 Okay, welcome to the second session of today. So there's one announcement that those are giving the short talks would have received a link, right? To prepare a poster on A, A zero size. Okay, so I think the deadline is today, so those are giving the short talks, kindly prepare, yeah, A zero poster. Okay, so we'll begin with, we'll just have 15 minutes session and a little time for changing. We want our questions during the short presentations. Okay, so our first speaker for today will be Deepankar Ghosh, and he will talk on non-linearity of regularity of TOR over complete intersections. So first of all, I would like to thank the organizers for giving me the opportunity to speak here. I will talk on non-linearity of Kasunabu Mumford regularity of TOR over complete intersections. So here are the contents. First I'll give the motivation of our research work, and then we'll see that these over graded complete intersection rings, this regularity of X-modules, asymptotically they have nice behavior in the sense that both the functions, these are asymptotically given by a polynomial of degree one. And similar results holds too for TOR when we have this additional condition that i-th TOR, dimension of i-th TOR that is at most one for sufficiently large i. And, but if we remove this condition, the dimension of i-th TOR it is at most one for sufficiently large i, then this behavior it can be far from being linear. And if time permits, then we'll discuss the probes also. So now let me first say a few words about this invariant. So what is Kasunabu Mumford regularity? It is kind of a universal bound for important invariant such as the degree of minimal generators of CZG modules and the maximum non-menacing degree of local homology modules. So if we elaborate these statements, the first one it says that the n-th CZG module of a finely generated graded module over a polynomial ring, it is generated by homogeneous elements of degree at most regularity of m plus n. In fact, one can define regularity of m by taking minimum possible such bound. So what does it mean that it, so regularity of m, it is nothing but the minimum of all integers m such that the n-th CZG module of m, it is generated by homogeneous elements of degree at most m plus n. And this thing holds true for all m. So one can take this thing as definition of Kasunabu Mumford regularity. So in terms of CZG modules or free resolution, this definition was given by Eisenberg and Goto, but originally it was defined in terms of local homology modules. And so we have the second application, it says that regularity of m, it helps us to compute Hilbert polynomial of m. So what is this? So it is basically if we know the regularity of a graded module, then for all n greater than regularity of m, we can have that Hilbert function value of m at n, it coincides with Hilbert polynomial value. And in fact, this is the main motivation to define this invariant Kasunabu Mumford regularity. So here is the definition. It is in terms of local homology modules. So here we can take any standard graded Noetherian ring and then regularity of a finally generated graded module, it can be defined as maximum of end of ith local homology module of m with respect to q plus plus i. So what is end of this ith local homology module of m? So if we consider a finally generated graded module m, so ith local homology module of m that is also graded and end of this thing it is just maximum non-vanishing degree of that graded module. So to get advantage of these two applications, one has often tried to find upper bounds of Kasunabu Mumford regularity in terms of simpler invariance, which are comparatively easy to understand. So there are many results in this theme. For example, it is known due to Karkovsky-Herzogchum and Kodi alum that regularity of powers of ideals, it is asymptotically given by a linear function. So what I mean that it is given by a polynomial of degree one, where the leading coefficient it also can be described as an invariant of that ideal, but the constant term and stabilization index, these are not known. So people are studying various classes of ideals to describe that constant term and stabilization index. So here I will recall a few results which are related to our research work. So first one, it is due to Eisenberg, Muneke and Ulrich. They prove that when our base ring is a polynomial ring over a field, then for two finally generated graded modules m and n, if the first star has dimension at most one, then regularity of ith star minus i, it is bounded above by regularity of m plus regularity of m. In fact, we have a stronger result that under the same condition, we have this maximum of regularity of ith star minus i. It is actually same as regularity of m plus regularity of m. For x model, we have this result that when dimension of m tensor n, it is at most one, then maximum of regularity of ith x plus i, it is regularity of n minus initial degree of m. So all these results are over polynomial ring over a field. So then naturally one can ask that what happens when we consider an arbitrary standard graded ring. So in that case also under this condition, the dimension of ith star, if it is at most one for after certain stage, then we have this bound that regularity of ith star, it is bounded above by i plus regularity of m plus regularity of n. So this is floor function, it is just the integral part of this ratio, i plus d by 2. The following result, it controls regularity of x modules over graded complete intersection ring with respect to both co homological degree and powers of iths. So keeping these results in mind, one may ask that what is the actual behavior of regularity of tau and regularity of x modules. So more precisely we have these questions to study that, so here we are considering all even indices x and odd indices x separately. So