 So, thank you very much for this invitation to all the organizers. I can imagine that organizing this conference in this COVID chaos, it was not easy. So thank you very much for all the work. So I'm going to talk about these motives of singularity categories. And, well, sorry, I have a discord, wait, I'm sorry, I lost my, how can I see again my share screen? Sorry, I did something happen here. I don't see my screen. Oh. What's? But at the moment it's shared. Can you? Alt-Tab? Yeah, yeah. Alt-Tab. Try hitting the zoom thing again on your laptop. Yeah. Come up. Well, it doesn't share. Yeah, I don't see the screen. So the screen is sharing now. It's sharing fine. You just have to make the correct window appear again. Or you can stop the sharing and start it again, maybe. Maybe it's going to be easier. Sorry. Sorry. I'm sorry for this. I don't know what is going on. So let me go again on this. Yes, you can see now. Okay. Yeah. I'm sorry. I'm sorry. I won't touch this again. So, yes. So as I was saying, thank you for organizing this school. So I will be talking about this story of motives of singularity categories. So I will need a bit of preparation to talk about this. So let's say more or less the contents of the talk. So first, if you have any questions, please stop me. Second, the more the results of this talk were obtained together with the Anthony Blanc, Bertrand Thouen and Gabriele Bizzozzi. Some time ago. And more recently, there are some progresses in this in this topic. And I want to briefly by the end to mention two. Progresses. Concerning this topic. So the first one is concerning the PhD thesis of Massimo Pipe. We just defended. And another is this project, this program of Thouen Bizzozzi about the block on the formula. So I will very, very briefly mention these two applications of the story outright. Okay. So here's the plan for the talk. So the first part of the talk, I will spend sometimes explaining the story of singularity categories and the, the relation to market factorizations. I will tell a bit of where these things come from. And I'll tell the motivation for what we're going to do next. So in the first part of the talk, I will talk about the motifs of singularity categories, but more general, generally motifs of digit categories. So this comes in the continuation of Gonzalo's Tabuada previous lecture. So there is an overlap between the two talks. I'll try to explain this. Then I will try to explain this. I will try to explain this. I will try to explain this. I will try to explain this. Then I will try to explain the connection between these singularity categories and its vanishing cycles. So that's the main theorem of the stock. And as I said, in the end, if I have some time, I just want to briefly survey some of these results and your processes. Okay. So let's start with this review of the singularity categories and the matrix factorizations. Okay. So let's start with how this one way to enter this story is to start with this theorem of Sech. That allows us to understand, to detect the singularities of a scheme just by looking at its derived category. So essentially what this theorem of Sech says is that a scheme is regular if and only if perfect complexes, the derived category of perfect complexes and derived category of coherent bounded coherent sheaves agree. So I want to say more concretely what this means. So this means the following things. So let's first make a remark. The first remark is that there is always an inclusion going in this way, meaning perfect complexes inside the derived category of bounded coherent sheaves. It's not exactly always. Well, you need some hypothesis on X, eventually co-connective, but in the cases of the stock is always. Then the complicated part is the other inclusion. And this is not true in all generality. So the key lemma that we have to look at is this lemma of Sech that allows us to gives us a criterion to test if a scheme is regular by looking at the shape of resolutions of coherent modules. So the first theorem is this one. It says that a scheme is regular if and only if any module coherent sheave on them, let's say in the heart of the derived category, admits a finite resolution by vector bundles, which is finite. So meaning I start resolving my module by projective models and at some point this stops and this stops at the dimension. So, and this is exactly what allows me to say that the derived category of this derived category, all objects here actually have a resolution that live in here. So the two categories agree. So this is the starting point for this story. But we are not interested in this case. We are interested in something a bit more general. So we want to understand what happens when X is not regular. So in this case, we want to understand the excess of this inclusion. So this is always living inside. That's a perfect complex as if inside the derived category of cooking sheaves. So we are interested in understanding this excess because the theorem tells us exactly that this excess vanishes if and only if the scheme is regular. So most of the time in this talk, I will be working with the scheme over is over overseas or over a field. So I will use regular and smooth. Say the same thing. So. So the first thing we can try to do when this is definition due to Orlov is to isolate this excess. So this excess called the singularity category is exactly the quotient of the co-heating sheaves on X by the perfect complexes on X. So the only thing that remains after this quotient are the things that give obstruction to smoothness. So we take the verdict quotient because here, everywhere here, I mean DG categories. Okay. Throughout the talk, I mean DG categories. Or actually in this case, in particular case, you can actually mean the triangulated category. So the first thing we want to understanding this talk is how to control this excess, this piece of information. So Eisenberg gave us a formula to control this, this excess in a very particular case. And this particular case is the following. We suppose that our X or scheme is actually given by the zero log of some function on an ambient space you. So I have X is just the zeros of some polynomial function on some U and U is smooth. And then in this particular setting, I can say something about the singularities of X. So what can I say? Well, I can say the following thing. So let's look at this formula. I'm sorry, I wrote it already, but let's try to isolate piece by piece what I'm saying here. So let's take again here a coherent module on X. So X is my upper surface is the zeros of this polynomial. Let's take M. So