This is a pure algebraic object, a 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, which was quite some time ago, maybe around 2001, something like this.

And it's a very mysterious object, which can be thought of as an analog of Keller class or ample class for a 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 is close to Jean-Pierre Hart. So the definition, it's kind of simple. But first of all, I'll just remind you what triangulated categories.

Example of triangulated categories is something which would be technical. Consider, let's say, bounded coherent shifts on some algebraic variety. And we can see the complexes of shifts with compact support.

Now, bounded, it means that comulgia only infinitely many degrees, and compact supported comulgia coherent shifts with compact support. Or you can see the modules of some algebra and consider complexes of modules, things like this, and get triangulated category, triangulated category C. And to say what is a stability condition, you need two things.

First, you need some kind of Chern class for each object of your category. You should define several numerical invariants. And I'll say that just suppose you get a map to homorphism to find and train

three abelian group, some coefficients of some components of Chern class. And then that you 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 Zn 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? Sorry? Q? Zn? Not Q. Not Q. No, no.

Zn. Like Chern classes. Like Chern classes, yeah. Yeah. Yeah. You just consider indices. You multiply your coherent shift by another shift, consider index. Things like this. Yeah. Or you can set the maps to some final, a billion group and kill our torsion.

Yeah. So for any stable object, if you apply to, first make Chern class, and then apply Z, you get complex number. It should be non-zero number. And this 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.

For any object f, change the grading by one, and then as this Chern class will change by minus, it will be minus this number, and the argument you add pi. OK. And then for any object, there exists an analog of kind of filtration,

which in turn will indicate you're just a sequence of morphisms. Any morphism, in a sense, is a monomorphism. Such that you consider quotients called Ei.

They all belong to Cs, C to i, and arguments are decreasing. So in the case of quotient shifts, yeah. It's very complicated. Even in the case of quotient shifts, it's extremely hard in 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. So people constructed in dimensions really recently, but it's the 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 bridges is that in the appropriate assumptions, if you vary a little bit the central charge z, you uniquely vary stability conditions. So they form a complex manifold. You form a complex manifold from any category. And manifold with kind of vector space structure. It's gone in a vector space.

So it has many wonderful properties, but as I said, it's very hard to construct them. And I'll explain 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 the stability condition. It is the following. 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. So essentially one can consider this quiver, because one can.

And if algebra is finitely generated, you can choose generators. And algebra will be pass algebra of some quiver. Maybe model finitely many relations, vertices of quiver labeled 1 to k.

So we'll have such algebra, but this algebra should be kind of enhanced to some h0 of some differential gradient algebra.

And I assume that all cohomology of this thing is equal to 0. So it made some kind of resolution, add 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, dg modules appropriately localized over this dg algebra

with finite dimensional cohomology, total cohomology. This form some triangulated category. 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 heart, which is abelian category, which is this representation of this algebra. And which maps to zn? Because if you consider any representation,

you can see that it maps to dimension of finite dimensions. It's just collection of finite dimensional spaces and look at the vector 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 homomorphism. And then all objects in abelian category will be a positive linear combination of some numbers in the upper half planes in a space. And then we define what a stable object we do not have sub-object with a larger slope.

So we immediately get such things. And you can see the shift of size thing is called a semi-stable object. So you get upper half plane to power n, maps to space of stability of this category. And the gen class.

Yeah, so you get a big domain. And one can also, there is a very nice game called tilting. Very often, one can make another algebra,

another digi-algebra. You can make another digi-algebra such that 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 products of upper half plane,

will be attached to each other that have common boundaries. So it can go from one to another. You get different algebraic descriptions, but you continue to this algebraic manifold. And what to do for tilting that are left and right tilting? The formula is like this. So this algebra with projectors, we can see,

it's kind of digi-category with finitely many objects. This will be just different bag-rated components will be homos between different objects. You get some object ei. And suppose you choose some generators, choose generators which are of algebra,

which is called maybe a alpha, some collection of generators. And I assume that arrows in this quiver. And alpha is equal to some projector for some ij depending on alpha.

And assume that you get a certain object. You choose a0, set it. There's no generator, there's no generator a alpha, which is equal to pi0 a alpha pi0. So there's no loop at vertex i0.

In this case, one can make new collection of this digi 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.

You put ei0 in degree plus 1. 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 digi algebra. Its cohomology in positive degree of this guy is again vanished. And then we can continue the game. Yeah, so there is some kind of combinatorial game

if you go from one algebra to another algebra with the same triangulated category. And stabilities are continued up to another. OK, so what I explained to you, it's pure algebra. But already we want to start with some differential geometry. And now the differential geometry should go to arbitrary field. Let's say field is complex numbers.

