 some pure algebraic object structure which appeared quite recently, it's called stability structure in the triangulated category, I'll say a few words. In the triangulated category it's due to Tom Bridgeland, around quite some time ago, maybe around 2001, something like this. And it's very mysterious object which can be thought as analog of Keller class or Ample class for very general framework of triangulated categories. And it has very nice properties but it's very hard to construct and I will explain that the best hope is through differential geometry, variation principle, convexity and all this. So the subject close to Jean-Pierre Hart. Yeah, so the definition is kind of simple but first of all I will just remind you about the triangulated categories. Example of triangulated categories is something which we technically consider, let's say bounded coherent sheaths on some algebraic variety and consider complexes of sheaths with compact support. Now bounded it means that commulgia only infinitely many degrees and compact supported commulgia are coherent sheaths in this compact support. Or you can consider modules of some algebra and consider complexes of modules, think like this and get triangulated category, triangulated category C. And to say what is stability condition you need two things. First you need some kind of churn class for each object of your category should define several numerical invariance. And I'll say that just suppose you get a map to homomorphism to find a joint free abelian group, some coefficients of some components of churn class. And then that's your fix once for a while and then there's something which you can change, which will be kind of like your Keller class. First you get a map from the n to C, so it's eventually you get a map to C from K group. And a class of stable objects, stable object with slope theta. It's just a collection of isomorphism class of objects, S is from stable and C is real number. And with the following there are several properties. If you get stable object, sorry? Zn. Not Q. Not Q, no, no, Zn, yeah. Like churn classes. Like churn classes, yeah. Yeah, you just consider index, you multiply your coherency by another, you consider index, things like this. Yeah. Or you consider maps to some finite joint abelian group and kill our torsion. Yeah, so for any stable object, if you apply to, first make churn class and then apply Z, you get complex number. It should be nonzero number and with argument equal to theta. And then if you shift by one, the argument will be shifted by pi. And then any object of your category should be decomposed in this stable or simple object. So for any object F, shift, change the grading by one. And then this churn class will change by minus this number and the argument you add pi. Yeah, okay. And then for any object, there exists the analog of kind of filtration which in triangulate category just a sequence of morphisms. Any morphism in the sense of monomorphisms such that consider quotients and called EI. They all belong to CS CTI and arguments are decreasing. So in the case of quotient, complex quotient shifts, yeah. It's very complicated. Even in case of quotient shifts, it's extremely hard and high dimensions to construct any example of such a guy. Yeah, so it's slightly related to killer positivity, but it's a very delicate story. Yeah, so people constructed in dimensions really recently, but it's a record. I think in dimension four it's very, very hard to construct such a thing. And there are many nice things about the stability conditions that the main theorem of bridging is that if in the appropriate conditions, assumptions, if you vary a little bit the central charge z, you uniquely vary stability conditions. So they form a complex manifold. They form a complex manifold from any category and manifold with kind of vector space structure. It's cone in a vector space. And so it has many wonderful properties, but as I said, it's very hard to construct them. Yeah, so and there are, I'll explain kind of two frameworks where we can hope to construct them. So the first framework is some easy situation when one can construct this stability thing. It's kind of quiver like situation. There's really no trouble to make this stability condition, which is the following. What will be, suppose we get some algebra, associative algebra over any field. And I assume it's actually finitely generated. And I assume that in this algebra I have a collection of projectors, commuting projectors. So PI squared is equal to PI. Some PI is equal to 1. And PI Pj is equal to 0 for I non-equal to J. Yeah, so essentially one can consider the quiver because one can, and if algebra is finitely generated, you can make, choose generators. And algebra will be passed algebra of some quiver or some quiver. Maybe modally finitely many relations, vertices of quiver labeled for 1 to k. Now, so we'll have such algebra, but this algebra is, and this algebra should be kind of enhanced to some, usually H0 of some differential graded algebra. And I assume that all cohomology of the thing is equal to 0. So I made some kind of resolution at something in degree minus 1, minus 2, and