 Так, спасибо большое, у меня есть дуэлс-статус. Так что, как один из организов, я должен сказать спасибо всем учебникам, кто уже остался, и кто будет верить очень скоро, для того, чтобы здесь приходить. Так, спасибо большое. Так что, secondly, как много лет, коллаборатором и другом Максима, я должен следить за традицией и сказать несколько слов, так что я должен считать, когда мы встретим, что случилось, но, наверное, я имею в виду, что это слишком много слов о том, как кто-то встретил Максим впервые. Так что, когда я подготовил этот ток, я просто пытался вспомнить несколько эпизодов, как бы снапшот из моего память, и я вспомнил три, которые демонстрируют Максима, broadness and power, and mathematics. Интересно, что все эти три реколекции, которые происходят в 1-й годе, я просто скажу в 1 или 2 фразе, и люди, которые знают, что произошло после, что был разработок, они будут реально понимать. Первая реколекция, это произошло в старом строении Макс Планкинститута, а не в новом, и... Да, да, да, да. Я также помню это по-другому. Я помню, Максим встретил меня в коридоре и спросил, если я знал, если я встретил формулуру для количества рациональных курсов и CP2, походящих через фиксированные поиски. Это был один. Окей, да. Второе, это было... Я был в Институте в Адвентстаде, и я discutал с Делиней деформационной кванитизации. Он был интересен в том, что в то время, и даже написал пейпер позже. Один день, я получил информацию от Максима, который говорит, что хохшелт, хохшелт, хохшелт, хохмологический комплекс от алжебровых функций по-сонному, должен быть формулуру, деференциал, алжебра. Итак, я сказал Делиней, он не верил, он сделал компьютер, он не верил, но, как мы знаем, все эти деформационные кванитизации, история, и Максим был прав. Да, я думаю, что он даст. И последний, это произошло, я думаю, что я был в Институте, и я оттензовал семинар, курс, который был от Юрии Манин, и в день, перед тем, я получил информацию от Максима, который basically contained just one phrase, I understood mirror symmetry. И так, это было что-то, как-то 1992, 1992 года, и я сказал Манин, что он был шокен по этому поводу, и 2 месяца позже, там был Арбай Стагнон, где Максим дал его話, о том, что сейчас называется хомологическая мировая симметрия. Итак, это 3 истории, и теперь вероятно, что мой話, который, конечно, тоже может быть, просто маленькая часть о том, что мы делали с Максимом. Это про хомологический холл-лжебраз. Репрезентация. Так что эта нотика была introduction в нашу joint с Максимом. Я думаю, это архивная референция. И я бы хотел покинуть этот субъект о моем репрезентации, соединяющейся к Максиму. Было несколько лет, Виктор Катс был в ВИЧ, и это была половина дня, к его 70-х годов. Да. Максим говорил о том, какие-то similarities с геометрией в Викторе, и с симметриками и с кусимметриками. Поэтому, он мотивировал эту notion of wall-crossing структуры, которые мы делали. Я бы хотел спекулировать, что хомологический холл-лжебраз есть что-то, которое появляется на скусимметрическом стороне, на симметрическом стороне, в котором у нас конвенсиональный холл-лжебраз конструктивный холл-лжебраз или холл-лжебраз или холл-лжебраз может быть просто неправильно, но может быть не так. Поэтому, я просто объясню немного, suppos we have a quiver like an oriented graph finite graph this is a set of vertices and suppos we are interested in finite dimensional representations of the quiver so say over complex numbers or a finite field that really doesn't matter to each vertex we associate vector space of some dimension and to each arrow a linear map so then we have dimension vectors which adjust collections of non-negative integers and the Euler form or Euler-ringel form as people say associated with this quiver so in general it's indeed Euler form on the k theory in this case the category is homological dimension one and it can be computed entirely in terms of the dimension vectors so this is collection say gamma2 is a similar one so that's the formula this is the number of arrows between vertices i and j so it's neither symmetric nor skew symmetric so there are two possibilities we can symmetrize or skew symmetrize so if we symmetrize so then we can associate the corresponding cut smoothie Lie algebra and it has triangular decomposition into the negative diagonal carton and positive part and one can quantize the positive part or the direct sum of carton and the positive part intrinsically this is ringel's idea for that one need to consider representations of the quiver over finite field and construct the whole algebra which is basically algebra of functions with finite support on the stack of the abelian category of this representation so if we have two functions with finite support for example delta functions so it's constructable version then their product is defined up to some normalizing factor for which there is some