 Thank you very much. And my talk will be kind of my mathematical birthday present to Professor Koshivara. You will see that actually several ideas which you introduce in various subjects will appear in quite surprising way. So I will speak about quantum difference equations. So the basic algebra is the following. Fix a complex number. And that's a Q is between 0 and 1. And denoted by algebra AQ will be 3-algebra generated by two symbols, invertible symbols x hat and y hat. Modular the relations y hat x hat is equal to Q x hat y hat. So as a vector space, it can be written as Laurent polynomials. So consider Laurent polynomials, let's say, in y first, or maybe in x first, and then in y second, you can put a canonical in this form. So first, and then one can go maybe to high dimension. One can speak about AQ to power n for large n. So for such algebra like for differential operations, one can speak about holonomic modules. Maybe you can tell in general what is a holonomic module. M or n module is called holonomic. If there exists a filtration, 0 and 1 and so on, such that compatible with natural filtration on algebra in x and x inverse of y and inverse of degree 1, compatible with natural filtration on algebra, and such that dimension of mn is capital O of n to power n. The model is finally generated here. It's automatic here. OK. And is there an inequality or that it can be that if the dimension is non-zero, that the dimension is for a non-zero module, the dimension is at least something? Yeah. No, I think it's a good definition. So in the equation, both x and x inverse have degree 1? Yeah, yeah. Degree of x, x plus minus i, y plus minus i degrees. OK, now I will first describe the case of n equal to 1. And I'll do it in parallel with differential equations. So we have q difference equations. It is algebra of q, aq. And in usual differential equations, you do something like this. You do polynomial differential equations in one variable. And also this notion of filtration and a similar definition of a polynomial module. So when you study differential equations, first you study it's kind of our formal puncture disk. So it means that you replace your ring by this guy. And automatically, holonomic modules after reduction to the things are finite dimensional spaces over Lorentz series, in doubt with the action differentiation, with the connection. No torsion. No torsion, yeah, because it's a field, yeah, plus connection. And here one can also do kind of formal quantum puncture disk. One considers c of x and then take the similar ring here. And modules, again, will be finite dimensional Lorentz polynomials. And the action of y hat means that instead of connection, you get isomorphism of a bundle and pullback of the bundle where psi is dilation. So it's a completely similar story. And among holonomic modules for a puncture disk, you get a very special case. It's called regular holonomic. And the definition of regular holonomic for usual equation means that there exists a E0 in E, which is 0 is free c of x module and such that E is equal to E0 obtained by localization and invariant under operation x over dx. Because, in fact, here one can write the same ring in this way. E0 is invariant. E0 is invariant under x dx. Sorry? E0 is free, finally generated. But it's finally generated. I think it's automatically free. It's called lattice. So one can repeat the situation here. You get E0 in E, which will be c of x hat modules, and invariant under y hat. So it will be a notion of regular singularity. And first, we want to classify regular modules. And the regular modules in the usual situation are the following. It's all irreducible in the composable modules are correspond to Jordan blocks. Where lambda belongs to c module of z. So it belongs to the quotient set. One can say that one can see the coherent shifts with zero-dimensional support on the kind of bed quotient stack. You divide a fine line by z. So it's kind of pure algebraic description. And here's a story that's similar in the composable modules correspond to coherent shifts. No, there's all regular modules as the same coherent shifts with zero-dimensional support on c star divided by q to power z. But essentially, the same reason. Yeah, you have a frangable basic model. You want to write equations something like y hat of psi is equal to lambda of some kind of z psi. You write this equation as it belongs to c star. But when I replace psi by x to power n psi, you multiply z to some integer power of q. And now it's kind of a transcendental map. This thing is isomorphic to elliptic curve. So we change the algebraic structure. And also we can make a transcendental map. It's isomorphic to c star, c mod z is isomorphic to c star by exponential map. OK, that's all very simple. Now there is also a level to retain the composition for d-modules of a formal puncture disk, kind of a holonomic d-modules of this formal puncture disk. This is a billion category. It's an infinite direct sum over blocks, kind of. And each block is equivalent to this coherent shifts with zero-dimensional support on c mod z. And blocks are labeled by some set which is an uncountable set. And the set correspond to kind of irregular types, or