 Explain what I'm really thinking about while giving this course, and about origins of this bridge and stability and all stuff. This story is the following. We have many triangulated categories from algebraic geometry, gives many triangulated categories, and particularly if we have algebraic variety, x over k, that's a smooth, proper variety. This is a category bounded complexes of coherent shifts. And it has some nice intrinsic categorical property. And to a great surprise, these things can have strange symmetries, non-geometric symmetries. I'll just give you two examples. The first example is by Mukai long time ago. He discovered Fourier Mukai transform. What is about? If x, suppose x is a billion variety of any dimension, and you have dual billion variety, which is a variety of line bundles of degree 0A. If it's Jacobian of curve, it's the same, but in general it's different variety. And then Mukai discovered was called Fourier Mukai equivalence. If you get category of complex of shifts in one variety, it's the same as in dual variety. And it's given by Kernel, namely, what is this, what is the function, which gives this equivalence. You have A cross check, it projects to A to a check. And also on A cross a check, you get universal bundle. Because this thing is crystallized line bundles, so you get universal bundle, and the function is E goes to direct image on the second projection. You take pullback, multiply by this line bundle P, and project high images. And it turns out to be equivalence. But it's, in particular, suppose if A is self-dual, like Jacobian of curve, then you get equivalence of things with itself. But one can take, for example, A is elliptic curve, to raise to some power n. And of course it's self-dual. And a variety is also self-dual. And then you have plenty of functors you can map by J and Z, removing, acting on, using additional law. Then you can multiply by line bundles. Then you get free, okay, transform, shifts, because it's a great category, you can shift. And then what you see at the end of the day, is that cover of group SP2 and Z, X on the indirect. What does this mean, this cover? If you consider group SP2 and R, it has universal cover. And you can see the premature of matrices, its integer coefficients. Okay, so you get action of nice group by symmetry of a category, which is not action on a variety at all, itself at all. So it's some kind of representation in algebraic geometry. Yeah, that's one example. And the second example, I think it's actually me. Very long time ago, around 1994, something like this. You have started with a Quintic 3-fold in P4. It depends on main parameters. And what I can do with coherence shifts, you can multiply by line bundle. You have multiplication by line degree 1 bundle. And also there is something called reflection function, associated with structure trivial bundle O. And what does it mean? It's given by kernel, X cross X. It's very simple. In degree 0 put O diagonal, which gives you identity functor, and you're modified by aiding of X cross X. And obvious restriction map is differential. Okay, this complex of shifts. And it turns out to be invertible. One can make little calculations. Okay. So there are two functors and they act on complex of shifts. So here they act on chain classes. These two functor act on chain classes. And chain classes belong to H0 of your variety, plus H2 plus H4 plus H6. It's three-dimensional variety. And these are just Q2. You can see the commod here with rational coefficients. It's Q2 power 4. And tensoring by O1 gives a following matrix in obvious basis. You just multiply by exposed chain character, not plus its character, of length 1. In order to get 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6. It's the beginning of series for exponent. It's for chain character. You get this matrix. And then you get, for the second functor, let's call it matrix D0. And for reflection of O, you get matrix D1, which is like this. It is this guy. And what is this number? 25, 26. It's actually one can call it using Krim and Roch, I can give you easy derivation. You can see the line bundle of degree M, where N is greater than 1. And you can see the sections, like polynomials of degree M, which vanish on this Quintic default. And to calculate dimension, you get dimension of all polynomials, minus dimension of polynomials, divisible by the equation of the Quintic, which will be here M minus 1 over 4, one of the coefficients. And then to calculate it, you get 25 over 6 M cubed plus 5 M cubed over 3 factorial, also 6. So these numbers appear here. OK. So you get these two matrices. And then you take product, raise to fifth power. And what do you get? You get identity matrix. Yeah, so I didn't make calculation by hand. I have to admit it using pari. Yeah. So that was a great surprise. And what do you see? These three matrices generate monodromia of some false-order differential equation over Cp1 minus 3 points. And the equation is well-known. It's mirror symmetry. One of the solutions is this hyper-dramatic series and it's branches. So you get exactly this monodromia. Yeah, so that's something remarkable goes on. So this Cp1 minus 3 point is model space of mirror dual Calabio 3-fold, which is pretty complicated here, but we have just one parameter. And it's fundamental group. You see it acts on middle commode. It will be exactly that how the symmetry acts on the function, on categories. So that's the story. And it was kind of a way to introduce what is mirror symmetry. There are many ways, many approaches, but one can see that it's something which produces tons of examples like this. You get P1 of model space of Calabio varieties. It's preval elements X by automorphisms of derived category of any Calabio variety from your family. And in fact it's something very, very striking. Here you can see the model space of complex numbers and you can see the fundamental group, which is discrete group. Usually on geometric geometry they use fundamental groups only to make profanity completion. These things, as an abstract group, have no meaning in algebraic geometry. But now these groups have some meaning. And here it's a Calabio variety from family, but could be defined as some finite field. So it's kind of strange interaction of complex numbers and arbitrary fields. In this story, I said Calabio. So Calabio means that canonical class is trivial, so they have a holomorphic volume element. One can ask why Calabio also means canonical class is trivial. It's the same as first-chain class of a cotangent bundle. It's the element of the curve group. The reason is that a theorem by Bondel and Kopranoff late 80s, if X-smooth projective, but