off with your random quadrangulation, ebs of n. And as I explained earlier, you can take a limit as n goes to infinity in the Gromov Hausdorff topology. And it was shown by Lagal and Miermont that what you get in the limit is this thing called the Brownian map. Now, if you have a random quadrangulation, this is close to being a Riemann surface. It's not smooth because it has singularities where quadriliterals are glued together. But you can still uniformize it. And you can describe a quadrangulation in terms of its metric. So you get a random metric if you parameterize this by the Riemann sphere. And then the conjecture would be that if you take a limit as n goes to infinity, that these rho sub n's that describe this metric are converging. So actually, let me make this a dotted line because this is still a conjecture. So the conjecture is that the rho sub n's converge to a particular value of gamma, the square root of 8 thirds, times the Gaussian free field. So this is one possible way that you could try to prove that these two theories are equivalent. You take the discrete approximations for the Brownian map. You show that when properly embedded, they converge to leave your quantum gravity in some sense. Now, this is something that we don't know how to do yet. So what we're going to be talking about is a different approach to this problem. It's going to be purely a continuum one. And what we're going to do is we're going to draw an arrow over here. So we're going to show that there is a way in the continuum to embed the Brownian map and get Liouville quantum gravity. And conversely, if you have Liouville quantum gravity, there is a way to recover the Brownian map. So what is the actual theorem statement? And this is based on a series of works with Scott Sheffield. And the theorem says that, well, very roughly speaking, the Brownian map and square root 8 thirds, Liouville quantum gravity are equivalent. And so what does this mean? So this means that if you're given an instance of the Gaussian free field, then there's a way to associate it with a metric. So there is a way to associate with your field. And I'm going to call this metric d sub h. And it has the property that if you look at the metric measure space where my set is given by the Riemann sphere, my measure on my metric measure space is the Liouville quantum gravity measure. So mu sub h here is going to be the square root 8 thirds Liouville quantum gravity measure. And the notion of distance is this d sub h. And the property is that this metric measure space is equal to the Brownian map. So these two things have exactly the same law. So again, what we're saying is that it's possible to make sense of this construction, e to the gamma h, where h is a Gaussian free field. Gamma is the value square root of 8 thirds. It can make sense of this as a metric space. And when you do this, you recover exactly the Brownian map, which is the scaling limit of your random quadrinculations. And this construction has two very important properties that I want to emphasize. Number one, so the metric measure space structure associated with a Gaussian free field instance is measurable with respect to that Gaussian free field. So in other words, square root 8 thirds Liouville quantum gravity, this determines its metric. So there's no extra randomness in this construction. And secondly, the Gaussian free field h, this is measurable with respect to the metric measure space structure. So in other words, this says that the Brownian map determines its embedding into Liouville quantum gravity. And so in this sense, these two objects are exactly the same. So they both encode exactly the same set of information. Because if you observe one, you just observe the Gaussian free field, then you can recover, in a measurable way, the corresponding Brownian map instance. And if you observe the Brownian map, but nothing else, then you can recover, in a measurable way, what the corresponding Gaussian free field instance must avenge. So they're just exactly the same. OK. So that's the main theorem statement. Now what I want to do is explain what the general plan is. So today, what I'm going to do is I want to make the objects which appear in the theorem statement mathematically precise. And one of the reasons that this is quite important is that in the theorem statement, you have to be very careful when you describe the particular type of Gaussian free field that you're using in order to make sure that the metric space you're constructing is exactly the Brownian map. And it takes a little bit of work to see exactly what that is. And today, that's what I want to do. And then in the later parts, I'm going to try to develop some of the tools that are used to prove this theorem. So in particular, I want to explain the relationship between the Schramm-Lovner evolution and Leaville quantum gravity. And the reason for this is that understanding exactly how the Schramm-Lovner evolution or SLE works with Leaville quantum gravity is what will ultimately lead to the construction of this metric. And it's also very important because it leads actually to many other connections between random planar maps and Leaville quantum gravity. OK, so that's one thing I'm going to do. Once I've developed that, then I'll actually be able to give the construction of the metric. And what we're going to do here is somehow we're going to start off with the relationship between SLE and Leaville quantum gravity. And we're going to build out of it a metric space structure. Once we build this metric space structure, it's actually not going to be obvious at all from its construction that, well, first of all, it is a metric. So that takes some work to prove. But it's also not going to be obvious that this metric space has anything to do with the Brownian map or scaling limits of random planar maps. And so in the last part, what I'm going to do is I'm going to explain why or prove this metric I'm going to build is actually the Brownian map. So that's the plan for the rest of the remaining sessions. So now what I'm going to do is I'm going to start to make precise the different objects which appear in the theorem statement. And the thing that I'm ultimately aiming at is the precise mathematical definition of the sphere in Leaville quantum gravity that we're going to be using to prove this correspondence. And in order to do this, what I need to do first is give a little review of the Gaussian free field. So the Gaussian free field is, for me, it's going to be an object which lives on a domain in the plane. And