expressions like this. You can see the set of all possible formal pieces of polynomials. Some more ck z to power kn. Consider a set of finite sums, such that for certain maybe m, maybe put, such that great common divisors of k such that ck is non-zero is equal to 1, or just second, what do I mean? We cannot, and m is equal to 1. We cannot divide by a common factor. And you're modelled by cyclic group action. ck goes to exponent 2 pi i k over m ck. You can see the cyclic modulable by free group action. So different branches with the same functions. And if you have such expression, you imagine this kind of, you consider all possible branches. You get this demodule, a typical example of demodule in a block. And here's the same story. I find that amplification. You get it, sorry. Then it contains it by a kind of regular singularity in m-sroot. And here's a very similar theorem in, sorry? x1 over m. Yeah, x1 over m, yes, you're right, yeah. x1 over m, yeah, then it will be right. And then you can multiply by a regular model in x1 over m and get this category. And here's the similar story. It was described by several people, I think by Bronovsky and Ginsburg, but not only. Then consider a halonomic Q difference modules over c, no, put it as x hat. It's again the sum of blocks. Each block is the same coherent shifts with zero dimension on my elliptic curve. Oh, c star for the museum. But a label set is countable. And in fact, it's just rational numbers. Yeah, what is geometric meaning of this? Yeah, so you have to see some kind of big difference here. It's countable sets here. You get continuous parameters. What is the origin of these two sets? In fact, it has a very clean geometric meaning. Again, I can do the following. In differential. Before when you did this thing, when you went to your first thing, you ended up with coherent sheaves. Just one of them, there were no blocks. How come it? No, no, it's regular sum, it corresponds to zero regular. It's one special block. It's one special block, kind of the most basic block. Now, what is the origin of these two sets labeling these blocks? The origin is actually the following one. In differential equation, one can see the spectrum of, let's say, c of x, multiplied by projective line. It's called lambda. Like this. Now, so you get a kind of product of very short interval, and it will be x coordinate, and lambda will be here to hit lambda is equal to infinity, in fact. Now, you can see the product. You can see this formal scheme. And you can see the Poisson bracket, dx. Or you can see maybe two form, dx over x times the lambda. This form has d lambda, sorry, what do I mean? dx over x times the lambda, yeah. Just a second. Yeah, this form has poles of order 1 and 2. It lambda equal to infinity has pole of order 1. And now, what you start to do, you start to make some blow-ups at various points. And sometimes, you see devices when the form has pole of order 1. And it turns out that these devices, which you see in all these things, are exactly corresponding to all possible blocks. Roughly speaking, if you have this Poisson series, you can see the graph of differential. You get certain curve, which maybe goes to intersection point. And you make many, many blow-ups. And eventually, you end to have devices all for the 1. But how do you get something to count? Yeah. Sorry? How do you get something to count? Because when you make a blow-up, you can blow up, and maybe you make some trees. You blow up in some real points inside. Not, yeah. And here, what we do, we consider spec over the, kind of again, spec of c of x hat. And maybe I put also times p1 with coordinate y, say. Or maybe I don't put hat. And consider form, which is the x over x, which dy over y. And this, my form, will have, I get two very short devices of poles, when y equal to 0 infinity and x close to 0. And everywhere, I have pole of 4 to 1. But now, if I make a blow-up, also exactly at the section point, again, get pole of 4 to 1, I make a blow-ups iteratively. And what I get, exactly all rational numbers, corresponding to, this explains the difference between situations, so I get countable set, because only countable many such devices. OK, so that's essentially all formal story. You see this big analogy, kind of one can try to describe in some more invariant way, what is this direct sum of copies of blocks. And in fact, it's direct sum of all rational numbers. You can see there's a following thing. You can see the, let's see, something like semi-stable bundles on Eq with slope A over B. Slope is rank over degree, maybe I'll put the same. Yeah, this category are all equivalent to category where Fourier-Muqai transform of shifts with zero-dimensional support by T-A theorem. And that's this kind of more invariant way to see it. OK. Yeah, now the next step, we should go to Stokes' phenomena, yeah? So we get things over formal power series. Now, what would we do the same things over convergent series? So it works also when Q is not a root of unity. It should be enterprise. Yes, yes, the same story works over any field of characteristic zero and Q is not a root of unity. But you don't use elliptic curve as a language, because there's no elliptic curve. And