if X was smooth, then all I would have would be this piece of information, a finite resolution by vector bundles. Only this. And this actually leaves is a perfect complex. So the theorem of Eisenberg tells us that, well, in the case of an hyper surface with singularities, what's going to happen is the following, the resolution does not stop. It continues all the way to infinity. But there is a strange phenomenon happening is that it's the fact that after the dimension, this resolution becomes periodic to periodic. So the term here is a term here. The term here is term there. So although it is infinite, we can control it by these two periodicity. So how do you prove this? Well, if you go through the computations, it amounts to just using the Auslander books amount formula in commutative algebra. But this is the idea. So essentially, if I, in the case of a hyper surface, any M coherent on X, I can resolve it by two kinds of data. One is this finite resolution by vector bundles. And another one is this infinite length with two periodic resolution. So let's try to axiomatize a bit the situation here. And this is exactly what Eisenberg did. It introduced a category, which is called a category of matrix factorizations. So U is our ambient scheme. F is our function. X is our hyper surface inside U. So we define the category of such strange objects consisting of a pair of vector bundles Q and P, two maps of vector bundles, okay, vector bundles on U. This is a point of attention. So a pair of vector bundles on U, two maps of vector bundles, such that when I compose them independently of the direction, the map I get is the multiplication by F. So you see that in particular, this is something that this leaves over U. But if I go to X, meaning I kill F, when I kill F, this thing here becomes a two periodic complex because multiplication by F just by definition is killed. So it just gets zero. So in a way, these objects of this category become presentations for such kind of complexes, infinite, but two periodic complexes. So the starting point of this construction I will mention today is this following theorem of Warlow that says the following. What perfect complexes can only be seen inside the derived category of cooking sheaves. This is always for the purpose of this talk. And then there is this excess, meaning the quotient of this by this. And the theorem of Warlow says that I can identify this quotient exactly with this category I just described, matrix factorizations. And how does the procedure goes? Eurystically, it goes exactly like this. If I have a module M, the finite piece of resolution by vector bundles leaves in this part. And at the other, from the other side here, I only get the two periodic piece of information. So in a way, I have an exact sequence of categories where I store these two kinds of information on the extremities. So this is the starting point of this talk, of course, by the theorem of Sir that I mentioned before, if this excess piece of information vanishes, then axis move. And this is the need for knowing. So this category controls all the singularities of the hyper surface. So this is the starting point of the talk. And now I want to briefly mention two computations one can do just to get a feeling of what these matrix factorizations, the kind of information that we can have stored in this category. So the first one is called as neuro-periodicity. So, yes? There's a question. So is there something similar for regular embeddings of higher co-dimensions? I will come back to this by the end of the talk. So this is exactly what I said. So this is the concerns, the progress in the thesis of massing people. So I'll come back to that at the end. Okay. Any more questions? No? No. So let me just, as an example, brief mention two different computations one can do that can give us an idea of what these things are. So the first example, this one, is the computation of matrix factorizations where the ambient scheme is just A2 and the function is x squared plus y squared. I'm working over C here. Sorry, there is another question. Does the category of matrix factorization have a structure of a triangulated category in a natural way? Yes, in a natural way. Yes. Yes. In a way, if you want, either you see it through this equivalence. This is the Breddy equation and the Breddy equation of two triangulated categories as a canonical structure of a triangulated category. And this equivalence is compatible with these equivalence. So, yeah, so back to this first computation, we can show that if I take the ambient space A2 with this function, then I have an equivalence of two periodic digit categories. So this is something I didn't mention here, but as you can guess already by the structure I put here, this is two periodic resolutions, this category is going to be two periodic. I didn't say anything about these two periodic. It is not obvious that this category here also has two periodic structure. Okay. At least from the description I gave here, but in fact it is true, it has a two periodic structure and these two things are compatible. It takes a bit more time to find out the two periodic structure on this category. So I was saying, first of all, we can try to compute this simple example and what we get is called nor periodicity that says that Mf on A2 with this function x square plus y square is just Mf on the point with the function zero. And if you see what Mf on the point with the function zero means, well, we just get exactly two periodic complexes. So complexes with maps going in different directions whose composition is zero. So this is an exercise we can do. And as another feature of this category is this Tom Sebastiani property that says that Mf of a product, A2 is just a product of A1 and A1, is actually the tensor product of Mf of each copy of A1 with the corresponding functions. So this is called the Tom Sebastiani theorem and the generality that I'm going to use here, it has been proved by Prego for matrix factorizations and Prego's thesis. So just two examples to give you an idea of what this kind of information these things contain, although it doesn't say much for now. Okay, so now I introduce you this categories of matrix factorizations. Now you know that they somehow encode the existence of singularities. So I want to give you an idea of where we are going and start to give you an idea of where we are going. I have to mention you this first result by Tobias Dikerov and at the same time by Anatoly Prego. Let's say it's the following thing. Let's suppose I take a regular local ring, R, with an element F, okay, a non-zero