 Thank you very much, Stefana. In fact, ICTP is a very special place for me because this was my first major international conference. This was in 1998, much before I went to PhD and somehow so this whole has many very special associations. So in fact, I would like to talk about palm measures for some special kind of processes, determinant point processes, but since this is a basic notion seminar, let me start with some history of theory of point processes and I follow here the exposition in Daily Veer Jones, the classical text on the theory of point processes, and in fact our story starts in the 17th century when John Grant investigated the London bills of mortality, so lists which were compiled in parishes of London, of people who were born, people who died and so forth. So his work is now on the World Wide Web. It's a pioneering work in many respects and let me just start with a little preface. Let me just start with a little fragment from the dedication. So the work is dedicated to Lord Roberts, member of His Majesty's private council, and so Grant starts, well, in a customary manner by saying that my Lord presenting some learning to you is useless because you already know everything. So everything that's contained in books, you know already, but continues, Grant, my... So to present you something, which is already in other books, we're about to derogate from your Lordship's learning, which the world knows to be universal and so on. But then he says, so what, what observations he draws from the bills of mortality? So, and he says, just okay. So it does not ill become a member of His Majesty's council to consider how few staff of the many who beg how wasting of males in wars do not prejudice the due proportion between them and females. I skip some items, that London, the metropolis of England is perhaps a head too big for the body and possibly too strong, that this head grows three times as fast as the body unto which it belongs. And so on and so on and so on, that the old streets are unfit for the present frequency of coaches. When you read this, you see that nothing much has changed and so the point in the study of Grant, what Grant really studied, was the frequency of somehow the statistics of deaths. So this concept was then continued by very many mathematicians. So Halley, Halley continued the investigation of Grant, then there was Huygens, then Moivre suggested the first very simple model. So the concept is very simple, one considers some population and then there is probability, S of X, probability that lifetime is greater than X. So one considers some sample population and well, the members of this population die at some point and one considers this probability that lifetime is greater than X and well, so Moivre proposed the model that S of X Yes Okay, better Okay So Moivre proposed a model whereby S of X decreases linearly between decreases linearly between ages of between ages of 22 and 86. So it becomes, it becomes obviously less and less probable and then well the probability after 86 was considered to be zero. Then this question was also studied by such mathematicians as Laplace and Euler and in particular one important quantity that is considered is Q of X dx which is probability that lifetime terminates probability, lifetime terminates terminates between X and X plus dx provided that it hasn't terminated provided that it did not it did not terminate before X, not terminated before X. So and in fact the model which is used even to this day is the model of Gomperz whereby this probability grows exponentially. So this probability density grows exponentially. Okay, so the whole point of this very preliminary discussion is that first of all the very concept of studying sequences of indistinguishable events which however which occur at random intervals such as precisely in this example deaths of members of a given population this goes back to very old very old well, Gronz book appears in 60 something and just after the restoration 69 I think and some of the best mathematicians in the history of our science thought about this problem. So new impetus arrived when telephone companies appeared and telephone companies were interested in reducing the waiting time for their customers. So in fact this problem was important as late as the 1960s. It is enough to think of the opera of Poulenc Lavois-Umen where in fact the dialogue of the characters is constantly interrupted by phone troubles. So just in fact it was engineers who considered who introduced the so-called Poisson process. So Poisson did consider the Poisson distribution but he did not really consider the Poisson process and it was introduced by engineers in Scandinavian countries who worked for phone companies. So we recall very briefly this part of the talk is basic and also not very rigorous but a second part will be much more rigorous. So we recall that in Poisson process which is considered on the line there is a sequence of events and just events arrive at random intervals. So for example a rival of phone call at phone stations or many other things and the main idea of the definition of Poisson interval is that the numbers of events in this joint interval are independent. Numbers of events in this joint interval are independent. Indistracted. And the process is stationary so it doesn't it is all the probabilities are shifting variant. So stationary. So this already allows in a relatively straightforward way to show that the