so on. Yeah, so we have this algebra. And then one can immediately construct a category and plenty of stability conditions. What will be the category? In fact, maybe this case should be equal to n. What is the category? The category should be, in a sense, GG modules appropriately localized over this DG algebra, this finite dimensional commulger, total commulger. This form a certain triangulated category. The k group, if you, the k group of this category C is the same as k group of category of finite dimensional A modules. So this category has something called heat heart, which is a billion category, which is this representation of this algebra. And which maps to Zn. Because if you consider any representation, you can see the maps to dimension of finite dimensionals. It should get such a collection of finite dimensional spaces and look at the vector of dimensions. And these dimensions are non-negative, if you consider orbit representation of this algebra. And now you choose arbitrary any map, such as base vectors, some number Zi, whose imaginary part is positive. You just pick arbitrary numbers in the upper half plane, you get the homomorphism. And then all object in your a billion category will be a positive linear combination of some numbers in the upper half plane in a space. And then you define what a stable object, object we don't have sub-object with a larger slope. So you immediately get such things and you can see the shift of such things called the semi-stable object. So you get the upper half plane to power n, maps to the space of stability of this category. And the gen class. So you get the domain and one can also, there is a very nice game called tilting. Very often one can make another algebra, another dg algebra. You can make another dg algebra such as this category, which gives the category c tilde. And the category c tilde will be equivalent to category c. So you describe the same category in different algebraic ways. And this space of stability conditions, these two kind of products of upper half plane will be attached to each other that have common boundaries. So you can go from one to another, you get different algebraic descriptions, but you continue to this algebraic manifold. And what to do for tilting, there are kind of left and right tilting, the formula is like this. So this algebraic projectors, we can see it's kind of, dg category is finitely many objects. This will be just different, back-graded components will be homes between different objects. You get some object EI. And suppose you choose some generators, choose generators, which are of algebra which called maybe i a alpha, some collection of generators. And assume that arrows in this quiver. And a alpha is equal to some projector for some ij, depending on alpha. And assume that you get certain object and choose i0, set it, there's no generator, there's no generator a alpha which is equal to pi0, a alpha pi0. There's no loop at vertex i0. In this case, one can make new collection of this dg modules. You just define EI tilde is equal to EI from i non-equal to i0. And EI0 tilde will be kind of two-step complex. EI0 in degree plus one. And you take direct sum over EI over all i non-equal to i0 and all arrows from i to i0 in this quiver. And then consider universal map. Yeah, and then very easy check shows that if you consider endomorphism of this direct sum of EI tilde, it gets some, again, new dg algebra. It's a homology in positive degree of this guy's, again, vanish, and then we can continue the game. Yeah, so there's some kind of combinatorial game if you go from one algebra to another algebra with the same term, related category. And stabilities are continued up to another. Okay, so what I explained to you, it's pure algebra. But already we want to start with some differential geometry. To start differential geometry you should go to from arbitrary field. Let's say field is complex numbers. And then one can describe what are stable objects in completely different way without just going to axiomatics and so on. What not stable object but what's called polystable objects, which are direct sum of stable object with the same slope, of slope theta. And the description is the following. It will be unitary classes of star representation of the following thing. If you consider pi, you can see the representation of the algebra in finite dimensional Hilbert space of the same algebra. Yeah, the representation of algebra in finite dimensional Hilbert space. Let's call it V. So each projectors are orthogonal, so it's a direct sum. And then the following condition holds. I remember I choose some generators. I consider some of commutators, A hat by hat. Get some self-adjoint operator. And then should be equal to imaginary part, exponent minus square root minus 1 theta. Zi, the complex numbers, times projectors. And take some from 1 to n. So this equation, actually this Alistar King discovered it many years ago. And where it comes from, it's come from the relation with geometric invariance theory. The stable object is exactly the same stable things in terms of Manford theory of geometric invariance. Then we get Hamiltonian reduction and we get this