 Thank you, thank you. It's been fantastic week and I'm really happy I got a chance to be here and listen to all those really inspiring talks that really synthesize all of the wonderful things Maxim has been teaching us throughout the years. So I gotta say an anecdote too, right? I mean everybody is telling one of those so I met Maxime z nami, da je tudi svoj tudi 20 let. In tudi sem tudi, da se počekajte o formacijenih kvantizacijovih K3 servici. In ne bilo vštih, da se ovo pričeš, da se nekaj se nekaj nekošli. In se da se počekajte, da se nekaj nekošli, da se nekaj nekošli, da se nekaj nekošli, da se nekaj nekošli, da se nekaj nekošli, da se nekaj nekošli, zato, da se je več vsema, da je boš češtje, tako to je pravdje, da je vsema, z kateriji, zači je. In nisam, da ne bilim, ali, da sem vsema, in, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, Nekaj. Zanimlj wasn't there, but then you gave a talk in horror. This was the first thing you said. He was really excited about that billboard and he really wanted to talk about that. Zato, da je tudi začusten, da je to. Tako, tako. To je taj pa. Zato, kaj je taj stor, je tudi o zvršenih folijacijov v drevjih geometriji in zvršenih folijacijov v drevjih geometriji. Zato, različno je, je, da je, In zelo izotropika in lagranja infuljace v števnoj zimplektivne geometriju zelo začeličnico s tem zelo potenšeljem. So, še je to pa vse, ki smo otekli, bari je matematik, začeljamo v matematiku, vzene, ki je, da modulno je vse zelo of geometric objects or a vacu or for whatever the physicists are interested in, in physics are always given in some global way as the critical loci of potentials, whereas for us there are some intrinsic objects that have nothing to do with any ambient spaces, any actions or anything like that, and in fact very often the spaces we describe may not even be describable as critical loci of potentials, and so in that sense they're not physical, and so there is this conceptual problem of trying to understand which moduli problems arise as critical loci of potentials, and this story with the fluctuations actually comes as a mechanism for providing an answer to that problem, especially isotropic fluctuations. And so, yeah, so I'll talk about these live spaces of fluctuations and their relationships to the derbut theorems in derived geometry, and the most interesting part of the talk will be the examples and two general constructions for producing these fluctuations and potentials, which are, I think, quite intriguing. So I got to apologize for about 20 minutes, it's going to be kind of dry because I need to set up the language and explain the concepts that we're going to be working with, but once we have that under our belt, we'll start actually seeing some interesting geometry, so bear with me for a while, don't fall asleep. So, OK, so here is the setup. I'm going to talk about fluctuations first in the affine setting, so suppose that we have non-positively graded, commutative differential graded algebra over the complex numbers, the same way it appeared in Bertrand's talk, and we're looking at the derived scheme, which is the spectrum of that. And so the most economic way to define a fluctuation, which is not the way you really think about them and work with them, but the logical reason is best to think about it this way. It's a mixed graded, commutative differential graded algebra, such that its weight zero piece is isomorphic to our algebra functions on the derived scheme, and there are two conditions, the weight one piece is a perfect module, so it's like a vector bundle on our derived scheme, and ignoring the mixed structure, ignoring the second differential, our this algebra is free, freely generated by its weight one piece. And since it's going to be relevant, let me just recall what the mixed graded, commutative differential graded algebra is. So it's a bi complex, it's a bi graded vector space, which is equipped with a graded product, which is associative, which is graded commutative with respect to either of the two gradings. So one of the gradings is written as a superscript, that's the homological grading, and one is written in parenthesis, it's called the weight grading. And it's equipped also with two derivations for the product, one is of homological degree one and weight degree zero, and the other is of homological degree minus one in weight degree one. And those anti commutes, so they give you a bi complex, so it's a bi complex, which has a homological differential and a homological differential, but also has an algebra structure. So that's what the mixed graded, commutative differential graded algebra is, and oh, that's the problem. Okay, thank you. So this is really what the affiliation is, but let me try to explain that and reformulate it in a way that's more familiar and makes it easier to work with. So since as an algebra, differential graded algebra with a homological differential but ignoring the mixed structure, the other differential, it's freely generated by its weight one piece, I can actually describe it in terms of that perfect complex, which is the weight one piece, but before I do that, let me just give you the two standard examples, which are really the model examples of what's going on. One is the trivial affiliation, which is given by the weight zero piece is our algebra that was part of the definition and there is no weight one piece. The weight one piece is zero. And then there is the tautological affiliation, which is given by the derang complex. So you have the cotangent complex, which is the complex of forms on our derived scheme and it has a homological differential because it's a complex and it also has a mixed structure, which is the derang differential. And so the only non-familiar thing here may be the fact that I've chosen a weird grading. I've shifted the form degrees so that all my forms sit in negative degrees and that's so that we have