 Prijezovalo občasno je pribrati na tježkega stabilitva, da ne lahko vse priberimo, da imamo početne. Prijezovalo pribrati na tježkega stabilitva, da imamo pribrati na tježkega stabilitva, da imamo pribrati na tježkega stabilitva. Kaj ga je pribrati? Ako ga početno pačemo, vseč vsečo. Vsečo, kaj je inštavna stabilitva, načo je vsečo vsečo vsečo. I zelo, druga vsečo je nekaj analitivne, nekaj je kajštavne, nekaj je kajštavne, nekaj je kajštavne, nekaj je kompleks in nekajče. Vsečo, nekajče, nekajče. Vsečo, nekajče, tačna, ki sem si čast na to, s boj, ki so vzelo na veli, ali ne zelo, da je to počudno, da se način način način način način, and it is better to start with an example. And the basic example to keep in mind is the following. We start with the following data. We have first arbitrary field, and we get some kind of finite quiver, ker lahko ingene sela in je odrožena in izgovorila, občinno da je sva oprimodnosti in počekno, odelo se, da tez si všelje vse da pa vsič nekakro residense na turnimi sečnih doličnih doličnih doličnih doličnih doličnih doličnih doličnih doličnih. kaj kajov. Tako, kaj, ko vse vektro spese, in bi se kot obječenje početke. Izvodno, je tudi kategorija abilija. Vse je kernolje in kternolje. Čudovje početek, in začne početek v svoj del, ne zelo ali. Vse je, ne zelo se krivor, i in c for i running the vertices of the cleaver, and with the only conditions that lie in an upper half plane. In fact, what it gives us, it gives a way kind of to put vertices in a clear and upper half plane. So we can really try to, of course, one can try to get some collection of numbers. Supposedly, all these things can just draw vertices now between points in upper half planes. That's a way to kind of put some geometry on the cleaver. OK. And then we have the following thing. Then what this collection of complex numbers will give us, it gives a map. Big z from k group of my category to complex numbers. Namely, if you have any representation, it goes to some of the vertices zi, small zi times dimension of i's component of my representation. And clear if e is non-zero, then zi is non-zero. It has also lines in the upper half plane. And one can speak about argument of this complex number. So define argument of e, which is just argument of this complex number, principle range, which is something between zero and pi. You get some vector in upper half plane. That's the direction. OK. So one can speak about argument. And the main definition is that non-zero object is called theta semi-stable. This sigma is number here, if argument of e is equal to v. And there is no sub-object, said that argument of e prime is bigger than theta. Or for any sub-object, all arguments lie on the right. Or argument is theta. Argument is theta. Sorry, I said something about this one. OK. So this is the definition. So some objects are semi-stables and not semi-stable. And kind of very basic theorem is, say, the following. First, that for any object in my category, there exists unique integer and a filtration by real numbers, decreasing filtration by real numbers, which has only finitely many steps. And filtration can write the following. So we get zero. So we'll get some n numbers when we get non-trivial jump. And we get a zero, which will be part of, when we get strictly bigger than c1, if e then contains not equal to fn, fhn in your, for which, sigma 1 and 2 be the same as bigger than sigma 2. And contains, it contains the last part. The last part will be f greater than sigma n, hne. Yeah, hn state from hardener symbol. And the first constructor for the case of bundles for carousage. And such that all associated graded is sigma j semi-stable. Yeah, so it's first basic fact. And second is that for any theta, just kind of first introduce some notation, for any angle theta denowed by a theta, it will be kind of full subcategory of a. Full subcategory means to have just collection of object and have the same morphisms in ambient category, consisting of zero and ceta semi-stable objects. And the claim that it, claim a theta is abelian. So we get a new abelian category. And inclusion in sigma 2a is exact. So it means that if you consider kernel, c kernel will be the same as in ambient story. And this sort property, if you consider home from some object e1 in eta sigma 1, to object e2 in sigma 2 is zero if sigma 1 is bigger than sigma 2. So there's no homes from left to right. Now that's essentially what basic one should know. And I'll just give you sketch of the proof. Yeah, first let's prove that such thing exists. This harness infiltration. Existence of h infiltration. And essentially can read from the definition. First, what will be number sigma 1? Suppose if is zero, then n is equal to zero, nothing to talk about. But now if is non-zero, then sigma 1 will be defined as a following. It's the maximal possible theta, such that there exists a non-zero sub-object, such that central charge has argument theta. Object is non-zero. Yeah, because there are only, finally, many possible dimension vectors, as we get one sigma 1. And obviously, all such sub-object will be sigma semi-stable, because it cannot have larger slope. All such sigma prime are sigma 1 semi-stable. And then this kind of one easy argument is suppose we have two such sub-object. Now one have short exact sequence, we have the intersection, goes to abstract direct sum, and goes to actual sum as sub-object in e. Get short exact sequence. The left side and right side are sub-object on e. So the argument of this guy is greater than sigma 1. Argument of this guy is less than sigma 1. Also less than sigma 1. But here argument is exactly equal to sigma 1. It means that they have to be equal here and here. And you see that sum is also semi-stable. So now we can declare this first term of iteration to be greater than sigma 1 hn of e to be kind of maximal such e prime is argument of e prime is equal to sigma 1. And then you can start the first term, you make quotient and the quotient you get obviously some larger number and then you proceed step-by-step. So you prove the existence in similar way. You prove second statement, third statement and then from this you can reduce its uniqueness. So you show us what it means of the zi in the literature? Yes, yes. That's its kind of abstract story. Now we should go to really concrete, more concrete examples. And yeah, I will start with, first I will start with idiotic example, but it's kind of, I want to start. When a quiver has just one vertex and the quiver itself could be just a bunch of loops and you get representation of free algebra. And when you get this central chart you get just one number and you get sigma 1, which is