 Yeah, first of all, thank you. I would like to thank all the organizers for, you know, giving me the post-native talk here. So, I will talk about threshold of filtration of ideals. Am I audible? So, this notion was introduced by Mustafa Takaki and Wander Nabe in 2004 for ideals, I mean the IID filtration. So, let R M be a regular local ring of characteristic P and I and A and I be non-zero proper ideals of R such that the A ideal is contained in radical of I. Then for every E we can associate an invariant mu A i that that is maximum of R is that A R is not containing P Eth, Eth Frobinus power of I. Then F threshold of A with respect to I is the limit E tends to infinity mu by P and then I mean I will not go much in detail about. So, this is an invariant which is associated with I mean singularities in prime characteristic. So, for let R be a Noetherian ring of positive characteristic P, then they define this C plus and C minus. So, basically the limit soup and limit of the quotient of mu with respect to P and they prove that like in many cases this these two are same and they call when they are same they call this as a F threshold of I with respect to A. And, essentially in 2018 D. Stephanie Pattencourt and praise they proved that this limit always exists. So, that means F threshold exists for an ideal A with respect to I if A is contained in a radical of I. So, I mean we extend this filtration of ideals. So, like if we consider A to be an ideal then consider this ordinary power filtration. So, that is the invariant introduced by Honke, Mustatan, Wontanabyan, Thakagi. So, we introduce for in general filtration and we ask the question when you know for which filtration this invariant exists and then mostly we focused on the symbolic power filtration in regular rings. So, let's recall the definition of filtration of ideals. So, let R be a Noetherian commutative ring of positive characteristic P, filtration of ideal is a collection of ideals A and indexed by the set of non-negative integers with A node to be the ring itself. And, this collection is actually a decreasing sequence of ideals with respect to containment and it satisfy the graded property. That means that mth piece, mth part of this collection times nth part of this collection is actually containing m plus nth part. So, the standard example of filtration are basically ordinary power filtration for any non-zero ideal in I, ideal in R, then symbolic power filtration and then we can take this integral closer power filtration and tight closer power filtration of non-zero ideals. So, let I be a non-zero proper ideal of R and A node, A bullet T node air filtration of ideal in R. So, we define I mean new A node I for every which is supreme of R where R is the non-negative integer such that AR, the RF piece of the filtration is not containing P eth Provinus power of I, eth Provinus power of I. Then, we define like similar notion which was defined by Honke and others that C plus is lim soup of the quotient of mu and the mu by P and C minus is lim. So, then and we say that if these two are same and finite, then we say that that is F threshold of the filtration with respect to I. So, in general like it's not finite like if that depend upon the filtration and I like there should be some sort of relation between them like for example, the lim soup P plus that that is fine I depend on if there exist a positive integer M such that for large E M P that that part of the filtration is containing E eth Provinus power of I. And if we assume this condition and we assume that R is let's say F pure ring then I mean this F threshold exists. So, basically for existence of this, we prove that if a filtration is linearly finer than the Iedic filtration, then these two lim soup and lim in both exists and they are bounded by that N times mu phi where N is that number for which a I mean NS piece of the filtration is containing IS for all S and if R is F pure ring then these limit exist and this is nothing but the supremum of the quotients. And in case of Noetherian filtration we prove that if the radical of the filtration radical of the filtration means radical of the first piece A1 if that is contained in radical of I then this F threshold of I filtration with respect to I exist and it is basically equal to R times the F threshold of AR I mean AR Iedic filtration with respect to I for some R. So, properties of F threshold filtration. So, in this case when suppose we have two filtration and each piece of the filtration is contained in corresponding piece of another filtration then we prove that these lim inf and lim so they satisfy the same relation that means C plus of A note is less than equal to C plus of B note if A note filtration is substitute inside B note filtration and if the result of B note is finitely generated module or result algebra of A note then these two are same. So, as a corollary we can see that I mean if we consider I to be a non-zero proper ideal and we consider this I closer filtration and integral closer filtration then these two are same if they exist. Okay, so now we will focus on F threshold of symbolic power filtration. So, for this section we assume that I mean this rest of the talk we assume that R is regular ring of positive characteristic that I mean A and I be non-zero proper ideal in R such that A is contained radical of I then we prove that the F threshold of symbolic filtration with respect to I exist and like this F threshold of symbolic filtration is not same as the ordinary filtration that was introduced. So, for example we can take this R to be polynomial ring in three variable over a field of positive characteristic P and if we compute this F threshold of okay I this A is an ideal which is generated by three monomials x, y, y, z and x, c then the symbolic F threshold is two but the ordinary F threshold of this is three three by two input in general like for square free monomial ideal we prove that this F threshold of symbolic filtration for any square free monomial ideal with respect to the maximal ideal is actually the height of ideal A and we gave a bound for F threshold of symbolic power filtration with respect to the maximal ideal. So, we prove that the this is actually bounded over by the big height of the radical of A right if M is any maximal ideal containing I in that case and if R M is regular local ring then in that case we prove that this is even we can restrict to the height instead of the big height we can replace big height by the height of the ideal and we prove that like you know in most I mean in nice cases this the symbolic the F threshold of symbolic power filtration is actually height of I like for example square free monomial ideal or we consider T by T minor of a generic generic matrix or Parfian matrix and in in these cases but we can see that for this example I mean if we take A 1 up to A n be any positive integer and I A is ideal generated by x 1 raised to power A 1 to x n raised to power A n in this case the F threshold of symbolic power filtration is 1 by A 1 plus up to 1 by A n. So, that means if A i's are bigger than like some of A i's are bigger than equal to 2 then it cannot be equal to height of the ideal ok. So, in k in graded setting we gave an upper bound of this of F threshold of a filtration in terms of the wall smith constant of the filtration. So, wall smith constant is suppose A note is A bullet is a filtration then it satisfy that graded property and this alpha of A note is the minimum degree I mean minimum of degree of F, F belongs to then this this sequence is sub additive sequence. So, this limit exists and we denote it by alpha hat and actually this is the infimum of this is numbers. So, we prove that if alpha hat is positive for a filtration then F threshold of the filtration with respect to maximal ideal is bounded over by n by the wall smith constant of the filtration where n is the number of variable in the polynomial and if we consider i to be non-zero proper homogeneous ideal and ordinary power filtration or symbolic power filtration. So, we got this this like I mean the ordinary F threshold is bounded by n by alpha of A and this for principal ideal this was proved by Karen Smith and his co-authors and this is tight bound actually we gave you know example where these two are actually attained and we give an family of ideals for which the F threshold is both of the F threshold of ordinary and symbolic power filtration are actually height of i and that's like class of F coin ideals. So, if A B F coin ideal in R so F coin means like there exists a regular sequence of height i such that the quotient of that regular sequence is F pure ring. So, in this case we prove that actually both the symbolic as well as ordinary power I mean the threshold of symbolic and ordinary power filtration is actually height of i. Okay.