compatibility with cyclic homologes and functions on the whoopspace. But it's not really essential. I mean it's just a convention. Okay, so these are the two standard examples of foliations and they really model what's going on. So you can say what the foliation is in this language in terms of the derang complex. So you can think about it as a triple. L is a perfect module over our derived scheme. We have an anchor map from the cotangent complex to that perfect module and we have a mixed structure on the free algebra generated by that perfect module shifted by one so that the algebra map induced by the anchor map is a map compatible with the mixed structure. So intertwine the derang differential on the derang algebra with the one on the foliation. So why is it called foliation? I'm explaining. It's coming. So there is a dual notion of that which is... So if you dualize this perfect complex to the dual perfect complex then the mixed structure dualizes differential graded Lie algebra structure. You still have the anchor map and again the condition is that it gives morphism of shifted Poisson commutative differential graded algebras. So this is more familiar. This is similar to the classical notion of a foliation when you talk about an integrable distribution in the tangent sheave. The only difference with the classical differential in the symmetric notion is that there is no injectivity condition because we are in the derived world. So any morphism works. So in the previous one, on the previous slide we had it as quotients of the derang algebra. Here we have it as Poisson sub-algebras in the algebra vector fields of our polyvector fields. And they are not really quotients or subs. They are just things to which the vector fields. That's the only difference that's specific to the derived setting. So these are all equivalent data. The data of the mixed CGGA or the one coming from the cotangent complex is actually better than the one coming from the tangent complex when you start doing descent. So the ones that's written in terms of forms it's much better adapted to writing gluing conditions. To was also foliation simple example of le algebra. So the previous slide, the thing that we have is really a le algebra in the derived setting. But le algebra can be understood again in some DGM manifolds, which can be contained. In le algebra give a special example that is a particular case of foliation in this sense, but may not be a foliation in the ordinary sense, depending on the behavior of the anchor map. OK. So this is what happens on affine guys. So this is what happens on affine guys. OK. So this is what happens on affine guys and then you can map foliations to each other by saying that you have maps between these vector bundles that generate the foliations and they intertwine the anchor maps and the mixed structures. So this is what happens on a given affine guy. And now the interesting thing is that if you have a morphism between affine derived schemes which comes from a morphism of CDGAs, you can ask how to base change and you base change the same way you base change le algebraids. So you pull back so there is this notion of a base change of foliation which is not the pullback of the vector bundle that generates the foliation but it's a pull push construction. So you pull it back so you base change by tensuring with algebra prime and then you take the push out under the co differential which is the map on the cotangent complexes and that's the base change of the foliation bundle. And once you define it that way then the mixed structure descents along the right morphism and that gives you a notion of a foliation. The anchor map is the one just thought logically defined by this diagram. So that gives you a notion of a base change of foliations and again let's look at some examples if you take the trivial foliation then when you base change it by a morphism you get the foliation which is tangent along the fibers of the morphism. And I'll talk about it, I haven't defined it I'll define it a couple of slides later but it should be obvious in the ordinary differential geometric setting what that one is just vector fields tangent along the fibers if you base change the tautological foliation you always get the tautological foliation. So there are complexes that are funktorial. And here is an amusing example if you take a point a foliation on the point in this derived setting is just a differential graded le algebra and the base change along the morphism is a split foliation it's the foliation that's this differential graded le algebra extended by linearity on your space plus the vector fields with the standard track. So ok, so hopefully this gives you some sense of what these guys are and the good thing about this base change is that it allows you to descend and you can define now foliations for derived spaces and derived stacks so you can look at the assignment that goes from derived affine schemes to the space, the mapping, the classifying space, the groupoid of foliations on the affine scheme and this is a stack in the tautopology with this base change notion and then you can define a global foliation as an element in the mapping stack if you have any stack derived stack in the tautopology on the stack affine schemes you can take the map from that to foliations and that's a pattern derived stack geometry the stackiness is always a problem because a lot of notions that we want to work with like forms and vector fields and differential operators they have bad descent properties in respect to smooth morphisms and you have to stackify every time you try to globalize and they make perfect sense on affine locals but once you start to globalize you need to do something quite homological descent on hypercovers and that's what this thing actually achieves so there is a I don't want to spend time, there is a way to actually write that in terms of local data on affine