And then one can describe what are stable objects in a completely different way without just going to axiomatics and so on. What not stable object are 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 representations,

of star representation of the following thing. So consider pi, you consider representation of your algebra in finite dimensional Hilbert space.

Oh, it's the same algebra. Yeah, yeah, yeah, representation of algebra in finite dimensional Hilbert space. Let's call it v. So each projector is orthogonal, so it's a direct sum. And then the following condition holds.

I remember I choose some generators. I consider sum of commutators, a hat by hat. Get some self-adjoint operator. And then it should be equal to imaginary part exponent minus square root of minus 1 theta, zi, zi minus complex

numbers, times projectors. And take sum from 1 to n. So this equation, actually this Alistair King discovered it many years ago. And where it comes from, it comes from the relation

with geometric invariance theory. The stable object is exactly the same stable things in terms of Manford theory of geometric invariance. And 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 scalar metric. And if you choose different generators, get different scalar metric on the same manifold.

So this choice of generators can be sort of choice of non-commutative scalar metric. It gives you scalar metric on all modular spaces, simultaneously. And it's a very interesting equation here. This thing you can sort of as a representation of some star algebra. We consider formally complex Hermitian conjugate variables.

And we see that you get different C star algebras associated to your associative algebra. And they have the same representation theory. So it means that this C star algebra, maybe it's a finite dimensional representation, but 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. OK, that's one thing. But it's a very simple story applied to this query. And it's very far from a situation of interest,

like coherent shifts and so on. In fact, it has something to do with coherent shifts, those kind of remark. I explained everything about finite dimensional representation. But one can try to formally do infinite dimensional

representations. And let's take algebra. It will be just, let's see, polynomials in d variables. Excuse me. Instanton modelized space as a query. For instance, you can get the same description. Actually, I can run through this.

But now let's consider algebra of polynomials. And module will be A or ideal finite 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 certain sense. Your projector will be only one. This identity. Yes. So you can formally write these things. And you write the equation. The generators will be accelerated. A alpha will be generated with algebra.

And you write the equation. So you get some unbounded operator in Hilbert space. And you write this equation as equal to, let's say. You always find the measure of Hilbert space. No, no. Now it's infinite dimensional. Now it's infinite. Maybe imagine completing the space of function with kernel exponent minus the square.

You consider entire functions with appropriate growth. And you take d of times identity. You write this equation. And it has some solution. For example, if you go to one case of one variable,

you can consider this just polynomial variable, just labeled by 1, 0, 1, 2, 3. And then you get an annihilation and creation operator whose commutator is 1. Yeah, so you consider xn squared will be n factorial. And consider multiplication by x.

And adjoint, you get exactly representation result where the commutator will be exactly 1. And then one can ask the equation how to regularize it at infinity. You say that good behavior at infinity will be such as commutator of x a hat xj will be identity operator plus some kind of small trace one, 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 you consider, for example, shift of ideals of finitely many points,

which will be a stable object because it's going to have really good bundles, should have canonical pre-Hilbert 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 to the limit, you

get a Young-Mills equation, a Hermitian-Young-Mills equation of Donaldson. But now if you have adventures, it works for coherent shifts, not for vector bundles. And that should be right point of view on relation with coherent geometry.

OK, that's one word. But physics suggests something completely different. All this axiomatic abridging comes from Mike Douglas and eventually from mirror symmetry. But physics suggests a completely different class

of examples, not just quivers. But let me see. Maybe I should remove. Sir? Bring it down. Yeah, maybe I just.

Yeah, what is the second framework?

It's actually from Caffron Phil's theory. It's about D-branes. So you start with Calabi-Yau variety.

It's of arbitrary dimension. It has scalar form and has top degree homomorphic form

and satisfying. It could be not necessarily compact. Then the category which you are such to it is actually is a category not over complex numbers, but over formal power series in one variable.

It's category which depends on parameter one can think. It's not one category. And this is called Foucault category, which is consist of some Lagrangian objects. Now maybe I'll give you just this definition. We'll say what are morphisms, what was it,

but what will be stable objects. And the conjecture, central conjecture is the subject that this category has very canonical stability conditions. It will be the main source of stability condition from geometry. This C has canonical stability condition.

As the object of this category, it's very complicated things that we want to solve some abstractions through. But one can consider a 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 you this answer.

