Fields Medalist: Wendelin Werner
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Werner, Wendelin
Lecture of Wendelin Werner, Fields medallist 2006.
 Good afternoon. It is a really great pleasure for me to introduce Ferdinand Werner for the final Fields Medal Lecture. Ferdinand received his PhD in 1993 at University of Paris-Sies under the direction of Jean-François Legault. Since 1997, he has been Professor of University Paris-Soud in Orsay. From 2001 to 2006, he was a member of the Institute Universitaire de France and he currently also holds a position at École Normale Supérieure in Paris. His prizes include the Rolo Davidson Prize, the European Mathematical Society Prize, the Fermat Prize, Jacques-Hubert Prize, the Leuven Prize and the Polier Prize. On a lighter note in trivia, he is also one of the few people I know who has finite identical Erdos numbers and Kevin Bacon numbers, something for people to look up later. Three in both cases. From a personal perspective, I just want to say this, it was about ten years ago that I first, I was giving a lecture about intersection exponents, the problem of brine intersection exponents, one case of which was the mandelbrot, equivalent to the mandelbrot conjecture about browning motion. When Bedlin was in the audience, and a couple weeks later I received an email with a very nice idea how to approach this problem. From the start of that email, I started a collaboration that has lasted a long time and I've seen many, many of his nice ideas. I'd only like to mention a couple of them today. One of them a couple years later was, as I think was mentioned this morning's talk by Odette, he was talking with Odette Schramm and he realized very early that Odette's beautiful SLE might be the key to understanding the browning intersection exponents. At that point, our two-person collaboration on this side became a three-person collaboration and suddenly I had two outstanding collaborators with giving nice ideas back and forth. Secondly, I'd like to sort of general idea, although a lot of the work was problem-driven by these particular problems. It's Bedlin, many times is the one who's actually sat back and said, that sort of looked and said, not only are we solving these problems that we're asking, but in fact, these structures that are done, both SLE and browning in motion, are really perhaps the keys to understanding putting conformal field theory on a rigorous basis and now that's an active study area being pursued by many people. With that, I'm happy to introduce Bedlin for his talk. Thank you very much. As you might imagine, this week has been a very special week for me and enjoyable one. I hope I'm not too tired to deliver a reasonable talk today. Since it's the first time I get a chance to speak, I have, of course, many people to thank from personal reasons to PhD supervisor, but I'll do that on personal level that would be too long to do now, but of course I'd like to, I'm very happy and glad that I'm speaking after Oded's plenary lecture this morning and that Greg was able to be here to also and as chair of this session because a lot of what I'm going to say and a lot of maybe part of the reason that I'm standing here in front of you is that I met these two nice people. So the topic of my talk is random planner loops and conformal restriction and, okay, I added a survey because of course this is supposed to be more an introductory lecture that gives some flavor of some of the ideas and some of the results that we obtained during the last years. And this talk is, of course, not unrelated and rather closely related to Oded's talk this morning and also to Stasman's talk yesterday, but I'm not going to assume that you all went to these two nice lectures. I think I'll try to make it self-contained today, but maybe with some partial repetition of at some points on some issues with what mostly what Oded said this morning. But I'll try to take another perspective which is more to try to look at description and understanding of continuous random structures in the plane using the, of course, the discrete models as a guideline but really focusing on properties of the continuous objects. So I think it's fair to start with some background about motivation and history in the physics community because this subject of research, one of the main motivation comes from physics and physicists have been given a lot of input to these subjects before we actually started to look at them. And actually I'd like to thank also take the opportunity to thank them, Michael Eisenman, Bertrand Duplantier and other physicists who also made the effort to come to math departments and actually say, here we have a nice subject problem for you that we know sort of know how to solve but which is a nice problem for you, mathematician, because the way we approach them is probably not the way one should do it mathematically. And then when we came up with some ideas, they also accepted these and didn't just say, well, we knew it before. So they acknowledged the fact that these were, was really new input. So I'd like to stress this. So I'm going to say a couple of very general statements and to start with, with the danger that it's too general so that you don't see what I mean. But in general, when you learn about physics, you learn that, well, the laws of physics is something when you repeat it twice, the same experiment, you get twice the same results. And that's the result of the laws of deterministic laws of physics. And it has been observed long ago actually experimentally that on macroscopic scale, when you are exactly at a point at which the physics is called a phase transition point, that means a point where, say, for instance, you can imagine the temperature at which liquid become vapor or there's an even competition between two possible states in the system, that when you're exactly at that point, this previous deterministic law doesn't hold anymore because you see some features that become random on macroscopic scale. So you repeat the same experiment twice, you, and you will get different answers. And part of the story is to understand some of these macroscopic random features or complex systems that you see on macroscopic scale. And a closely related question is that, well, how do you describe what phase transition point is? Well, usually it means that some of the physically deterministic quantities that you observe on macroscopic scale, when you are away from the critical point, go to zero or go to infinity. And it has been observed that sort of these quantities often obey certain power law behavior when you approach the critical point. So that some quantity, deterministic quantity, sorry, basically behaves like t minus tc to some power gamma. And this exponent gamma is called the critical exponent for this model or for this object. So here you see you have a description of the first random object at the critical point, and then below you have deterministic behavior of deterministic points, quantities near the critical point. And of course, well, not of course, but it has been observed also that these two, or argued that these two phenomena are very closely related with each other. So physicists have came up with, theoretical physicists have came up with a number of inventive and clever ideas in order to describe these problems and these questions. And the first of which has been developed by, okay, it's not to intimidate you, I just put down some names of physicists that have contributed to these issues. And of course, I am omitting a lot of names here. The first idea is that of a renormalization group. And that basically is, so of course, it's a very simplified explanation, but it explains roughly or gives you a convincing heuristic to the fact that different models or different gases or different experiments will give rise to the same exponents or to the same random behavior at the critical point. And the idea is basically to say that this random macroscopic behavior to interpret at this, at a fixed point of some renormalization function that basically you divide your system in a large system into smaller boxes and each box, this random system is created and you put them together, you create a large system. And what you argue is that at this critical point, the system is a fixed point of this because it becomes scaling variant, it will be a fixed point of this transformation. So here are three items that are specific to two dimensional systems. And so conformal field theory and also what is called Coulomb gas techniques or quantum gravity are ideas or that sort of based on analogies, based on explicit computations, based on many different facts that provides mathematical tools in order that enabled these physicists to predict the value of these critical exponents of many different systems in the case where the dimension is two. So we are looking at planar systems. And the exponents of the models are classified according to what physicists call the central charge of the model. That means each model or system you look at has a specific central charge. And this term central charge refers to the fact that some of the mathematical tools that are hidden behind the conformal theory scene have to do with the representation theory of some infinite dimensionally algebras. And the third item is explanation of the fact of the precise relation that there exists between the behavior, this random behavior at the critical point and the deterministic behavior near the critical point. And I think it's fair to say that apart from the last item which has been treated in the mid 80s by Harry Keston, that's one of the many things that Harry Keston did in all these questions. So he treated the case of percolation in two dimensions where he made sense of the scaling relations and explained that if you understand the random behavior of the system at the critical point, you also understand the behavior of the system near the critical point. And it's fair to say that apart from that, these three, even though conformal field theory was related to very deep and rigorous mathematics, the relation between these deep and rigorous mathematics and the actual questions you look at was a question mark. Okay. So I'm going to repeat very quickly so that you have one model in mine. I chose to repeat a model that has been introduced that you may be seeing once or twice in this ICM before, but, well, I guess that's the simplest model to explain so that you have in mind something specific and it's not too general. So the specific model is called two-dimensional percolation. And the idea is the following. You have a honeycomb lattice, sort of, you tile your plane with hexagons, and you toss a coin. Each hexagon is going to be black with probability P and white with probability 1 minus P and the state of each of the hexagons is going to be independent of each other. And the question you're looking at, of course, here you're just doing coin tossing, so you have to explain what question you are looking at. Otherwise, so the question you look at is you're interested in the connectivity properties