divisor. Then I can define this category Mf. So this category Mf is supposed, I expect it to give the singularities of the zero locals of F, meaning a spec, if I write it here carefully, I want to expect the singularities of spec R mod F. These are exactly the zero locals. So this sits inside spec R. And the result of Dikerov is, so in this case there are two results I will mention. So the first result is that this category here, we can show it has a compact generator, an explicit one. We can get very good control of this compact generator, meaning every object is generated by under shifts and called limits by this object. And the second thing you can know, once you compute explicitly this compact generator, is to compute the actual homology of this category, because it becomes just the actual homology of the endomorphism of this compact generator. And the computation of Dikerov shows that the actual homology of this category is Jacobian ring. So maybe I should write here Jacobian ring, where you quotient the ring by the derivatives of it, by the Jacobian of F. This way. And so this is, but the degree of this is, it lives in degree the dimension of R. So this is an important point. Okay. So if you want, this is the first sign that the story of how we are taking the story is the first hint. And the second hint explains. Yes. As a question, yeah. Do you assume R is essentially a finite type of a field? I don't think so. I don't remember. I have to look at it. I don't think so. There's also another question. HH star is graded. Yes. And the Jacobian ring is in general not graded. No, I'm saying that it's concentrated in degree D dimension. So this is not graded. I'm saying that the only piece is going to be this. This is the graded part of this in dimension D. Okay. So the question about essentially a finite type is because the, how do you define the partial derivatives? Ah. Probably need some condition of finite type. I don't have it in mind. I would have to look it up, but probably right. Probably right. If it poses a problem for the derivatives, then you probably need it. And there's another question. Do you apply the product structure of R? The product structure of R. I'm not sure I understand that question. Same. Okay, let's see. I think what this means is going to become more easy to understand in the next slide, I hope. So the way to go to understand this, it's a sequence of results and computations. I'm going to write this way. That compute the periodic signal homology of this category. Okay. So we already saw in the previous talks, this periodic signal homology appearing. So there is a computation. Due to several people. That connects this periodic signal homology of matrix factorizations to a homology of vanishing cycles. And this is going to be the starting point for this talk. I will come back in a few, in a few minutes to explain the vanishing cycles part of the story. I just want to use this result as a motivation for what we're going to do. I hope that what we're going to do makes this picture also a bit clear. So if you have questions, I suggest you ask in the end. And then maybe what I'm going to say next can answer your question. So the first part of this result identifies the periodic signal homology with the twisted the homology, meaning I take the complex, but I twist the differential by this wedge with the differential of F. So this computation was due to F MOV and to Decaroff. And then there is a second half of this computation that identifies the twisted homology precisely with the homology of vanishing cycles. So what we're going to do in this talk is to explain you why these two things, this one and this one, agree, but we will explain you from the motivic point of view. So here's the program for the talk. So here's the idea. We're going to try to conceive an object which you want to call the motif of the category MF. We're going to look at the theory of vanishing cycles from the motivic point of view and we'll try to establish a comparison between the two. So this motivic vanishing cycle side of story, this was developed in Joseph Iub's thesis. And this one, I will explain in a second what this means. But the whole idea as well I will explain is to go, explain how to go from this side to this side. And we will avoid this. Okay, we want to establish a direct comparison. Yes. So there's a question. HP is taken over Laurent power series here. It's taken over Laurent power series with the variable in degree two. There is a two periodic phenomenon that we have to take in consideration here. This is very nice. So this form of writing is a very naive. It's a very informal. Let's say there are some subtleties in how I take this HP and I was trying to put them under the carpet for the purpose of the talk, especially because I'm going to concentrate on this part. And this part I hope is going to be clear. So this is what I said. Let's hope this will clarify something. Okay. And there's a further question by Birken Huse. Is there a six-function formalism in the non-commutative setting? A six-function formalism in the non-commutative setting. You mean for non-commutative motives? Yeah. I guess. Well, the answer is, I don't know. And I don't know means that I thought for a long time about that and I could not prove it. So I tried to do it a long time ago. I could not prove the six-function formalism. Yeah. So the answer is I think it's, I don't know. Maybe someone has thought more about that. Okay. There is a four-function formalism for sure. Okay. The question is the shrieks. Okay. So let's try to explain what this means. So I will start from this side to try to explain what is the motive of the category of singularities or FMF. And this is where there will be some overlap with what Gonçalo explained in the previous lecture. So this is the first part. I'm going to give you a general regression on non-commutative motives. But I'll try to go very quickly. So this story of non-commutative motives started with some ideas on Konsevich. As Justin came from Konsevich. And later on Gonçalo developed a formalism and later with Gonçalo and Wada and Sisinski that developed a formalism for these non-commutative motives. But this formalism is mostly comological. Actually Gonçalo in this lecture he kind of explained this comological because in order to relate it to the stable monotopy theory of schemes he had to use duality. So in this talk we're going to take a more homological approach and by these I mean that something closer in the spirit to the original construction of Moral Vevovtsky and the motifical monotopy theory. So here's