not Calabio. So there are kind of two basic cases. The canonical class is positive, so it's called a variety of general type. So it means that the number of equations is bigger than the number of variables. Or it's negative, it's called funnel. And differential geometry could respond to get a metric with a rich curvature minus 1 plus 1, and Calabio's rich curvature is 0. But if it's not Calabio, then this category knows about a variety itself and the automorphism of this category is equal exactly shifts, automorphism of a variety itself and tensoring by line bundles. So everything is geometric. And for Calabio, you get a huge amount of extra symmetries. So the picture is that automorphism, its group is much bigger than one can see naively. And I want to say that it's definitely much bigger usually than the things which are proposed by more symmetries, this fundamental group, this automorphism group of Calabio category. Calabio category, it's just property, which formalizes this variety as Calabio, which may be n-Calabio category and its dimension. So for any object in F, if consider home from F to E and take dual, it's the same as home from E to F shifted by n. We get functionalized morphisms, it's called Calabio. And automorphism group Calabio, it's usually very big, much bigger than this fundamental group and this one way to see is the following. Suppose you have object in this category, which you don't by C, suppose it's spherical object. And spherical, yeah, it looks like why it's related to geometry, we'll see it later, but spherical means definition that if consider dimension of x-group of object to itself is equal to 1 in degrees 0 and n. By duality, it's the same in degree i and n-i, so it should have identity map, something degree 0, and it's kind of minimal possible thing, and zero otherwise. And you see, it's the same dimension of homology of sphere. Yeah, so that's why it's people called spherical. And then it can make reflection function, namely R for any spherical object, Rf will be the following. Object E goes to cone of the falling guy, you have f times r-home fe, it's a finite dimensional complex, multiply, get universal map and take cone. And the theorem is that it's invertible exactly for Calabio category spherical object. So you get plenty of invertible function for spherical object. And spherical objects are, at least if dimension is 3, should be plenty of them, typical object, if you take vector bundle on Calabio 3-fold, should be rigid, should be spherical object, should have only x0 and x3 and no x1, x2, typical bundle. Or one can make a curve, sitting in Calabio 3-fold and such that normal bundle is, it's something vector bundle of degree minus 2 and it's, again, typically should be o-1 plus o-1. And then it's again spherical object. And we know from Gromov-Veteran events there are really zealons of rational curves on Calabio 3-fold and we got a lot of commuting functions. So we get huge, huge group. So how stability enters in the game. I just want to say this is kind of highly hypothetical picture. We have this huge group, automorphism group of and Calabio category C, which could be equivalent to one of this DB Koch or something like this. And then this guy acts on some abstract stuff, space of stability conditions, which is, as I explained, vibration theorem, it's complex manifold, locally is a morph to cone in some finite dimensional space. So it's kind of complex manifold, but very stupid structure, very close to model space of quadratic differentials and have many common properties like action of jail to R and so on. So we get this space of stability. It's manifold, but nobody says that this guy is connected. And what is expected, that by one of model space of mirror, space of mirror Calabios should be equal to stabilizer of a connected component under the section. And that's rough picture. So this space of connected components should be something like quotient of this huge group by smaller this fundamental group. So that's actually my goal kind of introduced to framework to understand why fundamental group acts on a category and acts on category and one can look on concrete examples and see that it goes on in example by example, but what is general reasons? At the moment I have mathematical explanation only for categories defined over non-archimedian fields and kind of close to very degenerate. Now, so you see that by this picture that should act by symmetries of categories for the whole family or yeah, it seems over FQ and so on and family is even considered complex points and some kind of variety with casps. In what we kind of control only neighborhood on the casps we can see that for categories with parameters belonging to this formal neighborhoods you get the section in a way it extends everywhere it's complete mystery. So there's really no explanation but there are physical reasons namely what's called two-dimensional and two-two super conformal fields theory gives kind of some reason why should these things should act by the whole family not only near the casps. Now that was actually proposed by Mike Douglas The story was pretty convoluted about when we proposed homological mirror symmetry so the categories and Mike Douglas started to think why we should have and this response something called d-brains and why it's a category. So there was some he thought about this and then in this term category it gets z-grading on the homes and it turns out to be pretty, pretty tricky story when he analyzes z-grading and then starts to analyze what happens with d-brains when they change parameters when they kind of become unstable and so on and then Tom Bridgen formalized a rule which Douglas invented to absorb to get this definition of abstract stability. So the origin of the story is really strong string theory and what is rough picture? So there are such beasts they have some mathematical definition what people believe but it's impossible to work with so it's you leave it to physicists one can give definition but you cannot prove any theorem as this definition one can maybe check on computer but nothing can be proved So the model space of this n equal to 2 whatever it is it's kind of kind of comfortable disjoint union of some connected components and connected components typically should be complex manifolds of the form which are product of two smaller complex manifolds algebraic varieties a posteriori it's not does follow from physics from physics you say we have only complex geometry a posteriori both things are algebraic varieties defined over integers maybe sometimes roots of one but it doesn't follow from definition and these two guys each of them is a model space of some