the argument of z1 and this category is equal to this. So there is really nothing to talk about. Yeah, it's completely, nothing goes on at all. Yeah, it's kind of degenerate example. So this really first real example is when quiver is just map from one to two and in this case there are just three in the composable objects. Any, you have map to two vector space, there are three ways how it behaves. It's either called e10, it's k maps to zero, e01, zero goes to k and e11, k goes to k by identity map. And central chart of this guy will be z1, z2 and here z1 plus z2 because dimension one here, dimension one here. Ok, now there are kind of again g1. This number, z1, z2 could be one on the left, on the right. If they are aligned then we get again only one category, nothing to talk about. Now suppose they are not aligned. So this argument of z2 bigger than argument of z1. So the quiver looks like this. Now on my plane. Then e10 and e01 are semi-stable and e11 is not. Because it fits in some short exact sequence it has a sub-object, it has a quotient object that destabilizes it. So there are among these two categories. There are two non-trivial categories, two non-trivial k-sigmas corresponding to argument of z1 and z2. In both are epsilon-cabillion categories, kategories are finite-dimensional vector spaces. Both are finite-dimensional k-modules. And in the second case when z1 stays on the left of z2 and then all three objects will be semi-stable and you get all three in the couple of objects are semi-stable and you get three instead of two, non-trivial a-sigma and again all equivalent to k-modules. So in this way you can see that these depending on stability conditions get just two or three categories. And two in three in fact it's a pretty nice story in theory of Donald's and Thomas's invariants is just some quantum identity for quantum dialog for product of two quantum dialogs is equal to of three quantum dialogs. It's really based exactly on this example. Okay. And but this is too simple. We should go to a little bit more complicated story. Next example will be Kronecker Queer. Again have two vertices but now two arrows. And it was Kronecker Queer because Kronecker solved this question of linear algebra, how it goes to two vector spaces in two maps. And again there is this kind of stupid non-interesting case. In non-interesting case you get again only two directions when you get semi-stable representations and interesting case when you get opposite. And here I should draw the picture not what is going on. So we still on the upper half plane we get Z1, Z2. I take some of this positive coefficients so we get all possible charges of objects and the places when I think exist that kind of three three lines. Difference of dimensions is zero or plus minus one. And how one can write these representations. Usually people draw something like this in vertex one and two. For dimension vector one zero you draw zero and nothing. Or for dimension two one you draw something like this. You get two one-dimensional space minus two one-dimensional but arrow one type on the other. Or you get vice versa. Or conversely you can get things like this. And in the middle when both dimensions coincide you get for example multiply by x0 by x1 one-dimensional space. And this one is point in projective line. And it makes some important expansion in that kind of operator. So you get continuous parameter here what will be point in projective line and this picture. Ok. That's it's called thin quivers to something which we can understand. And there are wild quivers for which there is really no tansva. Just do three arrows. And three arrows what will happen the dimensions vectors which belongs to this square of semi-stable representations in interesting case arguments are in other ways again get to just two one-dimensional representations of the following form. It's either 3 minus 1, 1, 0 to some positive number supplied to vector 1, 0 which can correspond to left and right picture and all pairs of dimension vectors d1, d2 d1, d2 are positive and the ratio is between two quadratic rationalities and these are eigenvalues of these matrices. So it's solution of equation lambda square minus lambda plus lambda plus lambda. Yeah, here I can explain little bit why this matrices appear. There is something called reflection functions for example have just two vector space in three maps you can do things like this, you can map e2, have three maps to the cokernal map cokernal of just a second like this and what pictorially goes on with this stable case you get certain cone you get all integer points in the cone and then you get two approaching discrete things in fact the same pictures one can see when consider the same things appear in something called hyperbolic cut smoothie algebras when consider some cut smoothie algebras to some hyperbolic quadratic form then one get some kind of things inside convex cone and some discrete story going outside and in fact as a question what are dimension vectors can appear in this square arbitrary stability this subtle charges z is not known, should be some definitely some fractal picture pretty nice to set here it's very simple but and also right quivers but one can put some relations, some composition of some arrows should be equal to composition of another arrow you can mod out pass algebra arbitrary ideal everything works as well so it's kind of stupid to ask question just about quivers many more categories ok so this example we should always have in mind and now give kind of general definition so it's it's not a framework of abelian categories and frame of triangulated categories so suppose have triangulated category and again you can try to think as bounded drive category of some abelian category as this representation of quiver of pass algebra or some algebra so you get some triangulated category the complexes and there are two ways to give definition one can go to kind of more traditional way going to using normal t structure so the first definition is a funk the abelian stability the data is map from k0 of this triangulated category to complex numbers additive map and bounded t structure on c satisfying some constrain and what is bounded t structure of direct category abelian category can consider complex in certain degrees that's the way to speak of support in which degrees your complexes concentrated so this notion was I think first appeared in many years ago in this book about perverse shifts by ofer gabber and his collaborators yeah but there are many ways to give the definition what is bounded t structure t structure is the following and I can formulate in very similar through this sequence