the idea of what we'll try to do. So the motifical monotopy theory of Moral Vevovtsky it's a construction that takes a smooth scheme and assigns an object in this category we call SH. Well this SH is a very formal construction and actually I will try to explore what is the universal property of this construction and what we'll try to do is to mimic exactly the same steps of this construction but starting from DG categories instead of starting from smooth schemes. So the first thing I'm going to do here is I will define non-commutative spaces not the DG categories but DG categories with a knob and the only reason I'm going to do this is to have the same front reality. And then I will introduce this construction like a non-commutative version of the Moral Vevovtsky construction and no dualization is going to be needed to compare these two things. So how are we going to get this? Well the first step is just to tell you how this is constructed through an universal property. So what is the universal property of this Moral Vevovtsky construction? So I'm going to write here the following theorem the following theorem is just I mean this is more like a it's a characterization of the of functors out of this category and it says the following it says that every time you have a symmetric monodal functor going from smooth schemes to any category that is stable and presentable and such that f sends these Navigic categories to push out squares f is a one invariant meaning precisely what is written here and such that f sends the motive of p1 pointed at infinity to some tensor invertible object then there is a factorization so I forgot to write this so I wrote the theorem in such a way that this functor is the initial functor that verifies all these list of properties so any other functor sm to some d f sorry to some d that verifies the same list of properties it factors canonically through sh you have a question I think first so I I think you mean d10 sorry the monoidal infinity categories right yes this is a symmetric monodal yes infinity and there is also a question can you explain in what sense what approach is homological and yours homological the remark about these in the next so let me just write this yes symmetric monodal infinity category yes then the result well the first thing it has to do with this up here ok so it corrects I correct the duality in the beginning I correct it in the end so homology theories are usually something that go from smooth schemes up to some abelian category ok and in this case we're not taking homology theory you're taking an homology theory that goes to sh over s so it has to do just with the variance or counter variance of the of the theory and of course one is related to the other in the next slide if I can skip it is this piece of information here it says that the theory we're going to get I'll go back to the next slide the theory we're going to bet is actually dual in PRIL stable to the water's construction so meaning precisely that SH one of this talk is just to spectra from the water's talk to spectra so it exactly dual ok so the only reason why I'm mentioning this one is because we want to have a comparison that does not where the duality does not play a game so I'll come back to this in a second so but first let me go back to what I wanted to say so this gives you a result gives you a characterization of this construction so what we'll try to do is to define a theory of non-committative motives that in a way is the universal thing that satisfies the analogues of these properties for the digit categories so what would this be so the first thing is we define non-committative spaces to be the digit categories of finite type finite type just means that they're obtained by attaching finitely many cells up to homotopy so I have to tell you what is in this average square of digit categories of finite type so I have to, I mean I didn't say what was in this average square of schemes so in this case I will just have to assume that you know what is in this average square of schemes so I define this average square of digit categories to be pullback squares such as this one such that both these maps have the properties you would expect from an open immersion meaning a localization and the kernels of these open immersions have to be isomorphic which is exactly what we expect from the etal property from the niche average square so the interesting, the only thing you have to know about the niche average squares is that geometric niche average squares through the functor perf so perf of a niche average square is niche average in this sense so that's the only thing relevant for this story yes? there's no condition on the functor from u' to u it's a localization it has to be a localization only this one but what about the map from u' to u then? there is no condition exactly like you have a niche average square down you have open immersions and then the map is etal but there is no condition so I'm not asking for any condition because it's going to be automatic because of this ok there are all the questions what else that you can attach in the category of dg categories? you start with dg categories that have only so first of all if you have a simplex you can take the free k module with those simplex if you start with just a simplex complex you start with the free k module generated by those simplex complexes this is going to give you dg categories that have for instance one object zero and one object one k module and these simplexes and you start building from these ones so it's essentially you take the free dg categories generated by simplex and then you start attaching cells and building like that is it ok? more questions? yes I think so ok so essentially for the purpose of this talk we will be taking these non-commutative spaces to be these non-commutative motives to be what you get from dg categories up by forcing nisnevich by forcing a1 invariance and by forcing poincare duality meaning you force sorry you force this motive to be tensor invertible so what you get is something formal but for the purpose of what we want to do it's enough so let me tell you what we want to do so I just said that this construction is actually dual in this sense through the other talk, yes? what about the a1 invariance property how is it expressed? just this I have funtors from dg categories such that t tensor perf of a1 is perf of t I force all funtors to verify this property ok good ok so why we did all this? so the only reason we did all this is just to have this diagram and now it's completely what to say setting is clear we have