 So, what we have seen more than developing a theory of HK multiplicity, one thing is hard to compute as HK multiplicity itself due to various non availability of standard methods. So, each time we computed HK multiplicity we threw some big machinery or something. So, this is another approach in this talk I will talk, in this talk I will give and which is I give a notion of HK density function. So, this works of course in graded setup not generally so I will define it. So, here you have a ring R and an ideal I with the following property. Your R is a standard graded. In fact, later you can see it's even for graded domain this and graded domain it works. And I just assume R is standard graded right now and I is a homogenous ideal. So, here R0 itself is a K which you assume to be perfect and characteristic of R is of K is of course P positive. So, remember what we have done or how we have to define HK multiplicity this is defined as a limit. You take Q tending to infinity where you are running over all the colons of I upon Q to the power of D. So, here one more restriction is there you assume always D to be greater than or equal to 2. D equal to 1 we already know it's there's nothing much to say. But now in our setup R is a graded ring and I is a graded ring so I for when a skew is a graded ring a graded ideal. So, this quotient is a graded quotient. So, I could have looked this as a summation collected all the pieces and greater than or equal to 0. So, in fact, when we compute for like a projective curves and all in fact the computation goes by computing all the graded pieces and combining together. So, here the thing is idea is that instead of looking at the this total length and then going model of Q d why don't you just look at the and the graded pieces and take the limit in a normalized way. So, let me make it more precise. So, you fix n and take Q to be P to the power n and define a function fn. So, this is a function which depends on the pair R and I and is given by R positive to R positive. So, what do you do here so it is something like. So, fn of r i of X is you take the length of our IQ X Q upon Q to the power d minus one. So, suppose your, your X is say m mod of P to the power n or m mod Q whatever. So, this will go to the QM piece of the thing. So, it's going to basically going to form a step function so you're looking at a set of step function where so I hope I can draw a picture. So, for like one Q you have a step function like this. So, we further when I look at P and plus one so basically now you're defining this intervals further. So, so it is like this this this so here are three function like this and then I should I use a different color, but you'll understand what I'm saying here. So, similarly, you can define fn plus one so that will be a step function which added more steps final steps. What you have defined so far for a given pair R and I a sequence of functions. So, this is a theorem fn r i is a uniformly convergent sequence limit of fn r i where n tends to infinity, which henceforth hk d function hk density function. This is a compactly supported continuous function. Once I have proven once we have said that your fn r i is a uniformly convergent sequence. It's now it's obvious that if I take integral over this function f r i x or whatever it is. So, it's going to be your e hk of r i. So, now you're approaching the hk multiplicity in a very complicated way, you are looking a function instead, other than the number. So, but this contains more information about hk multiplicity and it turns out to be not only that you look at the properties so let me list some of the properties so the properties which holds for. hk multiplicity does hold which is like if I is in j. And our equidimensional here now. This is given, then. And so, so I and j. Now we are working in a great setup so I and j both are great. So j is contained in the tight closure of if and only if f of r i is same as a fortune. So this, this, this is nothing big deal because this is, it relies very much on the fact that J's continent type closure of if and only if hk multiplicity is same because this is, it's very easy to see that. Then other thing is associativity formula. So in fact you can define the notion of hk density function for any finitely generated graded module, and you'll have this formula so I won't write it here. Now, the third thing, which is important and which doesn't seem to exist for hk multiplicity is a following. Suppose you have this pair r and i, r is the standard graded and s and j, and j over the same field of course, over k. So, so product should make sense. Then recall, a segue product of R and s is our check s says that which is nothing but the ring, a graded ring and grade components are given by R and tensor SN over k. So the name is of course a segue product because if you have our project is embedded in PN1 and as a project is embedded in PN2 projective spaces then their segue product is embedded in another project. That's another project space and the homogenous quadrant ring will turn out to be this one. So, so here, so the what I'm trying to say here is that if I know the density function of this one, if I know the density function of this one, then I know the density function of the segue product. So how is it. So, recall, let us write this FRIX. Oh, I'm sorry, this is you define F of R, a polynomial. So given R you can define a polynomial ring, which is nothing