ambiguity so it's basically sum over all possible extensions geometrically it can be interpreted in terms of certain hacketype operators because here we consider flags of an object and sub-object so you can imagine this is a kind of hacketype construction and so what we get in the end that this whole algebra gives us a quantization it's quantum group in the sense you want me to be very pedantic okay so you consider the sub-algebra which corresponds to say simple representations with a dimension vector of this type simple representation you generate the sub-algebra so those give you shiwale generators in the quantum groups and so on so this is a kind of very very classical story it's an industry with many names starting with Ringel so I really do not want to give the full list of names and even the partial list of names but what I would like to mention this constructable whole algebra let me call this constructable whole algebra so it appears in real life in geometry not only in algebra when one it's impossible what people would say or physicists would say on the instanton side of geometric engineering namely one consider torsion free shifts torsion free shifts on p2 of a fixed rank with fixed framing at the infinite line so this gives some modular space actually collection of modular spaces which depends on the rank and on the second churn class for r equal 1 one basically get just a Hilbert scheme of c2 and so on the direct sum one get action of hecke operators kind of hecke operators and representations of algebras which are closely related to constructable whole algebras appear this is the work of many people including nakajima shifman basero i think maybe more recently akunkov probably i should mention kapranov in relation to the whole algebras associated with curves anyway this is what i would like to think of the story associated with a symmetric well it's kind of a nakajima quiver varieties so then we first should symmetrize so we basically in dimension 2 there are kind of a nearby kalabiyao so then our bilinear forms are symmetric now on the kalabiyao side geometric engineering in physics so the original idea was to compute i don't know to model the spectrum of certain 4 dimensional gauge theory in terms of the stream theory and it's not well understood mathematically but there are some nice coincidence of certain generating functions for example for like kind of a classical story for r equal 1 here one can consider the generating function but not just of the cohomology of Hilbert's scheme but equivalent cohomology of Hilbert's scheme with respect to to these torus and this equivalent cohomology generating function like that it's it coincides with the generating function of this partition function as physicists will say of certain 3 dimensional kalabiyao for r equal 1 it will be the resolved conifold the total space of rank 2 bundle o minus 1 plus o minus 1 on p1 so then it's a kind of a very active and stimulating idea yeah this run if you change the rank you change your kalabiyao so it's like you have something with i think a or maybe a r singularity and then you resolve so that's what so what's happened on the kalabiyao side so this scheme of course remembers very well but not everyone may be in the room know so let me brief you recall it's a cohomological whole algebra story the idea of cohomological whole algebra story it's a following it's very general so if we have a 3 dimensional kalabiyao so we have a duality a third duality in dimension 3 so then the Euler form alternating some of dimensions of x group is Q symmetric it's Q symmetric for any odd dimensional kalabiyao yeah alright and so basically 3 dimensional kalabiyao category it's kind of an analog of the algebra endowed with a scalar product it's indeed like if it had one object it would be kalabiyao algebra so then for any object here one can define the potential kind of a generalized чернсаймон where I have these brackets which is a scalar product or pairing kalabiyao structure the objects and these are higher composition maps because everything is a infinity in this story so this is a potential alpha is in general is an element of degree 1 so then it can be thought as a function as a partially formal function kind of defined in the it's an intuition it's an idea, it's not very precise but it's useful so as a function defined in the formal neighborhood of the space of objects so then one can hope that if one has a function so which actually for which objects belong to the critical set just look for the shift of vanishing cycles and so kind of informally the space where the product structure should exist should be the cohomology of the actually it's a stack something nice it's a stack of objects and this is the shift of vanishing cycles yeah that's so I compactly supported then you should dualize that's true but