the the motivic stable or motoveterial schemes we have whatever this formal construction is a stable motivical motoveteria of non-commutative spaces we have a functor, this functor which just comes from the universal property of this gadget and by a joint functor theorem it has a joint which is a symmetric luxuonoidal so all these are infinity functors and these two for free are a symmetric luxuonoidal there's also a last question, does not the last condition come for free? what last condition? can you go upstairs? I guess this this one? yeah it comes almost for free but you still have to invert the circle in this Morale-Vewacki theory you have these two circles, the topological circle and the algebraic circle in this case what happens is that as soon as you invert the topological circle the algebraic circle is invertible and the reason for that is that this category has a semi orthogonal composition where this becomes just perf k so this is Baylinson's description of the semi orthogonal composition on perf of p1 that makes this trick, so yes, you don't have to tensor invert all of p1 it's enough to tensor invert the topological circle so let's continue so the second thing I want to say it's this result so this result was first was proved by Tabuada, I just kind of did the computation in this setting but this is Tabuada's result so the result says that actually if I compute the homes in non-competitive spaces, this is k-theory maps to one, this is k-theory, so this is Schlichting and Waldau's Schlichting's non-connective k-theory yes, so again a precision asked by Marc Levin what do you think is built-in? Barbaricity is built-in, yes, it's built-in it's built-in from the fact well, it's built-in because it's built-in already on k-theory Marc's question? That's sort of backwards it's backwards, yes, I agree but it seems it's in your description of p1 you killed the GM you said? Right, because of this we are talking about the composition okay great, you're just making sure that that's what you were saying yes, yes, yes and there's again another question but that's good, there are so many questions so by unanimous he says when you are taking 3DG categories and simplices in this construction for a simplex delta n, take its image and have a left joint to a coherent nerve C delta n, which is a simplically enriched category then solve the ohms with k and take the associated complex and adult can yes, and then you have to do something else because you have to force modulated invariance so there is an extra piece of data, this would only give them the parallel objects in the theory of DG categories but as I'm working in morita theory I have to make one step further so someone asked to read, I have a weird symbol I think it's dNc inside when you in the theorem or sigma infinity of t and inside it's dNc no, it's a 1, it's a tensor unit it's 1, it's a tensor it's a unit non-commutative motive there's another question what is the additional thing to make it morita invariant it's already morita invariant, in the in that construction well I have to invert morita equivalence so I have to look at the categories of modules, so I have to okay, I have to take an important completion does this answer? I have to take all retracts of all retracts of other important morphisms okay again another question so what are examples of e1 and c contractible objects in sh and c that's a very good question I don't know maybe we can discuss about that later okay so where was I I was about to tell you this result that describes the home spaces as k theory so a consequence of this is that now that we have this machine that goes from motives to this gadget and comes back actually what this result tells you is that you need, so this is one again this is one if I take the unit non-commutative motive and I send it through this joint m what I get is a spectrum, is a motivic ring spectrum representing algebraic k theory and actually the nice thing of this machine is that for free I get the commutative ring structure on this spectrum it's a way to get it just by playing with the machines and another nice thing is that this m is construction because of this result that it sends one to k h it lands inside modules over k h so anything coming from whatever this is lands inside k h modules so let's go to motives of the g categories so here's the construction we're going to do the construction we're going to do is the following start with the digit category which you see as a non-commutative space some definition you send it to s h and then you take this home to one and then you send it again through this m to s h so in the end all this construction what is it doing let's pick a notation I'm sending t to what I call through this process this parenthesis t so what is this explicitly so let's look explicitly what is this doing so if I take any smooth scheme in s h so maps from the smooth scheme to whatever this is is by a junction is maps from perf to this and this is just these maps but I just told you that maps to the unit is KT so this construction is sending a digit category t to an object in s h which you can think of as a pressure of spectra an object in s h that does the following it takes a scheme and it speeds out the KT of t and it's on that scheme it's a construction and this is what we're going to call the motivic realization of a digit category for the purpose of this talk so let's just check some of the properties of this so the first example if you take a smooth scheme over a base s and if you construct this look at the motivic realization of the perfect of the category of perfect compasses on x what you get is just the following object so x lives over s you have the six operations so this is due to the work of Szynski degrees and Joseph I you so you can push forward this object k h that represents algebraic theory here so push it forward and the result is that these two things coincide so in a way this is computing k theory of x of x the global sections of k theory of x so I will not well explain you that how this computation go instead of I will go to this singularity category story so here's a setting of what we want to do I want to start with a scheme over x with a function I want to look at the zero locus of that function and I want to look at the singularity category of that zero locus so we will define this motive of the singularity category to be this this construction I just described so in a way it gives me k theory of whatever I eat whatever smooth scheme I plug in tensor this category so this lands inside k h modules and it gives me definition and this