 I start. Thank you. First and foremost, I'd like to thank the organizers for giving me an opportunity to speak at this platform. So as the title suggests, I'm going to talk about epsilon multiplicity and some density functions associated to it. So in part it'll be a continuation of the previous talk. So let's define what is epsilon multiplicity. So this notion, it came up in the works of Kleiman Ulrich and Validashi. So what it does is it gives a generalization of the usual Hilbert Samuel multiplicity. So the setup is as follows. You take any local ring r, take an ideal in there, right? And you look at r more i power n. That didn't be a finite length. So you apply this local homology thing to make it finite length, right? And then you can talk about its length. And then you divide by n power d, multiply by this normalizing factor, d factorial, and then you take the Limb-Soup. So the first question is why is it a Limb-Soup? Is because in general we do not know if it exists as a limit or not. So in particular if i is m primary, this process of taking H naught, it doesn't do anything, right? So it recovers the usual Hilbert-Samuel multiplicity, right? So now let's see some quick properties of this multiplicity. So the first thing is a non-vanishing result. So it says that the epsilon multiplicity is positive if and only if i has maximal analytic spread. So that is the cruel dimension of the fiber cone of i that is equal to the dimension of the ring. So for the most of our talk, we're going to implicitly assume the ideal i has maximal analytic spread because otherwise the epsilon multiplicity is zero. The next property, it's basically one of the motivations for defining this multiplicity. So it gives generalization of Ries' criteria for detecting integral dependence of ideals. So Ries' theorem says that for a suitably nice ring R, if you have two ideals in primary, one contained inside the other, their integral closures are the same if and only if their Hilbert-Samuel multiplicities coincide. So this basically gives a generalization of that statement through this new invariant. Now, if you recall the definition of epsilon multiplicity, it's defined in terms of limso because in general we do not know if it exists at the limit or not. So natural question to ask is when does it exist as a limit? So there is this theorem by Kudkowski which says that if you take R to be an analytically un-ramified ring, meaning the completion has got known importance, it's reduced, then the epsilon of i is going to exist as a limit. This example sort of presents the obstruction in working with this multiplicity. So in general, the length function which is associated to the epsilon multiplicity didn't have any polynomial behavior. It's partly due to the fact that the saturated Ries' algebra that didn't be Noetherian. That's part of the reason why this happens. In fact, there is an example by Kudkowski, Taiha, Srinivasan and Theodrescu. So they produce an example of an ideal i in a four-dimensional regular local ring whose epsilon multiplicity is irrational. In fact, if you accept Nagata's conjecture, then at least conjecturally, you can produce an example of an ideal in a three-dimensional regular local ring where you get an irrational epsilon multiplicity. So the question is what about dimension two? So we will answer the question in up to a few slides. So now, this brings us to the first part of our talk. So we're going to understand epsilon multiplicity through the lens of density functions. So this was already done by Professor Trivedi in her study of Hilbert-Kuhn's multiplicities. We're going to employ a similar approach to study this epsilon multiplicity. So what exactly do we mean by this density function? So this is part of an ongoing work with Sudeshna and Professor Trivedi. So the setup is as follows. You take the field K to be some perfect field, characteristic P, and R is something like a homogeneous coordinate ring of some projective variety, right? And then you also need this mild technical condition that is you need the depth at the relevant ideal to be at least two, because you sort of want to translate this problem to geometry and you need this condition to do that. And then you take any graded ideal and then you define this kind of function, right? So what this is doing is, maybe I'll use the board, you look at this bi-graded algebra, right? This is a bi-graded algebra and you're sort of looking at the diagonal components of this bi-graded algebra with slope approximately X, right? And then you want to take the limits. The first thing to show is that the limit actually exists. So basically we translate this problem to computing the growth of global sections of some appropriate line bundle and then we show that the relevant limit exists. So in translating this problem to algebraic geometry, we do not need anything on the field, but when we try to make those estimates, we really use that the field is characteristic P. So then we further show that this function that we have defined G of X, this is actually a continuous function. So you're allowed to integrate over some finite interval. And what do you get is if you integrate from zero to some finite integer C, you get precisely this limit, right? So this comes from the, basically appeal to some theorems from analysis which allow you to interchange the limits and the integration, okay? So that's the first part. The second part was already discussed in the previous talk by Hudeshna. So I mean, she probably discussed a