of the picture you obtain when you draw, when you do the simulation, you'll see one in a moment. And P plays the role of the temperature if you want. That's the parameter you're allowed to play with. And it happens that for these connectivity properties, the phase transitions appears to occur at PC equal one-half. And so the idea is when P is larger than one-half, then you have one infinite, you have a majority of black hexagons and you get one infinite black connected component called the infinite cluster and that has, if you want, a positive intensity or a positive density is actually the right word. Theta of, called theta of P, which is a deterministic function of P when P is larger than one-half. And when P is smaller or equal to one-half, you have no infinite black connected components, only small islands. And at P equal one-half, which is the critical point, so these random, what you would expect to be the random feature, you see clusters at any scale and their shape or the shape of these islands appear to be random. So that's critical percolation. So you should just think of it as a great television screen and you try to detect on this great, great television screen the connectivity properties. And you realize that, well, it's not a trivial question because our eyes are not well trained to, to detect if there's a left to right white crossing in this, in this picture. And actually this is a symptom of a more deeper question and the fact that the, the, well, the way the randomness is organizing and put together in order to create the macroscopic event of the left to right crossing is very subtle. And this is what a percolation cluster looks like in the previous picture. So that's one island. And I surrounded, I mean, the outer boundaries here are red, but forget about the boundaries for the moment. So this is one island, one cluster in the previous picture and shows you that when you look at your television screen, you will see clusters or islands of size comparable to the screen you look at, but you will not see infinite clusters. Okay. So here are predictions by physicists that are now mathematical theorems and I'm not going to explain you the root to that, to these predictions, to these theorems now because I would repeat a lot of what Oded said this morning. So here's one, the first prediction, I think this is due to Nienhuis, Denneis, and yeah, I think these are the two, first two who probably, maybe I'm mixing up things, predicted these five over 36 and five over 48 numbers. So the first one has to do with the behavior near the critical point, the intensity decays like P minus one half to some power when P approaches one half. And the second one has to do with a description of what happens of those random behavior at one half that tells you that the probability that there's an open path from a given site up to distance R is decays in a certain way. And so these are typical examples of these predictions that physicists made and that are maybe now accessible. Another model to keep in mind is that the so-called easing model, percolation has some very specific features because of the fact that what happens here and there on your television screen is independent. And we'll come back to these very specific features a bit later. And there's a model called the easing model, which is basically a model where you're going to, similar as the previous one, where you're going to bias the probability of each configuration depending on the number of, I mean, you look at the configuration, you count how many neighbors are disagreeing, how many pairs of points there are, which are neighbors, and such that one's black and the other one is white. You count how many there are and you're going to penalize the probability of a given configuration according to the number of disagreeing neighbors. So the easing model prefers to have, if you compare it to the disordered system of percolation, prefers to have neighbors that are of the same opinion or the same color. And this induces a long range or correlation between what happens at different various points. Okay. So one of the novelties sort of of this new mathematical approach that has started in the very late nineties is that instead of looking at the correlation functions, and here I just give you an example of what you might call a correlation function in the case of percolation. So instead of looking at the behavior when the mesh of the lattice goes to zero of probability that you see that far away points are distant at a given color, you're looking, you're going for to try to describe the entire random object and the actual random picture you see instead of some sort of things that look like finite dimensional marginals if you want. Of course this is a big simplification. I'm not, but if, so if you want to oversimplify a little bit more, you might say, well, that in complex analysis in general, you always have this magic trick that on the one hand you can write an analytic function as a power series. And this has a very local feature with some of the ANZ to the N and this looks like an analytic thing when you compose an analytic map you start to have playing with algebraic structures and you have this magic trick that this corresponds in fact to actual map from a portion of the plane to some other portion of the plane and that has some geometric insight. So we are going for the second approach if you want here and now I just try to be very quick on the second item because that was the topic of Odez's talk this morning if you want. So one of the main ideas is to use or to assume the fact that these models on large scale or these critical systems in two dimensions behave in a conformal invariant way. So what would that mean? Well, you should imagine the following. You take a system, so either you look at the critical lattice based system on a very, very fine mesh, so you, or you just go directly for the actual physical system in the, in the continuous and you look at the system in two different domains, D1 and D2 and which are conformally equivalent which means that there's a one to one map that preserves the angle from the domain D1 in the plane to the domain D2. So the way to describe these random systems is to say that in system in D1 you have a collection of clusters say that you call C1 index by K and in D2 you have another collection of clusters that you call C2 index by J here and that's these random systems that you want to describe and you could just going to say that these are going to be conformally invariant if basically the law, if you take the, your system in your first domain you map it onto, you map what you see in the first domain onto the second domain by a conformal map then you get a random system in the second domain that had exactly the same law as the system itself in the second domain. So in this morning's lecture they described the status of these, what is proved and what is not proved about the fact that the scaling limit of discrete lattice based models are indeed or not conforming invariant and if you've been to Smirnoff's lecture yesterday you know that there is some ongoing spectacular progress going on. Okay so the goal now is to describe these continuous structures that you see as scaling limits or not scaling limits but that have nice conforming invariance properties in the plane. So of course there is a very easy way to create conforming invariant, conforming invariant random structures. Well just pick your favorite domain say the upper half plane, choose any random object in the upper half plane or any random structure that is scaling variant or an invariant say under the Mobius transformation that will give you one random structure in the upper half plane and you define the random structure in any other domain just by taking the conformal image of what you've seen in the upper half plane in the other simply connected domain and that defines you a family of possible observables or possible systems once for each domain that satisfies conforming invariance. So conforming invariance itself is not a very restrictive condition, you need to add more condition in order to be able to pinpoint or to describe all possible such systems. So the first additional, I mean one possibility was explained by Oded this morning is that to focus on discrete interfaces. So here this is very schematic and actually has been drawn by my daughter when she left at my, the poor way I am handling her pictures on the, okay. So imagine that you have a domain with pre-cooked and nice boundary conditions say you assume that one part of the boundary here is red and the other part of the boundary is blue, I chose red and blue instead of black and white for obvious reasons. And basically if you are going to assume that here everything here on this boundary is blue, everything here on this boundary is red, you are going to have this random color television screen with blue and red colors in there and then it's very easy to see that you'll get one single interface that is going to separate one random line that is going to separate the blue cluster attached to this part of the boundary to the red cluster attached to the other part of the boundary. So this is a random curve and the point is that discrete interfaces can be explored and that's the basic starting point of Oded's trance that led into the diffusion of SLE which is we are going to explore the system just starting here and we start to explore here this interface. Because we are looking at, we are supposing that we have nearest neighbor interaction, system with nearest neighbor interaction, this tells you that once you know that this is the way the interface starts and you think what is the law of what remains to be explored here of this strange television screen that still remains to be explored, well, it's still now say percolation or easing in the remaining domain and now the domain is the circle with the slit and with new boundary condition where on this part of the boundary you can hear everything is blue, here everything is red and so if you assume conforming variance, in fact once you start exploring the boundary, I mean this interface, you ask, okay, now how do I continue? You still have, you still are in the, asking exactly the same question, you still are looking at, have to explore an interface between, in a simply connected domain where you are looking at one part of the boundary is red, one part of the boundary is blue and you are looking at the interface between these two things and so using this, Odech-Ram explained this morning that combining sort of classical ideas from complex analysis namely Leuveness theory with basic probabilistic insight, you get that there is a, exists at most a one parameter family of such random curves in the, in a simply connected domain that satisfy both this conforming variance property and this exploration property, this idea that you can sort of explore the curve progressively like this, okay, this is of course too short an explanation but just to mention that the output of this, of this idea that just that you can explore interfaces