definition will be interested until the end of the stock it seems a complicated definition but we can actually get some computations done so let's look at the so it's just a first remark I will tell you some properties of this construction so the first thing is that if I take so this construction of MF that sends a pair to MF of that pair it's actually lax monoidal so meaning it sends if I have two pairs and I tensed and I multiply them with the functions being the addition of the two functions it goes to the tensor product so this is the Tom Sebastian theorem I just mentioned okay and so in particular the unit so the point with the zero function goes to the unit which is two periodic complexes and because this is lax monoidal it tells me that for any pair the motive of this pair is a module over this category so there is an action of this meaning two periodic complexes on on this motive and this just comes back again for free because of the way we built this machine and so in fact so well this is the action is what I meant by the first piece of the slide so this thing of X zero is actually a two periodic K H module because of this action so can we compute so this is when we get the computation so can we compute this object so I will fix the following setting so the setting is I have X over a base scheme S with some P I will suppose that the genetic fiber so I will assume S is of the following kind either it's something of this kind power series in T or power series in T with coefficients in Fp or Zp or even more generally some excellent trade so in that case with these hypothesis but you can think of one of these three examples I'm going to consider the following setting I have my X I have my base scheme my disk is like one of these is the formal disk I have my punctured disk and I have my center of the disk and then I'm going to consider the function given by my protection and the uniformizer so the uniformizer just a choice of T or P here and I have this set ring I can make a several a bunch of constructions they might seem a bit disconnect for now but you will see how this comes into place so the first construction I'm going to build out of this setting is the comology of the punctured disk so maybe I should respect this way so what do I mean by comology of the punctured disk well I'm just going to compute this object in SH maybe I should write it in SH over sigma so meaning I take the unit object in motifs over eta I push it forward I push it back and I get something I call motivic homology of the punctured disk so the first observation is that this is a commutative algebra object and the second observation is that I can give an explicit formula for this homology of this punctured disk so it's not very different from what you would expect in topology from the comology of the disk you just have a generator in degree 0 and you have a generator in degree 1 but shifted it happens that here the shift has to be with a tight twist so there is this extra piece of information but essentially this is a circle so I'm going to call theta this generator that lives in this state degree so I'm going to call it a algebra to say something about what I want to find is this category of sin x0 so how does this go the first observation is that this comology of the punctured disk acts on the comology of the fiber of the smooth fiber so this guy acts in the comology of this guy by pullback so there is a multiplication by theta of this fiber acts under specialization on the comology of this fiber so there is a map of algebras from one to the other by specialization so I'm sorry if I cannot explain what this means but the upshot is that in the end I can combine the action of the circle with this specialization to produce a map of algebras like this so let's keep in mind that there is an action of the punctured disk on the action or on the comology of the generic fiber and then what we can also do is to repeat exactly the same thing but we work in the level of KU modules so we replace the unit somewhere I put the unit by the KU module the unit KU module when something happens, the same thing holds I have again an action of the punctured disk or this KH version of the comology so I can now tell you the main somehow the main technical result of this talk it's this one is an explicit computation of the motif of MF as the motopy fiber of this action so this might seem a bit dry just to look at this way and this is why I want to briefly mention the main ingredient that goes into this pool I think the main ingredient already gives some intuition so the main ingredient, the main idea of proving these two things are equivalent is actually to show that two certain algebras are equivalent so what are these two algebras we have the gadget I just introduced the comology of the punctured disk on the other side we have this sing of S0, so this is the category but because this is a unit of the entire two periodic categories, actually it is a symmetric model category so in particular when I take the motif I get a commutative algebra object so the first claim is that the comology of the punctured disk of the circle and this motif of sing of S0 are the same algebras so this is the iso of algebras so this is the main piece of ingredient that allows us to prove this formula I described before and once you have this once you have this theorem this one follows just by playing with exact sequences and I will not mention what this is you can check the notes after but that's the main idea and so using this result we can compute this, sorry, compute this motif of MF and for the time left I have, I'm sorry how much time I have left? I have five minutes okay you can take slightly more so I will try to explain you very quickly what is the relation with vanishing cycles to this story so I just told you how to define the motif of a DG category if you don't want to you can forget all the details the technical details, just think it's an object in SH it's a BH module in SH and now I have to explain you how the from the side, the vanishing cycle side of the story I can also produce a KUH module in SH so very briefly I will not have time to go into this as I expected but the story of vanishing cycles essentially tries to study the following, if you have a family of algebraic varieties parameterized by over the line then you can try to understand the complex line then you can try to understand how some cycles they generate once T goes to zero so this is just a heuristic picture of what can be going on cycles like these that collapse as the fiber moves to zero and you also have the monodromy