how is it defined? The Gaussian free field, H, you can construct it using a series approximation. And the way that you do it is you just take it to be the sum. N goes from 1 to infinity of alpha n times phi n. We're here alpha n. These are just IID normal 0, 1 variables. And the phi sub n, what are these? These are an orthonormal basis for a particular Hilbert space, which I'm going to call H of d. And this is the so-called Diraclet space. So this is the Hilbert space closure of the compactly supported C infinity functions in my domain with respect to the following inner product. So the inner product of two functions, f and g, is just given by 1 over 2 pi times the integral of the gradient of f times the gradient of g over d. So this is the Gaussian free field. Now, there are lots of different ways of describing the Gaussian free field. What I just described now was one construction. Let me just mention one other way of thinking about it, because this is also going to be important for the calculations that I'll do. And that's that the Gaussian free field is just the Gaussian distribution with a certain covariance. So alternatively, the Gaussian free field H, this is just the Gaussian distribution with covariance given by the Green's function. So if you have two test functions, let's say phi and psi, which are C infinity and compactly supported in my domain, then the covariance of the integral of H against phi with H against psi is given by the integral of these two functions against the Green's function on my domain. So here, g is the Green's function. And just to be pedantic, this is equivalent to saying that the Laplacian of g is given by minus 2 pi times the Dirac mass at x evaluated at y, and it has zero boundary values. OK, and to be even more pedantic, we can write down a formula for g. And g of x, y is just given by minus the log of x minus y minus another function, which I'm going to call g tilde sub x of y, where g tilde of x of y, this is harmonic in my domain with boundary values given by y maps to minus the log of x minus y. OK, so that's the Gaussian free field. I don't want to say a huge amount about the Gaussian free field beyond that, except for a few things. So there are going to be a few important properties that are going to be very relevant for the construction of things like, well, first of all, the Liouville quantum gravity measure, but also the mathematical construction of the Liouville quantum gravity sphere and other Liouville quantum gravity surfaces. So OK, so first of all, the Gaussian free field is not a function when you're working in two dimensions, but it's almost a function. So in many situations, we just sort of pretend it is a function and work with it that way. There are various ways to regularize it to get a function. For example, one way which is going to be very important for I'm going to talk about is what's called the circle average regularization. And in the circle average, what I'm going to do is I'm going to, for a Gaussian free field instance H, let H sub epsilon of z be the average of H on the boundary of the ball centered at z of radius epsilon. And this is a very convenient process to work with when you work with the Gaussian free field because this is a Gaussian process as a function of epsilon and z. And it's a Gaussian process with some very nice properties, namely, or one of them, is that if you fix your point z, then the map, which takes you from a time t to H e to the minus t of z, this is a standard Brownian motion. So if you average the Gaussian free field about concentric circles, which go to 0 exponentially fast, then what you get is a Brownian motion. And this is very helpful for doing different computations. It's also conformally invariant. So this means that if you have two domains, d and d tilde, and a conformal map, phi, which goes from d to d tilde, and H tilde is a Gaussian free field on d tilde, then this implies that H composed with phi is a Gaussian free field. Sorry, H tilde composed with phi is a Gaussian free field on d. And the last relevant property that I want to mention is the spatial Markov property of the Gaussian free field. And that just says that if you have a subdomain u in your bigger domain d, so this is a subdomain, then you can write the Gaussian free field h as the sum h1 plus h2, where h1 is a Gaussian free field on u. h2 is harmonic on u, and h1 and h2 are independent. So these are just the kind of basic properties of the Gaussian free field that we're going to be using. I don't want to provide any more detail other than what I've said now, but if you're not familiar with the Gaussian free field and you have any questions about any of these things, I'm happy to explain how you justify any of these statements mathematically. So now what I want to do is I want to start working towards constructing the Liouville quantum gravity sphere, which appears in the theorem statement. And so let me begin by now giving you a more detailed review of Liouville quantum gravity in a sense that we understand it. And so how is this actually defined? Well, what you imagine is you have a parameter called gamma, and gamma is some number between 0 and 2. Then in Liouville quantum gravity, we have a measure associated with a Gaussian free field instance. So the gamma Liouville quantum gravity measure associated with a Gaussian free field h, and let me call this measure mu sub h, this is just the limit of the approximations e to the gamma h epsilon of z times dz, where this now denotes Lebesgue measure. And I'm going to normalize this appropriately so I get a non-trivial limit. And the correct normalization is epsilon to the gamma squared over 2. OK, so the Liouville quantum gravity measure is the limit as epsilon goes to 0 of these measures here. Now, I'm not going to go through the proof that this construction defines an object which is non-trivial. You get some kind of non-trivial measure. This is then done many places in the literature. For example, there's work of Cahan in the 80s in which this is done. And more recently, this was rediscovered in work of Duplantier and Sheffield, et cetera. OK, so all that matters is that this limit exists, and it defines some kind of non-trivial measure from the free field. Now, one thing which is very important in Liouville quantum gravity, and it's going to be very important for constructing the sphere, is the way that you change coordinates. So the coordinate change formula is very important. And as I'll explain in a moment, it actually led to quite a lot of confusion recently in how you