 Thank you, Moshmi, for introducing me. First of all, I thank the organizers for giving me an opportunity to speak here. So, as it's visible, my talk is on Hilbert function of Artinian local computer sections. So, first I will say what are Hilbert functions, even if all of us are familiar. And then I will explain what is my problem, which I worked on and a result which we have got in this direction. Okay, so let me recall. In my setting, I'm assuming that my ring R is either a power series ring in R variables or a polynomial ring in R variables, where I assume that the ring is graded with the degree of xi is equal to 1. And for me, always the R0 part is a field. For an ideal i in R, we consider the ring A to be a percent of R, R mod i. And in the case of graded ring, I mean that my ideal i is homogeneous in the polynomial ring, instead of power series ring. And I denote by this R i, the i degree component of the ring R. And by ii, I mean the degree i component of the ideal i, which consists of the all homogeneous ideals elements in the ideal i. So, by the function of the graded ring, we mean a numerical function from natural numbers to natural numbers, which takes the integer i to dimension as a k vector space of ith component of A, which can be computed from the dimension of the k vector space R i minus dimension of ii as a k vector space. So, I write this m for the maximal ideal generated by x1, x2, xr in power series ring, also in polynomial ring. And I denote by this small m the maximal ideal in the local ring R. So, in the local case, what do you mean by the function of the ring A? I mean that it is a function of the maximal ideal, which is the function of the graded ring, a serial graded ring of i, which is given by this, which is the maximum of mi over mi plus 1. So, this ring plays an important role in singularity theory. I will not go in the details here. So, this is what we mean by the function of the local ring A. So, what is our problem piece? So, before coming to that, let me recall a nice theorem due to Macaulay, where he characterized all possible numerical functions that can occur as the limit function of some standard graded algebra. So, this criteria is very nice. So, by an O sequence, we mean a numerical sequence of integers h that satisfies the Macaulay's criteria, that is h0 is 1. And so, this, I am not defining this notation, h i upper bracket i. So, this is some condition on the, this sequence of integers, whenever we have this condition. So, Macaulay put that there is a graded algebra, which for, with the limit function h, whenever we have this criteria on h. So, this result of Macaulay raises the following question. So, suppose you have an O sequence to begin with. So, can we characterize the hybrid functions of graded or local k-algebra with additional properties like domain or radius or combinator section and so on. So, in this talk, we want to characterize the hybrid functions of algebras with the property being Gorenstein and combinator section rings. We want to study which numerical functions can be occur as a function of Gorenstein or combinator section algebras. So, this is a deep problem. So, before coming to my results, let me recall what we know in certain cases. So, as I just said before, this is a largely open problem, but we know in certain cases some results, what we know. So, before saying the result, let me set to the, let me reduce my problem to the Artinian case. So, I am reducing this problem to the Artinian case because sometimes we can reduce the apologetic to this case. And also, this is an important case which is open even in this special case. So, for the moment I am assuming that my ring is Artinian and by S, I denote the so called degree of A, by which we mean that the largest I, so that the power of M or to the power I is non-zero. Because A is Artinian, we know that some power of M is zero and the largest non-zero power of maximal ideal, we call this a circle degree of A. So, we assume that the ring A is Artinian with a circle degree S. So, by a CI sequence, I mean a combinator section sequence. By this we mean that if we have a numerical sequence H, we say it's a CI sequence if there exists a combinator section algebra with a rabbit function as H. And similarly, by a Gornstein sequence, I mean a numerical sequence which is realizable for a Gornstein K algebra. So, McCollet against the classical result classified the numerical sequences that can occur as the function of combinator section algebra or Gornstein K algebra because they are required in this case when we have co-dimension 2 here. So, he proved that the sequence of this form where we are in the co-dimension 2 case 1, 2 and this has to be 1 because I am assuming that my ring is Gornstein. So, then this is a Gaussian sequence if and only if the difference between the successive HI is at most 1. So, this is the result non-routine McCollet since long back. But so, when we go to the higher, any question? Yeah. So, when we go to the higher co-dimension instead of 2, if you look at the higher co-dimension, what we know? So, here there are many things are not yet known. For example, this result of in the when you take 1, 3 sequence, so co-dimension 3 case. So, here due to a nice result of customizing, but we can prove the following result in the graded setting by which we mean that our algebra that we realize is a graded ring. So, what we know is the following. Suppose you have a numerical sequence 1, 3 and so on, then this is realizable by a Gaussian K-algebra which is graded if and only if first of all this has to be symmetric H vector. Moreover, if you look at the difference of the numerical sequence obtained by taking the difference of successive terms here 1, 2, H2 minus H1 and so on, like more generally H1 minus H1 minus H1 minus 1, then this numerical sequence should be again an O sequence. So, whenever this is different on this criteria, that means whenever we have a numerical sequence which is an O sequence satisfying these two conditions, it is realizable by a graded Gaussian K-algebra. And once you go beyond co-dimension 3, so many things are open. And now once I go to the combinator section case, once you go beyond co-dimension 2, they are not equivalent to each other, but here in the case of graded combinator section rings, we know exactly how the function look like. This can be easily written using the degree of the of the gender of the ideal i and the embedding dimension. So, this is pretty easy, but now if I just remove the assumption of graded here instead of ask the question for the local case, this is completely open even in the co-dimension 3 case. So, that is what our problem is about. So, I am coming to the local case now. So, we can we ask the following question, which numerical sequence can occur as a function of RTN and local combinator section K-algebra