progressively leads just to actual concrete description of the possible interfaces for these, I mean interfaces for these special, especially precooked boundary conditions and you get the one family parameter of such random curves called the Schramm-Leuven evolution, that's a difference maybe with this morning's lecture, they call it Schramm-Leuven evolution and this one parameter is usually called kappa and okay, here are lists on properties that Odech described this morning, so there's, there are two types of curves, some of them are simple random curves and some of them are, have double points, so there's a phase transition at kappa equal four, this number will come up later and kappa equal six and a third are also very special, they have some, turn out to have the very special properties and I'll come back to that later. So now the, the, I want to focus on this so called conformal restriction property. So conformal restriction is, is another idea that is complementary if you want to the previous one but gives another approach or another characterization to these random curves or I say leakers. So actually this idea has been developed and refined in a sequence of papers with Greg and then Greg and Odech and the result I'm going to present now is a paper of mine but it's really just a continuation of our earlier joint work and the idea now is not to explore the curve if you want from, explore the curve dynamically but you explore it from far away. So you see well how does the curve look like from far away and one way to, to view, to say this is to say that you're going to compare the law of the shape of the curve into, when it's defined in two different domains. So here you have, you start having this idea that you're going to, I'm not going of course to explain this and anyway I'm not sure how to understand it really, fully that you're going to start playing with a variation of the law of the curve with respect to the domain you are looking at and that's where these Lie algebras enter into the game but I'm not going to say more about that. So here I'll ask a specific question. I give the motivation later, you'll see the motivation with the answer if you want and the question is the following. You're looking for a measure supported on the set of loops in the plane. So it's, you're, the set of, you're looking at is a set of single loops that you can draw in the plane. And you want this measure to satisfy this property that we call, that I call here strong conformal restriction property and this property is the following. You take any two conformally equivalent domains. You look at the measure restricted to the first one and you look at the measure restricted to the second one. What you want is that if you map conformally the measure restricted to the first one onto the second one via this conformal map, you get exactly without any scaling factor the measure restricted to the second. So whatever domain you look at if you want you're going to see the same measure. That's the idea. If you, if you ask this condition to be true for simply connected domains only, we call it weak conformal restriction, that means that whatever simply connected domain you choose you're going to see the same measure modulo conform equivalence. So at first you say, well, such a measure cannot exist, this is too strong a condition. And some, the first two items here are not only rather easy but just trivial consequences of the definition. Suppose you have such a measure, then of course it has to be scaling variant because some conformal map, I mean multiplications are conformal maps. So if you take a disk, you look at what you see in this disk or twice disk disk, you see the same measure. The measure is scaling variant, it's translation invariance for a similar reason and therefore it must have an infinite mass. So it's going to be a measure with infinite mass where this infinite mass is going to be supported on both very, very small loops and very, very large loops. And the third item is rather easy but not completely trivial. But I ask you to believe me that this is not a difficult statement that in fact you only have one, at most one measure satisfying weak conformal restriction. So already the weak condition is very restrictive and you end up very quickly with the fact that this is so restrictive that you cannot anyway have more than one measure satisfying this condition. And a more difficult theorem which involves SLE is that in fact such a measure exists. And not only satisfies weak conformal restriction but it satisfies this stronger conformal restriction property. So there exists a measure on simple loops in the plane that satisfies this property that whatever domain you look at it, you'll see the same measure, modular conformal invariance. And it turns out that there are three different, a priori completely different construction of this measure. I'm going to say a word about these. So the first construction uses Brownian motion. Imagine you take a planar Brownian motion. So this is the scaling limit if you want of a very, of a simple random walk on a very fine mesh or the trajectory of a crazy fly say that it's just moving around at random in the plane. And you condition to be back at the starting point at time one. So this creates a loop. Of course this loop is not self avoiding. We are looking for a measure, remember, maybe I've not insisted on that before. We are looking for a measure on loops that are self avoiding loops that have no double points. Just things that separate the plane into two connected components. So a planar Brownian loop has looked like this. This is a trajectory of this and it has been studied, this type of properties has been studied extensively. And it's known since the 50s that planar Brownian motion because it's related to, that's one way to view it related to Laplacian and related to harmonic functions is going to be invariant in some way under conformal transformations. And it doesn't take, didn't take us too much effort with Greg to actually see that this measure on Brownian loops here. We can turn it in the way you of course are going to natural way into a scale invariant and translation invariant measure on paths in the plane that are Brownian paths. And what you get is a scale invariant measure on Brownian loops which is translation invariant and satisfies conformal invariant properties. And the very definition of the way you are going to define it is going to tell you that if you focus just on the outer boundary of this loop, this is going to be a self-avoiding loop. So each Brownian loops define one self-avoiding loop and that the measure under which the self-avoiding loop is defined satisfies weak conformal restriction. Now I will not insist on that but if you think about the percolation model, the percolation model on the large scale is going to describe very naturally a measure on clusters. Basically each cluster you see in the continuous, I mean in the discrete picture counts as one, has mass one and then just take this counting measure on clusters defined by percolation and you get a measure supported on the set of clusters. If you imagine that these clusters have a scaling limit that is conformal invariant, the independence properties of percolation are going to imply in fact that as before this measure on percolation clusters that you get here is going to satisfy the same weak conformal restriction property and therefore this outer boundary, so again now you focus just on the outer boundary of this cluster, this red outermost path, will satisfy, will be a measure on self-avoiding loops that satisfies weak conformal restriction. So here you see without the help of SLE at all, you get the fact that outer boundaries of percolation clusters and outer boundary of Brownian motions are just the same in the scaling limit. Now it turns out that with the help of SLE you can say more because SLE, one of these SLE, the one with parameter 8 third, turns out to have a very special property. As I mentioned before, that makes it possible to define directly a measure on SLE 8 third loops and to prove not only that this measure satisfies weak conformal restriction but also the stronger version. And one of the big open questions is precisely to prove that this measure on self-avoiding loops that we are describing is in fact the scaling limit of the, if you want, I'm cheating slightly, the uniform measure on self-avoiding loops in the plane. That's one way to understand it. For how measure if you want it to be invariant. So what you end up with here is that these three constructions, outer boundary of a Brownian loop, outer boundary of a percolation cluster in the scaling limit, and these SLE 8 third loops are going to define exactly the same measure, the same random, well it's an infinite measure but the same random shape. And furthermore you know that the last one satisfies not only weak conformal restriction but strong conformal restriction so the outer boundaries defined in the previous case define also satisfy also this strong conformal restriction property. So this is one explanation why SLE is useful to solve this Manfred-Bröhrl's conjecture about the fact that the outer boundary here of a planar Brownian loop here or planar Brownian motion in general has dimension four thirds because I just told you that the outer boundary is exactly the same as an SLE 8 third loops. SLE 8 third you can perform computation so you can compute probabilities that you couldn't compute with the Brownian motion picture and then you can deduce from that computation that you get the dimension four thirds for this outer boundary. So you see these two different ideas the first one SLE gives random or continuous objects that allow to perform computations and you have additional sort of argument that tells you that anyway even two different a priori different models will give rise here to the same object in the scaling limit. And similar ideas led to the derivation of all what's called Brownian intersection exponent that had been predicted by Bertrand Duplantier before. So okay let me now continue another consequence here just to mention to show you that this strong conform restriction is not a trivial statement it shows you something like so you see the outer boundary of this red island has a certain random shape now if you instead you look at the inner boundary so now you take the outer boundary of this white island inside this red island yes so you look at the outer boundary here so of course this is very different because here when you are inside the Brownian motion is not inside of the loop but it's outside okay but the previous theorem shows in fact that the law of the shape of the inside thing of this white island is exactly the same than the law of the shape of the red one right so you have surprising inner outer symmetries if you want. So you see here that we started with this idea that you have you are looking for asking an abstract question about we are looking for a measure satisfying a certain property and we end up with the fact that well there is just one and it's not