define and make sense of some of these things mathematically. So how does this work? So if you have two domains, d and d tilde, which are in the complex plane, and you have a conformal map, phi, which goes from d to d tilde. Sorry, I'm going to take it from d tilde to d. OK, so you should imagine you have one domain here. Let's say this is d tilde. You have some other domain here, and you have this conformal map, which takes you from one to the other. OK, then if h is a Gaussian free field on d, so you have your Gaussian free field h over here, and you set h tilde to be equal to h composed with phi plus q times the log of phi prime, where q is equal to 2 over gamma plus gamma over 2. Then you have the property that if you look at the measure under h tilde of any set, then this is equal to the measure under h of the image of that set under the conformal transformation. So that's how you change coordinates on the evoke quantum gravity. And let me take a moment to explain what this means and how you see that this is actually correct. So when gamma is equal to 0, in other words, you're just working with Euclidean space, then this coordinate change formula is exactly the same thing as the Jacobian determinant formula that you're learning calculus. OK, now let me just quickly explain where it comes from when gamma is positive. So when gamma is positive, there's going to be an extra correction, which corresponds to that q I've written down. And this extra correction, basically where this comes from, is the normalization in the definition of the measure. So it just comes from the normalization in the definition of the levio quantum gravity measure. So let me call that star. And because the coordinate change formula is actually so important for lots of different things, let me just take a moment to explain why this is really correct. It's very simple to see. So if you start off with your approximately levio quantum gravity measure epsilon to the gamma squared over 2 and you form this measure, then if you take your conformal map and you bring this measure from a measure on d to a measure on d tilde, then the corresponding measure is just going to be epsilon to the gamma squared over 2 times e to the gamma h epsilon composed with phi of z times phi prime of z squared dz. This gives you a measure on d tilde. And you can rewrite this in a very simple way. So epsilon to the gamma squared over 2 times e to the gamma h epsilon of phi of z times phi prime of z squared. This is the same thing as epsilon over phi prime of z to the gamma squared over 2 times e to the gamma h epsilon composed with phi of z plus q times the log of phi prime of z, dz. And the point is that h epsilon of phi of z, this is almost the same thing as h composed with phi averaged about the circle of radius epsilon over phi prime of z evaluated at z. These two things are almost the same. And therefore, this thing here will converge as epsilon goes to 0 to the levial quantum gravity measure associated with the field h composed with phi plus q log. OK, and that's the change of coordinate formula in levial quantum gravity. This is actually a very powerful thing. Yep? Where the effects are epsilon to gamma squared over 2 Oh, where does it come from? So it comes from the fact that the variance of h epsilon of z, this is equal to the log of 1 over epsilon plus something which is bounded. And so when you have a Gaussian e to the, let's say, e to the h epsilon of z, this is a Gaussian random variable with this variance. And so its expectation is going to be e to the gamma squared over 2 times the log of 1 over epsilon. And so this is the correct factor to normalize by to get rid of this divergence. Yeah, right. And the reason that you see a log here is that the covariance function for the Gaussian free field is just the log function. OK, yep? When gamma equal to 0, q is infinite. But the important thing is that in the exponent you have gamma times q. So you have e to the, yeah, you can see it, right? Yeah, yeah, right, right, right. Yeah, so we interpret 0 times infinity as 2, 2. Any other questions about? And this page is clear that there are holes between 2 of them and gamma over 2? No, no, at this point. Yeah, no, no. That won't actually be. The theorem is that the log of gravity for n is equal to gamma prime over 2. So that is going to, it's not going to come up directly. It's sort of indirectly related in some deep way to the relationship between SLE and the Gaussian free field, which in turn is underlying the proof of this theorem. But it won't be in the most direct, just direct thing. Yeah, that's right. Yeah, so one thing I just wanted to mention as an aside is that this formula is actually very powerful, not just because it lets you change coordinates, but it's actually theoretically very useful. So for people that know about SLE, you can actually give Gaussian free field proofs without doing any estimates of things like the boundary of SLE is a holder continuous curve. There's a very short proof of this that just uses the change of coordinates formula and leave of quantum gravity and no other estimates other than that. OK, but that's just a technical aside. Now, so the reason that the change of coordinates formula is so important is that it leads to the following definition. And this is the notion of equivalence of quantum surfaces. So we say that if you have two h's, h and h tilde, we're going to say that these are equivalent as quantum surfaces if your two fields, h and h tilde, if they're related by the change of coordinates formula. So if they're related by the relation, h tilde is equal to h composed with phi plus q times the log of phi prime for some conformal map phi. Then we say that these things are equivalent as quantum surfaces because we want to think of these two fields as somehow parameterizing the same surface. And to get from one to the other, you just have to apply the change of coordinates formula. And more generally, a quantum surface, this is an equivalence class of h's with respect to this equivalence relation. So in other words, quantum surfaces are defined modulo conformal transformation. And a particular choice of h, so a representative of a quantum surface, so a particular choice from the equivalence class, I'm going to refer to as an embedding of the quantum surface. And this notion is actually very important because it's actually not always clear that two different representations of the same quantum surface are equivalent. And actually, one very recent situation where this came up and actually caused some