 Thank you very much for the invitation. Okay, so I will first start with this well-known result of the finality theorem, which Sasha just mentioned in 19 for 19th century. So it says the following, if we consider infinite binary sequence, the space for binary sequence, and we consider the action of finite permutations by just the finite many coordinates. So we have this group action, then the organic measures, so all measures in these mini-courses will be probability measures. So the organic measures of this group action are exactly those Bernoulli measures, which means we have p that zero plus one minus p that one and take power for p for p, from zero to one. Here I'd just like to recall that by organic measures, I mean that so if we have a group action, so the organic measures are those invariant ones, such that any G invariant measures so in fact, for a subset of X has full measure or negligible, so full measure means measure one or negligible, minus zero. And this is a very well-known theorem. The next theorem is the Schoenberg theorem. I think it's seen 53, something like, so we consider this group. Actually, we will consider for, okay, let us consider this group. Your infinity, which is just the union of all, here it's just n times m are unitary matrices and the union means the following. So actually every element here is written in this way. So we have here n times m blocks, this block and we add infinitely many one and this block is from inside unitary group and we consider the group action. So it's natural that this is a group. Take action of this, which is just by, in fact, so every element is a unitary group and the action will be, so here, suppose we have an element, here's gm and here's one one, then we write this as cm times cm and the action of this element on this space is just action on this part and leave invariant, does nothing on this part and it turns out for this group, the classification of unitary measures is also possible and it's given as follows. So unitary measures for, find a color, for this action, exactly those Gaussian measures for this action adjust Gaussian measures, which means that we take on each say, we take standard complex action not standard, we coordinate, we take a random Gaussian measure, complex Gaussian measure, then we take product of this. Here, sigma is from zero to infinity. Of course, the other inside because there's a fixed point. And so these are random result and in this course, we will explain our new result with recently, with Sasha, it's from the beginning of this year. Actually before introducing this theorem, let me first introduce our setting. So we will consider the following thing. We consider piatic number, the field of piatic numbers, piatic field and we consider, we'll explain in detail, we'll call it in detail what QPZP is. The ring integers, ring of piatic integers, which turns out to be actually a compact ring and we consider the general linear group over these compact ring. So this is the set of n times m invertible matrices over and in this case, matrices over and in very similar way, we consider the union over this compact group. We can see of this. Our theorem says the following, I will not have enough space here and write this theorem three times and in Chinese it is one saying that if one says it's important, repeat it three times. So okay, I'm not saying this is very important. So myself, the action that we are considering is gl infinity zp and gl infinity zp, gl infinity zp, the group action on the space over infinite matrices but with coefficient in QP. So here, by this notation, I mean that this is a set of infinite matrices, so of course this is just topologically just the product, the Cartesian product of QP and our results say the following. So we give a complete classification of the organic measures of such action. The organic measures of this action are exactly those probability distribution of some explicitly given random matrices for the distribution of the following we take. Some p x i j m plus p minus k z i j I will explain all this notation m. Here is for all those m greater than some k and where this k1, this sequence k1 are actually the parameters parameterization of this family of the whole family of organic measures. This is a sequence in z union minus infinity. And here k is the limit of km. It can be in z minus infinity and suppose if k here is in minus infinity then there's no restriction here and with convention of course p minus minus infinity x to p infinity x to 0 in QP and actually all these k1 have a meaning are later described as asymptotic singular values we will describe of a typical matrix with respect to this given organic measure. And let us first compare this result with another well-studied classification in Euclidean case compared with the theorem of originally due to peak rail and our version will be Oceansky-Vershik version. So we consider the same similar action. This is the union of compact group, U.M. And here it is also union of compact group which is actually infinite and infinite dimensional and which is not locally compact. So we consider the group action. I forgot actually to say this, what this natural action is. Of course this means B1, G2 is just, yes. I forgot to define it. So we define in similar way this group action and so the organic measures are exactly, so we will see that they have very similar structure of those probability distribution of the infinity random matrices given by a sequence XM and from the infinity. Here we'll write GI, GJ, GJ tilde plus delta Gij. I forgot also, sorry, sorry for that. Here XIM, YM, Zij are independent uniformly distributed on the integers, on the periodic integer distributed on ZP. And here X, we have a sequence of real numbers such that they are sumable and delta is positive and delta is positive. And GIM, GJN, Gij are independent Gaussian, independent Gaussians. Also, this classification was originally obtained by Pick-Rail but this version of Oshansky-Vershek has the advantage to give a precise meaning of these parameterization parameters as we did in Piatty case. So here X, if I write these random matrices as M, then the truncation, this is the N times N corner of M with singular values X1, XN, how they did them? Of course, since the matrix is random, so the singular values is always so random, then we get this random thing XM is the limit of normalize, this is almost surely with respect to, this is almost surely for any M. It is by this equality that I say that these XN are actually asymptotic singular values of these infinite random matrices and actually we have similar situation here but we have a similar situation here but we will write it later. And also data has a meaning, which is actually data will be, data will be actually the limit of trace M. I think this should do minus, minus something like that, this is also almost surely true. So since we have classification of organic measures and of course, yes, by the way, all these things, we have a symmetric motion, we can consider the action of this group, we can also consider in both case, symmetric or Hermitian case, this version acts on infinite symmetric but here P is different from two, action by GXGT, we have similar, but the presentation is more difficult, similar result and also here, action on Hermitian, infinite Hermitian matrices, action by U times X, UX and this is the same or it's a similar result and it turns out if we consider the natural unitary invariant probability on this space, then this invariant thing we would like to decompose into the ergodic ones and the natural one is the following, we have the K-lay transform which sends the harm measure here to some measure