kind of varieties of given dimension n and n is related to central charge of this conform field theory ok so we get this things we get product space and over this you get two families of triangulated categories called CA and CB depending on point on space and out with bridging stability condition also depending on a point on a model space this says that it's bridging stability condition translation to mathematics of physical language of d-brains so the stable objects something called d-brains which again can have mathematical definition with which you cannot work at all one can give you some Hilbert spaces operator product expansion but we can't really prove anything here and what goes on so if you have point X, Y in this product space then the category CA depends on X say holomorphically but on Y it's locally constant so this category locally doesn't change if it changes Y the central charge which maps from this category to complex numbers is conversely it's kind of locally constant in X and holomorphic in Y so it depends kind of of second parameter and similar story for if you replace A B it changes the role of the whole story so you get two family of categories and you know it's BPS BPS d-brains, yeah it's actually not BPS states in string series it's something called BPS states just d-brains, yeah yeah it's extension of conformity series to the boundary it's not BPS states or supersymmetric brains now so you get two family of categories and you can see the fundamental group of this product space it's product of two groups it's sex by automorphisms of samsic like homology or topological case theory of one category or another category and the story is it's the same action homology will be identified categories are completely different and what the goal is for example consider this A category go in parameter Y go around we got automorphism of categories that means that fundamental group in Y direction of Mb acts by automorphisms of category in depending on the context it could be and X you can fix so it's for all of them you get action simultaneously yeah so it's it's the story there's no actually callabeo varieties in this game it just it's some categories and that's a picture which is very far from being understood mathematically again typically this model spaces are not compact they have casps infinity so we can I don't know I can write like product of mA multiply by mB and here infinity of mA you approach these things or here approach infinity of mB you can so you can fix one parameter another parameter go to the to the casp and if you go either way go to the casps or to casp in one direction in one factor choose them then the claim then in this limit you many fault will arise from nothing from conformal field theory because in conformal field theory you get some space of fields some operator product expansion and then go to limits to the limit you will see that algebra fields became close to commutative and this operator product expansion gives a structure of complex remaining manifold with some scalar metric so you get kind of target space description so you get certain Calabio manifold with Richy flat metric it will be two types of manifolds depending on which point No, no, no the mirror symmetry I will not talk about the mirror symmetry if you go simultaneously to both limits then you will see some mirror symmetry station I am not talking about mirror symmetry so you see some kind of geometry you can write Lagrangian for your story inside there is no kind of Lagrangian you see some geometry and if you go to whatever in some limit then CB will be equal to like let's go to horizontal limit so fix point in B modulus space and then this category B will be derived category of coherent shifts on this manifold so it depends on the complex parameters and CA will be Foucaille category of real symplectic manifold like just infinity manifold if I forget about complex structure X and you put scalar form and strictly speaking there is something called B field which is kind of imaginary companion of symplectic form B field is commulger class of X with coefficients in U1 and one is H2 of X in R and if you add of them you identify with commulger H2 of X with coefficients C star so it gets some complex structure parameter space and both categories have some stabilities but the story is here that stability is kind of geometrically visible in this limit only for Foucaille category so things which are very geometry allow to this coherent shifts will be very complicated and hard to construct things there is no obvious differential equations here and here will be some differential geometry yeah, so there is a story about Foucaille categories which think associated to symplectic manifold and a posteriori seems to be the best way to think about triangular categories what people used to think it's complex of shifts it's not kind of the right think from non-commutative viewpoint here you can see all structure very literally in this thing and Foucaille category is defined when they approach closer and closer so it will be serious in small parameter and formally at the moment it's category over some non-archimedient field which is called Novikov field which is like Laurent series with complex coefficient but with real exponents so we just consider set of all possible infinite sums t is variable we see i complex numbers i are real numbers and if they are infinitely many they go to plus infinity yeah, in principle series should converge but it's not proven at all and people work with this form of our series now I will now speak about definition of Foucaille category which is yeah, the story is that it's really disastrous situation it's a disaster so for many many years we are still not able to give really satisfactory good definition and so it's people working Foucaille category form kind of like sect so they can talk to each other but it's really hard to make it really commonly accepted I think people are used to but if you are not in the subject it's impossible to work technically in this yeah, so there's no SGI yet so it's yeah, so can't you Foucaille I think it's just because the story is genuinely more difficult than the Sitalic homology it's next level complexity but I don't I don't agree, I think it's maybe just not enough, there's no grotting in this subject so Foucaille category almost all approaches you should make some choices and choices form some contractable set result is independent yeah, so first to start with you need to have some symplectic manifold and the moment I assume that it's compact it's infinity symplectic manifold and also I assume that first-gen class of tangent bundle is trivial now I'm doing