confusion in the community is that there were two definitions of the Lieve-Quantum Gravity Sphere. So we posted a paper together with Bertrand de Plentier that defined the sphere in one way. This is the construction of the sphere that I'm going to give in a minute. And a bit later, there was another paper that appeared due to David Coupillian and Rogin Vargas that defined the Lieve-Quantum Gravity Sphere in a completely different way. And when you see the formulas in the two definitions, it's not at all clear that they should be the same. It actually took a whole paper's worth of work due to some others to show that there actually is a conformal transformation which takes you from one setting to the other. And in general, it actually can be quite tricky to identify the law of a quantum surface because somehow you have to find this conformal transformation which takes you to a nice setting. OK. So I guess before stopping for a break, let me just explain now what we want to do. So what I'm going to do for you is I'm going to describe, I'm going to give you the construction of the quantum surfaces, let's see here, the quantum surfaces which are ultimately going to describe scaling limits of random planar maps in different settings. So these are going to be the quantum surfaces which describe or are equivalent to scaling limits of uniformly random planar maps. Actually, I should say that these constructions also describe the scaling limits of planar maps which are not chosen uniformly at random, but also with a statistical physics model like the easing model. But again, we've only constructed the metric sense of these surfaces in the uniformly random case. And let me just tell you what the names of these things are. So if you have, let me actually write it here. So just to lay out some terminology, if your discrete model is a quadrangulation of the sphere, so let's say you start off with, as you discrete model, a quadrangulation of the sphere. This is your discrete random planar map model. Then the limit in the metric space sense, the Gramov-Hausdorff sense, this is, in this case, the Brownian map. And the quantum surface that I'm going to construct for you, which describes this object, is going to be called, well, in this case, we're just going to call it a quantum sphere. But there are many other random planar map models. For example, you can take quadrangulations of the disk. In this case, the limiting object that you get is the Brownian disk. And the corresponding quantum surface is just called a quantum disk. You can also consider quadrangulations of the plane. Then the scaling limit is the Brownian plane. And the corresponding quantum surface is called a quantum cone. And then lastly, if you have a quadrangulation of the upper half plane, then the scaling limit of this is the Brownian half plane. And the corresponding quantum surface is going to be the quantum wedge. And it can be actually quite tricky to see what the right form of the Gaussian free field is going to be for each of these cases, because it's actually not something so nice. It's not just a Gaussian free field. It's more complicated than that. And so what I'm going to do is I'm going to explain first how you construct a quantum cone that will be the quantum analog of the Brownian plane. Once I've described the quantum cone, then it's very natural to construct a sphere, because the way that you build a sphere out of the plane is that you sort of pinch a bubble of mass off of it. So you imagine that you have some kind of surface, which is like a plane. And the way that you construct a sphere is that basically you just kind of condition on the event that it has a huge bubble. And then when you send the bottleneck size of this bubble to 0, you get a sphere. And so you can see exactly how to construct that. And then the constructions of the half plane and the disk are exactly the same. There's a very natural way to see what the right construction of the half plane is. And once you have that, the way that you make a disk is that you pinch off a bubble. So I think. Is it correct that for the case of the Liouville field theory, what you have constructed you cannot do S2, you cannot do C at the moment with the Liouville quantum gravity. Oh, we can do all of these. Well, with the definitions, no. I mean, the Green's function don't work. Oh, so I will construct for you exactly. So the way that you construct the Gaussian free field that corresponds to the sphere is not the Gaussian free field on the sphere. It's a different object. And I'll explain exactly what it is and where it's going to come from is we're first going to build the plane version, which will be kind of an indirect sort of thing. And then once you have that, the way that you make a sphere is that you kind of condition it on it having a big bubble. And then somehow the bubble will become the sphere. And you'll see that it's not the same thing as the Gaussian free field on the sphere. OK, so maybe now I'll stop for a short break. All right, so let me continue. So what I'm going to do now for the second half is I'm going to give the construction of the Liouville quantum gravity surfaces that appear here without boundaries. I'm going to do the case of the quantum cone and the quantum sphere. The construction of the quantum wedge is almost exactly the same. And the way that you construct the quantum disc from the wedge is almost the same way that you construct a sphere from a cone. And yeah, I should say that these particular constructions, these are described in a paper by Scott Sheffield where he looks at SLE on Liouville quantum gravity. So OK, so the quantum cone, the way that we're going to build it is that it's going to describe the local limit of a Liouville quantum gravity surface near a typical point, so near a quantum typical point. And so what I mean by that is that we're going to have some domain, and we imagine that we have a Gaussian free field in this domain. Then we can pick a point z uniformly at random from the measure mu sub h. And then the way that we're going to build the quantum cone is that we're just going to zoom in around this point and see what we get. So let me explain how that works in detail. So the starting point for the construction is what's called a rooted measure. And this is a measure on pairs, h comma z, where h is going to be something like the Gaussian free field, and z is going to be a typical point from the corresponding Liouville quantum gravity measure. And the starting point for this is an approximation of the rooted measure. And so