derived schemes that probe your stack but I don't want to go into that so I'm going to skip that part Excuse me, is it groupoid or high groupoid? It's groupoid means an infinity groupoid for me there are no trankated things there are no trankated things in this talk everything is infinity so the tangential that I promised suppose you have a morphism of derived stacks a local affine presentation the tangential is given so I didn't give you the description in terms of local data but it is vertical tangent complex the cotangent complex along the fibers of the morphism if you want to write it in terms of an anchor map and mixed structure you have to take the big cotangent complex on the big et al side because the RAM differential doesn't exist on the quasi coherent cotangent complex globally and the anchor map is just the restriction from global forms to vertical forms and so this is the relative cotangent complex you have the restriction map from global forms to vertical forms that of course if you take the quasi coherator acting on the big cotangent complex you get the quasi coherent cotangent complex and this gives you the restriction map and the only thing that doesn't exist on the quasi coherent version is the RAM differential which you have to keep there so that's the tangential foliation this is an important remark even though it's not gonna well it's gonna appear in one of the examples but it's not gonna appear essentially in this talk but it's really important so let me make it you can define the tangential for morphisms which are not morphisms of algebraic derived stocks and are not locally defined presentations and there are many important examples where you really need to do that so they make sense when the target is a formal derived stock and again this is slightly delicate so I'm gonna skip the technical definition locally defined presentation the formal derived stocks do not even need to be locally defined presentation they can be almost locally defined presentation here in kukumologi but maybe unbounded tor amplitude and and they can also be informal derived stocks almost a finite presentation locally and this actually have been studied extensively by Nick Rosenblum and there are many important examples coming from representation theory which are exactly of that type so I said in this case will be kukumologi in unbounded tor amplitude and let me just give you some examples of these guys to convince you that they are important geometrically they arise really naturally one example is a formal stock that Bertrand discuss last in his lecture is the drum stock which will reappear very soon in play a very prominent role in our discussion so it's a formal derived stock of locally defined presentation but it's not algebraic the derived Hilbert and Quartz hems of Chukan, Fontanine and Kapranov they are almost locally defined presentation and run spaces are informal and almost locally defined presentation so those are important and you would like to work with them and they do appear when you start dealing with geometric settings or the place ok, let me just since I'm talking about tangential fuljations let me just mention two examples that the trivial and the tautological fuljations are tangential the trivial fuljation is just a tangential and the structural morphism to a point gives you the tautological fuljation so those are particular examples of tangential fuljations and in fact with the exception of one construction in this stock all the fuljations will be dealing with will be tangential ok, now the one thing that we really want to extract from this formalism is leaf spaces so we want to be able to take a derived guy and quotient it by a fuljation and as in usual geometry the output of that process is not going to be a space but it's going to be or a stock and it's going to be a formal stock and so you have to actually depending on the generality of formal geometry you want to allow yourself you have to work a little bit for that so here is a definition which is actually a theorem so this is what Bertrand mentioned so these quotients are defined by universal property and they exist so if you have a derived stock luckily a finite presentation and a fuljation on the derived stock in the strict sense so given by a perfect guy then there exists a formal derived stock which is the leaf space of that fuljation and a quotient map and it's uniquely characterized by universal property universal property is that if you map x to any other formal derived stock so that the tangential fuljation to the map to the test stock maps to the given fuljation then that map on fuljations induces a unique morphism on the from the quotient to the test map and this is unique so that's the universal property and you can construct these leaf spaces which are formal stocks and we'll be using them very soon so again the reason maybe just let me just say a little bit about quotients they behave exactly as you expect them to behave first of all these quotients are really sensitive to neopotents or to derived structures I mean if you reduce everything so if you truncate to something which is non derived and pass to reductions the quotient by fuljation gives you an isomorphism as a higher under derived stock so the quotient by the trivial fuljation doesn't do anything, just gives you x the quotient by the tautological fuljation gives you exactly the deramstak which Bertrand defined it you attach to a commutative differential graded algebra the points of x or the reduction of that is first you kill all the derived pieces and then you reduce by neopotents and this is carvers definition of the deramstak and when you work out this universal property this is exactly what you get for the quotient by the tautological fuljation if you take the cotangent complex for the leaf space it is exactly what you expect it to be it's the cone of the map from the cotangent complexes from x