which appears before so we have full sub category which is in fact at the end of the day will be abelian and it's called heart of t structure and the properties are the following that for any two objects in a there is no negative homes between them and formal notation is that if you shift object by one then you get zero so it's no next minus one like in abelian category it's only positive effects and the second for any object there exist infinite chain of maps and the guys which you know it's the sub notation standard notation is called in terms of complex you can't tronkate you keep commold up to a certain degree and this thing is eventually here you get map 0, 0, 0 and here it's eventually e, e, e with identity map minus one, yeah so this thing is kind of stabilized and this can be since analog of filtration because monomorphism can speak about monomorphism in arbitrary maps and the property is that for any n the quotient map which is denoted by in terms of categories is in a in degree plus n which is notation shift by minus n, yeah so it seems something in the degree and then this thing is automatically unique in the category of the automatically abelian so this is the definition what is the constraint and the constraint is the following one let's now restrict z to k group of a which is in fact is equal to k0 of c and the property is that for any nonzero object in a is z of e we'll lie in slightly more it will be lie for example two ways to say it's plus positive numbers so it should be nonzero number and can again speak about argument belong from 0 to pi and then when you look of this theorem some induction procedure this procedure should be terminate h n filtration exists and then of course unique so it's kind of mixture of two things but in fact the same this equivalent definition the data will be central charge plus collection of subcategories c sigma full subcategories c sigma c subcategories c subcategories c subcategories c sigma full subcategories for all sigma in r and with the property and the but axiomatics a little bit longer but the following so for any object in c sigma which is nonzero the central charge of this class of this object is number of argument exactly cities particular it's not zero and then if you add n it's the same you shift by n for any integer and if sigma one bigger than sigma two and e one in one two belongs to the sigma sigma one two then there's no homes and and then for any object there exist n n filtration hn filtration just get the sequence of morphisms essentially like tau filtration but it will be kind of tau maybe minus sigma one e minus sigma two and so on and such it responding cones of morphisms are belongs to this category c sigma i yeah so it's kind of nice way to combine these two notions and the equivalence both as follows definitions for example if you start with definition number one or to definition number two you declare this sigma sigma plus n is using one of the sections is sigma shift at n for any sigma zero pi okay obtained by shift and conversely if you get definition number two then you define a sigma and I write zero pi it means that object when you make filtration you get only things in this argument all hn filtration in these arguments in this interval and yeah so it's why this second definition is better because what one can see immediately is that you have if you have one stability then you can make another by just shifting this number theta and rotating z yeah so this action of r which is universal cover of s o to r in fact activities you just take sigma mu will be if you get some real number you just declare to be c sigma minus sigma zero volt and central charge mu will be exponent of i sigma zero sigma I just rotate the whole things and in fact this s o to can be extended to action of universal cover of group of gel to r but this plus is positive determinant and the reason that in all these games what we really need for these complex numbers is to be oriented plane we need one raise on the left or the right and one can make kind of by shear transformation make on these stabilites and and hence one and also one get a lot of abelian categories first this category c sigma but also you can see this category from sigma to sigma plus pi or I can go semi closed interval in opposite direction and all three types of categories all are abelian some get huge abelian categories depending on continuous parameter I just want maybe to show some nice example we have this Kronecker quiver and we can see the complexes of representations of this Kronecker quiver it's actually the same from turn grade point of view as coherent shifts on p1 for example this this guy can go to shift shift whatever o and this guy can go to one but shifted in turn degree minus one this gives equivalence and for this category of representation of p1 one can try to think geometrically so the central charge let's take abelian category will be coherent shifts on p1 it's drive category of abelian category what things one can measure the central charge of e will be square root minus one times rank of e plus eventually this category will be something like 0 by I think plus minus degree of e and about coherent shifts on p1 nor everything we get o powers of line bundles every line bundle is got to be sum of line bundles but this sits in parallel line to but here we consider of x where x is a point or divided by maximal ideal of x or can go off square of maximal ideal so we get such in the composable objects and they will be all semi stable so what you get to get one infinite strip one half strip and here is the same thing and if you rotate should by one make it symmetric and rotate by 90 degrees exactly this picture so it's essentially the same stability condition just rotated one and you go from to algebraic language that's one very nice example and one can go to bit more complicated story one can see the db of coherent shifts on elliptic curve again take rank and degree and then by tier theorem one get all points integer points on a lattice and for each direction rational direction we get subcategory each c sigma is either zero or equivalent to something very nice coherent shifts with zero dimensional support on the elliptic curve just a bunch of points and in importance of them and here that's also kind of very nice we have action of SL2z and this SL2z acting on direct category of elliptic curve and everything is kind of nice and clean so that's kind of first glimpses of what our stability look like so we kind of can construct some case of coherent shifts on curves and then people had succeeded to do things in dimension 2 and 3 but in dimension 4 it's complete it's