a trivial one it's a measure you know supported on the set of loops that have dimension four thirds and so on however this it turns out I'm not going to elaborate on that but this measure because it's the only one satisfying this property is going to be probably this very natural property related to other features and maybe other parts of mathematics and useful there. Now I want to spend the remaining time discussing what we call conformal loop ensemble with Quarchefield and that's ongoing work but let me give a little motivation for that. As I explained you before SLE is going to give you the law of one interface you precook you have prescribed boundary conditions everything is blue everything is red here and you get the law of this random curve that is between them. Well you might say well that's enough we know once we know the law of this curve we have a lot of information about the system and indeed you get a lot of information of access to some critical exponents and many things. However if you want to describe the scaling I mean the entire picture that you are actually seeing you need to continue you need to say well okay what happens here and what happens there but once you did draw this curve here now you have a domain here which has monochromatically blue boundary conditions so you cannot really start the same you cannot iterate the procedure by starting a new SLE somewhere here because now the boundary conditions do not have these specific blue and red parts. So you try to you have to figure out something different and if one looks at the discrete models if you want and the properties of the discrete models well how are you going to describe what you see here in your domain well you are going to describe via a family of loops that correspond to the boundaries of the clusters that you are going to see in this picture. And these loops are going to possess certain properties that lead to the following definition in the continuous case. So here again this is a schematic picture so you are looking trying to describe what possible laws could be the scaling limits of these random collections of loops that you see there. So this time here imagine that we are looking at a configuration in our case is going to be a random collection I mean a collection of loops that are disjoint and that are not nested. So you may see this picture as some sort of a cantor set if you want if you fill in the interior of all these loops the idea is that you should see some type of random cantor set here. And these blue guys should be you can think of the outermost loops of clusters in certain models if you want. And the condition we are going to ask is the following. First of all we want it to be conforming variant so we want this to be defined in any domain in such a way that what you see in any domain is obtained by taking the conform image of what you see in the disk. And now if I cut something out of the disk so you cut anything and so you take any smaller domain of the disk. Now you see that you have two different types of loops you have those that go stay in the white region and those that go out. So let's go in black the interior of all the loops that touch that go out of this white region. So now you see you have okay I'm cheating slightly of course but you have a new domain which is this complement of all these black loops. The condition you are asking is that given the loops that intersect the outer boundary here the law of what remains to be explored here the law of what the loops in the remaining domain is the same as the law of what you started with. So in another words and this is a very natural this looks maybe a little bit surprising a condition at first sight but this is the condition you would ask if these loops correspond indeed to outermost interfaces if you want in lattice based models. So question what are the possible laws on collection of loops that you describe like this? Well we have a theorem with Scott Sheffield that tells you actually that again you only have a one parameter family of such objects and what happens is that for each of these objects the loops will look like SLE loops with a certain fixed parameter kappa. So for I haven't described to you what the parameter kappa was in the definition of SLE but you should imagine roughly speaking that for any dimension D between four thirds and three halves there exists exactly one such measure supported on this joint collection of loops like this that is supported on collection of loops with which have this dimension and you don't have any other such random collections than the ones I described to you. Okay you have three different constructions of these loops of these collections and let me just very quickly describe you one of them that I like a lot but that's just personal taste it's not a mathematical value. So the idea is the following you are going to use the previous conforming invariant measure on self avoiding loops that I've described to you this measure mu you are going to use it and use its property in order to construct a more elaborate measure which is elaborate measures which are these interacting loops now. Well you are going to imagine that it's going to rain loops on the screen and you are going to let it rain with intensity given by this measure mu and I'm not going for those who are probabilists or know about Poisson point processes they know what I mean and those who don't know also what I mean because that's just what I mean is raining loops according to the intensity given by the measure mu and you let it rain a certain time so here that's at the beginning you only see very small loops because and then you fill them in and then you let it rain a little bit more you have other loops that fall in and of course all these loops are going to overlap right because they are independent it's like if you have leaves falling on the ground if you want. Okay and you see this picture and then you continue the picture grows and what happens is after a certain time of course this is very schematic these loops you should think of these loops as being loops which have dimension four thirds and not things drawn with paint with a lot of paint on my PC. Well what you see is that at a finite time there exists exactly a finite time and you can actually prove it such that immediately after this finite time all the loops that did fall down before that are going to hook up and form a single connected component so the union of all the loops are suddenly going to crystallize in one single connected component. Now this in a way is a way to define fractal percolation or which is also sometimes called Mandelbrot percolation which in short is just defining random two dimensional kentosets that at each scale you are going to look at certain shapes and say well either we keep it or we throw it away and so that's what we are going here to do here that if you want we are defining a random kentoset that has conformal invariance properties because instead of we are going to remove shapes which have these measure mu as definition. Well it turns out that these outermost loops of the clusters you get here for any c smaller than this time at which everything crystallizes are going exactly to be these CLE loops. So it turns out that you let it rain these self avoiding loops or Brownian loops or whatever you define according to the measure mu a certain time they are going to create clusters and you look at the outer boundaries of each of those clusters they are going to correspond to define you sort of here you have one loop here like this and here you have another one and the picture you are going to see is exactly this describing for you the CLE loops. Why I am spending a little time describing to you this is that in this interpretation the time you are going to let it rain instead of calling it T you are going to call it C because that is precisely what physicists call the central charge of the model. So you have a very constructive and with bare hands construction of random geometric objects that have a lot of structure and for which you can now interpret these quantities here not as central charge of some representation but just at the time you are going to let it rain. So there is an alternative as in the previous case of the self avoiding loops you have an alternative description of these loops of these loop ensembles using other means and okay I have two. One other mean is the Gaussian free field construction which is very exciting and done by Schramm and Sheffield and Sheffield also partly and here again you have this feature that two different very different models in the continuum are going to define the same continuous I mean the same object even though a priority are very different just as the outer boundaries of percolation and outer boundaries of brownie loops were the same shape here you see that different objects are going to define the same CLEs. So one hope of course is that these CLE these collections of random interacting loops are going to define help you to understand better some of the mathematics that are behind or the relation that are behind this conformal theory field theory or okay Coulomb gas and tie links between these two because we can show these are the same. So here in the final slide here actually I forgot some many people I listed the names of people active currently active in this area and thinking or about such questions and you that you can if they are in your department you can just knock at their door and ask them more information about these questions if you want. So here you have mathematicians roughly speaking here you have physicists and I included also the list I mean in this list okay I forgot Michael Costron recent PhDs and postdocs and actually there's one Spanish one from a student of Greg. So if you want more information of this type of topics there's many surveys lecture notes ICM proceedings and even a book by Greg so there's a lot of introduction to the subject and I apologize I realize I wanted to focus really on this self-interacting loops and try to get so my goal was to get you an impression of this general feature that you are looking for a continuous object a measure supported on continuous object you are requiring certain properties that are natural if you assume that these are the scaling limit of physical models or if they are actually describing physical phenomena and you end up actually with a very restrictive collection of possible candidates either if you're focusing on one line you get one SLE curve if you focus on these more global things you get this collection of interacting loops and that just with sort of a simple arguments you get to actually rather description of a rich random objects that is and I had the time of course to describe that to you probably related to various other parts of mathematics sorry for being over time and thank you.