anyway ideally we would like to have such an algebra which exists even in a triangulated framework so this goal is not achieved but there are some special cases which appear in the nature in applications and it's a framework in which we did define this cohomological whole algebra it's very explicit so it's a quiver with a potential so quiver is an oriented graph and the potential let's work over complex numbers it's a polynomial in arrows of the quiver which is cyclically invariant if there are no cycles then the potential must be equal to zero and so this kind of wish or desire in this particular case it indeed becomes a theorem which says that if we consider the graded vector space it will be the sum overall dimension vectors so because the stack of objects is a disjoint union overall dimension vectors of quiverient cohomology compactly supported if you wish here I have representation space of dimension gamma with the coefficients in the shift of vanishing cycles and if I have a cyclically invariant polynomial then the trace is well defined so I have a regular function now some shifts so yeah introduces Euler form it should be the shift by the value of the Euler form on this vector and so I have dualize I think this is it so this is actually the same but now it has a meaning and so the claim is that these vectors this graded vector space in fact it has more than just the grading with respect to the weight latest with respect to the dimension latest but the claim is so h carries associative algebra structure but you jump from Calabria category to quiver and then quite explain so if you have you see in this case it's category of homological dimension one and you can use you can extend it I mean there is an algebraic way to do it you can add x3 and x2 and x3 such that this will be these two spaces will be dual these pairs will be dual it's like in geometry like if you have a curve you should know I mean physicists especially like it if you have a curve you can for example construct a three-dimensional Calabria starting with something one-dimensional so there are several ways to do it in geometry but anyway the product comes so the product on H it comes indeed from the Hecke correspondences so we consider so that's diagram maybe let's do it this way and basically okay so this is a space of flags of representations of these dimensions and this is a space of representations of dimension gamma one representation of dimension gamma two so then the projection this projection maps the flag into the quotient this is the left and the right is of course the projection to the middle term so exactly as in the definition of constructible whole algebra and so this pull back push forward construction gives associative product so of course one has to prove it and okay so then there are several versions of cohomological whole algebra so for categorical purposes one has to use the shift of vanishing cycles but in fact what we have if you see we have some algebraic scheme whatever x and the function only the regular function so any cohomology theory for such a pair will give you a version of cohomological whole algebra and we considered all of them for example you can take like relative cohomology something like this or maybe even just this okay so anyway yeah you can start the space objects is a function and then this cohomological whole algebra what do you want to say you see you have upstairs this I want to say that in general in order to define algebra structure I need to define the space and the space is cohomology with coefficients of shifts of vanishing cycles and some function in three dimensional it's a kind of in geometry you heard it from Richard Thomas talk that there is some perfect abstract so like space of objects if you can make it even in the modular space it's a space of critical points of some function so this is a sort of sticky version so I want to have some potential and ideally to define it in a very general framework for any inconstructible three-kalabia category with some conditions like orientation data so I don't know how to do it in general nobody knows where it can be done one example is a quiver with a potential it defines the three-kalabia category by means of Ginzburg construction you can even assume that the potential is zero then you still can make it into three-kalabia then you get just a cohomology an equivariant cohomology of the representation spaces which simpler but in all the cases for example if W is zero so you can use hacketype diagram to produce an associative product it's not really a subject of my talk it's just I didn't give you no I didn't give you any algebra there are two final fields ah, to constructible no relation it's similar yeah very very different no that's the whole point so it's something which sits on the it's not just in general it's an object