action on these vanishing cycles I thought I would have more time at this time so I'll just skip this part and just tell you directly straight to the point the point is that if I look at X0 my fiber I can construct a sheave called the sheave of nearby cycles and then another sheave called the sheave of vanishing cycles that captures exactly these kind of cycles like these that disappear when I go to a fiber with critical points so the main theorem is that I can do the following construction so let's look at my central fiber and let's take for each point on my central fiber like this one let's take a ball in CN and let's take a nearby fiber so it's a fiber very close to the fiber at zero and let's intersect this fiber with the ball so I want to say something like this so if I take a fiber at T small enough I can take this thing called the Milno fiber and as the point varies along my central fiber I'm going to have either nothing, trivial comology or I'm going to have something like this as I move to the critical point so essentially if I just start taking the comologies of these fibers of these small intersections as X varies this forms a sheave on the central fiber and this is called the sheave of nearby cycles this is the theorem you can find this Milno or you can find this in SGA7 and you can also take the reduced comology and this is kind of called the sheave of vanishing cycles so for the purpose of this talk what is important is that you have an exact sequence where you have the comology of the central fiber, the nearby cycles and the quotient is exactly the vanishing cycles so this is the workshop shot so let's go back to the motivic side of the story so in this talk we're going to have not up there I gave you the a Betty side of the story but we're going to use here the allatic side so we have vanishing cycles the sheave of nearby cycles and this also has motivic versions and in fact what happens well let's focus on the allatic side of the story you have this exact sequence I mentioned before where you have the comology of the central fiber vanishing cycles, sorry nearby cycles and vanishing cycles and all these come with an action of the Galois group so the Galois group in this case is the Galois group of the punctured disk and in particular to make the theorem true, the theorem I want to give we're going to have to look at the particular subgroup of the Galois group and it's called the inertia subgroup I'm going to write it inertia subgroup so we're going to be looking at inertia subgroup we're going to be taking the omotopy fixed points for this inertia subgroup and the way the previous part of the story comes in is this theorem of Deline, let's say it's the following thing if I look at the omotopy variants of this of this QL sigma what you get is the comology of the punctured disk this is just a very refined way of saying something in topology you're perfectly used to which is the following result, think of the perf of the punctured disk as a circle S1 and think of the choice of an algebraic closure of the punctured disk as the choice of the universal cover of S1 meaning just a point and think of sigma SC in this case this side of the story is just telling that it can take fixed points by inertia, but inertia is just automorphisms of universal cover and taking fixed points or derived fixed points is just taking omotopy fixed points for the constant action, the trivial action on C on the other side this is the comology of the disk which is this, so this theorem if you look at what it's saying in the topological analogy it's just saying that the comology of the disk is just taking fixed points of omotopy fixed points for the trivial action on C so this is how this this is how to interpret this result so another piece of ingredients we're going to need for this story the first one is this heladic realization functor so everything we built so far we're going to take the heladic realization and then there is this theorem of Ayub saying that the construction of vanishing cycles well first there is a motivic version and second they coincide under this realization so with all this in place I can tell you finally the comparison theorem so the comparison theorem works like this let's start with the DG category therefore I can land, I can produce this is a functor like called this from all the things we said before I can produce an object in SH which actually lands inside KH modules and now I'm going to take the realization and the realization lands inside the realization of KH and there is a result of Jouel Hayou that uses both DCT and a gamma filtration to compute this realization and it just says that the realization of KH is just the two pre-advised heladic homologies essentially it's the copies of heladic homology in all degrees even degrees with the shift by they twist by N so another way to say it's just the free algebra on the Tate object so now that we have all these I can tell you the main theorem and the main theorem is this, the main theorem says so let me go slowly here the main theorem says the following, it says that we started with this category of singularities we define this gadget called the motivic realization so it lives in SH and now we took its heladic realization so now this is an heladic SH over X0 actually so I can take global sorry I should take global sections to make this work so on the other side what do we have well we have the vanishing cycles SH I take the homology and then I have the two pre-advised and the theorem says that these two things are the same so again I will not give you the proof but I will give you the main isomorphism that makes this work so let me go like this so as I said before the main thing that made the first comparison that made possible the computation of the motive of singularities was the fact that S0 gave us the homology of the punctured disk so this we saw in the previous slide and now I also saw in the previous slide the algebraic version of the circle result that says the homology of the circle is actually the homology the invariance with respect to the reaction on C so the combination of these two isomorphisms is what allows us to show these equivalents so everything follows from these plus some gymnastics with the exact sequence, the exact sequence that defines singularity category and the exact sequence that defines vanishing cycles so I'm sorry if I had to go too fast here I think I went really too fast please tell me if you have questions or if you want me to go back I had a survey of recent results ready but I think I already went my time so I apologize okay so