what you do is you take the measure epsilon to the gamma squared over 2 times e to the gamma h epsilon of z times dh dz, where here in this expression dh is the law of the Gaussian free field. And dz is Lebesgue measure. This is just Lebesgue measure. And the point is that if you take a limit as epsilon goes to 0, the law of z is going to converge to the Liouville quantum gravity measure. So what we're going to do is we are going to, first of all, observe that the law of z given h under this construction. This converges as epsilon goes to 0 to mu sub h. That just follows from this construction. And so what we're going to do is we're going to describe the conditional law of h given z, and then send epsilon to 0. And that will tell us what a Liouville quantum gravity surface looks like near a typical point. And this is going to follow from a calculation, which is very standard in this Gaussian free field stuff. And it's also a very standard Gaussian calculation. So let me explain how this works. So the first thing is to observe that we can write the average of the Gaussian free field on this circle in the following way. So it's the same thing as the Dirac-Ley inner product of the function h with a truncated form of the Green's function. So here psi z epsilon of y is minus the log of the maximum of y and z minus y minus another function, which I'm going to call g tilde sub z epsilon of y, where this function g tilde sub z epsilon of y, this is harmonic in D with boundary values given by y maps to minus the log of the maximum of epsilon and z minus y. OK. So if you send epsilon to 0, then this is exactly just, oh, yep. Is that max epsilon and z minus y? Yep, that's supposed to be an epsilon. OK, that looks better now, thanks. Right, so if you send epsilon to 0, then this just converges to the Green's function under domain. And for positive values of epsilon, this is just a truncated form of the Green's function. And if you do an integration by parts calculation, it's not difficult to see that this Dirac-Ley inner product is equal to the circle average. OK, and now we're going to use this to do a standard kind of Gersanov Gaussian change of measure argument to compute the conditional law of the field given the location of the point. So OK, so before I do that, let me just observe that we can write epsilon to the gamma squared over 2 e to the gamma h epsilon of z dh dz. That's the same thing as epsilon to the gamma squared over 2 times e to the gamma Dirac-Ley inner product of h and this function dh dz. OK, and now to do this calculation, I just have to remind you of one very important fact, which again gets used all of the time in this Gaussian free field stuff. And that's that if you have a normal random variable with mean 0 and variance 1, and you take the law of this random variable, so if you have this random variable, and if you weight the law of it, if you weight the law of z by the Radon-Nickerdiem derivative, e to the mu x minus mu squared over 2, then you get the law of a normal random variable with mean mu and variance 1. OK, that's a very simple but very, very useful fact. And we're just going to use that in this infinite dimensional setting to calculate this conditional law. So in particular, remember, I can write the Gaussian free field h as the sum of alpha n phi n, where the alpha n's are iid normal 0, 1, and the phi n are an orthonormal basis of my space. And I can also write my function psi z epsilon as a sum. I can expand it with respect to the same basis. And so their inner product is just the sum of alpha n beta n, OK? OK, and now if we insert this formula into the density function for the rooted measure, what we get is that epsilon to the gamma squared over 2 times e to the gamma h epsilon of z dh dz. This can be written as epsilon to the gamma squared over 2 times e to the sum alpha n beta n dh dz, where here the alphas are just the coefficients of the Gaussian free field in the series expansion. Oh, yeah, there's a gamma here too. Thank you. Great. OK, and so this here, this is just the product. n goes from 1 to infinity of e to the alpha n beta n times gamma. And so if you apply this fact for the normal distribution term by term, so if you apply a fact for the normal distribution term by term or in each coordinate, what we see is that the conditional law of z given h just corresponds to a shift in the mean of the field. And it is equal to Gaussian free field on d. And then we have to shift the mean by what comes from this right on nicotine derivative. And that's going to be gamma times psi z epsilon. Yep, conditioned. Oh, that's right, that's right. This is h given z. Thank you. OK, so if you condition on the location of the marked point, then you're just shifting the mean of your Gaussian free field by this function. And if you send epsilon to 0, this converges to the Gaussian free field plus gamma times the Green's function evaluated at z. So under this rooted measure, in the limit as epsilon goes to 0, the way that the Gaussian free field looks like near this quantum typical point is just Gaussian free field. And then you have gamma times the Green's function centered at that point. OK, so I think I didn't. So let me just say what this means a little bit more informally. So if you sample, so given z, the conditional law of h is Gaussian free field plus or say minus gamma times the log with a singularity center that z plus a harmonic function, which we don't care about. OK, so near a typical point chosen from the Liouville quantum gravity measure, what you see locally is just a Gaussian free field with that log singularity. So now we're going to use this to construct the quantum cone. And after we have that, we can build the quantum sphere. So now I'm going to build the quantum cone. And the way that we're going to build the quantum cone is essentially by taking this construction here and then zooming in near z, but in the appropriate way. That's very important. OK, so I'm going to assume that I have a pair h, z as above. Then again, I know that h given z is equal to a Gaussian free field minus this log singularity up to a harmonic function. And so this tells me that if I look at my field and I average it about the circle of radius e to the minus t centered at z, what do I see? Well, from the Gaussian free field term, I'm going to get a standard Brownian motion b sub t. From this term here, I'm going to get gamma times t. So that gives me some drift. And then when I average the harmonic function about the circle, I just get the value of this harmonic function at z. So