to the fuljation bundle you can in fact ignore the neopotents in x you need to keep the derived structure but you can ignore the neopotents so you can kill the derived structure kill the neopotents take the reduced underived space sitting inside as a closed sub-scheme in your derived stock you can base change the fuljation now this is not going to be fuljation in the strong size it's not going to be perfect but it will be in locally almost a finite presentation and then the map on quotient is an isomorphism so we can't prove it in this generality we can only prove it for the strong fuljation for the perfect ones so this thing exists in this particular case in general we don't know how to prove it for general fuljations which are almost locally finite presentations ok now if you have a reduced algebra finite type nothing derived non non-dg and you have a fuljation derived fuljation on it then the algebra functions on that leaf space is exactly what you expected it is the total complex the negative cyclic complex of the completed symmetric algebra fuljation shifted by minus 1 so you just you have this mixed complex and you take it as total complex so again if you understand formal spectra properly mrtran was discussing in his lecture in the affine case the quotient by the fuljation is just the formal spectrum of this total complex so here is probably the most useful example if you have a morphism of derived stacks locally finite presentation and you look at the tangential fuljation then the quotient is the relative formal completion along the fibers of f so what you have to do is the derived stack whose value on committive differential graded algebra is the fiber product of the points of it's the fiber product of the points of the target over a with the reduced points of x over the reduced points of the target so if the target was a point roughly the deram stack so this is like the family of deram stacks along the fibers and there is a reason why we call it a formal completion because if the morphism is smooth morphism of schemes then you really get the relative deram stack along the fibers but if it's a closed immersion you get the formal neighborhood you get the functions on the formal neighborhood of x inside y formal completion of y along x so there is a slight caveat in this isomorphism but it's technical I'm not gonna discuss it right now but they are isomorphic if you interpret the isomorphic what isomorphic means correctly I mean there are extra structures which not quite matched but as algebras they are isomorphic ok so we have these very nice properties of quotients and we have this relative completion which is a quotient by tangential foliation and now I wanna bring the symplectic geometry into this game and see what can we say if we have not just foliations but we are Lagrangian or isotropic so if you have say a fine derived scheme and an n shifted to form which doesn't have to be symplectic just closed n shifted to form so in in algebraic language it's just a morphism from this mixed graded commutative differential graded algebra generated by a piece in one homogical degree to minus n and one degree to to the derame algebra and so that's what a n shifted closed to form is and an isotropic structure on such foliation is just a homotopy between the induced morphism to the foliation from the symplectic form with the anchor map and the zero map so that choice of a homotopy is a structure you need to keep track of it but that's what it means for the foliation to be isotropic and you can also define a Lagrangian foliation I actually wrote down here explicitly how you write isotropic structures in this derived sense you can write close to forms as infinite sequence in terms sitting in terms of the cotangent complex with the corresponding degree which are annihilated, which are co-cycles with respect to the total differential in the double complex and then h is just a co-chain that bounds that form in the double complex corresponding to the foliation so in the affine case it's something very explicit I mean it has a large amount of homological data but it's something completely explicit it becomes complicated when you start gluing things and then you can play that game with gluing again you can define a morphism of isotropic foliations, you can define a base change of isotropic foliations this is where you need the form not to be not degenerate because when you start pulling back non-degenerate forms can become degenerate but we don't care I mean the notion of being isotropic doesn't care about non-degeneracy so you can define a base change of isotropic foliations and again the same trick works the assignment that sends shifted symplectic or affine scheme with a shifted close to form to the group point of isotropic foliations is a stack in data topology and you can define a global isotropic foliation as an element in the map stack then you can put a non-degeneracy condition on the isotropic foliation once you have one for non-degenerate to form then the non-degeneracy condition is a purely algebraic condition is just the condition that says that the foliation is isomorphic as a perfect complex to the comormal bundle of the map so you just write down the composition with the symplectic form to the quotient by the foliation and you want this to be a quasi-isomorphism so it's exactly, I mean it's a complicated formula but it just the most nifty thing you can do it's the standard notion in think of granshin so the reason we care about these guys is as I said because so this is our granshin when it's a quasi-isomorphism and the reason we care about them is because we want to talk about potentials and this is related to the derbutiura so in classical symplectic geometry the local structure the symplectic manifold is given by the derbutiura which says that say locally in the c-infinity setting or formally in the algebraic setting the symplectic