complete block by element remains it's impossible to do anything so the question is why at all we are talking about all such stuff of course it's very clean definition there are various reasons first it's kind of curiosity many new exotic abelian categories which are not modules over some algebras completely different guys even for elliptic curve it would take irrational slope you get half plane you get some kind of crazy categories and this interesting question how what will be algebra behind it and so on but also you get model space of stability structures with some certain technical conditions which I don't talk about right now it's by region theorem it's a complex manifold which is locally isomorphic to a subspace in home from k0 to c and one can make it sexually finite dimensional so it will be some finite dimensional complex manifold but pretty stupid one I have to say just the union of convex cone and cn and end out with action of gel plus to r which is close to the heart of people in dynamical systems but these are still not real arguments if category is three dimensional calabio category so you have some duality home of ef will be finite dimensional will be home of f e shifted by 3 then there is story of Donaldson Thomas invariance which is very complicated which gives certain deformation of this manifold get some kind of nonlinear modification of this guy and the resultant manifold should be really great complex manifold kind of maybe some but which you expect from string theory and gravity m theory and so on so it's kind of really good counting and the main thing it's there should be a lot of original stabilities it's coming from again theory and it will super conformal field theory that if x is let's say compact calabio variety of any dimension then the modulus space of dual calabio varieties should give non-trivial stabilities on this so it's really challenging to construct way to some things and maybe this one life I think it's only modulus space of stability structure is interested if you fix stability structure then in reasonable situation then consider this category a c sigma and fix class in k theory and consider this is consider only semi simple objects in this category they form some nice modulus space and this modulus space will be quasi projective variety so you kind of classify object to some groups organize classification of object in your category in reasonable groups and this will be quasi projective varieties so it was short introduction to stability and maybe I have 5 minutes break and continue to really interesting stuff so now in the second part it will be less abstract story equation, which we introduced by Alistair King, who was a student of Donaldson in that time so in return to situation of two quivers now suppose my field is complex numbers and q is and we get these numbers z i in apahar plane which plays a stability game now let's fix some dimension vector just collection of numbers for all i inverters and assume that it's non zero so some representations are stable semi stable, some are not semi stable and how to deal it with this so sigma will be argument of this number and we define mu i is imaginary part of z i times exponent of minus square root of minus i sigma and then we have the following properties sum of mu i d i is equal to zero mu i real numbers in what is geometric meaning we get just bunch of vectors we get z i and mu i is a projection to perpendicular line to vector z we just put 90 degrees and make projections ok so we get we replace by real numbers and how one can think about representational quivers up to isomorphism of given dimension vector so first one if we have a presentation can choose the basis in each vector space and then the presentation quiver will be just bunch of matrices and denote by v it will be direct sum of all arrows of my quiver of c to power d i d j and on this vector space this group acting which is product of group gl d i c in power text so if it contains collection of g i and here contains t i j then t i j goes to something like d j d i d i d i inverse this will be action on the group so it change basis and see how the presentation looks in new basis and isomorphism class of representations i is a quotient space and then this manford theory of stability which I'll talk more seriously next time says that how to get kind of good part of this quotient and it was slightly generalized by kink and here is the theorem which says the following point v in this vector space corresponds to semi-stable representations if and only if one can formulate question about sub module blah blah blah but one can translate geometrically this action of group on vector space that for any one parameter subgroup g m c c star embedded to g suppose said that limit lambda goes to zero lambda is a complex numbers inclusion g of phi of lambda of v exist so it doesn't so you get some one parameter subgroup which goes to some point inside vector space then for such one parameter subgroup certain things should be positive lim of chi mu of phi of lambda positive plus infinity v order of chi mu in algebraic geometry people like algebraic characters but I write just continuous homomorphism group which collection of g i goes to some over mu pi log absolute value of determinant of g i so you get formal real combination of characters of my group so it's one condition and then it's equivalent to something else consider function on a group capital phi on group namely phi of g is equal to in my vector it will be square of length of g of v coordinate space plus the same thing which I wrote before sum of mu i sum of mu i sum of mu i sum of mu i sum of mu i sum of mu i sum of mu iIT is bounded below on the group how it could be non bounded this term is very positive so if you go to change your matrix go to infinity in the group this thing should stay kind of bounce in vector space but then this thing should be positive otherwise it can go to minus infinity so it's equivalent statement Pobrožena. Tukaj, je to izgleda na MAMFOLD. Včešljaj. Včešljaj. Vse bo vznik, ko je spremal, način ne bo vznik. Ako je vznik, ne bo vznik, ne bo vznik, ne bo vznik, ne bo vznik, ne bo vznik, spremovano iz tezda, neko je to, ko se pri lakovosti sreka. Tao je iznačen več nekaj tezda, da je, trzebno, praviko o performaciji skupilov na daje SEIME- intellecti in se na tezda se sreka po, ne pa ne ino SEIME- intellecti v se themselvesi. To je ne SEIME- intellecti, ale se po sreka bljede, to sem je SEIME- intellecti. vsega sevrim, nekaj ona potrebna, ko težna. Všeč to demo, že ti je všeč nekaj globalno, pri vsej prosim, da tko vsežim, da je svoj skor, nekaj kratik, nekaj kratik, da ta