let me just write that as x. So here in this representation, b is going to be a standard Brownian motion. And it's independent x. So x doesn't really matter. It's going to go away in just a moment once we zoom in. So this is just telling you that near a quantum typical point, if you look at the circle average of the field, it's exactly the same as before, except for you have this upward drift. Now when we construct the quantum cone, it's very important that one zooms in in the appropriate way. And how does that work? What we're going to imagine is that we have two numbers, s and c. They're positive, and they're going to be very large. And then what are we going to do? We're going to imagine that we add c to our field. So in terms of the level measure, this corresponds to multiplying areas by the factor e to the gamma c. That's what adding c to the field does. And then the other operation that we're going to perform is that we are going to imagine that we apply the change of coordinates given by w maps to what I'll call phi of w. And this is just going to be the map that corresponds to rescaling by the factor e to the s and then sending z to 0. So in particular, this sends the ball of radius e to the s centered at z to the unit disc. So I'm just, yep. Oh, yep, that's right. That's a minus s. Thank you. So again, we have our picture. We have our quantum typical point. Then we imagine that we're just going to multiply the area by some big number, and then we're going to center our field around this particular point. And we want to sort of do this all at the right scale. OK, so what happens when you apply these two operations? What is the resulting field? So let me call this h tilde. And h tilde is just h after applying the coordinate change formula. Sorry, this should be phi inverse prime plus c. And by the definition of the conformal map, this is the same thing as h composed with phi inverse minus q times s plus c. And so now what I'm going to do is I'm going to explain how we choose s and c in the right way to get something non-trivial in the limit. OK, I'm going to do it over on this side. So the point is that you can look at the circle average for the new field. And if you average the new field about the circle of radius e to the minus t centered at 0, then what do you get? Well, by definition, it's just going to be b of t plus s plus gamma minus q times s plus gamma t plus the constant c. And then we have this factor x that we don't care about. So this is what you get when you look at this circle average process. And we're in particular going to be interested in what happens when you take t equal to 0. So if you average on the boundary of the unit disk, then what does that equal to? That's just equal to b of s plus gamma minus q times s plus the constant c plus the factor x that we don't care about. OK, so now what we want to do is we want to just choose the parameters s and c in a smart way so that we actually are zooming in near this point while at the same time understanding how this zooming operation determines the law of the field. So in particular, because this drift I have on my Brownian motion is always negative, so it turns out that q is always bigger than gamma, for each value of c there almost surely exists a smallest number s of c so that h tilde of 1 evaluated at 0 is actually equal to 0. So this is just saying that I'm just adding a constant to my field, which multiplies areas by e to the gamma times this constant. And then I rescale the field so that the average of the field about the unit disk is equal to 0. And there always exists such a rescaling because this calculation allows you to relate it to a Brownian motion with a negative drift. And so what is the quantum cone? The quantum cone is just the limit of this as c goes to infinity. So this is the limit of this construction as c goes to infinity with s chosen appropriately. So s is always going to be this particular c dependent stopping time. And again, this is just corresponds to a very natural way of zooming in towards a quantum typical point. The nice thing about this is that as I'll explain in a moment, you can describe exactly what the law of the field is, the limiting field. So you can describe the law of a quantum cone explicitly. And as I'll explain a little bit later, after applying a change of coordinates, the law of the quantum cone becomes very explicit. And it's directly related to Bessel processes. So you can describe it just in terms of a Bessel process. OK, so the point of this is that you can describe the law of the quantum cone explicitly. And it's relatively simple to do. So what you do is you describe it in terms of its radially symmetric part and its orthogonal complement inside of our space. So let me just make the observation that you can always take the Dirichlet space and you can decompose it in a certain way. So you can write the Dirichlet space on the whole plane as an orthogonal sum of what I'm going to call h1 and h2 of c, where here h1, this is going to be the space of functions which are radially symmetric about 0. So these functions in this space are just going to correspond to circle averages somehow. And the orthogonal complement is just going to be the space of functions which have mean 0 on circles centered at the origin. So it's actually relatively straightforward to check. It's a very short calculation that this actually gives you an orthogonal decomposition of this Hilbert space. So this means that to specify a function, you just have to specify its radially symmetric part, which corresponds to its circle averages about the origin and the orthogonal complement. And so what are these going to be in the quantum cone? So the point is that you can write them down exactly with this particular embedding. So the construction of the quantum cone only dealt with the circle average process. It had absolutely nothing to do with the second part of this space. So what this means is that when you sample from the law of the quantum cone, what you do is you first, you describe its projection onto the second space, the space of functions which have mean 0 when averaged about circles centered at the origin. And you take this projection just to be equal to the corresponding projection for a Gaussian free field. And this, again, comes because when we constructed the quantum cone as a limit, we didn't care about this part of the space. So we didn't affect it somehow. And then we have to describe its projection onto the space of functions which are radially symmetric about the origin, which, again, corresponds