manifold is isomorphic to the standard symplectic structure on a cotangent bundle and you would like to have something like that in this derived symplectic geometry the problem is that you have more cotangent bundles more than one cotangent bundle more potential local models than just a cotangent bundle so the most main thing you can do is what Bertrand described in his talk is the shifted cotangent bundle so you just take the tangent bundle shifted by minus n, take the symmetric algebra and take the spectrum of that that carries a natural exact shifted symplectic form so that's a perfectly nice linear shifted symplectic manifold differential graded manifold or a stack the problem is that not every shifted symplectic derived stack is isomorphic locally to this so there are deformations of these guys which are not locally isomorphic to them so the derived critical locus of a function is equipped with n-shifted symplectic form and in general is not locally isomorphic to a shifted cotangent bundle so if you have a function mapping you to the n-post first shifted line so you can think about this as a degree n-post kuhomologi class with coefficients in the structure shift of n this it has a well defined critical locus which is equipped with a symplectic structure which is shifted by n so there is a dropping shift by one because this is a Lagrangian intersection picture so if you have a shifted function n-post one shifted function is derived critical locus is equipped with an n-shifted symplectic form if that function is constant then you get a shifted cotangent bundle or locally constant then you get a shifted cotangent bundle but if it's not constant then the derived critical locus is not necessarily isomorphic to a shifted cotangent bundle and that's very easy to see in examples so you have a choice here whether you think about these guys as your local models or these guys and since I said these are more general they don't specialize to this even locally it's better to use those as your local models and so if you have a hope for a derbut jurem you want to say that a shifted symplectic manifold is locally isomorphic to derive critical locus of a shifted function and so that's what we want to do and this leads to this remarkable derbut jurem that Dominic Joyce and his group proved if you have a derived Dylene-Mannford stack with an n-shifted symplectic structure where the shift is non-positive then at all locally it's isomorphic to the derived critical locus of a shifted function so this is Dominic Joyce and his postdocs graduate students and in fact in all schemes it's a risky locally so it's a very strong statement and now the problem we were trying to solve where these relations were needed was to try to understand what's the ambiguity in this the choice of these local structures and when can we hope for them to be global so this tells you that locally you always have a potential but maybe sometimes you have globally a potential so when can you hope for the potentials to exist globally and so this is what the fluctuations help you answer so the point is that potentials always exist if we have anisotropic foliation so here is the theorem suppose that you have a derived stack locally finite presentation and an n-shifted symplectic form on it and assume it's exact of course if you want to have a potential it better be exact the ones that are shifted symplectic forms on derived critical loci are always exact and suppose that it's equipped with an foliation foliation with anisotropic structure H then the claim is that on the leaf space of that foliation there is a shifted function m plus 1 shifted function and a map from x to the leaf space factors through the derived critical locus of that function and in fact not only factors but it's symplectic it pulls back the shifted symplectic form on the derived critical locus to the original shifted symplectic form and that's completely global if the foliation exists globally then on the leaf space you get this function and you can identify the whole space as a derived critical locus if in addition the the foliation is Lagrangian then that map to the derived critical locus is also et al it's kind of it's subsuscable super meaningful in that is it's it's a very natural statement and it's actually not very hard to prove once the technologies I mean where is the function coming from it's coming from the fact that the form omega is zero on the leafs of the foliation for two reasons it's zero because it's exact and it's zero because the foliation is isotropic so you can find a homotopy between these two zeros and that's the function so it's just as simple as that and it really connects with the result of Joyce and Company where there was no exactness condition but there was the condition that we were local with non-positive shift and the point is that if you work out through the Hoch theory it turns out that if you have a close p-form which is n-shifted with a negative shift it's always exact so that's something very peculiar for forms with negative shifts and this is not a tautology, I mean it's a statement with content because the corresponding derangue homology is not zero so this is really a property of this mixed complex of forms ok so let me show you examples and I want to really show you two constructions of these global potentials that come from these foliations which are cool so of course the basic example is a derived critical locus if you have a smooth scheme and a regular function so this is just an underived ordinary scheme then you can take the derived scheme x to be the critical derived critical locus of W it has a minus one shifted symplectic structure now this better be coming from a potential we know it's coming from potential but in fact this potential is coming from a foliation what's the foliation you have the map from this derived scheme to Z and you can look at the tangential foliation for this map