but the multiplicity of maximal ideal x into D, D1, so D1 is dimension of R. Why am I trying this mod D minus factorial. I don't know why I'm writing D1, D1 I should. So this is a polynomial function. I mean it's easy to compute. Then the result is F of R check s, which of course I can easily compute in terms of the multiplicity of R and s both this function minus this F of R check. S with respect to the ideal I check sj. So I check j is defined in a similar fashion here so I check j is the ideal in this ring was given by I in tensor. So here it is. So this function is nothing but f of Rx minus f of RIX multiplied by this is a product here. I'm sorry, this is sjx. So hk density function you can say is multiplicative. So, like whatever the example so far whatever we know of the hk multiplicity in graded cases, we know we can compute easily the hk density function for those examples because as I said generally it is computed by usually computing the great each graded pieces. So you're automatically are actually is in build the construction. So that's one property. Of course a tensor product is obvious. I won't write it here. So question. I think this question is actually asked by what I know I think he's believes it is FRI piecewise polynomial function. So, so whatever examples we had like a project you cover and also so in fact it's a he expects to be a piecewise polynomial function of degree. I mean he of degree equal to dimension R minus one. For example, in the case of project because it's a in fact of degree. These are linear or piecewise linear polynomials and other examples like a has a brook surface or Tori varieties. It's always a piecewise polynomial. So, looking at this invariant as I said, remember, FRI is a continuous completely supported function. One, one reason we think that it may now some analytic technique may help here. Why because F of our eye. If I look at the set of these ones. We're running over the set of such graded pairs. It sits injectively to the Fourier transform. So Fourier transform I won't define here so it's analytic function here. So, the definition of Fourier transform tells me if I evaluate Fourier transform at zero. So we. So, but I don't know much right now happening here. So, how much time how much time do I have. Until nine, sorry until the 50th or 50 minute lecture so from now on I think 35 minutes. 35 minutes. Yeah. Okay, that's a lot. Yeah. So, anyway, so, so f of our eye, as he said is a compactly supported function. Oh, it continues. So this we have made the remark about this. So, he, of course, it gives hk multiplicity so what about the other invariant perhaps it tells me more about some characteristic being variant. So here, we look at the support of the FRI so let me just draw figure so continuous here so this is so alpha of our eye. Maximum support. Look at the support function. So, on the other hand, there is a notion of what they call F threshold. So, this, I'll just define it. So F threshold of M at I, I'm sorry if I am goofing with the supermodel, it is F threshold of M at I I think. So, this is if you definition is the following limit, you look at this one. Another characteristic being variant very much depends on characteristic. So minimum of our M to the power r plus one is containing are your probate. I don't know why it's coming. So this limit does exist is proved by people. I suppose Stephanie and so this is a threshold. Probation is the following theorem. If our is a normal domain, or doesn't matter you can otherwise can go to normalization normal domain, normal graded domain. So my ring is standard graded henceforth so and strongly F regular. It did not be everywhere. It's enough on the puncture spectrums on say spec of R minus M. So it's equivalent to saying suppose proge R is smooth that's good enough for me so it's a it's a regular so it's a strongly F regular puncture spectrum. This alpha of R I is your threshold. So this this is a useful property for us because one question was this. If I look at this is varying. No threshold of M where I is varying. So this is so where see after take an ideal J and look at a threshold of J with respect to I where it should make sense. I mean I and J because this containment issue, then is this discrete set. So this was I think question, question, question, but I think mustada takagi what I know this question was so now you have interpretation so you just don't have to look at arbitrary J you have very specific J SMI. And this is same as in the graded setup here alpha of R I so you have so now this this alpha R I or in fact whole of your density function in the case when the dimension of R is two is given in terms of the stroke of the hardener cement filtration and there is a result of gizekar where he constructs a family of vector bundles where the hardener cement filtration vary in a peculiar manner so so I don't want to get into technical detail. So if you take that those set of what are bundles put them in a family normalize suitably back then what you get the support of this thing has a limit point. That's what you get. So, so that answer the question. So this one application so so you attach to given pair R I you another invariant call a threshold and you you have some information about that also. So, so this is so whenever you know eHK or f of R I does give a many characteristic P invariance that's one and coming back to eHK I know that smaller the eHK in whatever way like for as far as regularity you consider or f rationality or of regularity is smaller it is better that's a matter of smoothness but you saw in the case of vector bundles where the you are looking at a non actually you're looking at a smooth curves in fact this and it tells you about the vector bundles the same stability behavior of the vector bundle is smaller your eHK this is the key bundle is going to better behave vis-à-vis semi-stability behavior so so this is a this is a strange relation. Happened so. Same stability of bundles. So, now other thing. I have about seven minutes now. How much time do I have. I think. More than 20 minutes. 