wouldn't you write the slope A over B equal? Slope, it's rank over degree, maybe I should put slope here. Sorry? Is there anything in the right column that is similar to E to the power of Poisson polynomial? Yeah, no, it's the claim these rational numbers are analogues of E to the power of Poisson polynomial. It's some kind of object corresponding to this exponent of Poisson polynomial. There's no exponential, yeah, okay. Now, if you consider over another field, germs of meromorphic functions, then there's this kind of DeLine-Malgrange theory of Stokes' filtrations. Go to it, yeah, so it's deep result, which also Masaki reproved with Piersche-Pirain in different way, in a sense. But what goes on here, one can ask the same question in QD from the equation case. So I can see the modules over this ring. Then there was a very nice result by recent result by Ramesh and Zhang, which can be formulated as a following. This holonomic demodule over this thing is equivalent to the following. It's a billion category. And what is the billion category? It's a vector bundles on elliptic curve and out this anti-hardener-simmert filtration. What does anti-hardener-simmert filtration mean? First, I should explain what is hardener-simmert filtration. It's in general considered bundles on the curve. Then for bundles, one can define the notion of a slope from a zero bundle. And then this notion of semi-stable bundles and this canonical filtration so that the slope increases. And associated gradients will be semi-stable bundles. But now in those bundles with the filtration labeled again by rational numbers, but such that this associated gradients are semi-stable, but the order is opposite. So this thing is not unique. But it turns out that this guy is again a billion category. And this corresponds to this model of an analytic puncture disk. What is the idea? It's again very similar to Deline-Marquardt filtration. Deline-Marquardt filtration, if you have differential equation, then you study asymptotic of solutions along various rays and you get filtration by order of growth and it satisfies some properties. And here, what is going on? You have differential kind of difference equation on puncture disk. So it means that you have your meromorphic bundle, you have some maybe vector space at each point with x, and then you identify with vector space at point with qx, q square x, and so on. And then I approach zero. Then if you have a vector in this bundle, suppose you write some equation, something like c of qx is equal to x to power lambda sub x. You write some equation, let me admit it. You write this equation with lambda, say integer number or rational number. Then what you see is that if you approach many, many times, you get the following. You get qnx to power lambda times qn minus 1x to power lambda times x to power lambda c of x. And then you get something like q to power n square over 2 times lambda times capital of n. So you see that if you transfer to zero, you can have different growth labeled by rational numbers. You get to filter. So it means that you get certain filtration. Before doing filtration, first you can make a vector bundle electric curve because it's an equivariate vector bundle. You go to the quotient, you glue a holomorphic bundle electric curve. And then for each point of this electric curve, you can see the filtration by growth. It turns out to be a holomorphic filtration. A social gradient will be semi-stable, and it will be one-to-one correspondence. Non-trivial result. And the final thing is just associated gradient with vector. Yes, in fact, the formula is associated gradient. Yeah, so it's all completely parallel to the usual story. And finally, there's some kind of little formulation how to go to global result. It's like in Delemargar's theory. If you know things locally, you just add local system and complement you. There is some following theorem, which is essentially corollary of Ramius-Junk result, which is the following. It's that's holonomic AQ modules is canonical equivalent to the following category. You can see the objects. I say what are objects in the morphine, it's obvious. You can see the coherent shifts on electric curve with two anti-hardener oxygen filtration. Now, so coherent shifts are a little bit more than vector bounds to get torsion shifts. What is the geometry of all the story? If you have these things, if you don't do this, consider this essential completion neighborhood over zero, you will have, again, vector bundle on a coherent shifts on the electric curve, or if you mode out ignore torsion. Suppose my bundles equations without torsion. And then you can approach zero and infinity. You can consider positive and negative orbit. And both gives filtration. I don't talk to each other. And this is essentially two anti-filtration. It's zero in stock, phenomenon. Infinity, yeah. That's nothing else, yeah. And this is a little bit of torsion because my module will be not really free as a module over Lorentz polynomial in X. And altogether, one can say that and consider like p1 cross p1 with this form dx over x dy over y. It has first of the pole at four corners. And this terms of filtration will correspond to all possible