 Okay, so our next speaker, Yan-Chi Chiu, I apologize for the pronunciation. Thank you. He will speak to us on Patterson's Elephant Construction for Point Processes and Reconstruction of Harmonic Functions. So first of all, I would like to thank the invitation from the organizers and it's a great pleasure for me to have an opportunity to speak our recent work with Professor Alexander Buffett-Darth. So we will basically, roughly speaking, we want to generalize classical constructing which is called Patterson's Elephant Construction in dynamic system to random setting, namely to the setting of a point process and use it to reconstruct certain harmonic functions on unidisk or even higher hyperbolic spaces, high dimensioning. So the motivation of this work has already been mentioned in the talk of Professor Alexander Buffett-Darth. So let us recall briefly these motivations. So we will consider the space of all a homomorphic function on unidisk. So then for those functions, which is not only a homomorphic but also with certain integrity condition, we obtain the classical Bergman space. And in particular, we are interested in the Hilbert space case for the index p equals to two. So then we call a subset of the unidisk uniqueness set for this Bergman space. If it satisfies that all function from this Bergman space is uniquely determined by its restriction to this fixed subset. For instance, if this subset has an accumulation point inside the unidisk, then of course by the classical result for homomorphic function, of course such a set will be a uniqueness set for the Bergman space, even for homomorphic functions. But of course we are more interested in this situation when the set has no accumulation point inside disk, but of course it could have accumulation points on the boundary. So basically we are considered infinite subset. So this means exactly has the same meaning that any Bergman function in this space, if the restriction to our set is zero, then it has to be identically zero. So this notion is closely related to a notion called the Bergman zero set, meaning that such set is exactly the location of homomorphic function in our space. So simple remark set is a uniqueness set for fixed Bergman space, if and only if it is not a zero set for such Bergman space. Okay, now we're going to consider the random set given as the zero set for this classical Gaussian analytic series. So we just consider a formal series and we consider the coefficient to be very simple, just the standard complex Gaussian, mean zero and variance one. And it's an elementary fact that these formal series, it has radius convergence one. So it is not just formal, it's in fact define a true homomorphic function inside the unit disk almost surely because it's radius convergence equals to one. Then of course we can consider the zero set of this homomorphic function. I will denote it by this symbol. So by a famous result of Paris-Virac says a following, so which is already mentioned in previous talk by Bouffet d'Auf that the distribution of the zero set of this Gaussian analytic function is exactly the determinant point process induced by this classical kernel called the Bergman kernel, which is in fact the kernel of the orthogonal projection from the L2 space on the unit disk to the space of our Bergman space with index two. Then as a colliery of our solution to this, I will not mention this solution to this line space as a complete conjecture. I think probably Bouffet d'Auf will mention this tomorrow which says that in fact as a general result for any determinant point process, once it is induced by the reproducing kernel of just some abstract reproducing kernel help space in some L2 space, then almost surely this random set will be the uniqueness set for the corresponding reproducing kernel help space. So in particular, since by this result, this random, this zero set of the random kernel function is the determinant point process corresponding to the Bergman space. So almost surely this set is a uniqueness set for the Bergman space. So in fact, this is the study of the uniqueness that is quite subtle. Let us just mention briefly, we are really in a critical situation. So it's kind of when you first see this, it's kind of a counter-induction. So you have a holomorphic function and you concede the zero set of this holomorphic function and the result is this zero set of certain holomorphic function cannot be zero set of other holomorphic function. But because here our random holomorphic function is given by this thing, so it can be proved easily that this thing can never be, almost surely can never be the Bergman space A2. Actually with a little computation, this can be shown to be A2 minus epsilon B. So it's the integrability of this holomorphic function is just a little bit less than the critical situation. But then usually in the study of a holomorphic function, we have usually encountered a situation that although you have a very badly behavioral holomorphic function, you can probably divide by some factor, this factor never vanish. So this factor never vanish so it does not change the structure of the zero set, but it will probably improve the integrability. So this, which means that probably, so the question is, we ask whether we can like find a factor from this holomorphic function and to improve a little bit the integrability without just drawing the structure of the zero set. But this calorie tells us now, so we can really not improve any integrability without destroying the structure of the zero set. Okay, so then this comes to our further step. In fact, the proof of this theory is really in a general setting and we do not use at all the holomorphic structure in this concrete situation. So it's a very natural question to why not find a direct proof in this concrete situation. So this is the motivation of this work. So just to recall that almost surely, this set is a uniqueness set for our Bergman space. What does this mean? This means that for a generic realisation, so you, with probability one, we fix a subset, fix a realisation, then any Bergman function in A2 is uniquely determined by its restriction to our set. So natural question is how? So this is just theoretically, so it's determined by restriction, but how it is determined? Can we find an explicit formula so that we can reconstruct, recover simultaneously and moreover explicitly all Bergman function from just the data which is spent as the restriction of the