smooth topology, it's class on second-come-old with integer coefficients and strictly speaking it's not a condition, it's a certain choice here, it's a certain kind of data here, namely you use whatever some cell decomposition and write some two-core chain and you use one core chain whose boundary is two-core chain, you should prove that this is zero on the level of core chain and then yeah, this variety of dimension to n and also you can fix, but it's really minor point B field which you again look on core chain level not just common logic class but some two-core chain in this coefficients shift yeah and now what are choices which one should make first one should choose an almost complex structure so that it's point-wise G and omega it will be point-wise scalar, but the main thing is G is not integral and the point-wise space of choices is some quotient of SP model unitary group which is contractable space, so that's why you can do it without any obstructions and second choice come from the fact this first chain class is zero you choose omega which will be n zero form respect to this complex structure not vanishing everywhere but not closed comes from this first chain class equal to zero existence of such form okay and now just recall if you know, but if you don't know just to try to frighten you what are some objects of fukai category it was originally definition by Kenji fukai it take compact Lagrangian sub-manifold then should be oriented and have spin structure and then one make another choice which correspond to some gradient choice namely because it's oriented and you restrict this form on this guy it will be non-vanished everywhere and take kind of argument then you choose a lift to real numbers it's not does exist always abstraction to existence it's called muscle of class it's kind of rotation homologous class of this map belongs to H1 of L should vanish muscle of class should vanish and then you can lift it okay and then you make thing which is not realistic still I assume that there is no G holomorphic disk in X whose boundary is in L, it is not a point maybe I skip story about B field we see please Tim okay so the role this will be in few minutes I suppose have two such Lagrangians some manifolds and I assume that intersection is transversal so you don't can take twice the same space then you can make define something called floor complex it will be this Novikov field raised to set of intersection points which is finite set it will be vector of space on the field with the bases and you introduce certain Z-grading for this set layman base you should search some integer number and what goes on I get two or Lagrange manifolds you can rotate in the shortest way tangent space of L1 to L2 make an LT continuation of argument and see how the lift to argument changes one way to another so this and now introduce differential namely dp1 will be some and you take so very differential in the base will be certain coefficients and things at degree 1 and cp1p2 it's some huge infinite sum namely I have two Lagrangian get to intersection points and consider all possible homomorphic disks with boundary here and here and again if there is generic situation it will be isolated it follows from collabial conditions and this degree reasons kind of isolated disks and you take T to power minus integral of omega of the disk this will be some positive number and in principle one can add if it has B field you pair B field with a disk roughly you get element in U1 which sits in C multiply also by complex numbers absolutely value 1 and also you put some plus minus here because it's these things are isolated it's coming from some red volume operators and almost complex structures and you get this formula and it's a kind of D-square equal to 0 how it's prune and these subjects are pictorial proofs and technical proofs nobody reads the pictorial proofs is the following here now consider matrix coefficients of D-square so consider P3 degree raised by 2 and then these disks form one parameter families and then you can see the boundaries of these families so some of the boundary points should be equal to 0 in boundary points in principle are of 3 types degenerate to these things D-square but then you get 2 other types of priority you can have such picture this bubble appears from one side or from another side and this is exactly what is forbidden by this assumption which is pretty formal it's clear where this naturally come from natural geometric situation and then you can count polygons isolated polygons and this remarkably gives something called cyclic infinity structure so it's not clear what is the first chicken or egg here people kind of make algebraic definition to make this geometry work and it's so it's whole thing to develop simultaneously and geometry and algebraic formalism infinity structure so we make definitions to make these things work literally without any trouble so it's here this is kind of pretty unpleasant condition which is not terribly natural and there was some improvements yeah first there was kind of 1000 page manuscript by Foucaille Octa Honor E.O many many years and a lot of controversies and so on but the main kind of thing from this manuscript you drop the condition that there is no holomorphic disk sorry there is no holomorphic disk no disks so you allow all like one hundred manifold with natural story but there is a price to pay make some other choices which were not present before first all this one thing which one can do from the very beginning one can end up put on L complex local system finite dimensional presentation of fundamental group or complex of such guys it's a little modification but the most unpleasant think it's choose something called boundary chain which is kind of when you see such disks they form boundary for example get some loops on an L and you try to choose a chain on L whose boundary the things you can try to correct step by step and I eventually it was some kind of solution of power carton equation some to certain Lie algebra you get some series in positive powers of t and exponents will be exactly areas of the disk which appear in this thing so it's kind of pure algebraic gadget kind of to cancel algebraic in these disks but what is advantage of this thing one can move now Lagrangian manifolds before Lagrangian manifolds you don't have any morphism to itself because intersection wasn't transversal and it will be good to move Lagrangian manifolds and can move L by Hamiltonian isotope so in symplectic geometry if you want to move Lagrangian manifold there is some special class of movements called Hamiltonian isotope it's a following if you imagine a manifold move it then the