 Good afternoon. It is a really great pleasure for me to introduce Ferdinand Werner for the  final Fields Medal Lecture. Ferdinand received his PhD in 1993 at University of Paris-Sies  under the direction of Jean-François Legault. Since 1997, he has been Professor of University  Paris-Soud in Orsay. From 2001 to 2006, he was a member of the Institute Universitaire  de France and he currently also holds a position at École Normale Supérieure in Paris. His  prizes include the Rolo Davidson Prize, the European Mathematical Society Prize, the  Fermat Prize, Jacques-Hubert Prize, the Leuven Prize and the Polier Prize. On a lighter note  in trivia, he is also one of the few people I know who has finite identical Erdos numbers  and Kevin Bacon numbers, something for people to look up later. Three in both cases. From  a personal perspective, I just want to say this, it was about ten years ago that I first,  I was giving a lecture about intersection exponents, the problem of brine intersection  exponents, one case of which was the mandelbrot, equivalent to the mandelbrot conjecture about  browning motion. When Bedlin was in the audience, and a couple weeks later I received an email  with a very nice idea how to approach this problem. From the start of that email, I started  a collaboration that has lasted a long time and I've seen many, many of his nice ideas.  I'd only like to mention a couple of them today. One of them a couple years later was,  as I think was mentioned this morning's talk by Odette, he was talking with Odette Schramm  and he realized very early that Odette's beautiful SLE might be the key to understanding the  browning intersection exponents. At that point, our two-person collaboration on this  side became a three-person collaboration and suddenly I had two outstanding collaborators  with giving nice ideas back and forth. Secondly, I'd like to sort of general idea, although  a lot of the work was problem-driven by these particular problems. It's Bedlin, many times  is the one who's actually sat back and said, that sort of looked and said, not only are  we solving these problems that we're asking, but in fact, these structures that are done,  both SLE and browning in motion, are really perhaps the keys to understanding putting  conformal field theory on a rigorous basis and now that's an active study area being  pursued by many people. With that, I'm happy to introduce Bedlin for his talk.  Thank you very much. As you might imagine, this week has been a very special week for  me and enjoyable one. I hope I'm not too tired to deliver a reasonable talk today. Since  it's the first time I get a chance to speak, I have, of course, many people to thank from  personal reasons to PhD supervisor, but I'll do that on personal level that would be too  long to do now, but of course I'd like to, I'm very happy and glad that I'm speaking  after Oded's plenary lecture this morning and that Greg was able to be here to also  and as chair of this session because a lot of what I'm going to say and a lot of maybe  part of the reason that I'm standing here in front of you is that I met these two nice  people. So the topic of my talk is random planner loops and conformal restriction and,  okay, I added a survey because of course this is supposed to be more an introductory  lecture that gives some flavor of some of the ideas and some of the results that we  obtained during the last years. And this talk is, of course, not unrelated and rather  closely related to Oded's talk this morning and also to Stasman's talk yesterday, but  I'm not going to assume that you all went to these two nice lectures. I think I'll  try to make it self-contained today, but maybe with some partial repetition of at some points  on some issues with what mostly what Oded said this morning. But I'll try to take another  perspective which is more to try to look at description and understanding of continuous  random structures in the plane using the, of course, the discrete models as a guideline  but really focusing on properties of the continuous objects. So I think it's fair to start with  some background about motivation and history in the physics community because this subject  of research, one of the main motivation comes from physics and physicists have been given  a lot of input to these subjects before we actually started to look at them. And actually  I'd like to thank also take the opportunity to thank them, Michael Eisenman, Bertrand  Duplantier and other physicists who also made the effort to come to math departments and  actually say, here we have a nice subject problem for you that we know sort of know how to solve  but which is a nice problem for you, mathematician, because the way we approach them is probably  not the way one should do it mathematically. And then when we came up with some ideas,  they also accepted these and didn't just say, well, we knew it before. So they acknowledged  the fact that these were, was really new input. So I'd like to stress this. So I'm going to  say a couple of very general statements and to start with, with the danger that it's too  general so that you don't see what I mean. But in general, when you learn about physics,  you learn that, well, the laws of physics is something when you repeat it twice, the  same experiment, you get twice the same results. And that's the result of the laws of deterministic  laws of physics. And it has been observed long ago actually experimentally that on macroscopic  scale, when you are exactly at a point at which the physics is called a phase transition  point, that means a point where, say, for instance, you can imagine the temperature  at which liquid become vapor or there's an even competition between two possible states  in the system, that when you're exactly at that point, this previous deterministic law  doesn't hold anymore because you see some features that become random on macroscopic  scale. So you repeat the same experiment twice, you, and you will get different answers.  And part of the story is to understand some of these macroscopic random features or complex  systems that you see on macroscopic scale. And a closely related question is that, well,  how do you describe what phase transition point is? Well, usually it means that some  of the physically deterministic quantities that you observe on macroscopic scale, when  you are away from the critical point, go to zero or go to infinity. And it has been observed  that sort of these quantities often obey certain power law behavior when you approach the critical  point. So that some quantity, deterministic quantity, sorry, basically behaves like t  minus tc to some power gamma. And this exponent gamma is called the critical exponent for  this model or for this object. So here you see you have a description of the first random  object at the critical point, and then below you have deterministic behavior of deterministic  points, quantities near the critical point. And of course, well, not of course, but it  has been observed also that these two, or argued that these two phenomena are very closely  related with each other. So physicists have came up with, theoretical  physicists have came up with a number of inventive and clever ideas in order to describe these  problems and these questions. And the first of which has been developed by, okay, it's  not to intimidate you, I just put down some names of physicists that have contributed  to these issues. And of course, I am omitting a lot of names here. The first idea is that  of a renormalization group. And that basically is, so of course, it's a very simplified explanation,  but it explains roughly or gives you a convincing heuristic to the fact that different models  or different gases or different experiments will give rise to the same exponents or to  the same random behavior at the critical point. And the idea is basically to say that this  random macroscopic behavior to interpret at this, at a fixed point of some renormalization  function that basically you divide your system in a large system into smaller boxes and each  box, this random system is created and you put them together, you create a large system.  And what you argue is that at this critical point, the system is a fixed point of this  because it becomes scaling variant, it will be a fixed point of this transformation. So  here are three items that are specific to two dimensional systems. And so conformal field  theory and also what is called Coulomb gas techniques or quantum gravity are ideas or  that sort of based on analogies, based on explicit computations, based on many different  facts that provides mathematical tools in order that enabled these physicists to predict  the value of these critical exponents of many different systems in the case where the dimension  is two. So we are looking at planar systems. And the exponents of the models are classified  according to what physicists call the central charge of the model. That means each model  or system you look at has a specific central charge. And this term central charge refers  to the fact that some of the mathematical tools that are hidden behind the conformal  theory scene have to do with the representation theory of some infinite dimensionally algebras.  And the third item is explanation of the fact of the precise relation that there exists  between the behavior, this random behavior at the critical point and the deterministic  behavior near the critical point. And I think it's fair to say that apart from the last  item which has been treated in the mid 80s by Harry Keston, that's one of the many things  that Harry Keston did in all these questions. So he treated the case of percolation in two  dimensions where he made sense of the scaling relations and explained that if you understand  the random behavior of the system at the critical point, you also understand the behavior of  the system near the critical point. And it's fair to say that apart from that, these three,  even though conformal field theory was related to very deep and rigorous mathematics, the  relation between these deep and rigorous mathematics and the actual questions you look at was a  question mark. Okay. So I'm going to repeat very quickly so that you have one model in  mine. I chose to repeat a model that has been introduced that you may be seeing once  or twice in this ICM before, but, well, I guess that's the simplest model to explain  so that you have in mind something specific and it's not too general. So the specific  model is called two-dimensional percolation. And the idea is the following. You have a  honeycomb lattice, sort of, you tile your plane with hexagons, and you toss a coin.  