to describing the circle average process. And what you do here is you take its projection onto the first space to be given by the function whose common value on the circle centered at the origin of radius e to the minus s is a certain function a of s. And let me write down what a of s is. OK, so let me first say what happens when s is greater than or equal to 0. So let's think about that for a second. So we have our domain that we started with. And we're constructing our quantum cone by picking a random point. And then we kind of zoom in around this point. And the point is that this random scaling factor that we chose so that this sort of had the right size never looks into the future. So somehow it's a Markovian operation. And so when s is greater than or equal to 0, a of s is just going to be equal to b of s plus gamma of s, where b is a standard Brownian motion. And gamma is, well, gamma. So this is exactly what it would be in the case of a Gaussian free field on the whole plane. So the interesting thing is what happens when s is less than or equal to 0. But in this case, a of s is going to be a conditioned Brownian motion, which I'm going to write as b tilde. So it's b tilde of minus s plus gamma of s. And it's conditioned so that b tilde of s plus q minus gamma of s is greater than or equal to 0 for all s greater than or equal to 0. And this extra conditioning here just corresponds to the particular way that we chose to rescale and construct our field. OK. So the nice thing about, so that's the definition of a quantum cone. Basically, the way that one thinks about it is that a quantum cone, this is the same thing as a whole plane Gaussian free field with a log singularity at the origin, gamma times log at the origin, except the difference is that Gaussian free fields on the whole plane are only defined modulo additive constant. And that's a problem because that means that the Liouville measure is only defined multiplicative constant. And so in the definition of the quantum cone, you have to somehow fix that additive constant. And you have to do it in the right way. So a quantum cone is just a whole plane Gaussian free field with this log singularity with the additive constant fixed properly. All right. So now in the last bit, what I want to do is I want to explain how to construct and describe the sphere from the construction of the quantum cone. And if you believe that what we did really should correspond to the scaling limit of a random planar map on the plane, then hopefully you also find this construction convincing to describe, well, what's ultimately going to be the Brownian map. There is no quantum thing that is translation invariant. Well, this is translation invariant, but in a special way. So like, for example, in the sphere version, the natural sort of operation to perform is you resample a point from the measure and then you re-center at that. So because if you perform a translation in Euclidean space, somehow your translation can depend on the embedding of the quantum surface. And if you scale things different ways, translations mean something different somehow. And so one has to be very careful when defining these things and thinking about them so that you only perform sort of quantum operations, like re-sampling from the measure or later with SLE. Right. Yeah. So there's just a different notion of translation, which leaves everything invariant. OK. Right. OK, so now let me just quickly explain the sphere. And where the sphere comes from is it's just going to come from the quantum cone. So the quantum cone is kind of like a plane sort of. And here's our special point, which is the origin. And the way that we're going to build the sphere is that we're going to somehow condition on the event that near the origin you have a lot of mass. There's sort of a lot of a mass really, really close to the origin. And then this is going to sort of lead to the formation of a bubble. And when you condition on the mass to be really, really large, the bottleneck somehow just collapses and you're going to get a sphere. So that's kind of the intuitive idea for how this construction works. To make it mathematically precise, we're going to relate now the construction of a quantum cone to a Bessel process. And the operation of going from the cone to the sphere just corresponds to people who are experts in Bessel processes reflecting the dimension of the Bessel process about 4. So it's like conditioning a Bessel process to be positive in some sense. OK. OK, so to make this rigorous, what I want to do now is I want to describe a quantum cone in a slightly different setting. So what I'm going to do is I'm going to, I can also parameterize my cone not by the whole plane but by the infinite cylinder. And once you make the change of coordinates to the infinite cylinder, you start to see the relationship between quantum cones and Bessel processes. And that's what leads to the sphere construction. So I'm going to apply a change of coordinates, which will take me to the infinite cylinder with the top and bottom identified. So OK, so I'm just going to apply this change of coordinates now and then write down what you get. So the map, which I'm going to use, which goes from the complex plane to the cylinder, is just going to be the map z goes to minus the log of z. The reason I'm taking minus the log of z is that for some reason I think of it easier to think of the origin as being plus infinity rather than minus infinity. Now what happens when you do that? Well, your field H is going to get mapped to the field H composed with, so if this is the map phi, phi inverse plus q times the log of phi inverse prime. And for this particular choice, you can write down exactly what this derivative is. And if you work it out, it's just going to be minus q times the real part of z. OK, now let's see what happens when you do that. So when you perform this mapping, circles are going to get mapped to vertical lines. So if you have this is the complex plane and you have a circle of radius e to the minus s, then under this mapping, phi, it's going to get sent to the vertical line, which has real part equal to s. So that's the infinite cylinder. So the circle average process and the definition of the quantum cone is going to get translated into the process which corresponds to averaging the field on vertical lines. OK, and so under these coordinates, what is the law of the cone given by? So the quantum cone can be described here as follows. So you take its projection onto the space of functions with mean 0 on vertical