thing. And this is actually very beautiful. In fact, it will be a complex analytic algebraic variety. Maybe singular, the space of such equivalence classes. It will be coarse modular space. But we describe it in terms of Hermitian geometry and it will carry certain calorimetric. And if you choose different generators, get different calorimetric on the same manifold. So this choice of generators can be sort of a choice of non-commutative calorimetric. It gives you calorimetric on all modular spaces simultaneously. And yeah, it's a very interesting question here that this thing you can sort of as a representation of some star algebra. Consider formally complex Hermitian conjugate variables. And you see that you get different c star algebras to your associative algebra. And they have the same representation theory. So it means that c c star algebra, maybe it's a finite dimension of the representation theory. But it's maybe some certain completion should be equivalent. It would be very nice to have canonical c star algebra associated to associative algebra. But it's one thing. And you see that we get some kind of differential geometry here. Okay, that's one thing, but it's a very simple story applied to this quiver. And it's very far from the situation of interest, like coherent shifts and so on. In fact, it has something to do with coherent shifts, those kind of remark. I explain everything about finite dimension representation, but one can try to formally do infinite dimension representations. And let's take algebra, it will be just, let's say, polynomials in d variables. Maximus, excuse me, instanton modelized space as an example. Yeah, for instantons you can get the same description. Yeah. But now let's consider algebra of polynomials. And module will be a or ideal finite co-dimension. Kind of functions vanishing with some multiplicity at some points. Which is not vector bundle, it's coherent shifts. But the claim one can write this equation in a certain sense. Your projector will be only one, just identity. Yes, so you can formally write this thing. And you write equation, the generators, they will be, actually this is my A alpha, it will be generator with algebra. And you write equation, so you get some unbounded operator in Hilbert space. And you write this equation is equal to, let's say... You always find dimension in the space, why... No, no, no, it's infinite dimensional. And now it's infinite. Yeah, imagine kind of complete the space of function of this kernel exponent minus the square root. You can see the entire functions with appropriate growth. And you take times identity. You write this equation. And it has some solution. For example, if you go to one case of one variable, you can see the polynomial variable just labeled by 1, 0, 1, 2, 3. And then you get a negation and a creation operator whose commutator is 1. Yeah, so you consider xn squared will be n factorial. And consider multiplication by x. In our joint, you get exactly representation of this algebraic commutator will be exactly 1. And then we can ask the question how to regularize it at infinity. If you consider that good behavior at infinity will be such as commutator of x a hat xj will be identity operator plus some kind of small kind of trace 1, trace class or whatever. Yeah, consider something at which infinity looks like identity. And then actually it was Nikita Nikrasov who invented these things about 10 years ago. The big conjecture is that if you consider, for example, shift of ideals of finitely many points, it will be stable object because it couldn't have really good sub bundles, should have canonical pre-heal bit structure when you get this equation. Mathematicians never studied it. It's a very difficult question. And then if you can go to some limit, you put some constant here and go to the limit. Then you get the limit to get a young mill equation, a Hermitian Young Mill equation of Donaldson. But now if you have advantages, it works for coherence shifts, not for vector bundles. And that should be right point of view in relation with coherent geometry. Okay, that's one word. But physics suggests something completely different. All this axiomytical region come from Mike Douglas and eventually from Mirosimetry. But physics suggests completely different class of examples, not the squivers. But let me say, maybe I should remove. Sir? Bring it down. Yeah, maybe I just... Yeah, what is the second framework? That's actually from... It's about d-brains. So you start with Calabi-Yau variety. It has... It's of arbitrary dimension. It has scalar form and has top-degree holomorphic form and satisfying... It could be not necessary compact. Then the category which you associate to it is... Actually, it's a category not over complex numbers, but over formal power series in one variable. It's a category which depends on parameter one can think. It's not one category. And this is called Foucaille category, which consists of some Lagrangian objects. Now, maybe I'll give you just this definition. We'll say what are morphisms? What is it? But what will be stable objects? And the conjecture, kind of center conjecture is the subject. This category