distribution, the probability of K calls arriving in given moment of time in given excuse me interval of length T is in fact has Poisson distribution. So this is probability K calls arrive in interval zero. Okay, well lambda is the frequency of calls and then very many computer Poisson process is very convenient in that many very many things can be computed and in fact with this process is this process has the waiting time paradox or it has many names waiting time paradox or inspection paradox and so on. So waiting time paradox is the following statement. So let us imagine that these are not just calls arriving at phone station, but maybe these are buses. Buses arriving to a bus stop in Grignano. Well in Grignano they arrive very punctually, but maybe if it is bus stop in the south of Italy then maybe it's not the same. So for example, let's imagine some bus stop where the buses arrive according to Poisson law. Then imagine that we come to the bus stop exactly at 8 a.m. every morning. Here we are at 8 a.m. So then we can ask ourselves so lambda is the frequency of the bus. We can ask ourselves how long shall we wait for the next bus? So it can be computed that waiting time for the next bus is lambda. Okay, we can ask ourselves how much time elapsed before the previous bus? But the previous bus before the previous bus the time that elapsed is also on average lambda. We can ask ourselves what is the interval between two consecutive buses? But this interval is also lambda. So this seems to yield a contradiction. So let us consider the opposite problem. So in Grignano where buses come exactly on schedule like every let's say 15 minutes. Imagine now we come at random time. So we come uniform random time. So what would be average time of waiting? It will be half of the interval. So if buses come every 15 minutes, we will on average wait seven and a half minutes. Sometimes we will come and the bus will just arrive. Sometimes we'll come we'll wait for long. The bus will just have left, but these things will compensate. And so interval between buses will be equal to average time until next bus plus average time since the previous bus. So if buses go exactly on schedule, so then there is no paradox, but here there is this paradox. This is lambda, this is lambda, this is lambda. So in fact, the solution to the paradox lies in the fact that when I say average waiting time, how do you say? It's like many paradoxes in probability theory in that there is expectation is taken with respect to different probability distributions. And this is why the sum formula doesn't work. So in fact, this lambda is expectation taken with respect to the palm measure, whereas this is expectation taken with respect to the initial measure. And this is the solution to the paradox. In more how shall I say, in more common sense terms, if I come exactly, every day I come exactly at 8 a.m. It is more probable so I will more often hit long intervals than short ones. Long intervals will come my way more often than short ones. And this is why waiting time is so long. Okay. So studying this waiting time paradox, by the way, waiting time paradox arises is not just, is not specific to the Poisson process. It is, for example, it holds in full generality for all so-called renewal processes. I don't want to talk about renewal processes which one can think informally as a process whereby, say, I have some device which breaks up and then has to be replaced by a new one, but it is the same device, but it breaks at random intervals. And so I must know how often I have to effectuate these replacements. And the same waiting time paradox is holds. And in fact, a very short and elegant argument is on Wikipedia in the, under the heading renewal process. So it's possible to, it's possible to explain, it explains completely why this interval is always longer than this interval. The interval which I hit is always longer than the interval between two consecutive occurrences. So thinking about this paradox, one of the main characters of the story, Konrad Palme, who was an engineer working for Ericsson before he was, he joined the Royal Institute of Technology and who lived a tragically short life. So Konrad Palme suggested to consider the following function, which he called Palme function. This is function phi naught of x. So he considers stationary flow, they always consider stationary flow, stationary flow. So he proposed to consider phi naught of x, which is probability that in interval, interval, let's say, let's say, let's say phi naught of u, it doesn't have phi naught, it will t, t plus u. No event has occurred, has occurred. And now comes this important point, provided an event occurred at time t. So it is a separate question why such probability can even be defined. It is usually defined by limit transition by taking interval, by taking two intervals. So here is t, here is t plus u, then I take small interval t, t minus delta, and I consider conditional probability of event here and nothing here conditioned on this event here, and I take the limit. This is how Palme, well Palme in fact was not even interested in questions of justification, but this is how one can give rigorous, rigorous treatment of Palme functions. So this is the discovery of Konrad Palme, that the importance of studying such conditional functions and but the fundamental work of Palme, it was all, however, of course, on physical level of rigor. It was not rigorous, not rigorous