that has to be known by M but this measure turns out to be by Hualou-Ken K-system, this measure, we define the same thing for M plus one, the truncation, this is just cutting, cutting the corner, so yes. Okay, space here and this is pushed forward, so we have, in this way we define measure on infinite Hermitian matrices and it turns out the ergodic decomposition, we have M infinity, the ergodic decomposition of this measure is actually described by determinant point process, I will write DPP for some abbreviation, this is by Borodin Janski and with some extra work for myself. Okay, now it is termed to present our result, several results in determinant point process which will probably help us to understand the ergodic decomposition and by the way we also have similar construction of this Hualou-Ken under the rating type measure on periodic case and the decomposition is still open, so let us now talk a little bit more, so all these announced results we will discuss later, so let us first give an example of determinant point process, we consider first indeed this case, the binary sequence and it turns out this, the following kernel, the following function, two variable function on Z minus Y, which by the way is the spectral, the kernel of the spectral projection of the Fourier transform on the torus, so these two variable function defines our probability and later we will see that is in our second which we call determinant point process, defines our probability measure on this space satisfying the following property which uniquely determines the probability that has to denote this by the sign, the field in the set, the Fourier field in the set on Mika K1 equals to Mika KL equals to Mika K1 is given exactly by a determinant that is why we call it determinant K, KJ. By the way, in fact this is a gothic probability measure with respect to the shift, but we are actually interested in another property of such measure, we have a theory of proof it, of such a, which says the following, this action, of course, this action probably was a variant in the sense that the action preserves the measure class of this measure and the recycle means that later we continue derivative, derivative is given explicitly as follows, so we write PQ, so for two PQ in Z, PQ in Z, the transposition between just change T and Q, PQ, then the rather than derivative of the action of this type of generating elements of the group is given by a formula which looks very similar to Gibbs property, so here let me write probably omega as the product of omega i, actually I don't like to write, I'd like to write X, X, so omega in Z actually correspond to a subset of Z, just the, the, the, the sub set of the sub set of Z, this is the indicator of a subset X of Z, so this will be X minus P X minus Q square for those X, in X a different P when Q is in X, but P is not, of course when both PQ are inside, then the action does nothing to the disconfiguration, so now let us describe more general, by the way, so similar result for gamma kernel, which we will not describe here, for gamma kernel we will not describe here, for gamma kernel instead of sine kernel is a result of Oshansky in 2011, and okay, before starting and to present our result, new result let us discuss further a property of Gauche pair as rigidity, it says the following, so this, let us just intuitively give this rigidity property, it says the following, so for any subset, finite subset, okay, we will, in fact, define totally, the number of partition inside gamma is determined, is in fact measurable, with respect to the information of the sigma algebra generated by the information of the configuration outside, so it says the following, so almost surely, so we have a configuration Z, and here we have a subset, then if we know what happens outside, we actually know what, how many particles inside this finite window, okay, how many time? 10 minutes, 10 minutes, okay, 10 minutes, that has briefly generated the, give a definition of theorem of determinant point processing, general case, and theorem is due to several people, Makisoshiko and Gahashi, so we just want to describe a simple case, so we replace Z by C or R, and we replace here, it's just a subsets of Z by, replace it by the space of configuration, which is defined as locally finite subset of C, locally finite means in every finite wall, there's a finite number of points, then for any operator pi C mu, this will be actually be at this pi sign, similar to this pi sign, here the mu is a measure which can be probably, in our case, the big measure or Gaussian measure, for any operator, bounded operator, which is positive, positive operator, and contractive, in particular, the orthogonal projection, and given by, given by an integral operator, why? So I use the same notation for the operator and the kernel, given by, with a kernel, a continuous kernel, then we actually define a probability on the configuration space, then this pi x, y defines uniquely a probability measure the configuration space, satisfying the following property that characterize this, satisfying the following, so defines a probability that has to know this by P pi, so of course, given a probability what we want to do is to compute the expectation of some testers, things, so x is a random configuration with probability P pi of some function, this function is nice, function compact supported, the continuous, et cetera, it's given by an integral, of course, determinants, so there's a determinant, again, here's for i j from one to n, with respect to the measure d mu by one d mu m, so in particular, we want to, we have the following theorem, so our setting is the following, consider pi, which is a function, real function c, that is c2 smooth, such that the Laplacian is bounded from below and above, and consider the reproducing kernel, so which is called weighted Bergman kernel for the Fox space reproducing kernel, that has to know i phi over the Fox space, define Fox space, the following Fox space, we will come back, so the notation here will probably, holomorphic functions on the complex plan such that it's integrable with respect to this measure, this d lambda is the big measure on complex plan, then our theorem says that this kernel, this kernel generally defines a probability on this configuration space, we have several steps just last week, we obtained the following, so the defilmorphism with complex support defilmorphism acts on this configuration, this probability space quasi invariantly, the action is quasi invariant, first step, second thing, p m of phi is gosh, I will describe later the cosine code for this action, gosh perhaps, which it means in that the, with respect to this probability or configuration, okay, or configuration outside the inside the particles, so the, for simplifying our notation, that has for this moment, at this moment, just stay in the result in radial case, in radial case, psi v is just psi, the following limit exists in pq minus p x minus q, this exists in, in fact, exists in the other, or p in fact, that turns out to be existing l1, one of configuration with respect to closely related probability, to this one, where p is the projection onto the subspace of the following subspace, f in psi such that fq is to zero, and the lateral negative derivative of, we can define here for p for q, psi p over is given by normalize, which actually, this constant is calculated also explicitly in terms of p c and the pi, the following constant, normalization constant, under the co-cycle of the action of this action is given explicitly, I will not have time to write explicit now in terms of this above and also on the c and the psi, I think I run out of my time.