nekratik vseži vseži vseži vseži vseži vseži vseži vseži vseži vseži. Vseži vseži vseži vseži vseži vseži, kaj je tudi v pridu. Všem je všem zelo, da je zelo, da je srečen v trenetih v zelo. Srečen v trenetih v zelo? Vse je tudi trajno, tudi trajnje z vsej metri. Zelo je zelo, da je srečen v trenetih metri. Zelo je srečen v trenetih metri. Tamo je determinen. Kaj nekaj se veče, nekaj se nekaj se se reprezentuje, nekaj se je vsega standardna metrična hrmitrona o vsim vsej veče elektro-spejskih, nekaj se danes se reprezentuje kako se zelo probila vsej, plus hrmišnja norma na svoj EI, in plus octogonalne base, pa izgledajte kodnečne spasje. In izgledajte, je to početnega, početnega na svoje vrtečnega, nekaj je komentant, početnega na svoje vrtečnega, je to vrtečnega, in je to početnega na svoje vrtečnega, of mu i, rejection of EI, and both left-hand side itself are joint operators in total space of the representation of the quiver. So you get this neat equation for the existence of minimal point, and this is something really very deep, because, for example, let's consider case when all mu i equal to zero. In terms of representation of quivers, it's corresponding to things that all the i are parallel, so let's call my number system one ray, and when the projector gets zero. And then we just think about semi-simple representation of quivers, at least those, which admits admitting norms, so that the sum of committance is equal to zero. And it's a very nice equation called Hermitian Young-Mills equation in matrix models, and one can see the following. So there are two algebras, which will be pass algebraic of your quiver, so find pass, and representation of quiver is the same as representation of pass algebra, which has, let's say, generators p i, projector i for all vertices, arrows t alpha for all arrow, and relations, some of these orthogonal projectors, some of projectors is equal to one, and t alpha goes from one to j is equal to j t alpha for i. We got this algeba, pass algebra, which is kind of generators relations, and then we get another algeba b, which contains a, and it will be star algebra. And you add extra relations, that projectors are self-adjoint, and this Hermitian Young-Mills equation. And then we see that same simple representation of a, finite dimensional representation of a, and the same star representations of b up to isomorphism. When we do this identification, we have some kind of freedom, we can make different algebras b, different star algebras b contain a. You just start to rescale t alpha, for example. And instead of this equation, I'll write this equation, so you kind of secretly rescale t alpha, you see alpha arbitrary real numbers. So you get different star algebras with the same representation theory, because the representation theory will seem a simple representation of original algebra, it doesn't depend on the Jesus. So you get many, many bs, and the natural question, I don't do any completion, because representation goes to some star completion, and one can ask whether star completions are canonically identified. It's actually old question I have, some conjecture that star completions are identified. And if you want to make its kind of precise question without looking for other completion, one can do, for example, as a following question, consider a class of functionals on b, t of a a star is greater than zero, t of a b is equal to t of a, t of 1 is equal to 1. Let's see, there are just traces called states in quantum mechanics on this algebra. This seems maps to homomorphism of vector spaces on a model of commotant to c. Small a is not necessary element of small of capital a, it could be anything. And you get the image, for example, the conjecture, the image doesn't depend on the choice of whatever, like for this constant choice of b. Yeah, that's the first case. Same one can do, add mu i to the games, and you should, for some you should have something non zero sometimes you have traces, sometimes do not traces and the kind of the conjecture if my algebra chose the sparse algebra of this wild quiver for which set of pairs mu 1, mu 2 such that trace exist when we consider ratio of mu 1, mu 2 when trace exist should be direction of rays, the following picture. If you remember I get this kind of sector where all rational directions were really present as finite dimension representations some kind of discrete series approaching it of rigid representations. In finite dimension representation I have only rational directions here because otherwise no finite dimension rational dimensions, rational integer numbers but for this thing it can go to factors and what I expect it can get all real rays in this domain so you get interesting question in star algebra ok, now this king's equation one can formulate as the following way consider operator on this space given by sum over committance plus sum over mu i projective mu i, it's not zi, now I put instead of mu i zi, these numbers now it's not self-adjoint operators because numbers are complex but this operator is normal operator, it's commutivist with mission code so you can write uniquely as product of argumentant for some self-adjoint argumentant b and this king's equation I equivalent to the fact that this argument of p which is self-adjoint operator is equal to sigma times identity that's the condition and now I will go to the main definition what is Hermesiem's law it act on what it's a flow action in positive direction it's sex on the space of metrization of e which is product of i a metrization of ei, it's a space of Hermesiem matrix in ei and this is my unity group now I kind of change gear, I don't consider basis I consider abstract fix representation, now change Hermesiem matrix on it and on this I consider the following flow so the element here will be collection of Hermesiem matrix on each piece I take h inverse h dot I get some self-adjoint operator is equal to minus argument of this p so here depends on h bar secretly hidden in the notion of Hermesiem conjugate so it's minus argument of sum of commotant and here depends on Hermesiem conjugate plus sum of i so I get a certain flow and the claim that this flow will kind of analytically produce me hard on assumed filtration in original description how we do, start to search so all possible subrepresentations and so on and here some kind of flow which by kind of depon of exponents will produce you hard on assumed filtration so depon of I will explain what is going on so consider if you also claim this solution exists for all times and consider solution t which goes to plus infinity and now consider ellipsoids collection of vectors I get some ellipsoid depending on