work. And so then, Alexandr Jakovlevich Hinchin in fact in a fundamental work, which has not lost its importance and which is really remarkably well written. I have some trouble translating its title. I will modernize it in English terms. The Russian title, Theory of Massive Obslusion, in English is translated as queuing theory. So I will give modern English translation, queuing theory. This is proceedings of the Stackloff Institute of 1955. It's a separate volume, volume 49. So this fundamental monograph gave rigorous foundations to the theory of, in fact, queuing theory and, in fact, to theory of stationary point processes. So the idea of Palme was that one can, from this probability, get the process back, get the stationary process back. In fact, it's not enough. This finite is not enough. It is necessary to consider more general palm functions. So Hinchin gave to these functions the name of palm functions. And so it is also function phi k of u, which is probability. So it's the same. Okay, let me just write it here. Phi k of u is probability that in an initial, at most k events occurred. At most k events occurred, provided that no event occurred at time t. Excuse me. At most k events occurred, provided an event occurred at time t. No, it's written correctly. Excuse me. Here it is. Yes, so phi k. And the collection, so the Palm Hinchin theorem, is that the collection of these functions, phi k u, determines the stationary process. The stationary process. So this is the Palm Hinchin theorem. The stationary process. So in fact, one can write very simple, very simple equations. If this probability, this one denotes as vk of t, these are basic functions of a stationary process, which determine it. Uniquely, then in fact, the Palm Hinchin equations, so here are the Palm Hinchin equations, say the following that v0 of t, well, this is hardly very surprising that the probability of no calls is found by integrating the conditional probability. It is, in fact, very much to be expected. And then vk of t is just lambda integral from 0 to t, phi k minus 1 of u, minus phi k of u. So and the proof of, I follow the notation of the book by Hinchin and the proof can be, the proof occupies more or less a page and can be also found in the book of Hinchin, which had this fundamental importance for development of the theory of point processes. So now the very concept of this Palm measure, how do I say? This was the discussion from the point of view of engineering applications, but it is desirable to construct a mathematical framework for the study of these Palm functions or Palm measures and of course, this definition is not satisfactory from many points of view and in particular also because there is this limit transition, which is a little bit awkward, it is not clear for what processes can be defined. It is not completely clear how to carry it to many dimensions and so I would like to now, now I stop with the informal part of the talk and I switch to more rigorous part. Now I would like to explain the modern approach to Palm measures. So Palm measures through Campbell measures and I am not completely sure how to properly assign credit for these beautiful constructions, but so Palm measures through Campbell measures. So but my reference is the work of Olaf Kallenberg. So Campbell, by the way, was also one can say not exactly a mathematician, so he was engineer and just physicist and philosopher of science. Okay, so I follow Kallenberg. So we let us first explain ourselves. What do we mean by point process? So we have so far had a very informal definition. Now let us have formal definition, which will also be suitable for many dimensions. So we consider phase space E, which is in our situation, especially for the purpose of the talk. Today it is one can think that E is something like R, Z or C, so nothing more fancy. Formally, it's a complete separable locally compact metric space. Locally compact assumption is useful. So there is then the space of configurations on E. The space of configurations on E is the space in fact of subsets of E. X is a configuration. X is a subset of E. Such that X intersected with any bounded set is finite. So I will denote cardinality of a set by the sharp sign and so B bounded, this is fine. Okay, so I will denote, I will write, it will be convenient for me to introduce notation. This one, HB of X is precisely number of... So points of the set X are called particles. I can even write it like this. X is the set of some small X and these X are called particles. So these are configurations of particles and every bounded set contains only finitely many particles. In fact, we do not expect infinitely many phone calls to arrive at a phone station at the same time. Number of number of particles in B, X in B. So since this is a basic notion seminar, I will be light on formal details, but let me point out that these functions determine the Borel structure on this space and also that assigning to every X the corresponding radon measure, we also obtain complete... We also turn the space of configurations in a complete metric space, but complete separable, but not locally compact metric space, but this will not be very important for us. Measurable structure will be more important. So a point process is nothing but a Borel measure on... Point process is nothing but a Borel measure. P is Borel measure on the space of configurations. So this is quite normal on the space of configurations. The