speed is section of conormal bundle which is the same as cotangent bundle in symplectic structure and so L dot is one form on L and if the movement preserves the properties of Lagrangian it means that it's closed one form so the speed of movements is given by closed one form so you have differential c infinity of L f goes to df or h goes to dh and if you move by exact forms it's called Hamiltonian isotope and this some sequence so this procedure one can for Hamiltonian isotope can transform solution at least for a small time of this bounding cut chain a long Hamiltonian isotope and it will be kind of the same object it will be isomorphic object so we can choose different models of the object and achieve transversality and then in fact I was also behind this project secretly long time ago but then eventually there was a different idea it's kind of like a skeletal approach and about it's something which I proposed maybe 10 years ago and then about a year ago this young servile man we kind of break through but it's not completely finished yeah it's a different way to define 4k categories and on skeletal things there are a lot of activity so the idea is the following the main idea just we drop the conditions Lagrangian manifold are smooth at all so we start with arbitrary thick L will be arbitrary maybe closed singular Lagrangian subset I can't be too precise here because singularity should be some kind of reasonable type yeah but at least kind of analytic singularity should be okay and with this singular subset and proceed in two steps kind of step one you construct a finite z graded quiver on the subset plus differential and it looks that all six will be defined over integers without completely things that can work in any characteristic and depends only on small neighborhood of L and this things is defined again up to derived maritic equivalence so from representation theory should get the same story so it's not one object but depends on some choices which I give the equivalence for example we sell the composition of L sell the composition of L if you choose sell the composition then vertices of the quiver will be the set of top n dimensional cells dim x is equal to n dim L is equal to n so in principle the story when can see not Lagrangian subset but something smaller and the smaller part will contribute nothing or if it can be contracted then this whole things will be nothing so it's only n dimensional cells will be vertices and edges will be things coming from cells are attached to each other, something related to singularities of this guy so you get this things in principle one can calculate you will see later examples but then holomorphic disks in X whose boundary is in L should deform pass algebra of this quiver of pass algebra of this quiver and deformation will be over ring of integers of the null field and given by some higher order terms exactly t to power areas of these disks this was kind of a rough idea and originally this technical difficulties how you count what is the formula what contribution of disk was kind of enormous and we found some kind of way to circumvent it and to get some answer and we define FL which can sort its objects of fukai category with models like supported in A that's a rough picture and FL will be to the category of ring of integers without negative powers it will be finite dimensional dg representations of the deformed of the deformed deformed pass algebra In concrete terms you said that it's very similar to what I explained in the first lecture about how one can treat these categories you get finitely many arrows which are enumerated first, second, so on each arrow with some expression previous arrows and now you can see representation each arrow will go to some finite dimensional matrix and then you want to calculate step by step finitely many equations and many variables I told that the holomorphic disks deformed pass algebra implementation of deformed algebra so you get these things in particular because of the representation q in each vertex you get certain multiplicity, some vector space or maybe finite dimensional complex of vector space you get some integer number such to each vertex and then you get kind of earlier characteristic map from object of this category to n chains because for each n dimensional cell you get certain multiplicity but in fact it will be closed n chain at the end of the day and this is the same because it's top degree part it's the same as n's homology of L with Z because there's no n plus 1 chains to speak about it's top degree homology so you get map to middle homology and which will be let play role and what is expected it's like interesting shifts with support on some subsets on larger subsets will be one full subcategory of another if L1 is closed subset of another L2 then FL1 will be full subcategory of FL2 so it means that you get a functors object for small category of homes here the same as homes here so it's just add more and more object and then you can see that limit of all possible subsets of L it's o-ring of integers and now you add inverse powers of T get all things and this will be definition of the full category and maybe now it makes small break for 5 minutes one can make the following conjecture suppose my form omega which is was my n0 form assume is form is closed in fact it implies it's complex structures integral so we get honest complex manifold then should have a canonical stability structure I will write you in the second water central charge and stable object stability structure is invariant under continuous changes of omega preserving homology class same condition whatever so geometrically there are infinitely many parameters because you use try to see how things interact choose some scalar matrix on complex manifold and there are infinitely many parameters but at the end of the day will be finitely many parameters because servicing depends on homology classes but geometry of the story depends on actual choice of scalar matrix and complex structures and so on so first of all what is the central charge you just look this definition has a representative which is a single Lagrangian guy which is e in f of L and then it has earlier characteristic most earlier characteristic of E which is belong to top degree homology group of L which maps to homology of X and then you integrate omega complex numbers this will be a central charge what will be stable objects semi stable objects they should have representative supported on some L those which have representatives f L for certain L and the picture is a following L is special Lagrangian of course it's union of some smooth pieces like n cells you can decompose the pieces