has very canonical stability conditions. It will be the main source of stability condition from geometry. This C has canonical stability condition. The object of this category, it's very complicated things. Everyone should solve some abstractions here, but one can consider kind of limiting object when t goes to 0. Limits as t goes to 0 of stable objects are the following creatures. I'll just explain this answer. First of all, objects are, it's a pair, L, rho. L is, let's say, oriented, I'll fix theta. I wonder what it says, it's a theta stable object. It's oriented Lagrangian sub-variety. I don't say it's sub-manifold. It's actually could be singular with singularities in co-dimension 2. And axis is symplectic manifold? Axis is symplectic manifold. Yeah, it's a Calabria, it's a symplectic manifold. It's purely a real object. Then, what happens in good acts? Nobody knows, but I expect that I'll put some conditions. No, no, no, here it's absolutely essential to have singularities. But of special kind? Of special kind, yeah. No, no, no, not any singularity, yeah. Yeah, but I think here actually I don't have to specify. I will say, I didn't finish the description. So there are several things here. First of all, this X, L is compact, yeah. L is special of slope theta. What does it mean? It means that if you restrict this D0 form to L, yeah, calibrated, yeah. Two kind of L minus singularities, outside of singularities. Then it will be, belongs to, again, argument of this thing. Tt is parameter of stability, yeah. Yeah, tt is the parameter of stability, yeah. Times some positive density. But one that, I mean, if it's special argongin, it's going to be calibrated, so it's absolutely minimizing locally. So the singularities are always cogenting, at least, too. Yeah, co-dimension of at least two, yeah. Yes, yes, yeah, but yeah. This singularity can appear for the singular argongin. Yeah, it's very special singularity, definitely, yeah. And rho is irreducible representation of fundamental group of the L minus singularities to some J L and C of local system. Such a determinant of this thing belongs to U1. And the last thing which is very mysterious, L is spin, yeah. Which is, come from some question of orientability in Foucaille category. And, but I think it's actually, means it should have some dear, dear operator should play some role, but nobody knows how to do it. And the right homology class, it's actually not homology, but class in case theory. Because it's spin manifold. Yeah, and Z of L will be integral, will be rank of the local system multiplied by integral of this D0 form which will be. But you claim they always exist. Sorry? You claim they always exist. They may not exist. Okay, but then the category will be zero. No, the claim, this form is exactly the least of stable objects. Yeah, and if we believe in mirror symmetry, Foucaille category is equal to coherent shifts on some other varieties. That will be, the category is very non-trivial. Definitely, yeah. Yeah, so that's... So this endless mirror symmetry, the stability condition on one side corresponds to this. Of course, yeah, it should be the kind of the only feasible way to construct stability condition on this category of coherent shifts here at the end of the day. Yeah, but here, of course, we say that it's Calabio variety. I think this Calabio, it's too rigid condition. What one really need? One need one one form Foucaille category differently. But omega in zero, one can replace by complex valued closed N form. Closed D form of middle degree. This sum non-degenerous condition should be kind of, should not vanish on all real aggressions of space. And it's very soft condition. It should be real, now everything is real. But form is complex valued. I'm complex valued, but you can't forget complex structure, okay? Yeah, it shouldn't have complex structures. That will be... Because the picture, that one should kind of minimize some... One can put the same condition for non-necessary complex form. Just forget about remaining metric. No, but then do you know meaning of special agrarian? No, no, no, it means that this... I think it's equal to... But the equilibrium property will be different. No, no, no, no. No, the story is the following. It will be still the same story, because one can always integrate... No, no, no, it will be no metric. There will be no metric on the space. It will... But you get a formula, it's just called omega's form. You can always integrate absolute value of omega with Lagrangian manifold. And you call it area. And this area minimizing will be special Lagrangian. Just because it will be kind of some of complex numbers with different slopes and better angle and quality. Yeah, so it's without metric, but one can also write kind of area functional and minimizing... And this thing will be minimizing this area functional. Okay. Because I believe that in this region or dimension the only possibility for this is to re-brain and work special Lagrangian. No, if you start with remaining metric, but if you measure area of things by some different means, not by remaining metric, but something else... She may have... Yes, yes, absolutely. No, exactly this