30 minutes around. 28 but you can. Yeah, I mean. Whenever you want me it's okay. Yeah, so so okay so. So now we come to a case of a project to Toriq variety. So now. The reason I'm sticking to project to Toriq variety because the corresponding ring will be standard graded ring if I take any affine Toriq Toriq ring. The ring won't be a standard graded but graded ring and now we have a notion of density function for arbitrary and graded rings also. So one can work it out that. So we have not looked at it. So now look at this project to Toriq variety. So what is happening here. So you have a X pair pair X and delta so here X is a project to variety project to Toriq variety and. It's an ample. It doesn't matter if you don't know but I'll just. Guard the divisor. So what I'm getting it that this pair if you have a project to Toriq variety in an ample divisor it will give you a ring. So which we call RXT maybe. I think some people call section ring. I think coordinate ring. So what about it so. So what one studies is the growth of HK multiplicity here as you take higher and higher power of that. So of course that you had a little bit normalized. So say KD so I'll just explain very soon the significance of this. So perhaps. KD minus one. So so in fact this this kind of growth of the HK multiplicity M to the power K. This has been studied by Watanabe Yoshida and hence also I think so. So here we are saying that this this exists of course that's okay but that's not the point here. What I'm saying. Okay. It exists too. Just bear with me. This is a funny number D minus one D. Very complicated. So the point is this you on the left hand side you have the growth of the HK multiplicity. This number is well known as a polynomial ring it's well behaved and so this is the main thing. So looking at the growth of the HK multiplicity how it grows. So it's always bounded by some number which is which is characteristic free which is the multiplicity of the ring and the dimension. D is the dimension of X is D minus one I think D minus one should be greater than or equal to two I think that is there. But the interesting part is when the equality holds. I'll explain this. Oh sorry if and wait a minute this is not very interesting what I want to say. So if you look at the Toric variety you you'll say there is nothing Toric varieties don't take care of characteristic actually there's no characteristic involved but still even in that case HK multiplicity has something to say. So it's saying that it achieves minimal the growth actually is normalized growth is actually minimal if and only if the PDL just explain this this there's a con convex polytope which will tile the space Rd minus one. So what happens. Given any projective variety with if you look at and fix T Carter divisor it it means it corresponds to rational convex polytope. So and vice versa if I have a rational convex polytope for example let rational convex polytope then by multiplying suitably M. Into P becomes a integral convex polytope so it has all the vertices integral convex polytope further you can choose. So call it M one now M to be say D minus two into M one. So this this is a very ample. There's a notion of very ample polytope convex polytope. So point is given a polytope suitably multiplying by M I can make it very ample convex rational polytope. So once you have such a thing that means very ample that means there is a projective variety X and D such that your PD is actually your polytope MP that is there. Then you have a corresponding ring ring of course XT and you have a notion of normalized multiplicity. HK multiplicity EHK. So so growth of this is smallest which is given in terms of thing if and only if the convex polytope P tiles a space. So this this property doesn't depend on M actually so this is well defined so doesn't depend. So I haven't just I haven't said what is called a polytope ties a place so for example if I take X to be P2 and D to be OX1. And if I take OX2 it will just magnify the same polytope so it's so so it will corresponds to the polytope corresponds to the triangle. And when I said tiles are things so basically there is a lattice. It's a bad picture. So at each lattice point to this start your polytope. So here it is each lattice point so on each square and so and let it grow so when you multiply by a lambda or something or something it will grow here it will grow like this. But after it grows beyond this point it will start overlapping with another one so this doesn't tile the floor. On the other hand if I take X to be say P1 cross P1 so with your O1 cross O1 I think that's your then your P P of B is actually a. Square if I take O1 cross O2 that will become a rectangle not a square so this will definitely will tile the floor because I can start here and for high enough this