Each hexagon is going to be black with probability P and white with probability 1 minus P and  the state of each of the hexagons is going to be independent of each other. And the question  you're looking at, of course, here you're just doing coin tossing, so you have to explain  what question you are looking at. Otherwise, so the question you look at is you're interested  in the connectivity properties of the picture you obtain when you draw, when you do the  simulation, you'll see one in a moment. And P plays the role of the temperature if you  want. That's the parameter you're allowed to play with. And it happens that for these  connectivity properties, the phase transitions appears to occur at PC equal one-half. And  so the idea is when P is larger than one-half, then you have one infinite, you have a majority  of black hexagons and you get one infinite black connected component called the infinite  cluster and that has, if you want, a positive intensity or a positive density is actually  the right word. Theta of, called theta of P, which is a deterministic function of P when  P is larger than one-half. And when P is smaller or equal to one-half, you have no infinite  black connected components, only small islands. And at P equal one-half, which is the critical  point, so these random, what you would expect to be the random feature, you see clusters  at any scale and their shape or the shape of these islands appear to be random. So that's  critical percolation. So you should just think of it as a great television screen and you  try to detect on this great, great television screen the connectivity properties. And you  realize that, well, it's not a trivial question because our eyes are not well trained to,  to detect if there's a left to right white crossing in this, in this picture. And actually  this is a symptom of a more deeper question and the fact that the, the, well, the way  the randomness is organizing and put together in order to create the macroscopic event of  the left to right crossing is very subtle. And this is what a percolation cluster looks  like in the previous picture. So that's one island. And I surrounded, I mean, the outer  boundaries here are red, but forget about the boundaries for the moment. So this is  one island, one cluster in the previous picture and shows you that when you look at your television  screen, you will see clusters or islands of size comparable to the screen you look at,  but you will not see infinite clusters. Okay. So here are predictions by physicists that  are now mathematical theorems and I'm not going to explain you the root to that, to  these predictions, to these theorems now because I would repeat a lot of what Oded said this  morning. So here's one, the first prediction, I think this is due to Nienhuis, Denneis,  and yeah, I think these are the two, first two who probably, maybe I'm mixing up things,  predicted these five over 36 and five over 48 numbers. So the first one has to do with  the behavior near the critical point, the intensity decays like P minus one half to  some power when P approaches one half. And the second one has to do with a description  of what happens of those random behavior at one half that tells you that the probability  that there's an open path from a given site up to distance R is decays in a certain way.  And so these are typical examples of these predictions that physicists made and that  are maybe now accessible. Another model to keep in mind is that the so-called easing  model, percolation has some very specific features because of the fact that what happens  here and there on your television screen is independent. And we'll come back to these  very specific features a bit later. And there's a model called the easing model, which is  basically a model where you're going to, similar as the previous one, where you're going to  bias the probability of each configuration depending on the number of, I mean, you look  at the configuration, you count how many neighbors are disagreeing, how many pairs of points  there are, which are neighbors, and such that one's black and the other one is white. You  count how many there are and you're going to penalize the probability of a given configuration  according to the number of disagreeing neighbors. So the easing model prefers to have, if you  compare it to the disordered system of percolation, prefers to have neighbors that are of the same  opinion or the same color. And this induces a long range or correlation between what happens  at different various points. Okay. So one of the novelties sort of of this new mathematical  approach that has started in the very late nineties is that instead of looking at the  correlation functions, and here I just give you an example of what you might call a correlation  function in the case of percolation. So instead of looking at the behavior when the mesh of  the lattice goes to zero of probability that you see that far away points are distant at  a given color, you're looking, you're going for to try to describe the entire random object  and the actual random picture you see instead of some sort of things that look like finite  dimensional marginals if you want. Of course this is a big simplification. I'm not, but  if, so if you want to oversimplify a little bit more, you might say, well, that in complex  analysis in general, you always have this magic trick that on the one hand you can write  an analytic function as a power series. And this has a very local feature with some of  the ANZ to the N and this looks like an analytic thing when you compose an analytic map you  start to have playing with algebraic structures and you have this magic trick that this corresponds  in fact to actual map from a portion of the plane to some other portion of the plane and  that has some geometric insight. So we are going for the second approach if you want  here and now I just try to be very quick on the second item because that was the topic  of Odez's talk this morning if you want. So one of the main ideas is to use or to assume  the fact that these models on large scale or these critical systems in two dimensions  behave in a conformal invariant way. So what would that mean? Well, you should imagine  the following. You take a system, so either you look at the critical lattice based system  on a very, very fine mesh, so you, or you just go directly for the actual physical system  in the, in the continuous and you look at the system in two different domains, D1 and  D2 and which are conformally equivalent which means that there's a one to one map that preserves  the angle from the domain D1 in the plane to the domain D2. So the way to describe these  random systems is to say that in system in D1 you have a collection of clusters say that  you call C1 index by K and in D2 you have another collection of clusters that you call  C2 index by J here and that's these random systems that you want to describe and you  could just going to say that these are going to be conformally invariant if basically the  law, if you take the, your system in your first domain you map it onto, you map what  you see in the first domain onto the second domain by a conformal map then you get a random  system in the second domain that had exactly the same law as the system itself in the second  domain. So in this morning's lecture they described the status of these, what is proved  and what is not proved about the fact that the scaling limit of discrete lattice based  models are indeed or not conforming invariant and if you've been to Smirnoff's lecture  yesterday you know that there is some ongoing spectacular progress going on.  Okay so the goal now is to describe these continuous structures that you see as scaling  limits or not scaling limits but that have nice conforming invariance properties in the  plane. So of course there is a very easy way to create conforming invariant, conforming  invariant random structures. Well just pick your favorite domain say the upper half plane,  choose any random object in the upper half plane or any random structure that is scaling  variant or an invariant say under the Mobius transformation that will give you one random  structure in the upper half plane and you define the random structure in any other domain  just by taking the conformal image of what you've seen in the upper half plane in the  other simply connected domain and that defines you a family of possible observables or possible  systems once for each domain that satisfies conforming invariance. So conforming invariance  itself is not a very restrictive condition, you need to add more condition in order to  be able to pinpoint or to describe all possible such systems. So the first additional, I  mean one possibility was explained by Oded this morning is that to focus on discrete  interfaces. So here this is very schematic and actually has been drawn by my daughter  when she left at my, the poor way I am handling her pictures on the, okay. So imagine that  you have a domain with pre-cooked and nice boundary conditions say you assume that one  part of the boundary here is red and the other part of the boundary is blue, I chose red  and blue instead of black and white for obvious reasons. And basically if you are going to  assume that here everything here on this boundary is blue, everything here on this boundary  is red, you are going to have this random color television screen with blue and red  colors in there and then it's very easy to see that you'll get one single interface that  is going to separate one random line that is going to separate the blue cluster attached  to this part of the boundary to the red cluster attached to the other part of the boundary.  So this is a random curve and the point is that discrete interfaces can be explored  and that's the basic starting point of Oded's trance that led into the diffusion of SLE  which is we are going to explore the system just starting here and we start to explore  here this interface. Because we are looking at, we are supposing that we have nearest  neighbor interaction, system with nearest neighbor interaction, this tells you that once you  know that this is the way the interface starts and you think what is the law of what remains  to be explored here of this strange television screen that still remains to be explored,  well, it's still now say percolation or easing in the remaining domain and now the domain  is the circle with the slit and with new boundary condition where on this part of the boundary  you can hear everything is blue, here everything is red and so if you assume conforming variance,  in fact once you start exploring the boundary, I mean this interface, you ask, okay, now  how do I continue? You still have, you still are in the, asking exactly the same question,  you still are looking at, have to explore an interface between, in a simply connected  domain where you are looking at one part of the boundary is red, one part of the boundary  is blue and you are looking at the interface between these two things and so using this,  Odech-Ram explained this morning that combining sort of classical ideas from complex analysis  namely Leuveness theory with basic probabilistic insight, you get that there is a, exists at  most a one parameter family of such random curves in the, in a simply connected domain  that satisfy both this conforming variance property and this exploration property, this  idea that you can sort of explore the curve progressively like this, okay, this is of course  too short an explanation but just to mention that the output of this, of this idea that  just that you can explore interfaces progressively leads just to actual concrete description  of the possible interfaces for these, I mean interfaces for these special, especially precooked  boundary conditions and you get the one family parameter of such random curves called the  Schramm-Leuven evolution, that's a difference maybe with this morning's lecture, they call  it Schramm-Leuven evolution and this one parameter is usually called kappa and okay, here are  lists on properties that Odech described this morning, so there's, there are two types  of curves, some of them are simple random curves and some of them are, have double points,  so there's a phase transition at kappa equal four, this number will come up later and kappa  equal six and a third are also very special, they have some, turn out to have the very  special properties and I'll come back to that later.  