function to our set? And once this question is just, it's very natural to ask a similar situation, why not just ask this question for this particular random subset? We can ask this question for more general random subset, and we can also ask for this question for not only the Bergman space, but for probably other even Banach space of a holomorphic or even harmonic functions on unit disk. Then the first step, so yes, I will really try to focus, try to emphasise that the most difficult part of the work is really this work, simultaneously. Simultaneously means that, so you have, because this is improbability setting, we have a fixed realisation and this realisation works for an uncountably many function usually, so. But before going to that step, we first started even, this first step is not clear, so for a fixed function and for a fixed point, so this is not really a restriction, so we can, because we are a reconstruct a holomorphic function, once you can reconstruct a holomorphic function on a countable many points, so you can reconstruct the function all, so but the really problem is, so this is a drawback, so for a fixed, but let us start at this problem first. So for reconstruct a fixed Bergman function, we obtain the following results, saying that, so first we record the Lopachevsky hyperbolic magic and we have, this will be used later, the hyperbolic real area on the unit disk. So we prove that if we fix a Bergman function and fix this point, then using, so forget for some time this expectation on the denominator, so we obtain here, this is part of some discreet version of mean value, so this is weighted average, so this weighted average it's when, so for a sequence of weights, when this weight goes to the critical point as goes to one, then it is very close to the value of this function, so let us look at the picture of this formula. So we have a unit disk and a lot of set points, so this is our fixed realization of the point process, which is given by the zero set of, sorry, zero set of this holomorph function. Now we know the value of a holomorphic Bergman function on these points and we want to obtain the value of this point, which is probably not in our original set. So what we do is the following. So we first want to obtain certain, so basically speaking we are going to do the following. So we draw the hyperbolic disk and average, average using some weighted average of values inside this hyperbolic disk. And then we take, we let the hyperbolic unit disk to go larger and larger, then this value, the average will be closer and closer to the value at this center, origin of the disk. So as a corollary, this we have, this is just a Borel-Cantelli argument from the previous results. So we have exactly this, the value of our Bergman function on this position will be the weighted average of this thing. So there's some subtlety here. So I take the average really from any line and take the average here, the summation here, they take larger and larger. The fact we use this is because in fact if we use just one summation to take the sum over all points, then this is usually divergent. So this is really necessary to obtain. So this is in fact, when you find this result, it's very easy to check, but before you find this result, it's really difficult for us to realize we should do this discrete version, mean certain principle value procedure. So this is explained, so what I explained is really this. So that by summation is needed because in general, this is not known. Then this is as I said, this is just the first step. We want to construct just one function. Now, of course, one function you can use in just one, it's okay, almost surely, so you can use accountable in many functions, again okay, just by intersection. But the problem, the difficulty is really our formula is not continuous with respect to the Bergman norm. So you cannot extend this explicit formula just by saying, ah, okay, and this holds for then subset, then it holds for all space. This is just not true because this formula holds for instance for polynomials. So, but you can never hope that this can work for any, because polynomials can be dancing like AP space, not just two, you can work for a two, a four, a three, but this is just not possible. So then we want to analyze what is really going on in our proof. So in fact, this homomorphicity, we are really using the harmonic assumption, which basically we are using the mean value, area mean value property, or in certain steps we are really using the spherical mean value property of the function we want to reconstruct. And the function, the complex value function can really be replaced by vector value, or some Hilbert space vector value function. And so these two steps are just trivial. So you look at the proof, it's very simple. But these steps really non-trivial, less trivial generalizations, we can not only reconstruct the function for the Bergman space, but in fact we can reconstruct more. More in the sense that we have, so if you look at this space, it seems to be very close to this space, just a little bit less integrable. So because without this epsilon, so this thing goes to, when z goes to the boundary, this goes to zero. So with this weight, this, it will be more possible that this integral be finite. So defining this, of course, you have bigger space, and we add some this growing condition on the norm of our function with respect to this weight. And this small, this really very small gap, we achieve a little bit, will help us a lot later. Okay, now we come to the original question. Whether we can do simultaneous reconstruction for this thing, for this Bergman space. As I said, as a corollary of our solution to line space conjecture, theoretically this is true, you can reconstruct, but the fact here I said it's impossible to reconstruct simultaneous all function using actually linear procedure. So probably there exists some non-linear way of the construction all, but this result tells us, so not only using the exponential weight, so I use compact supported because there's some issue of convergence. So for simplicity, we work with compact supported weight, then any weight you use here, just for if you want just reconstruct one point, it's not possible by using this average method. So here this constant is really universal for all radial weight on the unit disk. So this, so when we see this probably we will even ask whether the supremum here it's measurable or not. So the