definition of consider form which is not vanishing on any... But still we can point to the kind of variable structure. It's just a function, minimize which is not area, but it's supposed to function. Yeah, it's supposed to function which looks essentially like area plus... And we absolutely minimizing because of this form. Yes. And through the right example... Yeah, of course you just vary a little bit here. Yeah. But itself you see that the equation maybe I've never solved before. The question of the resolution is kind of... Yeah, but you don't have to... Explicit representative, it's not relative. Like Kellen matrix you can change for in geometry the same stable bundles like explained for quivers you get different equations that give the same solutions. And here should be the same picture that you change a little bit. Omega's you get different equations. No, it may be no solution. No, no, no. No, it's elliptic equations of index zero. It's completely... The reason is... No, no, no, calibrated for Lagrangian manifold has index zero. The reason is the following because if you have Lagrangian manifold and now deformed is by exact Hamiltonian transformation. So deformed at L plus graph of differential of some function. So what you can see, you can see the functions on L, model a constant which will be deformation space. And what you have, you get an argument of the things and argument should be constant. Argument is a function again with average value equal to zero. So you get map from space of one dimension to the space of the same dimension should really index zero. This equation for being calibrated. If you consider deformation... And even if you get a large families because you consider non... If it's not one connected, then consider... first-come-old, then consider non-exact deformation. So you get really modular spaces of such things. And it's very important to put this local system. Altogether, the smaller space of this... maybe could not reduce but polis direct sum of reducible of such type. The claim that the model space of such things will be compact. One can imagine some kind of compactness for minimal manifolds. But now what's about local system? Local system can depend parameter and go to infinity. But in fact, secretly, if you get a local system, you can always have, because it's reducible, can have harmonic metric on the system. Get canonical, suppose it's collabial, you have harmonic metric. And then when harmonic metric... when local system goes to infinity, then harmonic metric will be very, very fast and then you get kind of real Higgs field. When a row goes to infinity, you get real fixed field. And which can be thought as a point-wise, you get familiar of commuting Hermitian operators. And the spectrum will be... which can be interpreted as kind of multivaluate... you can analog of spectral cursals, multivaluate real harmonic one form on L. This limit of... when representation goes to infinity, you get multivaluate harmonic one form. And this multivalent harmonic one form will be deformation as a special Lagrange and multivaluate as kind of multiple cover of the things. So limit of representation will be kind of limit of multi... coincide with limit of multiple covers from other side and the whole thing will have no boundary. Yeah, okay. Yeah, so that's kind of very vague picture and I think I'm not an analyst, so I cannot really judge how hard it is. But it looks to be true. But now we have kind of two different situations and what else can happen? And in fact, I want to say that there could be a possibility of mixture of these two situations. Just the reason is roughly the following. In general, Kaffon-Philsier predicts you that you have a category and stability. But no geometry. Just category and stability. And that's what Kaffon-Philsier says. And to have a geometric description should go to some limit. And if you go to some limit, you get some target space, but it should be maybe not the whole target space. Some part of your theory will be degenerate, something will be not degenerate. And roughly speaking, you can think like you have a Calabi or manifold Y, which is fibrous over some Caler manifold of small dimension and with small fibers. And then, and X is just Caler manifold. You can see the such kind of like elliptic vibrations and so on. Like K3 surface degenerate to projective line with very small elliptic curves. And if you want to describe what is Foucaille category on Y, you can try to see what are stable, what are special aggressions. And then you get some kind of the following situation. We're just exchanging. Kind of mixed situation is the following. At least some example of it. You get let's say Caler manifold. And then you get constructible shift of triangulated categories. So it means essentially kind of local system of triangulated categories with maybe some jumps. Here you can think about categories CX. You can think about Foucaille, if you get Calabi fibrations, generic fibers again Calabi Yaw, you can see the Foucaille categories of fiber. And because the Caler class doesn't change, you get local system of Foucaille categories of fiber. And then the stability, then each