First of all, objects are, it's a pair l rho. l is, let's say, oriented.

So it's theta. And I want to say that it's theta stable object. It's oriented Lagrangian subvariety. I don't say it's submanifold. It's actually could be singular, with singularities in co-dimension two.

And x is a symplectic manifold? x is symplectic. Yes, x is a symplectic manifold. Yeah, it's a Calabi-Yau. It's a symplectic manifold. It's a real object. A has singular, what happens in grads? So what kind of singularities?

Nobody knows. But I expect it's, I'll put some kind of conditions. And there's a reason to love the Lagrangian, saying that you're using grads in Lagrangian with the cube as the whole thing. No, no, no. Here, it's absolutely essential to have singularities. Then what about your special kind? Of special kind, yeah. Of any singularity. 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, collaborate it here. Two kind of L minus singularities, outside of singularities.

Then it will be, belongs to, again, an argument of the singularity. All t is parallel to the stability, yeah? Yeah, t is the parameter of stability, yeah, times some positive density.

Well, one that, I mean, if it's special Lagrangian, it's going to be calibrated so it's absolutely minimizing locally, so the singularities are always cogent in at least two. Yeah, codimension of at least two, yeah. By ongoing regulators. Yes, yes, yeah. But, yeah, this singularity, she can appear for the singular Lagrangians. Yeah, it's very special singularity, definitely, yeah. And rho is an 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 has come from some question about orientability in Foucault category. And, but I think it actually means it should have some Dirac operator, should play some role, but nobody knows how to do it.

And, is that right? Homology class, it's actually not homology, but class in K theory, because it's spin manifold. 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 such object exist. In the condition, but they may not exist. OK, but then the category will be 0. No, the claim, this form is exactly the least of stable objects. Think of Foucault category in material, there, there.

Yeah, yeah. And if we believe in mere symptoms, Foucault category is equal to coherence shifts on some other varieties, that will be, the category is very untrivial, and. We have a few examples on the taken side, I guess. Definitely, yeah. Yeah, so that's. So this end is mirror symmetry, the stability condition on one side corresponds to stability.

Of course, yeah. It should be kind of the only feasible way to construct stability condition on this category of coherence shifts, yeah, at the end of the day. Yeah. But here, of course, we say that it's Calabi-Yau variety. I think this Calabi-Yau, it's too rigid condition.

What one really need? One need one form Foucault category, definitely. But omega and 0, one can replace by complex valued, closed, and form. Closed D form of middle degree. With some non-degenerous condition should be kind of, should not vanish on all real Lagrangian subspace.

And it's a very soft condition, which will. But form is complex valued. 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 not necessary complex form, just forget about remaining metric. No, but then you need no meaning of special Lagrangian. No, no, no, it means that this, I think it's equal to. Well, but they can embrace the property of difference. No, no, no, no, no, the story is a fault. It will be still the same story, because one can always integrate.

No, no, no, there will be no metric. There will be no metric on the space. But you get formal, it's just called omega's form. And you can always integrate absolute value of omega over Lagrangian manifold. You get, and you call this area. Yeah. And this area minimizing will be special Lagrangian, just because it will be kind of sum of complex numbers

with different slopes by triangle and equality. Yeah, so it's without metric, but one can also write kind of area functional and minimizing, and these things will be minimizing this area functional. OK. Well, I believe that in this region or dimension,

the only possibility for this, primarily was 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. You may have. Yes, yes, absolutely. No, exactly this definition.

Consider a form which is not vanishing on any. But so still we can point to the kind of value for structure, right? The function minimize, which is not area, but suppose the function. Yeah, suppose the function which looks essentially like area plus. And we absolutely are minimizing the point of response. Yes. To the right example. No, of course, you just vary a little bit here.

No, except you see that the equation maybe if I've never solved all that. The question there are solutions. Yeah, but you don't have to, yeah, but the claim that it's explicit representative, if it's not relative. Like Kelvin metrics, you can change for, in the 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. Omegas, you get different equations. No, there may be no solutions. You can just have a solution. No, no, no, no. It's elliptic equations of index 0. It's completely, the reasons of.

One minimizing and one being calibrated. No, no. It has index 0. The reason is the following, because if you have Lagrangian manifold, and now deform is by exact Hamiltonian transformation. So deform 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 a 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 0. So you get a map from space of one dimension to the space of the same dimension. So really index 0 is this equation for being calibrated.