So now the, the, I want to focus on this so called conformal restriction property.  So conformal restriction is, is another idea that is complementary if you want to the previous  one but gives another approach or another characterization to these random curves or  I say leakers.  So actually this idea has been developed and refined in a sequence of papers with Greg  and then Greg and Odech and the result I'm going to present now is a paper of mine but  it's really just a continuation of our earlier joint work and the idea now is not to explore  the curve if you want from, explore the curve dynamically but you explore it from far away.  So you see well how does the curve look like from far away and one way to, to view, to  say this is to say that you're going to compare the law of the shape of the curve into, when  it's defined in two different domains.  So here you have, you start having this idea that you're going to, I'm not going of course  to explain this and anyway I'm not sure how to understand it really, fully that you're  going to start playing with a variation of the law of the curve with respect to the domain  you are looking at and that's where these Lie algebras enter into the game but I'm not  going to say more about that.  So here I'll ask a specific question.  I give the motivation later, you'll see the motivation with the answer if you want and  the question is the following.  You're looking for a measure supported on the set of loops in the plane.  So it's, you're, the set of, you're looking at is a set of single loops that you can draw  in the plane.  And you want this measure to satisfy this property that we call, that I call here strong  conformal restriction property and this property is the following.  You take any two conformally equivalent domains.  You look at the measure restricted to the first one and you look at the measure restricted  to the second one.  What you want is that if you map conformally the measure restricted to the first one onto  the second one via this conformal map, you get exactly without any scaling factor the  measure restricted to the second.  So whatever domain you look at if you want you're going to see the same measure.  That's the idea.  If you, if you ask this condition to be true for simply connected domains only, we call  it weak conformal restriction, that means that whatever simply connected domain you  choose you're going to see the same measure modulo conform equivalence.  So at first you say, well, such a measure cannot exist, this is too strong a condition.  And some, the first two items here are not only rather easy but just trivial consequences  of the definition.  Suppose you have such a measure, then of course it has to be scaling variant because  some conformal map, I mean multiplications are conformal maps.  So if you take a disk, you look at what you see in this disk or twice disk disk, you see  the same measure.  The measure is scaling variant, it's translation invariance for a similar reason and therefore  it must have an infinite mass.  So it's going to be a measure with infinite mass where this infinite mass is going to  be supported on both very, very small loops and very, very large loops.  And the third item is rather easy but not completely trivial.  But I ask you to believe me that this is not a difficult statement that in fact you only  have one, at most one measure satisfying weak conformal restriction.  So already the weak condition is very restrictive and you end up very quickly with the fact  that this is so restrictive that you cannot anyway have more than one measure satisfying  this condition.  And a more difficult theorem which involves SLE is that in fact such a measure exists.  And not only satisfies weak conformal restriction but it satisfies this stronger conformal restriction  property.  So there exists a measure on simple loops in the plane that satisfies this property that  whatever domain you look at it, you'll see the same measure, modular conformal invariance.  And it turns out that there are three different, a priori completely different construction  of this measure.  I'm going to say a word about these.  So the first construction uses Brownian motion.  Imagine you take a planar Brownian motion.  So this is the scaling limit if you want of a very, of a simple random walk on a very  fine mesh or the trajectory of a crazy fly say that it's just moving around at random  in the plane.  And you condition to be back at the starting point at time one.  So this creates a loop.  Of course this loop is not self avoiding.  We are looking for a measure, remember, maybe I've not insisted on that before.  We are looking for a measure on loops that are self avoiding loops that have no double  points.  Just things that separate the plane into two connected components.  So a planar Brownian loop has looked like this.  This is a trajectory of this and it has been studied, this type of properties has been  studied extensively.  And it's known since the 50s that planar Brownian motion because it's related to, that's one  way to view it related to Laplacian and related to harmonic functions is going to be invariant  in some way under conformal transformations.  And it doesn't take, didn't take us too much effort with Greg to actually see that this  measure on Brownian loops here.  We can turn it in the way you of course are going to natural way into a scale invariant  and translation invariant measure on paths in the plane that are Brownian paths.  And what you get is a scale invariant measure on Brownian loops which is translation invariant  and satisfies conformal invariant properties.  And the very definition of the way you are going to define it is going to tell you that  if you focus just on the outer boundary of this loop, this is going to be a self-avoiding  loop.  So each Brownian loops define one self-avoiding loop and that the measure under which the  self-avoiding loop is defined satisfies weak conformal restriction.  Now I will not insist on that but if you think about the percolation model, the percolation  model on the large scale is going to describe very naturally a measure on clusters.  Basically each cluster you see in the continuous, I mean in the discrete picture counts as one,  has mass one and then just take this counting measure on clusters defined by percolation  and you get a measure supported on the set of clusters.  If you imagine that these clusters have a scaling limit that is conformal invariant,  the independence properties of percolation are going to imply in fact that as before  this measure on percolation clusters that you get here is going to satisfy the same  weak conformal restriction property and therefore this outer boundary, so again now you focus  just on the outer boundary of this cluster, this red outermost path, will satisfy, will  be a measure on self-avoiding loops that satisfies weak conformal restriction.  So here you see without the help of SLE at all, you get the fact that outer boundaries  of percolation clusters and outer boundary of Brownian motions are just the same in the  scaling limit.  Now it turns out that with the help of SLE you can say more because SLE, one of these  SLE, the one with parameter 8 third, turns out to have a very special property.  As I mentioned before, that makes it possible to define directly a measure on SLE 8 third  loops and to prove not only that this measure satisfies weak conformal restriction but also  the stronger version.  And one of the big open questions is precisely to prove that this measure on self-avoiding  loops that we are describing is in fact the scaling limit of the, if you want, I'm cheating  slightly, the uniform measure on self-avoiding loops in the plane.  That's one way to understand it.  For how measure if you want it to be invariant.  So what you end up with here is that these three constructions, outer boundary of a Brownian  loop, outer boundary of a percolation cluster in the scaling limit, and these SLE 8 third  loops are going to define exactly the same measure, the same random, well it's an infinite  measure but the same random shape.  And furthermore you know that the last one satisfies not only weak conformal restriction  but strong conformal restriction so the outer boundaries defined in the previous case define  also satisfy also this strong conformal restriction property.  So this is one explanation why SLE is useful to solve this Manfred-Bröhrl's conjecture  about the fact that the outer boundary here of a planar Brownian loop here or planar Brownian  motion in general has dimension four thirds because I just told you that the outer boundary  is exactly the same as an SLE 8 third loops.  SLE 8 third you can perform computation so you can compute probabilities that you couldn't  compute with the Brownian motion picture and then you can deduce from that computation  that you get the dimension four thirds for this outer boundary.  So you see these two different ideas the first one SLE gives random or continuous objects  that allow to perform computations and you have additional sort of argument that tells  you that anyway even two different a priori different models will give rise here to the  same object in the scaling limit.  And similar ideas led to the derivation of all what's called Brownian intersection exponent  that had been predicted by Bertrand Duplantier before.  So okay let me now continue another consequence here just to mention to show you that this  strong conform restriction is not a trivial statement it shows you something like so you  see the outer boundary of this red island has a certain random shape now if you instead  you look at the inner boundary so now you take the outer boundary of this white island  inside this red island yes so you look at the outer boundary here so of course this  is very different because here when you are inside the Brownian motion is not inside of  the loop but it's outside okay but the previous theorem shows in fact that the law of the  shape of the inside thing of this white island is exactly the same than the law of the shape  of the red one right so you have surprising inner outer symmetries if you want.  