if we take L equals to zero, then this is just giving all even indices x and L equal to one, then this is odd indices x and we are expecting that regularity of these x modules, it is given by a linear function like this, where this leading coefficient it is negative integer. And the regularity of tau modules, it is given by this linear function where we are expecting this leading coefficient that should be positive integer. So in a joint work with Mark Chardin and Navit Nemati, we proved that the answer to the first question, it is positive even in more general situation, we will see that later. And the answer to the second question, it is negative in general, we found examples for that. However, if we have this additional condition that dimension of ith tau, it is at most one, then this second question also has a positive answer. So here is the main result, here is the main result. So it says that, so this is the more general setup, we just need that, we just need this vanishing. And this ring, it can be deformation of any standard graded Noetherian algebra. So in particular, if it is polynomial ring over a field, so it is satisfying this condition. So if Q is a polynomial ring over a field, then it is satisfying this condition. So particularly, we have the result when this is graded complete intersection. So we have proved that this regularity of even indices x and odd indices x, these are given by this linear function where this leading coefficient we have described completely. So it is just one of the degree of Fj. And what is this Fj? It is just coming from this regular sequence. And for tau, we need this extra condition, then only we have this behavior asymptotically. So here also this leading coefficient, it is one of this degree of Fj. Then we have some examples. It says that, see in this theorem, this leading coefficient, it actually depends on the base ring, but not on the module. So it is like we have some finally many choices here in this case. But if we remove this condition, then next example we can see that the theorem no longer holds true. So in this case, we are just considering a graded complete intersection ring of co-dimension two. And module is just this one. It is co-carnel of this map. And n is just a mod y comma z. Then we can see that regularity of ith tau, it is, though it is given by linear function, but this leading coefficient, it can be arbitrarily large, depending on the module m. So in this case, this theorem does not hold true. But also we can see that in this case, we have this dimension of ith tau, it is two asymptotically. So we have another example. It says that even this regularity of tau, it can be far from being linear. So in this case, we are considering a polynomial ring in six variables over a field of characteristic two. And this is a complete intersection ring of co-dimension three. And again, this m, it is co-carnel of some map. And n is just a mod x, y, z. So in this case, we can see that initial degree of nth x and regularity of nth x, both are given by minus n. So particularly it says that nth x, it is concentrated on a single homogeneous component. And on the other hand, regularity of nth tau, it is just given by this n plus f of n, where this f of n, this is a step function. So in this range, it is given by this fixed value. And when n equals to two power l minus one, then it is given by this value. So in particular, we can see that this ratio, because we are expecting linear behavior of regularity of all even indices x. But if it is linear, then this ratio limits would exist. But in this case, we can see that this sequence limit does not exist. Even limit infimum, that is two. And limit supremum, it is three. So in this case, even this sequence, it is dense in this interval, two comma three. So we can see that regularity of nth tau, it is not linear in this case. And here we need this characteristic two, because we have verified in Macaulay two, that when this characteristic is zero, then we are getting linear behavior. So surprisingly, one may ask, one may expect that whether we have linear behavior when this characteristic of the base fail, it is zero. Because we have verified other examples also, and there also we have seen that it is linear. So only when this characteristic positive, so characteristic two, then we are getting this thing, non-linear behavior. So how much time I have? I think it's, okay. So then I can state, so I can give the sketch of the proof of our main result. So we just consider this general setup, where this Q, it can be any standard graded Noetherian algebra. And this A, it is deformation of Q by this homogeneous regular sequence upon up to Fc. And we are setting this degree of Fj, it is some Wj. And we just need this condition, Mn finally generated graded module, such that we have this vanishing. As I said that when this is Q is regular or finite dimension, then anyway we have this condition. So in that case what we have done, we have just considered direct sum of ith X module. And it is due to Eisenberg operators, we can give a graded module structure on this direct sum. And also on direct sum of Torr modules. And to get the result on Torr modules, we just have to consider this one. So what we have done, so it is due to Gullixian that we have this result that this direct sum of X modules, it is finally generated, graded module. And using that result, we can have the linearity of regularity of X. And using this result of Gullixian and some spectral sequence argument, what we have proved that these two graded modules are also finally generated. And here, okay, so using this theorem, we can get the result of regularity of Torr. And also we need to use this result. So one result of Bagheri, Sardin and Ha. Okay, so that's all. These are the references. Okay, so let's thank the speaker. Okay.