parameter t it will be kind of like maybe longer and longer in one direction and shorter in this direction and that automatically gives you a flag so you get things direction when ellipsoid will be long it will be subspaces and so on and the theorem which we have with Fabian Haydn, Ludmila Katsarkov, Pandit and me that it's not written in archive paper but it's still true theorem that behavior of this metric for t goes to plus infinity gives hard on assumed filtration the following way namely let's consider set of vectors u such that logarithm of norm of u respect to h of t is equal to minus sigma j t plus small o of t as t goes to infinity and the claim is term vectors belonging exactly to one specific term of hard on assumed filtration so this will be filtration it will be like exactly in sigma j subspace with filtration that's the theorem and let's explain briefly what's the proof, if we start with already kind of harmonic metric solution of the equation then the flow will be just rescaling with constant speed so you get automatically this story now there are some kind of tools we have monotonicity if h1 of 0 is less than h2 of 0 then for any positive time it will bounce preserved and also if you multiply by constant it will be multiplied by constant for all times it's completely obvious properties but they show that up to o1 the behavior doesn't depend on the initial condition because it can sandwich one metric between another shifted multiply by one constant another constant so if you start to analyze not polystable representation but semi-stable and it's a pretty complicated story because it's small o of t but in real life it will be linear combination of log log t iterated logarithm of t plus log log t so it's very very slow deviations and then at the end we get o1 for which you don't know but it's sufficient for all purposes by some L1 estimates one can prove that it's kind of ansatz if you know solution of o1 in terms of equations it will be so it's pretty sophisticated proof and I just want to say that there are this iterated logarithm so it's really pain in the neck because this is a very closely related equation from which you don't have ansvar essentially the same equation but it's slightly more general consider function of n variables plus exponent minus x1 minus x2 exponent minus x1 minus xn it's function from rn to positive numbers now consider gradient flow x dot is equal to minus gradient of f we don't know how to prove that xi of t is log log log i times so it's a big hope one I don't know how to prove it, I can prove for n equal to already because it's some explicit formula and leponov functions there is a method to prove such inequalities so horrible already in this case it's no way to generalize it so that's and now I briefly finish what is non-archimidian analog of it oh sorry, no I have a lot of time, sorry we'll have non-archimidian version and this non-archimidian version will be much simpler no iterated logs although in principle one can try to add them by hand but not here so now we should consider non-archimidian fields so I'll start lesson of non-archimidian geometry now so K will be a non-archimidian field what's the definition so you get field norm map from K to real numbers set it norm of AB is equal to norm A norm A plus B is less than maximum K is complete with respect to metric given by distance difference now so this if you get such things you can first make kind of 4k will be set of elements less than 1 it's a ring and contained ideal set of A is less than 1 and the quotient it's a field it's called residue field and also get valuation map from K star goes to R and A goes to minus log of norm A the image is a subgroup and I assume that valuation is on trivial image is not equal to 0 so there is some kind of really stupid situation norm of everything non-zero is equal to 1 and I want to exclude this stupid case and so the image is certain on trivial group discrete valuation and the image is constant times integers or could be completely continuous say valuation and the image is all real numbers it could be something in between like rational numbers so what are examples origin is to be introduced for chaotic numbers or you can take Lorentz series in one variable and valuation instead of norm it's the same story of sum of C and N and greater than 0 it's kind of leading degree of leading coefficients or is this a discrete valuation and as a continuous valuation form of power series we expone to real numbers so we take sum over C over lambda i t to lambda i lambda i belongs to resistive, lambda i real numbers and either finitely many or the limit is equal to plus infinity and by the way the same thing one can do with periodic numbers one can consider sum over C lambda i p to lambda i just complete formal notation where C lambda i are digits so right periodic numbers are written as digits i placed in integer places but now they are placed in real places it still can multiply and add such things get complete so get we have this non-archimidian fields and if you have vector space one can speak about norms ometerizations it's a set of all possible maps from v to positive numbers so it is thanks to axioms which you can clear v1 plus v2 v1 v is bigger than zero if v is no zero v1 plus v2 is less than maximum and if you multiply by constant in your field one can also do infinite dimensional spaces and functional analysis is quite tricky but we will do just finite dimensional like in usual complex vector space one can make normal form one can write things explicitly and there is a claim that for any complete flag in v one dimensional space in two dimensional zone in flag basis means identify coordinate space norm of any vector is equal to maximum of certain constants in kids norm so it is given by standard form and if my relation is continuous then one can compensate this lambda i by norms of some elements of the field you just get simpler form, maximum of all xi and then you see that all in continuous case all metrizations are isomorphic to each other and the space of metrizations if dimension of v is some number n is just glnk divided by analog of unitary group which are transformations preserving the standard norm in coordinate space so it is completely parallel to usual complex geometry and in not continuous space the story is a little more complicated save word in the case of discrete one can analyze what is the norm and it is the following it is the same as a choice of you can make a unit ball of your think you can see the unit ball so it is a free module over the ring of integers like periodic numbers submodule in v which is generates vS over k so it is something