and for each cell each L alpha should be oriented say now it's oriented and you choose in the argument of restriction of form omega restricted to L alpha is equal to theta which is constant and moreover representation of quiver which you set which you deform should be concentrated in degree 0 particularly Earth risk of the kind of positive non negative dimension of some vector space so these things are called special Lagrangian subsets and here we see calibrated geometry these are really like so bubbles which minimize the area in given homology class for such objects if you consider absolute value of the central charge will be the same as usual area counter trans multiplicities and in fact it's less than equal than remaining volume for any integer chain in class image of image of E in middle homology of X it's called DPS inequality because these things will be equal to integral over this class of real part of exponent minus E theta omega some real value 10 form and this real value 10 form is bounded below by remaining volume on each Lagrangian subset and the quality appears only get kind of right direction the main tool in differential geometry to prove this variety is minimal you found some close differential form and this point was in equality respect to remaining metric and remaining metric is actually which is remaining metric it's not calor metric it's calor metric multiplied by some function metric for omega and j multiplied by some kind of function bigger than zero and functions chosen in a way that's a volume form remaining volume should be equal to omega bar there will be no relation with omega yeah it's not what physicists say in physics you can see the solution of monchampere equation kind of rich flat things really equal to these things it's very complicated equation you have to solve and then strings what Mike Douglas wrote it's applicable only for calor understand manifold and Mike and Jack Schitt's match software says that any calor metric will work Hall-Morocco form shouldn't interact in any way with volume element it's closed yeah In fact in the next and last lecture even more general things even more complex structures the whole thing still should work it's very very soft and but let's stay in this generality so this questions why original axiomatics yeah there are several we have description of what are semi-stable objects then we can check the vanishing of ax it's very easy to prove but the main thing is why hard-on-assumed filtration exists on an object and like for queers I said you have some flow here I can imagine some flow which is the following it's defined only on outside of singular part so you get maybe a singular guy and you start to move it and L dot should be closed and form I said should be Hamiltonian isotope should be gradient of some function minus differential of function and function is one can best choices argument of restriction of omega to L which is was a choice of real valued function so get Hamiltonian isotope and one can show one can do much larger generality and that's the area which defined this integral real volume form which is kind of bound upper bound of the central charge this area decreases by this flow so these things cannot go wild and it should kind of stabilize but when it's stabilized it means the differential function is zero they get a special Lagrangian and then the hope is to prove that it's okay to get really hard noise infiltration I think it's it's a big project for many years it's not very far from in achievement and this one of major difficulties what to do at singular points and again in next lecture I will explain approach to this major difficulties you can have some graph and you start deform that the whole thing should replace by some kind of complicated so bubble inside and how to invent the so bubble it's a completely not trivial issue so next time I formulate some very general conjecture even in dimension 2 how such things appear and it's related to honykomp diagrams the whole story actually 4k category can be defined for non-compact varieties it was also some major development and in the non-compact case what you need roughly the following you try to contract your manifold by some boundary and in boundary choose open part and moreover this open part should be some special class called stops but one should have a Hamiltonian vector field which is defined non-trivial only in the neighborhood of this part and maybe I can draw the picture so I get this u other part of the boundary then I get vector field which is very very small starts very very slowly and very slowly ends so it goes infinite time from left to the right and the whole thing is kind of here as well but the whole thing is kind of completely zero, completely frozen outside of small neighborhood the whole thing could be defined in very very tiny neighborhood of this u and then you allow Lagrangian manifold or maybe let's speak about smooth Lagrangian manifold which are closed in X but not compact and this property is this bounded infinity of this guy belongs to this u so I allow things going to this u and when you can see the homes on partially wrapped focaya between L1 and L2 you can see there is a following thing you take you have two guys and then start to move one of them according to the flow and then the intersection points will be kind of stabilized will be certain finite number of intersection points so I define home of one is limit which is, which kind of stabilizes it's some finite moment t goes to infinity of the snowy field and you start to intersect exponent of this field t of L1 intersect with L2 and eventually it will stay the same and get some number yeah, so the main story is that it's no longer calabiocategories because if you start to move another guy's intersection, number of intersection points will be different yeah, so you get in some sense this kind of main main example of such particular focaya category when x is R2 burn the infinity will be S1 and you choose N plus 1 interval union of N plus 1 open intervals you get things like this here one can make a skeleton which will be skeleton will be this union of rays will be non-compact and corresponding quiver is derived equivalent to quiver AN quiver AN has N vertices this guy has N plus 1 vertices but when you can derive the quiver you can change number of vertices for quiver yeah, that's basic example for example when N is equal to 2 you get just air of 1.2 N but it's the same as exact triangles in exact triangles will be these three spikes of this picture and here it's a very nice example all this prediction with stability and Hallomort quantum forms worked perfectly although it goes a little beyond