measurability of this part or in fact the proof of this inequality is reduced to this factor. This supremum actually is the norm of the following function, the L2 norm of certain holomotiv function. So okay, yes. So now we have this result. We cannot simultaneously reconstruct our all-Bergman function. So it's natural to ask probably because we can reconstruct really the fixed function, probably we can reconstruct some smaller space using that method. So the smaller space, in fact, we obtain some abstract condition on the space so that we can reconstruct. So this space is just to give the several examples, which kind of space we can obtain the simultaneous reconstruction. So we work first with the weighted Bergman space with some rapidly growth in condition, growing weights, so think of this. So by this I mean some holomotiv function on the unidix, which is L2 integrable with a weight. So if this weight is the constant weight, this goes back to the original Bergman space. If this weight grows rapidly on the boundary. So of course this will force your function to be more regular because the integrability is better. You have these integrabilities with respect to bigger weight. So your original function should be, the growth on the boundary should be more controllable. But in fact, with this function, I will, why we really want to focus on Bergman space? In fact, there's one thing that I want to explain now is the following. So in fact there's this space, Hardy space, which is also possible to reconstruct our last, a substituation of the last case. So this is just those holomotiv function on the unidisk with A and ZN, with some square integrable coefficient. Then for such a very classical function space in functional analysis, we know that for any sector, from, for any sector, this is the rate, this is okay, this is one over square root of two. So for any sector of this, for almost all sector, you take the, there will be this limit. So the value on this, the value of our function on this position, then you take, you can take a limit here. Then you obtain a function on the boundary. And with this function on the boundary, you can solve digitally problem, then you will find the original function F, provided that this measure mu. So here I am really speaking of the, the big measure here, provided this measure is absolutely continuous with respect to the big measure. But however, even this procedure, if we consider singular measure, then although it can be, we can make a sense so that this has a meaning, that there's a boundary, boundary value, but this boundary value will be almost surely zero with respect to the big measure. So in fact, this procedure can never reconstruct the original function. So our result is the following. So if we consider the first two, the weighted Bergman space, or a general, more generally, some reproducing kernel helper space with certain assumption on, gross assumption on the reproducing kernel, then for all function in this space B, this partisan salivar reconstruction formula holds almost surely. So almost surely this holds for all function. And for the last space, which is a generalization of the classical hardy space, and the result can be improved a lot because here, this formula holds not only for all function, but also for all points. And also, this along a subsequence can be dropped. So it's just really a limit. So in fact, all this work, when I speak of the zero set of this random holomorphic function, it can be generalized to more general point process on the unit disk, such that the following assumption is certified. So first we needed some conformal invariance in average, meaning that the particle, the number of particles inside any disk, inside any subset is proportional to the hyperbolic area of this subset. And we need some small variance assumption of the linear statistic. So this linear statistics, the variance of linear statistics is controlled by the linear statistics of the square value. So in fact, all these assumptions are in particular satisfied by some point process and determine the point process with permission kernel, which we just mentioned, and more generally for negatively correlated point process. So in fact, this result is a little bit complicated. So in fact, we can also generalize this result in other direction replacing these... So if we really want to focus on reconstructed functions from this space, we need some, we need more dense assumption on the point process. So suppose our point process are more dense. So this assumption, if beta equals to one goes back to the original problem. If beta greater than two, then this is much larger than the zero set of this random holomorphic function. Then we do obtain a concrete formula for obtaining the original function. So there are some very natural problems arise in this work. In fact, it seems to us that even the existence of such, of the existence of subset of unidisk satisfying this formula seems to be new in the literature. But our works really prove the existence. So we really, it's really natural to ask whether we can construct a deterministic subset satisfying certain these kind of reconstruction properties. And it will be more challenging to give sufficient condition or some geometric criterion or a subset so that it will satisfy a given reconstruction property. So in fact, all this work, not it's not restricted to the unidisk, it can be worked on real hyperbolic space and complex hyperbolic, quaternion hyperbolic space and on these also locally finite, infinite connected graphs, et cetera. And in fact, on the level of major theoretical thing, so what we are doing here is formula if we understand these as some weight B, X, Z, that X. So if we look at not this function, we look at very small function, just the harmonic function, which can be extended to the closed unidisk. Then this formula says exactly that this measure almost surely converted to the Poisson measure, with respect to the point, harmonic measure with respect to that point. So using such, such result can be generalized to some more general situation for some negatively correlated, negatively curved remaining manifolds. And I will just mention that in complex hyperbolic space, the harmonic function will be replaced by the harmonic function with respect to the curved Laplacian. So this is the Laplacian for complex hyperbolic and this is the Laplacian for real hyperbolic space. So that is, yes, that is all. Thank you.