If you consider deformation, and even you get a large family, because you consider if it's not one connected, then consider first homology, then consider non-exact deformation. So you get really modulus basis of such things.

And it's very important to put this local system all together, this modulus space of this, maybe not reducible, direct sum of reducible of such type. The claim that the modulus 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 a reducible, you can have harmonic metric on the system. Suppose it's Calabi. Oh, you have harmonic metric.

And then when local system goes to infinity, then harmonic metric will be very, very fast, and then you get kind of real Higgs field. When r0 goes to infinity, you get real Higgs field, and which can be thought as a point twice,

you get family of commuting Hermitian operators. And the spectrum will be, which can be interpreted as kind of multivalued. You get analog of spectral curves, which will be multivalued, real harmonic one form on L.

This limit of representation goes to infinity, you get multivalued harmonic one form. And this multivalued harmonic one form will be deformation as a special Lagrangian. Multivalued is kind of multiple cover of the thing. 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, OK. So that's kind of very, very good picture. And I think I'm not 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 reasons roughly the following. In general, conform field theory predicts that you have a category and stability,

but no geometry, just category and stability. That's what conform field theory 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 should be maybe not the whole target space. Some part of your theory will be degenerate,

some will be non-degenerate. And roughly speaking, you can think like you have a Calabi-Yau manifold Y, which is fibered over some Keller manifold of small dimension and small fibers.

And x is just Keller manifold. Consider such kind of elliptic vibrations and so on. The case resurface degenerate to projective line with very small elliptic curves. And if you want to describe what is Foucault category on Y, you can try to see what are special Lagrangians. 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, Keller 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 Foucault. If you get Calabi vibrations, generic fibers, again, Calabi-Yau, you consider Foucault categories of fiber.

And because the Keller class doesn't change, you get local system Foucault categories of fiber. And 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, which should be top degree of the tangent bundle at point x. So there will be stability. Well, it's not a 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 on point. What happens again in this big example, you integrate holomorphic volume form along Lagrangian varieties

of fiber. You get volume four 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 singularities co-dimension one.

And you get singular Lagrangian in x, which is union of some kind of L beta. 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 the x to the x, you get holomorphic volume form near Lb.

And then this thing should be special with slope c with respect to this holomorphic form, with respect to this. So the argument, this thing should be equal to theta.

So you get some bizarre equation. And then it's in vertical dimension one, with 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 the main structure. Yeah, so you 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 like a kind of natural framework to define for chi categories, even for this not constant shift of stupid categories, so it's point to point. But any category and the whole things

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 consider something

called preprojective algebra, associated with some dinking diagram. You have arrows, some kind of dual arrows, like for this quiver n. You get some arrows, and you get dual arrows, and sum of commutators, but not complex conjugate,

will be equal to 0. You get, this will be a 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 special stable object. Very beautiful things, which I discovered by several people,

and also by Guyote and Murnetzky. It's on a complex curve. You can see the multivalued holomorphic form. Again, it's called spectral curve multivalued

holomorphic form. There's frankly many values in some verification. And then you can see the three value 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 value graphs. Yeah, because Lagrangian manifold is a graph in two dimension. And then you get some kind of nice conditions, some gradient lines, and just on the surface.

Just graph drawn on the surface, one can draw a 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 geodesic. There's no metric.

It's a gradient if consider a holomorphic form and consider rotate by angle theta and consider level set of real part. Yeah, so it's beautiful objects. And this is a stable object in some three dimensional Calabia category. Yeah, so there's some kind of high dimensional six-wing

code in simple language here. OK, thank you. Are there any questions? Any questions?

Sorry? Why do you call it stability structure? It's not me. It's Bridgeland. I know, but why is his name? It's a bit strange name, yes. No, it's called stability, even. Stability condition. He called stability condition, which is even. Stability condition, yeah, I don't know. It's the condition.

Yeah, to call something some structure condition I found it's a bit odd, yeah. Maybe it's because he took it as a definition then, I suppose. Yeah, but it's not like stability polarization, you say, no? It's like polarization, yeah. It's analog of polarization, yeah. I mean, it's all starting with the buckled down curves,

so stability condition is part of the schema. Yes, yes, yeah, of course, yeah. Yeah, but a curve in the sense analog of quivers, but go to high dimensions is really hard, yeah. Is this example related to kitchen system? Yes, yes, yeah. So these graphs are? These spectrometer graphs, yeah.

In fact, this is stable over concern of three dimensional caliber category. Are there more questions? I think we have time.