point is rather... The point in comparison with this representation that we don't fix our attention on random variables such as number of particles in zero T or something like this, but just so to speak of the configuration itself. And I will always assume that all these integrables, all these random variables are integrable. And in fact, this expectation is back to P of this random variable is the intensity of the process. I will denote like this psi pi of B. This is intensity. This is precisely the average number of particles in some set. So for example, for Poisson process intensity is just Lebesgue measure. Okay. Excuse me. P is probability measure. Yes, excuse me. Borel, thank you very much. Borel probably... But this one, no. This one is not finite. Borel probability measure. Yes, thank you very much. Borel probability measure, thank you. Any further, maybe questions? So if not, we proceed. So then the Campbell measure to this... So configuration is an unordered collection of particles. For a finite collection of particles, this is immaterial. One can order them in any way one likes. For an infinite collection of particles, this is important. It is not possible to choose a particle out of a configuration in any nice way. So in fact, the construction of Campbell measure is the construction of natural lift. Natural lift of my measure to these configurations, to configurations with a marked point. So in fact, we consider the space E, time space of configuration on E, and the Campbell measure C, Campbell measure of P, takes a subset B in E, takes a subset Z in the space of configurations and assigns to them the following number. So this is definition of the Campbell measure. So this definition, it's easy to represent geometrically. So if I horizontally represent the space of configurations and vertically represent the... or E, well my E will mostly be R, but this is E. Then just on the space of configurations, I have my initial point process P and over every configuration, I just put the configuration itself, which is an excellent measure. And so this, the resulting object is Campbell measure. At this point, palm measures are palm measures are just conditional measures of this measure. This measure is infinite, but it is locally finite. If we restrict it to any bounded set here, it is finite. So because on here it is finite. So conditional measure of this measure with fixed particle here is just called the palm measure. It is also possible to write it in the following way that Z is the Radon-Nikadim derivative of what? Of the Campbell measure where I fix this as a placeholder and I fix Z. So this is now a measure on E, on the face space. So this is the variable. With respect to the intensity of my process, at the point Q. So it is possible to write this formula, but I think it is more convenient just to say that palm measure is just conditional measure in the sense of rochlin. So just this is very nice space. There are very nice measures. There are conditional measures and this the conditional measure is precisely called the palm measure. So this simple and beautiful construction gives a very general description of palm measures. Of course it is clear that this coincides with this. The new construction coincides with the old one because in fact this is just a formula for computing the Radon-Nikadim derivative. So it's quite clear, but this object, this construction is much more convenient to work with. So and one last thing that I want to explain in this level of generality is the concept of reduced palm measure. Reduced palm measure. So in fact, reduced palm measure. So in fact, what happens, there's something, sorry, again. There's, I shouldn't stay here. Interesting. Okay, so what happens is that just it is not convenient for me to have this particle at the point Q. So palm measure is conditional measure with respect to the event that I have this particle at Q. This is formal definition, but it is convenient for me to erase this particle at Q. So just I consider, so I consider palm measure. Palm measure is supported. Palm measure is supported on the set of configurations, is supported, such that Q belongs to X. But it's convenient to kick this Q out. You might think, how is this possible? If here I put these Qs, then if I kick them all out, what is the fiber? But in fact, no, it becomes precisely much more delicate. The fiber starts to consist of those, the fiber becomes, maybe in fact, I should just switch the mic off because like this. Sorry? I'd state. Okay, so just, okay. So just it is the so if the fibers of this Okay, excuse me. Let me just say so in fact I will do so it is convenient. Now remove the particle. Remove the particle. The particle at Q and obtain reduced palm measure. Reduced palm measure. So at this point again, the fiber over given configuration will be those configurations from which you can remove by removing one particle fall into this configuration. So it becomes very subtle conditional measure of this reduced thing. Okay, now I will do something really terrible. So in fact in what follows, I will not need actual palm measures. I will only need reduced palm measure. So I will change notation in the middle of the talk. So I will put hat. So palm measure is with hat to obtain reduced palm measure and this will be PQ without hat. Because in fact, it's the reduced palm measure, which is important for us. Okay. So now palm measures will now I will explain I will