isomorphic to k to power n n is dimension of v and again the submodule is called v0 and then you make divide by maximal ideal you get vector space over residue field called it is kind of the vector space of the residue field and in this vector space you introduce a filtration by numbers from 0 to 1 and this is something which people in other areas called parabolic structure like you have a bundle on a curve and at some point you fix filtration by real numbers between 0 and 1 it is fine in many steps so this thing is different so that is the picture what will be non-recommended analogs of norms and now I want to define analog of normal flow one can do it in general but I will assume that I have continuous norm it is kind of opposite to what people usually like discrete relation and for me it is much simpler to do in this continuous one can translate from one language to another so the basic example I want to see is this ring which is kind of called Novikov ring so what will be the key ingredient for defining normal flow again will be first question general linear algebra suppose given metrization and point of metrization maybe I will denote something like nu it will not hear mission form it will be kind of like norm I don't know given nu metrization I can I already had this reduction to special fiber language of jremche which will be set of vectors in v norm v nu less than 1 divided by the same thing this norm strict less than 1 and in case of continuous variation it will behave very well it will be always n-dimensional vector space of the residue field because any continuous relation can move to the standard form then we get a maximum of xi without lambdas then we get just coordinate space under the quotient it's also a vector space for residue field of the same dimension now imagine the following I have this guy of the residue field suppose I get certain flag maybe not complete flag zero sitting in one certain length m is equal to be reduced in definition space and suppose I get a bunch of functions which are jerms of functions of continuous functions lambda 1 of t, lambda 2 of t lambda m of t v t is real number very close to 1 so it will be defined only as a germ it's equal to zero for positive t so I have a pass then I get canonically a pass in a germ of pass in metrizations of v there's no choices should be made here how this goes the construction is the following choose any basis of v reduced compatible with the flag it means that subsist of flag will be space drained by first several vectors in the ordered basis and lift it to basis of v so we get some coordinates and now what we do we just oh, I forgot one important property these functions should be one bigger than another and then define depending on metrization nu of t, which we want to define t goes to nu of t in the following way it's the maximum of all indices i x i times to e to lambda appropriate some index depending on i of t this counts to which terms of the flag i ellent so just consider diagonal and the claim doesn't depend on choice of the basis for small t and a lift what you do, just look what you get coordinate change and then it varies to see that it doesn't at terms of kind of high degrees and for small t do not interact don't change this norm so that's something pretty funny one can say it in the following way that kind of tangent cone whatever it means to metrization of v at some nu is a set of all r filtrations on we reduced so it's so the story is that the space of metrization is something very, very non smooth but still has it's like in simplest example looks like a tree and so on and so the tangent space it's not a vector space but it's still something which just a cone can multiply by positive real numbers and this set of r filtrations it's something which you can multiply positive real numbers you can multiply terms of filtration by any constant bigger than zero and it's collapsed to zero when all terms of filtration has zero and the claim that this path of the tangent space it's kind of canonical isomorphic to neighborhood so you get kind of exponential map by this construction yeah okay so that's r filtrations now go to actual construction now fix a quiver and e will be representations of quiver over my non orchimidian field and actually maybe I have to do the following what will be my abelian category A it will be consist of all representations of quiver over K such that there exist a model over presentation of ring of integers and and this can be formulated in the following way there exist collection of norms such that for any arrow corresponding operator from EI to EJ is contracting so it's T A of U normal of U so operator norm is less than one and then the unit both will form a model over ring of integers in this continuous relation and also same as models okay so that's a definition and for representation so we get something non empty for E in A define matrization of E this will be collection of all all such norms UI so it is like this norm of T alpha normal it's not arbitrary norms but main thing it's not empty it's product of I matrization of EI of course we get this constraint so this category it's not all representation of quiver so RSD field for example if you have a loop then all eigenvalues should have norm less than one so I put some constraint like quasi important in complex case next lecture explain how to go beyond this constraint this will be really the easiest situation so I get this a billion categories space of matrizations and now I want to define a flow it's not expanding yet contracting in not string sense it's not steep inequality by the way this if I get E and in object of A and norm of T matrization of E then I get because it has a model I can get representational quiver of the residue field for vector space I denote something V reduced but now I introduce kind of notation which will be next time J up a star nu of E to be representation of quiver of the residue field which we take again vectors of lengths less than one and divides by steep less than one and just to say some notation this kind of tradition in algebraic geometry when you get closed embedding like point in a formal disk denoted by J pull back to point point to formal disk it's just notation just convenient to have this thing so I get representational quiver of the residue field and the formula for the flow is for non-archimedian h and flow is the following like u dot or maybe and one can try to think in terms of Hermitian matrix in universal dot it's tangent vector should be given by some filtration and what is the filtration it's minus increasing by decreasing by minus numbers harnessing filtration on this guy it