Категории, тендер-категории экспоненциал-ходж-структуры, потому что, видимо, есть ходж-структуры в Кахамулахе. У них не только один грейдинг, но и много грейдинга. Для Квиво-Рой-2 есть много грейдинга. Да, для Квиво-Рой-2 есть много грейдинга, но это что я хотел сказать. Есть очень несколько случаев, когда это не explicitly. Очень несколько случаев, когда это может быть компьютено. Я имею в виду, что это очень страшно, потому что один из моих мотиваций в этом интервью было обзорвать идеи от физики, например, от Харвиа и Мур, и конструктировать, что они называют БПС-элджебра. Элджебра от БПС-элджебра. Есть дефинит, но, несмотря на это, мы не можем компьютать в экземплярах, которые интересны для физицей. Да, это... Окей, но в любом случае, что мы знаем? Если мы имеем Квиво-Рой-2 с тривиалом потенциалом, а х-элджебра от Харвиа-элджебра есть шафел-элджебра. То есть, продукт может быть компьютен с локализацией Торус. Например, потому что я нужен этот пример позже, если у меня есть эквивер с 1 вертекс и n-лупс, это, опять-таки, это экземпляр с нашим пейпором, а потом корреспондент, так называемый qn, а также алджебра, которая есть шафел-элджебра, но, в принципе, она может быть компьютена в любом случае на что-то, которое нет. Так что, если n-элджебра есть даже, например, если нет лупсов, то это инфинит-грасман-элджебра, и если n-элджебра есть от, например, эквивер с 1 лупсов и нет потенциалов, и тривиал-потенциал, это инфинит-симметрик-элджебра. Опять-таки, variables, но, ок, инфинит-симметрик-элджебра. Сейчас... Да, это вектор спейсов полиномиалов в один variable. Это мой следующий... Да, да. Для n-элджебра 0. Ок, так, дай мне его explicitly, потому что это мой главный пример. Так что это A1-элджебра, которая на обычном конструктивном алджебра сайт, это correspond to the l-элджебра SL2, и это quantization of... just quantization of the plus half of SL2, it's a polynomial in one variable, because there is just one nilpotent... A1, I mean just A1. You see, Sasha, one point. Ок, let me make it bigger. This is A1, it's not... But for co-homological... You see, the point is, that as a vector space, it's a direct sum of a covariant co-homology of the point. So it's already... it's polynomial for each n. So it's infinite dimensional, it has nothing to do with conventional whole algebra. So it's very big. One, three, five, seven. Ок. Yes, ok. So if you have a symmetric weaver, more general, the number of arrows from i to j is equal to the number of arrows from j to i, then this is a super... it's super commutative here. It was proven free super correct, free super commutative. So it's symmetric algebra of some graded vector space. Free super commutative, it was proven by Sasha Efimov, and more recently it was generalized by Ben Davison. So here the potential is trivial, and in Davison's story it can be non-trivial. And for him he needs a kind of what he called symmetric structure. It's an isomorphism of the quiver with the quiver with all arrows reversed. But the result is the same. It's generalization of your story. So it's still super commutative. And also there is like a special computation in our paper where we have Jordan quiver, which means quiver with one vertex, and potential which is polynomial of an arbitrary degree. So then depending on degree the algebra, the co-homological whole algebra is a tensor power of one of these. No, for any n, it's a tensor power of... if you have a potential... Yeah, but it can be polynomial. You forgot your own paper. It's too many things. I think it's power of this exterior algebra. Yeah, tensor power of this... Ah, okay, sorry, yeah, yeah. The one... So you remember, it's very good. Yeah, it's the power of this exterior algebra, yes. And actually I think we proved it just not for the shift of vanishing cycles, but then it's using the methods of Davidson. It can be proven for critical co-homology as well. So anyway, I think this is the end of kind of a reminder. And yeah, I can show you. So it took quiver. It's this one. So in this case, the algebra is neither... It's not a super commutative. And now we have two vertices. And for each vertex, we have... You see, it's a subquiver. So if you know the answer for the quiver with one point, so you know that inside we have infinite Grassman Algebra. So they sit inside. And there are non-trivial relations between these generators, which I don't remember, but I can look if you'd like. Would you like me to... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Okay, now comes a question. So this is the end of the reminder part. And actually two questions, two natural questions. So we introduced this cohomological whole algebras for the purposes of the Motivic Donald's and Thomas' theory, which I'm not going to talk about. But there are two questions which I believe are interesting. So first question is a natural representation theory for cohomological whole algebra. And as kind of a motivation, which I can even refer to one of the talks at this conference, I remember there was a talk of Gukov at the beginning of the conference and he mentioned something like that algebra of closed BPS states should act on Gukov's talk on the space of open BPS states. And if you look at his paper with dosage, you will see pictures like this. So this represents open BPS states probably in some sense which should be understood. Physicists think about the maybe real analytic shifts supported on surfaces with a boundary, which are complex analytic on the complement of the boundary with some boundary conditions. So there should be a modular space of such, but this theory is yet to be developed. And so this is the open part, this is the closed. And this is a more familiar story, for example, in Richard Thomas' talk, he heard about stable pairs. So he spoke about shifts, pure shifts supported on curves. This is a classical story. Actually some people still think that Donald's and Thomas' theory is just counting ideal shifts. It's something about shifts with one-dimensional supports and this is a sort of an action. If you smoothen this somehow, you still get something supported on the surface with a boundary and then there should be an action of this part is understood better. And the candidate for this is a cohomological whole algebra. And this is geometrically not understood, but we can try to look algebraically what should be here. Especially since in some geometric cases we can model the space of closed BPS states in terms of the quiver with potential. This is the first problem. And actually I wanted to write down the second one on the same page. And the second one is a double of the cohomological whole algebra. So what do I mean by double? So people who are familiar with constructible whole algebra they know that whole algebra gives you the quantization of the nilpotent part of how it was devoted to GQ plus just one-half of the cut smoothly algebra. You can add a zero part, but the construction of the whole thing, the whole quantum group, it requires some extra efforts and I think it's still not very satisfactory I mean all the proposed solutions. So basically even in constructible case whole algebra gives one-half of what we need. And the question is how to construct another half. Fortunately we know the definition of the full thing which we want to get. It's a quantized cut smoothie algebra. But in the case of cohomological whole algebra we know one-half but we do not know what the whole thing is. So then the question is are there any methods which allow us at least to guess what will be this double? Okay, and I think that these two questions, these two problems they are connected and in particular studying the problem A we can give some guess at least in some cases of what should be the double. And the idea is kind of a rough idea. It can be traced even in classical papers by Nakajima. So the idea to B, if we know the answer A. So recall that in one of his very first papers Nakajima recovered Heisenberg algebra by looking at certain hecke operators acting on the cohomology of Hilbert's scheme on C2 or we can take equivariant cohomology if you'd like. But the point was that he constructed two representations of the polynomial algebra of infinitely many variables. Basically two representations of the same algebra which as we know in the end they constitute two halves of infinite Heisenberg algebra. But to see the commutation relations one has to go to the representation space to compute the representation operators and what we get should be called if we do not know any algebra and no just geometry should be called infinite Heisenberg algebra. So then we can try to do the same thing. We know what cohomological whole algebra is. So let's try to construct representations some natural representations but we would like to have kind of a two type representations in the same space by erasing operators and by creation and annihilation operators. Then commuting we get something twice bigger. I don't know the general answer so but let me illustrate it in the case kind of illustrate the idea for a one quiver. In that case we know what cohomological whole algebra is so we can recognize the answer. And the idea in general is to use so in general construct representations of cohomological whole algebra in cohomology spaces in cohomology of moduli of stable frame representations. So differently similarly to this story with PT stable pairs in that case we have smooth schemes not stacks so the cohomology are finite dimensionals and the representations can be handled. So how it works in the case of quiver A1. The framing so the representations of such a quiver is given just by a vector space of arbitrary dimension. So the framing is given by an