explain the computation of palm measures and the use of palm measures for study of a very specific model. Namely the model of Determinant point processes arising in random matrix theory. So let me start by specific example. Specific example. And this is the most classical example of random matrix theory, the sine process. So I will, the sine process is a point process on R and let me give its definition in a slightly short way, but equivalent to just so to speak in a somewhat reduced way, but in fact equivalent to the full definition. So in fact, so sine process is point process on R is measure measure on space of configurations on R. So and to define this measure, I need to define some expectations of some events and events I will take are the following. I will take some disjoint intervals, disjoint intervals in R, disjoint intervals and then I will take the expectation of the product of these numbers. So in fact, it is necessary just so and this is where I'm skipping some details, but in fact point process is completely determined by such expectations. So differences that I am not taking, ideally I should also take powers. So not just so these are some random variables, but to determine them completely I should take not just their products, but also products of their squares of their third powers and so on and I don't do this but in fact this is enough to define term the point process and so this is determined given by the determinant of what is called the sine kernel. So excuse me integral, excuse me integral integral over I1ik of the determinant dx1 dxk. So determinant is from ij, ij from 1 to k. So it only is necessary to explain what the sine kernel is and in fact the sine kernel is just this function. So it takes considerable effort to show, to prove even that such process exists. This was established following a pioneering work of Mackey. This was established independently and simultaneously by Soshnikov and by Shiray and Takahashi. And just this process is absolutely fundamental object in the theory of random matrices in that it describes the local behavior of eigenvalues of random Hermitian matrix. The important difference between the sine process and the process that we considered before is that the range of interaction of particles is very long. Particles interact at infinite distance. So as opposed to for example Poisson process where particles so to speak don't see each other, calls arrive independently. Here particles interact at arbitrarily long distance. So in fact sine process in some sense models the gas of charged particles. So particles repel each other. Particles have the property of repulsion. And so now I'm ready to formulate the main result of the talk. So the main result is description of conditional measures of the sine process with respect to fixed exterior in interior or in a bounded interval. So okay, so for the formulation of main result one can forget everything that went on before. It's so to speak fresh start. I will explain the connection with palm theory a little bit later. So conditional measures of the sine process, conditional measures of the sine process, of the sine process. So what conditional measures do we consider? Again we take interval I and we fix the exterior. We fix our configuration in the exterior of the interval. So this is fixed. Fixed. And then we're interested in distribution of particles in this bounded interval I. And this is the result that I will now formulate. It's on the archive, the archive in May of this year. And just let me just introduce some notation. So this conditional measure, I will imagine that I have some configuration X in the whole R, X configuration on R. X is configuration on R. So then the measure, the measure P, R complement I is the conditional measure with, how do you say configuration? Configuration in exterior in R without I fixed as X restricted to R without I. So this is the definition. This is the definition. Okay, so now I give the description. So the result says that for almost every, for, so this measure, let me denote the pi S will be the sine process. So for pi S almost every X, the conditional measure is an orthogonal polynomial ensemble, orthogonal polynomial ensemble. That is, that is as the form, the following form. So it's important to note that the measure is supported on configurations with fixed number of particles, with fixed number of particles. So if X had let's say 10 particles in I, then the measure will be supported on the set of configurations with 10 particles. So fixed number of particles. Then there is interaction. So this property is, can be understood as some analog of Gibbs property. So then there is interaction between particles. What is interaction between particles? It is in fact the interaction as in log gas. So potential of the interaction is log of the distance between particles. Okay. And then there is how to say one particle potential. So rho, which I denote by rho, I, X, XI, DXI, I from one to this number. And of course there is some normalization constant, just so that this be a probability measure. So this orthogonal polynomial ensemble, now I just need to explain what is the weight. So the weight, and this is the main part, the weight, rho, I will be determined by the following formula, rho, I of X, pi. Equal to the product, Q minus X, Q minus X square over X in precisely R minus I. So this is, it is important to understand only in what sense product is understood. So