the map, the inclusion map from the derived scheme to Z and it has a Lagrangian structure which you can write explicitly in terms of the function W and the quotient is exactly the formal completion of Z along the underived critical locus and the potential is just the restriction of the function to that formal completion and the map is just the map from the derived critical locus to the formal completion and the pullback of the symplectic forms is the fact that the pullback of the function is the pullback of the function so this is a tautology but it should be there as a check that what we are doing is correct and you can do this with shifts so this was for an unshifted function you can do it for a shifted function and again you get the same foliation gives you a shifted function and again you get the potential is a pullback formal completion another example if you take a cotangent bundle so you take a smooth manifold it's cotangent bundle with the standard symplectic structure then you have a projection from the cotangent bundle to M with Lagrangian fibers the tangential foliation is Lagrangian the quotient by the tangential foliation is of course as a underived space as the space underlying a stack is just the total space of the cotangent bundle but the fibers are completed in the deram sense so it's the deram stacks of the fibers the function is just zero but viewed as a one shifted function and the derived critical locus is the shifting of the one shifted cotangent bundle which is just the cotangent bundle so again in this case you get exactly what you expect more interestingly you can do twisted cotangent bundles so suppose that you have a smooth manifold and you have a symplectic twisted cotangent bundle so those are classified by elements in the hypercuchomology of the stupidly deram complex in degree one so they are affine bundles over the cotangent bundle equipped with a symplectic form corresponding to that class the tangential foliation mapping to the base is still a grandian the problem is that the symplectic form is not necessarily exact anymore twisted cotangent bundles are not always exact depending on the twisting if the symplectic form happens to be exact then you should be able to find a twisted function one shifted function on the deram on the deram completion of the fibers and it's actually not hard to identify if you see where this function should live it should live in the degree one homology of m with coefficients in the structure shift and it is just the function that bounds the class 8 so it turns out that 8 will give you an exact twisted cotangent bundle exactly when it's in the image of the deram differential with respect to the hotch filtration and whatever bounds it is exactly the function so and then you get identification of this with the derive critical locus and let me just point out that this is of course interesting on m's which are not compact because this function f it's not gonna be unique, it's gonna be unique only up to a function which is locally constant only up to something that's coming from h1 with coefficients in c and of course if we had something locally constant then there is no shift because dff is 0 so what you really get is a shifted cotangent bundle again or the original shifted twisted, unshifted twisted cotangent bundle so you cannot have m to be something for which the hotch theorem holds if you wanna get interesting examples but for affine m's you'll get interesting examples ok, one purely derive example and then I'll give you the how am I doing with time, 10 minutes ok, great so purely derive example which is neat so suppose that you have a symplectic manifold which is a point zero symplectic structure but you view it as a one shifted symplectic structure the zero symplectic structure you can put in any shift so on the point you can talk about shifted symplectic structures of any shift so we can actually describe Lagrangian foliations if we have a finite dimensional vector space and you look at the map from the point to the classifying stack of that space so the stack classifying affine space is modeled on that vector space differential foliation is given just by the dual space with the zero anchor map and the zero mixed structure anisotropic structure is a self-homotopy in this negative cyclic complex that has a zero differential so it's actually an element in the second wedge power of the dual space and it's Lagrangian if and only if this element is non degenerate so if and only if V with this two form is a symplectic vector space so Lagrangian foliation for this so a situation in which this map from the point to the classifying stack of a vector space is a Lagrangian foliation it's a it's just a choice of a symplectic structure on the vector space now the quotient by this Lagrangian foliation is just the classifying stack of the formal group which is the completion of V at zero and by the theorem there should be a two-shifted function from that formal group from that classifying stack to a1 two-shifted function so that the point with the zero-shifted symplectic structure is the derived critical locus and so in order to find that function you need to compute the second cohomology of bV hat c in o and you can actually do that because we have the Brin calculation which in characteristic zero was redone by Carlos and it says that if you have a finite dimensional vector space and you look at the Albert McLean stack kVn then the height cohomology of kVn with coefficient c in o is either an exterior algebra or a symmetric algebra depending on the parity event or it's zero if the degree over the over the homotopical degree is not an integer so in particular in this case when we are doing something two-shifted the answer is exactly wedge to a Vdual and the function is just h so so it's it's a complete circle okay so I have another example about integrable systems but I'm going to skip that even