Good afternoon. It is a really great pleasure for me to introduce Fedler and Verner for the final Fields Medal lecture. Fedler received his PhD in 1993 at Université Parisis under the direction of Jean-François de Gaulle.

Since 1997 he has been professor at Université Parisoud in Orsay. From 2001 to 2006 he was a member of the Institut Universitare de France and he currently also holds a position at École Normale Supérieure in Paris. His prizes include the Rollo-Davidson Prize, European Mathematical Society Prize,

Fermat Prize, Jacques Herbin Prize, Loeb Prize, and the Polia Prize. On a lighter note and 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 10 years ago that I first that I was giving a lecture and about intersection exponents, the problem of Brownian intersection exponents. One case

of which was the Mandelbrot equivalent to the Mandelbrot conjecture about Brownian motion when Bedlin was in the audience and a couple weeks later I receive an email with a very nice idea how to approach this problem. From the start of that email I've started a collaboration that's 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 Oded. He was talking with Oded Shrom and he realized very early that Oded's beautiful SLE might be the key to

understanding the Brownian intersection exponents and 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 we solving these problems that we're asking but in fact these structures that have been done both SLE and Brownian motion are

really the 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. So with that I'm happy to introduce Bedlin for his talk.

Thank you very much. So 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 and since it's the first time I get the chance to

speak I have of course many people to thank from personal reasons to PhD supervisor to 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

I mean after Odette's plenary lecture this morning and that Greg was able to be here to also and share of the 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 plan loops and conformal restriction and okay I added a survey because of course this is supposed to be more an introductory lecture that in

order that gives some flavor of some of the ideas and some of the results that we obtained in the last during the last years and this talk is of course not unrelated and rather closely related to Odette's talk this morning and

also to Stas Monov'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 point on some

issues with what mostly what Odette 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 the 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 in the physics community because the 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 to thank them Michael Eisenman, Bertrand du Plantier and other

physicists who also made the effort to come to math departments and actually say here we have a nice sub problem for you that we know sort of know how to solve but which is a nice problem for you mathematicians 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 is 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

repeated twice the same experiment you get twice the same results and that's the result of ways the laws of deterministic laws of physics and it on macroscopic scale when you are exactly at a point at which we the

physical 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 load 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 this these quantities often obey certain power law behavior when you approach the critical points 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 points and of course well not of course but it has been observed also that these two or argue that these two phenomena are very closely related with each other so physicists

have came up with the 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 I it's not to intimidate you just put down some names of the physicists that have

contributed to these issues and of course I 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's explained

roughly or gives 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 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 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 the 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 planner systems and the exponents of the models are classified according to what physicists call the central charge of the model that mean 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 seen 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 Kestan that's one of the many things that Harry Kestan did in all these questions so he treated the case of percolation in two

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 question you look at was a question a question mark okay so I'm going to repeat very quickly so that you have one model in mind I chose to repeat a

model that has been introduced that you've maybe seen once or twice in these in this ICM before but well I guess that's the simplest model to explain them 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 you're playing with hexagons and you toss a coin each hexagon is going to be black with probably with probability P and white with probability one 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 are interested in the connectivity properties of the picture you obtain when you draw when you do this 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 occur appears to occur at PC equal one-half and so when so the idea is when P is larger than one-half then you have one infinite you have majority of black hexagons and you get one infinite black connected

component called the infinite cluster and that has is if you want a positive intensity or positive density is actually the right word theta of calls theta of P which is a deterministic function of P when P is larger than

and when P is small 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 we 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 percolations up 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 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 on the left to right crossing is a 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 boundary here is our 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 our predictions by physicists that are now mathematical theorems and I'm not going to explain

you the route to that to these prediction to these theorems now because I would repeat a lot of what Odette said this morning so here's one the first prediction I think this is due to Nienhuis then nice 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 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 are 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 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 pairs of points they are which are neighbors and such that one is 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 come if you compare it to the disordered system of percolation prefers to have neighbors that are of the same opinion of 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 90s is that

instead of looking at the correlation functions and here I just give gave 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 faraway points are

distant of a given color you're looking you're going for to try to describe the entire 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 a some of the ANZ to the end and this looks like

analytic thing when you compose analytic maps you start playing with algebraic structures and 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 geometrics inside 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 let'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 all 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 met 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 angles 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 this 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're 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 that 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 conform invariant and if you've been to smell off 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 variance properties in the plane so of course there's a very easy

way to create a conformal invariant conformal 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 Moebius 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 variance so conformal 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 Odette this morning is that to focus on discrete interfaces so here I this is very schematic and actually has been drawn by my daughter when she laughed at my poor way I am handling her

pictures on the okay so you imagine that you have a domain with precooked and nice boundary conditions say if you assume that one part of the boundary here is red and the other part of the

boundaries blue I chose red and blue instead of black and white for obvious reasons and basically if you if you are going to assume that here everything here on this boundary is blue everything here on this boundary is red you'll get 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 Odette Schramm's that led him to the division of SLE which is we're going to explore the system just starting here and we start to explore here this interface because we're looking at suppose we're supposing that we have nearest neighbor interaction system with