more general theorem but this is what we need. So if you, so here we're looking at the density function for just the attic filtration, sparse of ideas. So here the field is absolutely general and the heart of the proof is the vector partitions, right? And here also, but here we could say something more. I mean, it's not just any continuous function, it's actually a piecewise polynomial function, right? Now with this setup, we are ready to define this density function for epsilon multiplicity, right? So basically we want to be in the setup where the previous two theorems work which is why we all these hypothesis in the beginning. Now you define this epsilon of X in the following manner. The point is it will be a continuous, almost piecewise continuous function. Point is it's compactly supported. The compact support comes from a theorem of Swanson. So we have that this ideal and this ideal, right? The quotient is a finite length. So this means that eventually all the graded components are same, right? But what Swanson's theorem is telling us is that you can find a linear bound after which the quality happens. So basically says that this is true for all m greater than equal to some c times n. And that c in Swanson's theorem would be the interval of the compact support, zero to c. And of course if you integrate this density function, this is what you would expect, the epsilon multiplicity. So this is the main result of our joint project. But we suspect that we can make this theorem characteristic free, meaning k could be any field, I mean not necessarily characteristic p. And we could deal with a finer subsequence, not just powers of p, but rather, I mean, i power n, all integers n. But we have made some progress in the case of polynomial rings, but we do not have the general picture at hand right now, okay? So I'm gonna skip this. Okay, so what about integral closures and density functions? Now let's be in the setup where we know that the epsilon density function exists. Then we show that if you have containment of two graded ideals, right? So it's that they have the same integral closure, then the density functions are actually same for all values of x. But we do not quite have a converse to this statement. Just have one direction, okay? Now, this brings us to the next part of our talk. So this is an ongoing work with Saipriya Shudesh and Professor Verma. So basically the objective is to study epsilon multiplicity in dimension two. So this is the statement we have. So it sort of lies at the overlap of the two work that we're doing. So again, but notice the difference. We are not in characteristic p, we are in characteristic zero. So r is basically a homogeneous coordinate ring of some projective curve. But here we're assuming Cohen-McAulay, I mean not normal. So it can have some singularities. We show that the density function, I mean you can define it in an analogous way, right? We show it's actually a piecewise polynomial. So this is where the dimension two actually helps, right? So it's a piecewise polynomial. If you integrate, you get the epsilon multiplicity and it's in fact a rational number. So why is it rational? It's linked to the fact that if you take the volume of a line bundle on a projective surface, that's a rational number, right? So Zariski's theorem, it's linked to that. Basically you can represent epsilon multiplicity as volume of some appropriate line bundle minus a term which is an integer. So this is what we have. Now what about an explicit computation? Okay, but even in this very specific setup, I mean we can have non-noetherian saturated Ries Algebra. So this goes back to an old example of Ries. So this was probably one of the first instances where he gave an example of a non-noetherian symbolic Ries Algebra. So you take this, this has come up in previous talks before. So it's basically an elliptic curve, right? So you take a point which is a non-torsion point in the class group, so that's going to have a non-noetherian saturated Ries Algebra. And motivated by this example, we wanted to give a computation. So you sort of look at these pharma curves, right? And then you take these kind of ideals. So basically these are what's called the fact points, the ideals defining fact points, right? And for such things, we have an explicit formula for this epsilon multiplicity. So this is what we have. So the connection is, I mean, I mean these p i's sort of correspond to points, right? So you can look at this divisor, a i p i on Roger Farr, and then this ideal sheaf which is associated to i, this is nothing but o x of minus t. So that's the association and we take off from there. Yeah, so basically we apply some Riemann rock to get there and yeah, there is something there. You get it, and yeah. Okay, so I mean in whatever computations that we have done in the examples of irrational epsilon multiplicity, the density function is always turning out to be a piecewise polynomial. So the points where the polynomial changes, that could be an irrational point, but at least it's a piecewise polynomial in that interval. So so far we do not have any example where the density function fails to be a piecewise polynomial. In dimension two, we have proven that, but again, I mean, there is a slight catch. We do not know what happens in characteristic p. I mean, we do not have an analog of this theorem in characteristic p. We only know it's a piecewise continuous function, but I mean, what's the nature of the continuous function? We don't know. The good place to stop.