though it's interesting I want to show you two constructions of isotropic foliation and one of our grandchef foliation which are interesting and give you new new realizations of modular spaces of critical loss of potentials so suppose that we have oriented c infinity manifold of odd dimension choose a more smell function so it's a self indexed more function where the critical values are the indices and choose a regular value between the two middle critical values so now if you take the half of the manifold that corresponds to everything smaller than that regular value you get a sub manifold with boundary which when you include it in M you get a homotop equivalence on the k-dimensional skeleton in fact any such guy will do this is a way to construct it now if you have a complex reductive group and you look at the derived stack of local systems m so these are maps from m to bg as a derived module i then this guy carries a 2-d shifted symplectic structure as Bertrand explained by this ak-zik formalism and if k is greater than or equal to 1 this 2-d shifted symplectic structure is negatively shifted so by this theorem that I mentioned about the how it's exact so you have the derived stack of local systems on an all-dimensional compact-oriented manifold as soon as the dimension is bigger than 1 it is an exact shifted symplectic manifold or shifted symplectic stack and the claim is that tangential variation for the restriction map from local systems to local systems on this half of it given by the Morse function has a nitroisotropic structure depends only on the orientation data in the shifted symplectic form and so you can find a shifted function 2-2k shifted function on the quotient by that foliation which is the relative derived stack and the module of local systems with its shifted symplectic form is the critical locus of that so for three manifolds this just gives you an incarnation of the transimons function if you choose a Higgs splitting and this gives you a generalization of the transimons function to higher dimensions but unfortunately the off-shell space has no natural interpretation of a space of connections it's really very derived so that's one construction the other has to do with the non-o-bilian hotch theory so it's very similar maybe I didn't say that so you can prove that this structure is isotropic unfortunately we cannot prove its Lagrangian I believe it's Lagrangian so that this map is really a towel and in fact when any simply connected we can prove its Lagrangian when in any case under simple logical conditions we can prove its Lagrangian but we cannot prove it in general and I think it's just stupidity but it's certainly isotropic so it is really a pullback of a derived critical locus symplectic form of a derived critical locus is there a shift of vanishing set both you mean for shifted functions? you can define shifts of vanishing cycles for shifted functions but because these are formal stacks you really have to be very careful with your you know the boundaries properties of your complexes so we can formally define it I'm not sure that it actually exists because the cons may not be well defined in complex so we haven't tried really you can try to mimic the standard definition but maybe if you don't have bounded below bounded above complexes maybe you're not going to be able to do it so okay but let me do this example so suppose that we have a smooth projective variety of dimension D and a modular stack of rank and local systems again as a derived stack so this is now equipped with 2-2D shifted symplectic structure and as Bertrand pointed out this stack X it has a tangent complex and the tangent complex has a natural hot filtration from non-nuclear hot theory coming from the modular space of lambda connections so if you look at the relative modular of lambda connections in fact the shifted symplectic structure exists relatively in the twister space and this the C star action on the modular of lambda connection gives you the hot filtration on TX and the map given by the construction with the symplectic form actually is a filtered quasi isomorphism for the hot filtration and as a consequence if you look at the middle degree the middle step of the hot filtration middle then the degree step of the hot filtration when D is odd dimensional again this middle step of the hot filtration has a canonical Lagrange infoliation structure for the shifted symplectic form it also works on the Higgs module I but here it's more interesting because on the Higgs module I it's everything is split and so you can you can take this fuliation quotient by it and get the potential so you can write the module I of local systems on an odd dimensional projective manifold as critical locals of a potential and in fact this fuliation given by the middle step of the hot filtration is again a tangential fuliation so you have the local systems the derived stack of local systems and you can look at the derived stack of modules so this is like the DG modules over the full derang complex of M which as all modules are of rank N but you can look at DG modules over the truncated derang complex in the middle step of the hot filtration and you have a restriction map from those and the middle step of the hot filtration is just a tangential fuliation for that so if K0 if we are doing one curve then this is just the ordinary map from the module I of local systems to the stack of bundles but the moment we go in dimension bigger than one this map actually on truncations so this is just the truncation but this map on truncations is an isomorphism so you don't get a new space but you get a new derived structure and the full truncated stack is recovered as a critical locus of a shifted function on the relative derang stack and you can actually identify that function explicitly there in terms of the symplectic structure and I think I'll stop here in this case thank you very much