nearest neighbor interaction this tells you that once you have 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 can here everything is blue here everything is red and so if you assume conforming variance in fact once you've started exploring the boundary I mean this interface you ask okay now how do I continue you still have you still are in the say asking exactly the same

question you still are looking at have to explore an interface between in a simply connected domain where you're looking at one part of the boundary is red one part of the boundary is blue and you're looking at the interface between these two things and so using this Odette Schramm explained this morning that combining sort of classical

ideas from complex analysis namely Levene's theory with basic probabilistic insight you get that there is a exist at most a one parameter family of such random curves in

the in 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 pre-cooked boundary

conditions and you get to one family parameter of such random curves called the Schramm-Levene evolution that's the difference maybe with this morning's lecture and I call it Schramm-Levene evolution and this one parameter is usually called kappa and okay here I'll

list some properties that Odette described this morning so there's there are two type 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 equals 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 the so-called conformal restriction property so conformal restriction is is a another idea that is complementary if you want to the previous one but gives another approach or another characterization to these random curves or

SLE curves so actually this idea has been developed and refined in a sequence of papers with Greg and then Greg and Odette 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 you 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 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 your 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 restrict 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 right 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 this condition to be true for simply connected domains only call it weak conformal restriction that means that whatever simply

connected domain you choose you're going to see the same measure modular conformal 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 I mean multiplications are conformal maps so if you take a disc you look at what you see in this disc or twice this disc you say the same you see the same measure so the measure is scale invariant it's translation invariant 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 a 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 is just moving around at random in the 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're looking for a measure of 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 looks 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 the laplacian and related to harmonic functions is going to be invariant in some way on the 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 you can turn it in the way you of course are going to natural way into a scale invariant and translation invariant measure on path in the plane that are Brownian paths and what you get is the

scale invariant measure on Brownian loops which is translation invariant and satisfies conformal invariance properties and the very definition of the way you are going to define it is going to tell you that if you focus just at 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 in the in the continuous I mean 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 clusters on percolation clusters that you get here is going to satisfy the same weak conformal restriction property and therefore its outer boundary so again now you focus just at 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 a third turns out to have a very special property as I mentioned before that makes it possible to define directly a measure

on SLE a 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 that's one way to understand it or how measure if you want it to be invariant okay 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 a 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 Mandelbrot conjecture about the fact that the outer diamond 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 a third loops SLE a third you can perform computation so you can compute probabilities that you

couldn't compute with the in 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 actual 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 du okay let me now continue another consequence here just to mention to show you that this strong conformal 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 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're 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 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 Scott Sheffield and that's ongoing work but I 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 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 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 do not

that are disjoint and do are not nested okay so you may see this picture as some sort of cantor set if you want if you fill in the interior of all these loops the idea is that you should see some 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 the condition we're going to ask is the following first of all we want it to be conform invariant 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 conformal

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 type of loops you have those that interest go stay in the white region and those that go out so let's draw in black

the interiors of all the loops that touch that go out of this white region okay so now you see you have okay I'm cheating slightly of course but you have an oops you have a new domain which is this complement of all these black loops and 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 other words and this is a very natural this looks maybe a little bit surprising condition

at first sight but this is the condition you would ask if these loops correspond indeed to outermost interfaces if you want in 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 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 conformal invariant measure on self-avoiding loops that I've described to you this measure mu

you're 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're 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 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're 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 the paint with a lot of pain 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 mandible percolation which

in short is just defining random two-dimensional cantor sets that at each scale you are going to look at certain shapes and say well either it's we keep it or we throw it away and so that's what we're going here to do here that if you want we are defining a random cantor set that has

conformal invariance properties because we instead of we're going to remove shapes which have these measure mu as definition well it's 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 this loops according define according to the measure mu a certain time then you are going to create clusters and you look at the outer boundaries of each of those clusters

they are going to correspond define you sort of here you have one loop here like this and here you have another one and the picture you're going to see is exactly this describing for you the CLE loops why I'm spending a little time describing to you this is that in this interpretation the time you're going to let it rain

instead of calling it t you're going to call it c because that's 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 or some representation but just at the time you're going to let it rain so there's 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 to one other mean is the Gaussian free field construction which is very exciting and do it done by Schramm and Sheffield and Sheffield alone now also partly and here again you have this feature that two different a priori very different models in the continuum are going to define the same continuous

I mean the same object even though a priori they are very different just as the outer boundaries of percolation and outer boundaries of Brownian loops were the same shape here you see that different objects are going to define the same CLEs

okay so one hope of course is that the 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 show can show these other 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

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 Kostron recent PhDs

and postdocs and actually there's one Spanish one former student of Greg so if you want more information on this type of topics the exact there exist 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 argument you get to actually rather description of a rich random objects that is and I hadn't time of course to describe that to you

probably related to various other parts of mathematics sorry for being over time and thank you