such product is understood in principle value. So product is understood as convergent over increasing sequences of intervals. Over increasing sequences of intervals. So again, it is just normalization constant. So that it is probability measure. So conditional measure is probability measure. It is just constant. I don't know. This is not so. Oh, excuse me. Okay. Oh, you're completely right. C inverse. Okay. C inverse. Okay. C inverse. Okay. Okay. Okay. Yes. So the point is that the measure is supported on particles with fixed number configuration in this interval. But in fact, this is fixed number of configure a fixed number of particles in this interval. This fixed number is just the statement is due to gosh and Paris, but also in fact for this kind of stationary process. It follows from very old theorem of Kolmogorov in 1927. So for stationary for stationary processes, it is the fact, this fact that the number of particles is fixed, which might seem surprising that just it's not possible to add one more particle. In fact, can be traced back to a very old result of Kolmogorov. Kolmogorov was interested in the following problem. We have stationary process Xn and indexed by some integers. And we're interested in the situation when X0 is in linear span of Xn for n not equal to zero. So X0 is in linear span of X1, X-1, X2, X-2, X3, X-3 and so on. And so in fact Kolmogorov gives sufficient conditions for such situation to arise. These sufficient conditions are given in terms of spectral density. And this can be without much trouble verified for the sine process. In fact, this is an approach to rigidity for stationary process that we take in joint work with Dabrowski and Schiu. So from 2015 and just from this theorem it's possible to deduce these rigidity results. However, the approach of Gaussian pairs is much more flexible in that it can be used for many non-stationary processes. Okay, and now the main point is once the number of particles is fixed is how to find precisely the interaction between different tuples of particles. And this is exactly where the PAM measure, the PAM theory comes in. Okay, I need to erase something. So let me erase the basic definitions. So how to find conditional measures. Let us assume that we already have this property of rigidity of Gaussian pairs in this case. So how to find the conditional measure in my situation. So we know that the number of particles in the interval is fixed. The number of particles in the interval is fixed. So the exterior is fixed also and the number of particles is fixed. So we are interested in ratio of probabilities. So of particles in these positions and maybe of particles in those positions. So these are P1 and so on PL and these are Q1 and so on QL. So at this point we have just the following. That in fact, let me assume that my conditional measure has density lambda P1PL Dp1 DpL. Then in fact lambda P1PL over lambda Q1 QL is essentially equal to the ratio of the corresponding PAM measures considered precisely at the restriction of my configuration to the complement of my interval. This is some sort of bias formula. Just fixing this and moving this is similar to fixing this, the P's and then fixing the Q's and taking the ratio at this. It's very instructive to check this equality for finite sense. Even for finite sense it's not difficult but it gives an idea of what happens here. So in fact the main step in the argument in the proof is precisely the computation of this equivalence of PAM measures which uses, so heritization of PAM measures of such process which is due again to Shira and Takahashi who described PAM measures, gave description of PAM measures for determinant processes and just from this description it is then possible to obtain this resultant conditional measures. There are some technical issues about which I must speak very briefly. The technical difficulty is that in fact it requires proof that this measure does indeed have a density. A priori it contradicts nothing that the conditional measure is just atomic. The measure is just atomic but these atomic measures average in such a way that the resulting measures are absolutely continuous. Or put in other way it is different to say that for fixed P and fixed Q these measures are equivalent. And to say that for this fixed exterior these measures are equivalent for all P and all Q. There is just some interchange of quantifiers which is difficult in probability theory because one has to take continuum union of events of measure zero. It is completely not obvious that this should be indeed of measure zero, event of measure zero but this precisely is what goes into the proof. And let me close by an open problem so it is natural question. So this property can be understood as analog of Gibbs property for the sign process. In fact this form of conditional distribution is similar to the form that arises in Gibbs theory. The problem however is of course the question of the uniqueness of state. So is it possible to prove that in fact this description of conditional measures and maybe some other additional information for example here in this description of conditional measures the frequency of the sign process, the frequency of particles does not appear. So for example this description plus frequency of particles. So is it possible, is it possibly true that this description of conditional measures in fact determines the sign process uniquely. This is not known. Thank you very much.