calibrary geometry one can prove the stability category of this thing this example is equal to space of holomorphic forms about killer class you don't have to worry it will have infinite area so it will be holomorphic one forms on C of the form exponent whatever PN plus 1 W Z WDW we can see that this one forms PN plus 1 is polynomial of degree N plus 1 so you get this holomorphic form and the claim is each holomorphic form gives you stability structure on this category of quiver AN and the construction is the following if you have this holomorphic form then omega is one form non vanishing on C and you can see the omega cross omega bar you get flat metric you get flat metric on C but it's strange flat metric it's not usual R2 if even polynomial of degree 2 how it looks like I can consider draw a part of this of this manifold with flat metric you can glue from two pieces of paper such thing which is flat but it's not compact you can understand here and start to rotate these things infinitely many ways this ray also infinitely many rays the same at this point so get very hard to imagine flat metric on R2 and these points I do not these singular points are not parts of R2 you remove them they are not parts of R2 so this metric will be not complete and if you want to complete this metric you can add n plus 1 points infinity and stable objects will be geodesics connecting these points connecting these singular points so it's very clean nice one dimensional example miraculously space of stability conditions coincides with space of holomorphic forms в принципе мы будем быть так оптимистиками в высоких размерах так, я хочу попросить к further development так, эта история в 4k категории всегда меняется есть какие-то новые идеи и это какие-то идеи, которые можно definить в 4k категории с коэффициентами вот, вы можете может быть, вы можете помолчить с другими коэффициентами и что это геометрия геометрия supposing you get 1 killer manifold maps to another killer manifold and the top guy is Calabi Yao, но не Richer Flat metric я не использую Richer Flat metric just some killer metric and volume element so you get omega and suppose like complex dimension of y is n plus m and complex dimension of x is equal to n and omega is belongs to n plus m zero form we have map is not compatible with killer metric at all the arbitrary killer metrics then first of all consider generic fiber it's again Calabi Yao variety of dimension m you divide volume element on the total space by volume element on the base and and also is m dimensional Calabi Yao the base is not Calabi Yao at all it could be projective line or whatever then we do the following you can see the omega x omega y plus epsilon everse pullback of omega y omega x it's again killer form because it's strictly positive it's non-negative some of it will be strictly positive it's again killer metric and if you rescaled by epsilon you see that the fibers became very very short it will collapse to the base it's called killer collapse and then one can try to look at the question what are special Lagrangian submanifolds in this epsilon is very small one can argue that special Lagrangian in y when epsilon is very small will look will be special Lagrangian in the fiber and kind of special Lagrangian will be product special Lagrangian in fiber and some kind of maybe L in X which will be kind of special Lagrangian and what really goes on suppose the argument is some angle theta so it solves this differential equation argument is theta then on fiber it's Calabi Yao but volume element it's not really defined you get map from X if you get X sitting in Y X and if you integrate omega over L X you get not a number because it's N plus M form and you get element in VH N face to X at point X so it means if you fix homology class in the fiber on the base you get N form and then the things on the base should be special Lagrangian with slope theta with respect to this N form so form on the base depends on choice of homology class in the fiber it's not unique one and then the idea was the following here we have kind of description when we kind of special class of Calermatic of special Lagrangian manifold but now what we see here we get kind of stable object in the category in the fiber and now we can replace by pure algebraic notion region stability so the idea is just go back from geometry to algebraic formulation and try to axiomatize the situation in the following way so axiomatization is the following we get X, omega maybe J, calermanifold not kalabiyaw and on X I see with just configuration on X I get local system of triangulated categories so for each point we get category CX which in fact will be Foucaille category of the fiber in example and stability then I get stability structure stability condition on CX but with little twist usual stability conditions map from K group to complex numbers but the whole thing it's kind of in variant respect to C star you can map to one dimensional complex space it maps to VHN or tangent space XX and that should be halomorphically dependent on X so it's isomorphic to C but not canonical and it's halomorphically dependent on X and then kind of conjecture here says that in such situation first I forget about stability you get just local system of categories one can make agree with coefficients in this local system of categories which will be total space in the example like this and and then we'll have stability the stable objects are described in the following way which will be kind of spectral generalized spectral networks so it will be a single Lagrange subset on each on each an alpha use which will be kind of cells it will be a dimensional cells in which you get locally constant family of objects in the fiber which should be semi stable for all points and then you get volume form and this guy should be a special Lagrange respect to volume form yeah so service gets translated remarkably one can replace this geometry back by algebra and get an ocean and yeah that's the picture so here there are kind of same stories there are this representation of quiver and also bounding chain but kind of bounding chain which are kind of some series in solving some equation should play no role in the property of stability so it will be some kind of algebraic decoration not related to differential geometry and this category could be and then we try to think it could be very very general this category could be over any field and this new category will be over series extra parameter yeah so let's find it field to get automatically local field so it's this kind of very regular project and a lot of difficulties here and if dimension of axis 2 everything can be understood up