gets representational residue field it has some canonical filtration by some angles take minus instead of labeled by minus numbers and then we get tangent vector and it will be speed so now it's equation of analysis so the space is horrible what does it mean, what is this ODE theory what does it mean we can solve equation with initial condition in the speed in this case in this case actually it's doable but we have more complicated situation when we ran to some Michael Britches with some horrible problems maybe in model theory that like infinitely many things happens infinitely often and it looks kind of nightmare but here one can describe solution in the following way if you fix the point mu mu zero and I have a filtration filtration of this guy sorry for notations I get this filtration and now I start to rescale them so I can see the I get a pass drum of pass t greater than zero and much less than one so filtration by labeled by minus t sigma j t is a time labeled terms of filtration by this minus t sigma t to j t to j come from this hardware story and I get kind of straight pass this my functions will be just constant linear functions one greater than another it's less than tangent cone but it's actually a pass in a tangent cone straight pass only lambda i will be minus sigma i t minus sigma t these numbers were decreasing but minus will be increasing so we get a straight pass and then we get a straight pass of by this reversal construction gives it canonical germ of a pass in space of materializations so it looks like we can start to move but this procedure if you get any set and then for each point you get germ of a pass it could be something could go wrong here like imagine that for each point you produce then consider nearby point but this should produce maybe some different pass so this will be not compatible and it's easy to check this thing is kind of self compatible so it means that if you start so consider germ of a pass and now start apply procedure to some point on this pass you get the same pass with shifted time and that's good that means that we really this germ gives me really solutions of ODE not some curves which start with the right first speed ok now I get good notion of ODE I get a speed but this solution this is only for some short time there is no long time because the construction is not canonical for big times so change of pass very near to zero because it depends on chairs of bases choose different bases things start to grow straight maybe just move here and now what are nice is on space of matrizations of E there exist a distance it's a metric space namely if we have two matrizations nu1 and nu2 in distance will be supremum of all vectors whatever you consider maybe sum of ui and u is not zero and you consider log of maximum ratio u nu1 I can make distance and this is a complete metric space and now when I consider this pass it moves with the speed less than pi because my argument is between zero and pi this pass is lepschitz continuous so distance between nu t1 nu t2 less equals than pi which is times you implicitly are assuming some written condition in depth no no no I have representational cure I have numbers and my numbers gives me automatically arguments between zero and pi for quiver so it's even less than pi because it's numbers so it's kind of continuous pass and now if I have a solution try to make a solution for my equation then there is some kind of maximal open interval to which it exists then take limit point then when I take limit point I can again repeat the procedure and then by kind of induction or whatever transfinite induction one sees that there is a solution up to positive time because the space of solution should be closed and open subset like in elementary analysis you can always set point this solution for all positive time for all t greater than zero so you get complete pass and solution is unique because for each point now canonical is a germ in positive directions if we set it will be again closed and open subset so get some elementary analog of ODE theory and you get unique solution and now what goes on the speed of trajectory stabilizes for large t what does it mean the speed of trajectory at each point for each moment t the speed of trajectory will be some hardness filteration so you get a bunch of numbers theta and also get some multiplicity some vector spaces sitting in this dimension and if I get to about exact remember only dimensions and the claim the whole things became kind of eventually will not change number of t how to to see it then I get this reduction g t of e is a representation of q over sd field and has some hardness infiltration hardness infiltration I recall you that you split mixed sub-object with increasing thing and in particular it means that your vector the central charge of this vector e you write to the sum over vectors in it gets sub-objects next quotient zone so you write to get some kind of convex polygon or half polygon and then one can consider something called mass it will be sum over pole j theta j absolute value of z of vrth theta j of this hardness infiltration so it will be length of this polygon and what one can show that if you move a little bit this polygon polygon doesn't change but if you move immediately that it will be if you go to the limiting point then these things will only decrease so this mass claim mass decreases and because we have kind of only finely many dimension vectors we get only finely many choices so for some point it stays constant I didn't talk about it but for quiver it's kind of fabulous you have only dimension vectors so you get stabilizes and then then stays the same and and the claim if you get path then speed stabilizes is the same as the filtration of the residue field yeah that's again little thinking you such picture and the conclusion that and this will be hardness infiltration so and then speed stabilizes and I already remove this prediction story over complex numbers and here one can get the following things set of vectors which belongs to sigma j terms of hardness infiltration over big field this set of vectors is exactly set of vectors such that norm nu of t is equal to minus sigma g t plus constant for t is efficiently large so those no log log terms log log log it really goes like linear and that's the end of the story so harmonic matrix in this case this matrix such that reduction is semi simple object over the residue field okay yeah so it's kind of very easy example and yeah so next time will be more and more complicated analysis and at the end of the day will go to really impossibly hard hardcore analysis coming from this even normal commision geometry