thing it will cover up so. Now given any D. There exist finitely many rational polytopes I mean which tiles a space so the given D of course a number will grow exponentially depending on D so so like for in case of R2 there's a square and some this kind of hexagon I think like this there only two of them which will tile the space so that means given. There will be only finitely many projective varieties with the Carter divisor which has a minimal hk modulate city growth. So so that's another example. And of course this polynomial this polytopes to help me computing the hk modulate city so you can compute it is not looking at the convex polytopes so you have. But this is actually not unknown this is already known by I think ito. This thing in fact it for any touring and so what and I'll be also I think so what we do hk density function for this one. So this is a very nice here so you you have a convex if you take a project to touring variety it comes with I said up convex polytopes I'm sorry sorry this thing so this is your convex polytop in fact and you take a you lift it to say XY plane and take a cone over Z so this is so this is blown up. So this area here is the multiplicity and the hk modulate city you take the the same coin shifted to all the lattice points if it has a lattice point here and here so you shift all these cones. Same. So the ethos theorem says that you take the this area which is bounded area that is your multiplicity and the density function says if I take the cross section of these areas this is your hk density function. So that's a very new visual way of seeing it. So I think what should I do and perhaps I think I improved on my writing skills so become a little faster because I tend to write very slow otherwise. So I'll stop here I think so. All right, thank you very much for the lecture vjm listing the speaker please. Are there any questions. Yeah, I have a question real quick. Okay, go ahead Carl. Okay, so I was very interested to see all the analytic techniques you demonstrated for hope of multiplicity. In your opinion, do you think that that's kind of the next frontier to make public multiplicity work more nicely is to try and investigate the analytic properties of these functions. You mean hk density function. Yeah, yeah. I think so, actually, so, so what happens sometimes you quite don't compute hk density you may not be able to compute hk density function itself, but the very property of I don't I'm not saying it right but hk very probably the way it behaves. You cannot compute it but you know how it goes up and down in various ways you can say something about the hk multiplicity itself so one example I would like to give is here is this one so I think. Quadric hyper surface I think. Abirbakh and in a school have worked a lot in this so they'll know. So, this is the simplest example. So this is a quadric hyper surface. So there is a famous conjecture of what I mean you should. So you know every every if I take any local ring. I know the hk multiplicity is greater than equal to one one is the smallest number, but their thing is suppose you take a ring a with a characteristic P dimension be such that is of course formally unmixed is assume that formally and mixed we don't want anything says that then. And not regular if it is regular of course it is. As much as it is one, so suppose then he hk I'm sorry hk of a PD. So let me give this ring one name our PD is greater than equal to he hk of a PD. And this is equal to greater than equal to one plus md plus one I'll explain it so so md plus one so you take a sec x plus 10x. And then you can write it as mn x and whatever and make it so that's a coefficient of that so this is so that's a that's a part of the conjecture I think this up to D less than equal to five or six has improved I think. In a school in Aberbach had been part of this I think so. So I'm mainly interested in this one. What do you want to say here so the problem with characteristic P computation is one thing that suppose even if you somebody asked how do the SP tends to infinity or something how we hk behaves or something this kind of question. Just think of it when we start thinking about the question, means we need to have some formula for hk multiple in hand to compare we need to have some ground. If you have no for nothing then how do you compare. So here what one has done you come come look at hk density function. So this is an example I'm giving you off our PD. Okay, so this requires all maximal Cohen Macaulay module whatever factorization of the quadric hypersurface structure she walk or the hypers etc spin our windows and all. Whatever it is, the thing is this example tells me I don't know the hk hk density function entirely of this one. But I know up to some point so there will be something and I'm just so. Here is this. So I know that I can compute the hk density function for up to here so let me change the color. So here this reason I know the density function this I know, but when I go here at this reason I don't know. But then if I take P greater than P prime. So this is for P prime suppose then for P prime. I'll know up to this my movie. But I still don't know I have no information about this, but I know this this function is bounded this is so what I'm able to prove one is able to prove that you can still say that he hk multiplicity is there is a comparison ground between them. So this is, so this reason keeps shrinking so