devices on the blow-up. So the filtration is on the bundle obtained by modding out by the torsion? Yeah, it's a bit, no, it's a bit delicate story how we get coherent shifts and two-filtration. But if my equation was a bundle over cx over x minus 1, then you get a bundle with two-filtration. And just a little story with torsion which I don't want to dwell. So it's not obvious a torsion in X corresponds to torsion on an elliptic curve, right? Yes, yeah, yeah, something like this, yeah. So you get this small story. Now, how to understand this two anti-hard-on-sink filtration? This is a kind of very general remark. Suppose we have a series of triangulated category and we have a bridging stability condition on it. So bridging stability condition means the following. You get a map from k0 of category 2c, which is essentially R plus IR, which is essentially oriented plane, not the complex numbers. It's additive map. And we have a class of semi-stable object with some extract semantics, which, for example, one of the way to say what is it, you say that you have a bounded t structure. And if you consider the heart of this t structure and apply z and remove 0 object, you lend to something set of complex numbers, which is imaginary part is positive or z is negative real number. And then you can define the slope as argument of this thing. And you want to ask that every object has filtration by semi-stable one with kind of canonical slopes. So that's a notion. If you have a category filtration, then I can associate canonical Artinian-Ibelian category, namely consisting of object in the heart of t structure plus 2 anti-hard-on-sink filtration. Object in the heart plus 2 anti-hink infiltrations. And when you consider associated gradients with this hard-on-sink filtration, you get some semi-stable object and apply center charge. You get some numbers. And what you get? You get a bunch of vectors, some bunch of vectors in the upper half plane and some bunch of vectors in the lower half plane. Equipment to one filtration, another filtration. Artinian, you mean object of finite length. Obra object of finite length, yeah. And particularly, if I apply this construction to the coherency of scientific curve, we get this category, which we obtain in this way. There is something nice about this category, namely the following. On the set of stability structures, there is action of a universal cover of p-jail plus 2R, or drill plus 2R. Universal cover of drill matrices is positive determinant. It's a kind of essentially rotate your central charge, and you really find heart of destruction appropriate way. And this action identifies corresponding kabilian categories. How to see it, why it's identified corresponding kabilian categories? If you fix some number of rays, you get maybe N plus and N minus will be number of rays going up and down. You can see it as a representation of function from the quiver to your triangulated category. You consider quiver A and quiver N as a total number of 6 minus 1. And on this quiver up to shift to get N plus and plus and minus specific objects. And you want this object to go to semi-stable objects, some condition on a factor from the quiver to your category. And then it's things in variant under rotation group. And the corollary of this thing, it's the following one. On this algebra, on Aq, x, sl2z, maybe semi-steric product pc star squared, just by monomial transformation, replace x hat by y hat by monomials multiplied by powers. So it gets action of some group. The question, why this group acts on this kabilian category? Yeah, it's not clear at all. But when you have elliptic curve, then the space of stability structures, it's essentially one orbit of gel to R. That is well known. And hence, you get essentially canonical kabilian category associated to derived category of elliptic curve. That's the corollary of all this picture. So it explains this symmetry in a nice way. SL2z, acting by automapism of the algebra, it goes from one halonomic model to another. And I want to, in this description, to see. The man in the pacing x and y by monomial. Yeah, you get Fourier transform. x goes to y, y goes to x inverse. So when you consider those things, there's two artichokes and the same confectrations. Is it always true that it's a nabilian category? Yeah, it's a nabilian category, yeah. It's not obvious. Yeah, but it's not difficult here. OK, yeah, so that's a nice story. And you can essentially no conductors here. Now I want to go to high dimensions. But surprisingly, in the meta-predictive version, you have the double cover. Yeah, no, no, it hits universal cover because it will give you shift functions. It's not too periodic. Now my goal will be to formulate Riemann-Hilbert correspondence for a q tends to power n for any n. Yeah, I will not prove it. Yeah, but I don't know how to prove it, I have to say. But the formulation is already to get exact formulation. What is the right-hand side? It's already on a trivial task, and you'll see it's something much, much simpler than a usual differential equation when you get various, this again, stocks phenomena in high dimensions. And here are the stories and the following. First of all, when you consider