extra vertex with a number of arrows. So definition I do not want to give a general definition. There is a very general definition of what the framing is. So stable frame representation so which representations of the quiver should be called stable frame. So we put a one dimensional vector space here so we have a bunch of arrows. For example let's consider just one arrow for simplicity. So then the representation to define this in general for general quiver what's a stable frame representation one need to choose a stability structure which in the case of quiver is just a stability function which is the same as a bunch of points in the upper half plane. And then we have the notion of slope stability in this case. So the representation of quiver is called stable frame. If there are no sub-representations which contain the image of this one dimensional vector space under all arrows and which has a slope which is bigger than the slope of the whole representation space. In the case of quiver A1 this is empty so the only stability condition is a trivial one. So then basically stable frame representations are just surjective maps. So then the space of stable frame representations is isomorphic in each dimension to the Grassmannian think gamma d. It's a modular stable frame. And so we are looking for representation. In this case there will be just finitely many summands of the infinite Grassmann algebra in this finite dimensional vector space. So and then it's quite easy to see. So this has a dimension 2 in the power d and the basis vector. So basically in general where we get such representations we get them similarly to the definition of the product on cohomological whole algebra we get them by considering short exact sequences. For example we consider short exact sequence where 2 representations are stable framed and this one is arbitrary. Then again we have 3 projections 2e1 e2 e3 and the pull back push forward constructions give an action of the cohomology of this stack on the cohomology of this modular space. So then we get representation in this particular case we get representation which is naturally realized in the finite dimensional motion of this Grassmann algebra by the ideal consisting of or spent by by wage products with the last with more than d more than d factors. No it's finite dimensional. No no it's No. No. No no no no All possible. The basis of the quotient maybe I just Only your polynomials. If d is 1 you have only polynomials in too many variables. It's still infinite. Maybe it's the last guy. The last guy. You are right. I don't imagine why he started from d plus 1. From starting from starting from d plus 1 I guess. I don't know. Okay. You are right. CD plus 1 and so on. So then indeed the basis will be just these things No. You are right. K We are K K from 1 to d I guess. Yeah. Agree? Okay. So and the action is a natural one. So it's just a product. And similarly we can project to the last factor which gives us which gives us so this gives us a kind of creation operators if you like. Texts like this and projecting on the last factor this is a correspond to the projection on the middle. Projection on the last factor gives annihilation operator contraction. So then one can commute what we get in the end what kind of representation we get and the answer is that we get representation of the Lie algebra of the orthogonal Lie algebra. So then the natural guess is that the double homological Hall Algebra in general in this case it's a Cliffhart Algebra infinite Cliffhart Algebra with two types of generators and the central element so this is one half this is one half so this is infinite grassman this is also infinite grassman and this is a constant times one of course when we consider finite dimensional representation this is finite dimensional story then the central charge must go to zero and of course it's well and it's naturally again to guess that if we have a quiver within loops and no potential then we have a similar story if we have even number of loops infinite grassman infinite Cliffhart Algebra even if we have odd number of loops we have infinite Heisenberg Algebra so very much similar to the story to the story with Nakajima alright so and of course well one can square is the quiver A1 and if you look for the one half you'll see that this XCIs plus they correspond to the positive part of the affine Lie Algebra SL2 hat so A11 quiver so maybe this is where the dependence on the quiver is hidden plus they in fact come from the corresponding affine Lie Algebra okay so do I have any time? theoretically not okay so then okay so then it's better to stop I know that there is a Russian tradition that you cannot wish anything to a person but and so for Maxim it will be in August but we are in France so I can break this tradition and wish you like happy birthday Maxim so thank you