when I hope I'm making sense so he hk of our PD as you take limit p tends to infinity. It exists. It of course it exists I mean, this is stupid that is already proved by Monsky but it's greater than equal to one plus mn plus. Because as p tends to infinity this this reason will come to some. I mean it keeps shrinking and you can assume you know everything for P infinity case. So you can, and this this reason on the P for the P prime is bigger than this zero case the P infinity case so this all the regions are bigger than that. So I, at least I know this without without actually comparing I know that this bound will happen. So that's the one place even you entirely don't know density function, you can still predict about the thing. So, that's one thing. So, I, yeah, so I hope I made some sense of my statement. There can be places where you don't know hk density function enough but it's still enough data means you can go outside the measure zero or small set, which is good enough for you. That's it. Any other questions. I have a question. You guys have a lot about the multiplicity of k power of maximal ideal minus multiplicity of m to the k over the factorial. Yes, yes. Yeah, so in this formula, do you expect that the multiplicity of powers of maximal ideal. Does it behave like a polynomial for large is it expected to this is a graded situation you have. Yeah, this is entirely graded situation. Yeah, so what is the expectation of hk m to the k is it a polynomial in k for large k at least. And say even even even for Tory case actually I don't for arbitrary graded case of course no I don't think so, but even for this case, I'm not able to give a very satisfactory answer because this is polynomial in k. But, and this is of course a so it's of a small order of kd minus one generally. So, but I really I don't I can't. I often I can't vouch for correct the statement actually here. Of course in dimension one that is a trivial question because it is multiplicity. No dimension one the thing is I this this graded string know I had to. Work in greater than equal to two actually, because the uniform convergence thing doesn't work for dimension one. One thing that you have to that this limit exists for this. This limit, this limit exists for graded any graded ring left hand side yeah. So what he's asking is different. Yeah, yeah, that is one can ask the same question in local rings and there is a paper of what rabbi and you should. Where they prove that they suppose you take a rational similarity of dimension two. Yeah, then this is a polynomial. Dimension to art. Okay, dimension to rational similarity. Yeah, so. ESK MK or pseudo rational they even proved it for dimensional pseudo rational local ring. The statistic P then they will go to multiplicity of all powers, it's a polynomial in key. But they don't say anything about higher damage. Actually, even a two dimensional local ring. You don't know much about HK multiplicity is correct. So, yeah, they have what they have done as this this thing what they have said this difference is of big order KD minus one that's what their thing was I think I'm correct. So, this is more. This particular integrated situation that this result is more refined of course but local ring things are harder for two dimensional there is no satisfactory answer so yeah. So, yeah, actually it will be nice if of course I'll be very happy if people walk in. HK density function and find something I do expect to. I was trying to see more analytic connection but not that much. In the, in the game going to the SK multiplicity of powers of maximum mousineous ideal might simplify or give a better hold on the problem. Oh no not power of I'm talking about this in general HK density function. This. And what are the other invariants I'm talking about that powers actually I don't beyond that we don't have much clear answer right we I mean projected to require it is a simplest case right so we tried that one simplest miss it took works a lot of work but I mean, it's a canonical thing to see, but. Yeah, it would be geometric interpretation for it's game at SK density function of powers of maximum idea in graded ring. Yeah, yeah. Define that. No, no, no, no, no, no. It is not clear nothing. I don't think so then fact there might be examples there. Nothing good happens there. The Torah variety it is nice because it keeps blowing up basically a blowing up Torah variety means you're looking at a cone, and that cone keeps enlarging. So that's how we managed to control this growth of the m to the k but I was hoping for any grading the similar kind of trick should work and we can just look at the points and delete but it. I think it was just wishful thinking I couldn't do anything about it and then I was just telling the people who are many students here so I was hoping that density function. Maybe it may give something more also. In fact, Karen Smith's student I think I'll open or something he has looked in this kind of thing in a reverse way means he's looking. You can say about the Fourier transform from that side and coming back here. So, yeah. All right. Are there any other questions. All right, I hear no additional questions. So let's thank Vijay one more time. Thank you very much for your thoughts. Thanks for listening. Yeah. And then the school will continue next Monday with the tutoring session for Vijay Trivedi with regard to Hilbengos multiplicities. So, see everybody done. Okay.