maybe we'll see that for questions so first of all thanks for the talk Marco and so we'll go now to the questions so I have a question I would like to know a little bit more about the Toen Vezosi approach to the block to the block conductor formula yes this is one of the things I had in the survey so the approach very briefly very sketchy the idea is the following so if you are overseas let's take a family overseas and you have the central fiber and the smooth fiber then you can compute this the link number so this is the link number by definition the dimension of the vanishing cycles or the piece of commode of vanishing cycles and you can compute it as the difference of Euler characteristics between the Euler characteristics of the smooth fiber and the Euler characteristics of the central fiber but because of this difference of Euler characteristics if you think the way in vanishing cycles you define by means of this exact sequence they have the homology of the central fiber nearby fiber and the excess is the vanishing cycles so Euler characteristic sends exact sequences to sums so in particular the difference between these two has to be the Euler characteristic of vanishing cycles so this link number is actually the Euler characteristic of the Shiva vanishing cycles so this is the starting point of the story so this works overseas so the overseas this is known so now you want to go to other disks like ZP and in this case you have to correct the formula so this is block conjecture you have to the formula is true but there is a correction to be add which comes from the presentation theory and this is called this extra term it's called the swan conduct okay so the approach that the and the results are trying to develop and already have results for the block conjecture is first of all using the observation that the Euler characteristic of vanishing cycles is the same as the Euler characteristic this is because the symmetric model function and because of the theorem that I just mentioned the Euler characteristic of this and the Euler characteristic of MF F to coincide so the whole idea of the program is to compute explicitly the Euler characteristic of this MF so by MF here I mean the motive and to show that you get for free the left-hand side of the block conjecture formula so the new tool here if you want is the fact that you can compute this approach this number through MF so this is a theorem already when i is acting when you point into it so this is essentially the problem and the way they are tackling it I don't know if this answers your question okay yeah good so there are other questions so the answer is yes I expect them to be to exist I expect so it is yes so people so there are versions of non-commutative motives where there is a problem where essentially you can replace dg categories by other forms of dg categories or rather n dg categories or symmetric en monodal dg categories and essentially you can build versions of non-commutative motives on this and as you go higher on the n you should go higher in the chromatic tower also this is something I I mean this is a work in progress I'd say with the Gabriele Vizzoggi Maru Porta it's been in progress for quite a long time we discussed this last year also with Eldon Elmanto so now we kind of got somehow locked in this but the idea is that there should be yes there should be so if you have more we can discuss this after this talk if you have more questions it would be nice to talk about this sorry so there's another question by by Mark Levine is the so it's in the main theorem is the new power HL an etal sheaf or an object in SH of the central fiber which one so it's exactly this thing you have shown here this new high the right hand side where's that living the thing inside before you take the H star QL where's where's that thing living on the inside that thing yeah new hi minus it's on the stage of the central fiber okay so then then the question is do you get do you have an isomorphism somewhere maybe after making the right hand side into a KH module before taking etal realization so let me so you can make this new hi that this new hi this is just a UBS this is a UBS yes yeah so yes yes so I think yes I think this is possible I just have to be careful about this inertia variance in the I you setting but I think this is true if I think this is true oh yeah okay I can prove the theorem directly in motives right great thanks okay and there is a last question so someone wants to know someone wants to hear you say something about recent results oh the block yes so I mentioned the blog and someone asked the question about what happens when you have multiple functions so I just want to briefly mention this the thesis results of massing pp you can find an archive so the idea is that it extended part of the results I just mentioned but to the case where you have multiple functions and you intersect all their zeros so starting from some ambient scheme x with several functions it produced in this thesis a new definition of matrix factorizations and singularity categories in this context it also extended the results of our love and about his equivalence between singularity categories and matrix factorizations and finally he also computes explicitly this motive of mf with multiple functions using this result of or love and work in a walker that allows you to if you have a scheme with multiple functions this result allows you to pack all the multiple functions in a single not a function a single section of a line bundle in some projective space so essentially it allows you to reduce the problem of multiple functions to a problem of one function and using this it also gives a description explicit description of this motive of matrix factorizations with many many functions so yes so this is I mean I will not give you the details but you can look at his thesis on archive he posted the thesis on archive and it's it's very easy to it's very well written you can you can see the you can follow it over the complementary question does the sequence have to be regular or can I just take the right vanishing look I think it is not to be regular you can take the drive you can take the drive version yes okay so I went too fast at some point so in any case I'm going to post I give you the slides ah good thanks so we'll put that on the YouTube page of your talk also okay so no more question yes so let's thanks again thank you Marco for a nice talk and thank you for the for organizing December school in these conditions it works okay and so we meet tomorrow for at 1pm for the next day okay okay thank you bye