category will have a stability. The X, which maps from K group of this category, just it's a bit funny coincidence to the fiber of canonical bundle at point X. It should be top degree of Cartesian bundle at point X. So it will be stability where it's not complex numbers, but in one dimensional vector space, which actually makes sense. You can multiply stability by constant. So it shouldn't be any complex line here. And this should be holomorphically dependent, holomorphically dependent on point. What happens again in this baby example, you integrate holomorphic volume form, a long Lagrangian varieties of fiber, get volume form, holomorphic volume form on the base, depending on this choice. And stable objects will be the following. Now we get singular Lagrangian manifolds, which now have singularity of co-dimension 1. And it gets singular Lagrangian in X, which is union of some kind of l-beta. I will explain in what sense are special. Yeah, that's really funny. On each l-beta, you get a locally constant family of stable objects. In this category. But now if I apply zx to bx, you get holomorphic volume form near l-b. And then this thing should be special with slope city with respect to this holomorphic form, with respect to this. So argument, this thing should be equal to theta. Yeah, so you get some bizarre question. And then in singularity co-dimension 1, we will have something like three branches, typically coming to each other. And then you should get here at this point exact triangle in triangulated category, which is main structure. Yeah, so get this mixed things. And you can imagine these fibers, you can describe algebraically using quivers. Kind of very simple game. But then you get these things. And it looks it's a kind of natural framework to define four categories, even for this not constant shift of stupid categories. So it's going to point, but any category and the whole thing should be lived together. This is a very simple example, which I finish. This x will be a complex curve. And what will be category? Yeah, the categories, you can see there's something called preprojective algebra associated with some dinking diagram. You have arrows, some kind of dual arrows, like for the square a n. You get some arrows and get dual arrows and some of commutator, but not complex conjugate, will be equal to zero. You get this will be relation to this algebra. Then you double it to make star to this star equation. So you get this category where stability is very simple and stable object correspond to the following things. Stability correspond to collection of points in a complex plane and stable objects are possible intervals connecting this thing. And then one can make things together and get what's something called spectral networks. This mixed stable object are very beautiful things which I discovered by several people and also by Gaiot and Murnetsky. On a complex curve, you can see the multivalued holomorphic one form. Again, it's called spectral curve, multivalued holomorphic one form. There's finally many values in some verification. And then you can see the kind of three valent graphs for each graph. You get two indices, two values of this form. And on this form, the argument of alpha i minus alpha j will be equal to theta. And then you get indices i, j, j, k when they meet each other and you get some kind of three valent graphs. Yeah, because Lagrangian manifold is a graph in two dimensions and then you get some kind of nice conditions, some gradient lines. And just on the surface. Just graph-grown on the surface, one can draw computer program and see how this things. And this is a central object in WKB methods one can study asymptotic situations. So it's a very beautiful geometric object. They're not jades, this is not metric. Or it's gradient if considered holomorphic one form and considered, rotate by angle theta and consider level set of real part. Yeah, so that's beautiful objects. And this is stable object in some three-dimensional kalabia category. Yeah, so there's some kind of high-dimensional six-vein coating, simple language here. Okay, thank you. Any questions? Sorry? Why do you call it stability structure? It's not me, it's Bridgend. I know, but what is this thing? It's a bit strange name, yes. No, it's called stability even. Stability condition. You call it stability condition which is... Stability condition, yeah, I don't know. It's a condition, yeah. To hold something, some structure condition I found it's a bit odd, yeah. Maybe it's because it took it as a definition then I thought... Yeah. It's not like stability polarization, you say, no? It's like polarization, yeah. It's a lot of polarization, yeah. But is all static with the bubbles on curves? Stability condition and part of your system? Yes, yes, yeah, of course, yeah, it's... Yeah, but... Curve, in a sense, analog of quivers, but go to high dimensions, it's really hard, yeah. This example related to kitchen systems and then this one? Yes, yes, yeah, it's... So these graphs are... These pattern numbers, yeah. In fact, this stable overcurrent of three-dimensional kalabia category. Any more questions? Yeah, I think we thank you very much.