to very last detail but bit hard to write it real dimension 2 complex dimension 1 dimension of rx2 everything can be described rigorously have rigorous definition or to be of global 4k category and rigorous description also of semi-stable objects but there is no proof yet it's really stability so this things with flow should work but here at least everything is now rigorously defined I'll just explain the situation suppose x is a torus and because we use this complex structure it will be elliptic curve let's say c divided by c plus tau c z plus tau z and with some coordinate w and it has one form w and let's take constant family of categories just nothing just product situation constant family and stability which takes values in this funny vector space it will be just the following guys zx will be kind of 0 which will be kind of c value with stability which is the same area multiplied by one form dw yeah completely stupid transition invariant picture but one can kind of play some game killer form just killer form compatible with this structure just any positive one one form it will be certain some r of zz bar which is positive any c infinity positive function times dz dz bar so we get non-flat killer metric the result should depends only on the area so we get total area a tot will be integral of x omega which is positive number some real number the whole category in this case should be kind of like product situation we know what is it the whole category should be tensor product of original 4k category and multiplied by 4k category of the torus which is elliptic curve by mirror symmetry and it will be t-elliptic curve so db of coherent shifts on t-elliptic curves on non-alchimedian field so we get kind of gm analytic divided by powers of z of t to total area like divide complex numbers by powers of q we get usual elliptic like c star divided by q to power z get description of elliptic curve q will be number of normal less than 1 and we do something same in non-alchimedian case yeah so the answer we know but what is stability on this category so for this we should draw all this picture and then now we should draw all possible graphs you can see the various one dimensional subsets in x and I assume that one skeleton of cell decomposition of x now before I have cell decomposition of ellen but now I have cell decomposition of x so there will be certain kind of two periodic picture with certain cells finally many cells and then one can try to see what here goes on what is qrql in this case it's something dependent only on this graph yeah I told that approach when you have singular gram space you get some quiver whose vertices will be edges of this thing yeah some complicated quiver and in fact it depends only up to derived equivalence only up to number of two cells it's up to derived marita equivalent marita equivalence marita equivalence because quiver categories finally many objects it will be categories little bit edges it will be db of coherent shifts of singular elliptic curve yeah so there are many elliptic curves this complex elliptic curve will be single elliptic curve of integers and it will be union of certain f2 copies of p1 glue it in cyclic way when f2 is number of two dimensional phases two dimensional cells in the cell decomposition yeah in fact it's fukai category of torus minus f2 punctures and when I put infinite volume near punctures it's just the same story so the identification is not canonical because like punctures are points in the middle of cells and you can move them and on this thing x-simplectomorphism which is the techmoly group m1f2 in the mental group of more the space x-simplectomorphism f2 punctures acts on these things and if you move in some standard way you get one identification one category another yeah so it's a way to define this I think one can do something more explicit but now what you do what is deformation of the story of this QL deformation depends on n numbers which are areas of this domains depends on numbers area i area i is integral of omega by i's domain in i from 1 to you get these numbers sum of area i or ai sum of ai is equal to total the total area of the thing and the deformation is given by the following way if you get in some model you you can see the double points of the things which are given by coordinate cross you deform by xyyi equal to t2 ai so you get a kind of small hyperbola and you move together you get a smooth curve and the smooth curve will be exactly the state curve yeah so that's yeah and if you combine the whole things together you get now from this picture you can construct all this inductive limit for all graphs and so on you can identify the whole things and then you get a candidate what are stable objects you should draw straight graphs in this flat metric related to omega not in keller metric flat metric related to complex coordinate and this is description how the things should work together and then you can iterate the construction so one can multiply category by elliptic curve another elliptic curve and if you iterate it then you do then you get the following thing you get for any finite sequence of complex complex curves complex elliptic curves and this omega i will be one zero forms and when you check one zero forms like this d z here I construct a stability structure on d-dimensional abelian variety over iterated novico fields so it will be something like u d1 to power r it will be multi-dimensional local field and this abelian variety will be kind of products of state curves with small and smaller parameters various t i for a i also I should put some numbers a dot 1, a dot d greater than zero choose any real numbers so it will be like product over t i a i dot to power z in corresponding coordinate and stability structure will be the following a group of db coherences on a will maps to z to power 2 to power d which will be tensor product over h1 of my tori and then each guy maps to c by product of integrals of my form omega i so you get kind of complete mess between abelian varieties and get stability conditions on abelian varieties and stability conditions are kind of very non-trivial so stable option will be if you try to think what is literally it means that my lagrange manifold will be like piecewise linear I construct step by step by vibrational graphs but that will be kind of nice complete mathematical description and conjecturally it should extend to all abelian varieties because here I leave near very deep cusp in the model space of abelian varieties and conjecturally it should extend to all of them and for this it is completely open question ok so yeah but at least everything is formulated now the question to brood ok