or more generally Gaussian K-algebras. So, this is a widely open problem, of course, for any co-dimension and nothing is non, almost nothing is non in co-dimension 3 also. So, what we do is yeah, so what we do is that we assume this is a difficult problem because by definition the element function of A is the element function of the associated ring of A, which need not have nice properties even if the ring A has nice properties. For example, it can happen that A is girlfriend the ring associated ring of A may not be gone of time. So, this is the difficult, this is the reason why this problem is very difficult. So, what we do in this work is that we assume that the simplest case which we can think of after co-dimension 2 is take ideal I in the power seasoning in 3 variables and suppose I generated by 3 elements FGP is a combinator section ideal and the order of the elements here is 2. So, this is the next simplest case we can think of. So, that by this we mean that our elements, the function of H of A will be of the form 1, 3, 3. So, instead of taking arbitrary H2, we are now assuming that our H2 is also 3. So, under this assumption we classify all possible functions of combinator section k algebra. So, before splitting myself let me use some notation here. So, I denote by max H the maximum of H i occurring this numerical sequence H and by delta H I mean the maximum of the difference of H i minus H i minus 1. So, maximum jump we have in the numerical sequence H. So, once we have this one we have the following result. So, let me explain this result. So, what we prove that a numerical sequence H suppose it is the over sequence and the assumption our is that we are in co-dimension 3 and plus our H2 here is 3. So, directed by I will contain three elements of order 2. So, then the following are equivalent these three conditions. So, we say we prove that a numerical sequence H is a combinator section sequence if and only if this is a also a co-dimension sequence which is usually not true surprising. So, H is a combinator section sequence if and only if it is a wasna sequence and moreover we have a numerical criteria for the sequence of the form H to be noticeable by a C i algebra. What is the numerical criteria is the following. So, so first of all by Macaulay we know that once H2 is 3 Macaulay's bond will tell us that H3 can be at most 4 and in the case of H3 at most 3 we prove that any O sequence which has H3 at most 3 is noticeable by a Gaussian K algebra and not this is a this is a first case and the second case is that the number H3 is 4 and the jump of the terms in the sequence is 1 then also it is noticeable by any Gaussian K algebra and when the jump so we prove that the jump of the terms here cannot exceed 2. So, the jump has to be at most 2 and when the jump that jump is equal to 1 it is noticeable by both hand sequence a wasna algebra when the jump is 2 then we show that they can be at most 1 jump by 2 and not just that the position of the jump by 2 is the peak position of the sequence. So, there are this criteria so this completely classifies the numerical sequences that can occur as the function of Gaussian K algebra and so so that this this sequence we can have especially how it looks like. So, it will have suppose D is the maximum of integers that occurs in the numerical sequence H then then there will be it will increase up to up to it reaches D and then there is a drop by 2 at the peak position that means at the after D it descends to D minus 2 and after that there is no jump by 2 in the numerical sequence. So, this is how the numerical sequence will look like if it is reliable by a Gaussian K algebra. So, let me come to some examples before commenting on its proof yeah maybe I will just say examples. Suppose I take the sequence here now here we have 9 terms so the so called degree of the sequence is 8. So, here this is example which we have taken from the paper of Erobino and a Marques. So, they have certainly proved that this algebra is not reliable by a Gaussian K algebra. So, they use different methods to prove that there is no Gaussian K algebra with a little bit function given here. But this example can be easily because you say that this is not a Gaussian sequence using our result why because so here we have a numerical sequence of the form which we have considered 1 3 1 3 3 okay and here we have 4 here. So, where is the jump by 2? So, the jump by 2 is occurring from 4 to 2 here okay. So, there is only one jump by 2 in this sequence. But our theorem says that the jump by 2 cannot can occur only at the peak position. So, the jump by 2 can occur only at from 5 to the next position. So, it is not a Gaussian sequence using our theorem. So, that we can easily say that this is not a Gaussian sequence using our result. Similarly, let us look at this example okay. So, here also we have a 1 3 3 4 sequence with the jump by 2 again this is not a Gaussian sequence because the jump by 2 is occurring from here to here where the peak position is here. Okay. So, that this is again a Gaussian sequence this example also taken from the book of Erobino where he gives some other argument to say that it is not a Gaussian sequence. So, let me now come to some example where we actually construct a ideal with the Hilbert function given. So, as I remarked before once we have a 1 3 3 sequence with H 3 at most 3 then the ideal we construct is constructive proof and here we give the inverse system of the ideal I and so that the ideal we get here is a combinator section ideal with the Hilbert function H okay. So, we give an explicit inverse system of in the divided power ring so that the narrator of the ideal has the Hilbert function this and is combinator section ideal. And in this case of 1 3 3 4 our proof is again constructive when we have a jump at the peak position we actually construct a combinator section ideal with the Hilbert function H using a different method. So, here our idea is to reduce to the co-dimension 2 case and then use knowledge of results which we have in co-dimension 2 and again use this construction to come back to co-dimension 3 so that the ideal which we get has the Hilbert function H and is also a combinator section ideal. So, that is what how we construct I am just giving you a formula for the ideal I here against the combinator section ideal with the Hilbert function H here okay. So, I will stop here I will just skip this sketch and thank you. Yeah, in our theorem we actually can construct the ideal with the required Hilbert function not in general not in general but in our result yes we can yes there is an algorithm yeah no no actually H2 is equal H2 equal to 3 was very special case where where we can also also see how the I the I2 part will look like the FGP in the ideal I there of particular nice form so that's where we can do many reductions to co-dimension 2 case.