So you see here that we started with this idea that you have you are looking for asking  an abstract question about we are looking for a measure satisfying a certain property  and we end up with the fact that well there is just one and it's not a trivial one it's  a measure you know supported on the set of loops that have dimension four thirds and so  on however this it turns out I'm not going to elaborate on that but this measure because  it's the only one satisfying this property is going to be probably this very natural  property related to other features and maybe other parts of mathematics and useful there.  Now I want to spend the remaining time discussing what we call conformal loop ensemble with  Quarchefield and that's ongoing work but let me give a little motivation for that.  As I explained you before SLE is going to give you the law of one interface you precook  you have prescribed boundary conditions everything is blue everything is red here and you get  the law of this random curve that is between them.  Well you might say well that's enough we know once we know the law of this curve we have  a lot of information about the system and indeed you get a lot of information of access  to some critical exponents and many things.  However if you want to describe the scaling I mean the entire picture that you are actually  seeing you need to continue you need to say well okay what happens here and what happens  there but once you did draw this curve here now you have a domain here which has monochromatically  blue boundary conditions so you cannot really start the same you cannot iterate the procedure  by starting a new SLE somewhere here because now the boundary conditions do not have these  specific blue and red parts.  So you try to you have to figure out something different and if one looks at the discrete  models if you want and the properties of the discrete models well how are you going to  describe what you see here in your domain well you are going to describe via a family  of loops that correspond to the boundaries of the clusters that you are going to see  in this picture.  And these loops are going to possess certain properties that lead to the following definition  in the continuous case.  So here again this is a schematic picture so you are looking trying to describe what  possible laws could be the scaling limits of these random collections of loops that  you see there.  So this time here imagine that we are looking at a configuration in our case is going to  be a random collection I mean a collection of loops that are disjoint and that are not  nested.  So you may see this picture as some sort of a cantor set if you want if you fill in the  interior of all these loops the idea is that you should see some type of random cantor  set here.  And these blue guys should be you can think of the outermost loops of clusters in certain  models if you want.  And the condition we are going to ask is the following.  First of all we want it to be conforming variant so we want this to be defined in any domain  in such a way that what you see in any domain is obtained by taking the conform image of  what you see in the disk.  And now if I cut something out of the disk so you cut anything and so you take any smaller  domain of the disk.  Now you see that you have two different types of loops you have those that go stay in the  white region and those that go out.  So let's go in black the interior of all the loops that touch that go out of this white  region.  So now you see you have okay I'm cheating slightly of course but you have a new domain  which is this complement of all these black loops.  The condition you are asking is that given the loops that intersect the outer boundary  here the law of what remains to be explored here the law of what the loops in the remaining  domain is the same as the law of what you started with.  So in another words and this is a very natural this looks maybe a little bit surprising a  condition at first sight but this is the condition you would ask if these loops correspond indeed  to outermost interfaces if you want in lattice based models.  So question what are the possible laws on collection of loops that you describe like  this?  Well we have a theorem with Scott Sheffield that tells you actually that again you only  have a one parameter family of such objects and what happens is that for each of these  objects the loops will look like SLE loops with a certain fixed parameter kappa.  So for I haven't described to you what the parameter kappa was in the definition of SLE  but you should imagine roughly speaking that for any dimension D between four thirds and  three halves there exists exactly one such measure supported on this joint collection  of loops like this that is supported on collection of loops with which have this dimension and  you don't have any other such random collections than the ones I described to you.  Okay you have three different constructions of these loops of these collections and let  me just very quickly describe you one of them that I like a lot but that's just personal  taste it's not a mathematical value.  So the idea is the following you are going to use the previous conforming invariant measure  on self avoiding loops that I've described to you this measure mu you are going to use  it and use its property in order to construct a more elaborate measure which is elaborate  measures which are these interacting loops now.  Well you are going to imagine that it's going to rain loops on the screen and you are going  to let it rain with intensity given by this measure mu and I'm not going for those who  are probabilists or know about Poisson point processes they know what I mean and those  who don't know also what I mean because that's just what I mean is raining loops according  to the intensity given by the measure mu and you let it rain a certain time so here that's  at the beginning you only see very small loops because and then you fill them in and then  you let it rain a little bit more you have other loops that fall in and of course all  these loops are going to overlap right because they are independent it's like if you have  leaves falling on the ground if you want.  Okay and you see this picture and then you continue the picture grows and what happens  is after a certain time of course this is very schematic these loops you should think  of these loops as being loops which have dimension four thirds and not things drawn with paint  with a lot of paint on my PC.  Well what you see is that at a finite time there exists exactly a finite time and you  can actually prove it such that immediately after this finite time all the loops that  did fall down before that are going to hook up and form a single connected component so  the union of all the loops are suddenly going to crystallize in one single connected component.  Now this in a way is a way to define fractal percolation or which is also sometimes called  Mandelbrot percolation which in short is just defining random two dimensional kentosets  that at each scale you are going to look at certain shapes and say well either we keep  it or we throw it away and so that's what we are going here to do here that if you want  we are defining a random kentoset that has conformal invariance properties because instead  of we are going to remove shapes which have these measure mu as definition.  Well it turns out that these outermost loops of the clusters you get here for any c smaller  than this time at which everything crystallizes are going exactly to be these CLE loops.  So it turns out that you let it rain these self avoiding loops or Brownian loops or whatever  you define according to the measure mu a certain time they are going to create clusters and  you look at the outer boundaries of each of those clusters they are going to correspond  to define you sort of here you have one loop here like this and here you have another one  and the picture you are going to see is exactly this describing for you the CLE loops.  Why I am spending a little time describing to you this is that in this interpretation  the time you are going to let it rain instead of calling it T you are going to call it C  because that is precisely what physicists call the central charge of the model.  So you have a very constructive and with bare hands construction of random geometric objects  that have a lot of structure and for which you can now interpret these quantities here  not as central charge of some representation but just at the time you are going to let  it rain.  So there is an alternative as in the previous case of the self avoiding loops you have an  alternative description of these loops of these loop ensembles using other means and  okay I have two.  One other mean is the Gaussian free field construction which is very exciting and done  by Schramm and Sheffield and Sheffield also partly and here again you have this feature  that two different very different models in the continuum are going to define the same  continuous I mean the same object even though a priority are very different just as the  outer boundaries of percolation and outer boundaries of brownie loops were the same  shape here you see that different objects are going to define the same CLEs.  So one hope of course is that these CLE these collections of random interacting loops are  going to define help you to understand better some of the mathematics that are behind or  the relation that are behind this conformal theory field theory or okay Coulomb gas and  tie links between these two because we can show these are the same.  So here in the final slide here actually I forgot some many people I listed the names  of people active currently active in this area and thinking or about such questions  and you that you can if they are in your department you can just knock at their door and ask them  more information about these questions if you want.  So here you have mathematicians roughly speaking here you have physicists and I included also  the list I mean in this list okay I forgot Michael Costron recent PhDs and postdocs and  actually there's one Spanish one from a student of Greg.  So if you want more information of this type of topics there's many surveys lecture notes  ICM proceedings and even a book by Greg so there's a lot of introduction to the subject  and I apologize I realize I wanted to focus really on this self-interacting loops and  try to get so my goal was to get you an impression of this general feature that you are looking  for a continuous object a measure supported on continuous object you are requiring certain  properties that are natural if you assume that these are the scaling limit of physical  models or if they are actually describing physical phenomena and you end up actually  with a very restrictive collection of possible candidates either if you're focusing on one  line you get one SLE curve if you focus on these more global things you get this collection  of interacting loops and that just with sort of a simple arguments you get to actually rather  description of a rich random objects that is and I had the time of course to describe  that to you probably related to various other parts of mathematics sorry for being over  time and thank you.
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