lines to be the same as the corresponding projection for the whole plane Gaussian free field. So for the Gaussian free field on the cylinder. And so to finish describing its law, we just have to write down what its projection onto or what its average is on vertical lines. And what is that? So we're going to take its average on the line with real part equal to s to be the function a of s, where here, if s is greater than or equal to 0, a of s is just going to be a Brownian motion plus q minus gamma times s, where b is a Brownian motion. And when s is less than or equal to 0, a of s is equal to another Brownian motion plus q minus gamma of s, where b tilde is a Brownian motion conditioned so that b tilde of s plus q minus gamma of s is greater than or equal to 0 for all values of s. So this is just exactly what you get when you apply this change of coordinates formula. So the way that you think about this is that you have your cylinder, which describes your quantum cone. The point which was 0 before is now at plus infinity. And the point which was at infinity is minus infinity. And you have this process, which is describing the average of the field on line, vertical lines. And the way that it looks is that it's just a Brownian motion with negative drift gamma minus q. And this conditioning that you see here just corresponds to the fact that we're just taking the horizontal translation so that this process hit 0 for the first time at 0. And that's how we think of these things. Now, in the last few minutes, let me now explain how you see the sphere for this, and the sphere from this construction. OK, so let me draw the picture now of the quantum cone again. So that's this cylinder. And again, the way that you think about it is that you have this Brownian motion, which is drifting down as you go to plus infinity. So Brownian motion plus with drift gamma minus q, which is negative. And what we want to do now is we want to take this picture and we want to build from it the sphere. And remember, the way that we're going to build the sphere is we're going to imagine we have our cone, and we're going to condition on the event that it has a big bubble like that. OK. So we have to think about what could it mean to create this big bubble. And then I'll describe what it is very explicitly. So how do you create a big bubble? So you imagine you have your cylinder. And basically what you want is you want this process to at some point reach a very small value. So Brownian motion plus gamma minus q times s. So if it reaches a very small value, and then it becomes big again, and then it gets small again after that, this thing here is kind of like a bubble. Because when this gets really, really small, it's like saying there's a bottleneck, a bottleneck here. So this is how you, in some sense, create the bottleneck. You condition on this Brownian motion doing something weird like getting really, really big after it's gotten small, even though it has a negative drift. OK. So that's sort of the idea. And now let me very quickly, in the last few minutes, explain how it works more explicitly. And this can all be seen if you translate this into the language of Bessel processes. So in the definition of a quantum cone, you can describe this process A very explicitly. So A sub t is the same thing as can be sampled from in the following way. And people who are experts in Bessel processes will recognize this instantaneously. You number one, take z to be a Bessel process with dimension 8 over gamma squared. And note that 8 over gamma squared is bigger than 2. So this is a Bessel process which goes off to infinity. And then you take the Bessel process and you take its log, because the log of a Bessel process is a Brownian motion with drift. And then you scale it appropriately. And then you're parameterized by quadratic variation. And if you do that, this will exactly give you the same process A that I wrote over there up to horizontal translation. And so you just have to choose the horizontal translation appropriately. OK. And now let me finish by explaining what the sphere is. The sphere is constructed in the same way. It's the same as the quantum cone. But you replace the Bessel process with dimension 8 over gamma squared by a Bessel process with dimension 4 minus 8 over gamma squared, because that's going to describe what happens when you condition this Brownian motion to get small and then big again before going off to minus infinity. And let me just very quickly convince you or show you something which is kind of a good sign. So for people who are experts in Bessel processes, let me remind you that the length of a Bessel excursion of dimension delta is taken from the measure t to the delta over 2 minus 2 times dt. And here, our value of delta is equal to 4 minus 8 over gamma squared. The magical value of gamma that we like is the square root of 8 thirds. And if you plug this value in, you get 1 for your Bessel process dimension. And in terms of this infinite measure, it's the measure t to the minus 3 halves dt, which should be a very familiar measure to people who work on the Brownian map. Because this is the infinite measure which describes the size of the doubly marked grand canonical ensemble on Brownian maps. So you can see from these constructions that some of the exponents start to match sort of from the very beginning. OK, so what I did for you today is I constructed, I explained how to construct the quantum cone in the quantum sphere. The actual definitions of these objects are a little bit annoying because they're not exactly Gaussian free fields. They're Gaussian free fields with some kind of funny drift with additive constants fixed in some way. But when you work with them in practice, you just imagine that they are Gaussian free fields with log singularities. And so it's a little bit, it's not quite as annoying as it seems. And so next time what we're going to do is we're going to look at what happens when you take a quantum surface and you put on top of it one of Schramm's SLE curves. And what this is going to describe is it's going to describe in some sense the continuum analog of what happens when you put a statistical physics model on top of a random planar map. For example, percolation or the easing model. And what I'm going to do next time, I'm going to try to make it somewhat independent of what I described today so that if at the end some of this became a little bit more difficult to follow, you won't have to worry that next time you'll be lost from the beginning. OK, so I think I'll stop there.