holonomic aq2n modules, it is inductive limit when you get of certain subcategory and get fully-faithful embeddings of smaller categories. And indexing set will be something which can be called analog of singular support. Consider bound support by some larger, larger set in a certain sense. What is support? In this case, the definition of support will be closed Lagrangian polyhedral subset, rational polyhedral subset, polyhedral cone, even called L in R2n, which contains rational lattice, q2n, with standard symplectic form. What means it's just closed subset, which is finite union of a simplicial Lagrangian polyhedral cones. So you have maybe n vectors in q2n. So it's a linear independent. And the symplectic pairing is 0. You can see the sum of what it will be. Where is the subrational network? With rational, yeah, it will be non-zero vectors, which are linearly independent. And to your general basis, you take all possible subsets. In case of n equal 1, what you will do? In case of n equal 1, you get the same picture which you have before. You get just finite collection of rational rays, rays in n equal to 1. So what's the answer to the similar single support? Yeah, I will explain. Yeah, it's some better notion of singular support. In a sense. First of all, if you have a model, why you get such a gadget? It's essentially you follow various filtrations. Maybe I'll take the following analogy. Suppose we have an algebraic Lagrangian subr... Algebraic Lagrangian subvariety, L. City can see start to power n. Instead of polynomial demodule, they take Lagrangian variety. Then, this is start to power n is log Calabi-Yau. Oh, and in fact, it's log symplectic. What is the definition of d x, d y, d y over, d x over h? Yes, yes, yes, yeah, it's a standard form. We have this form. It's a log symplectic manifold. In particular, it's log Calabi-Yau. It has a volume element. And then, one can see the various toric compactifications of this guy. And then, if you have toric compactification, you see what intersection of device that's infinity intersect. And you get subset, conical subset, just for any algebraic variety. And for Lagrangian, you get conical Lagrangian variety. I think it's maybe called Tevelev construction or something like this, yeah. It's similar to a projection of Berkowitz spectrum, which yesterday was heard from Tony Youtok, yeah. Yeah, so you got kind of, you consider maybe curves going to infinity. You consider logarithm of norms and you get various thing, yeah. In the same story, it works for holonomic demodals. One can work with appropriate filtration. Holonomic demodals says that it's supported at infinity, in a sense, on this union of Lagrangian con. Yeah, by the way, the story, I have a suggestion what to do for usual differential equations. What will be the same look of better single of support. In general, if m omega is algebraic symplectic variety, you can call an analog of this possible support as a following sets. First, we can consider some huge set, divisorial valuations on field of rational functions on m, puts complex numbers, but it's kind of irrelevant. Is that differential equation or differential equation? No, in general, I want to say something which works both for differential, differential equations and even more. You consider divisorial valuations, so it means that you kind of glue device at infinity to m, set at near generic point. My form is, look like this, it's a logarithmic form and it has the pole. Yeah, so you get this set of divisorial relations. It's in a kind of trigonometric case, it will be rational race. Rational race, and for the case of cotangent bundle, if m is to start to part in, d is to be a rational race in q to n, and if m is cotangent bundle, it's uncountable continuous parameters. It's exactly correspond to this irregular terms for singular points in the case of dimension one. And then one considers a following analog of polyhedral cone will be the following object. Suppose you get some compactification, your top form has the following shape. You get intersection of n divisors and form has local coordinates looks like this. So you get n divisors, normally crossing divisors. And then you can start to all make, you get n various n points in these things, but also you can start make blow-ups and you feel rational points in a simplex. Get rational, so you get analog of this n dimensional simplices and you consider maybe n minus one dimensional simplices, it will be impracticization of the story. And then you consider finite unions of such subsets will be analog of support, yeah. Okay, yeah, but it's- So that discovers support in both cases. Yeah, but for differential equation, it's kind of supported some more delicate ways. You treat all irregular paths and it's kind of, yeah, not stock's phenomenon, formal classification, yeah. Court is it related to the support of the associated graded of a good filtration? No, usual conical support obtained by some, you remove a lot of information from this, I think it's, it's kind of better notion. Yeah, but I have to, sorry? Sorry? If you want to, you say it applies to the AQ to answer n. Yeah. What's the simplex? She start to power to n, yeah. Yeah, yeah, yeah. But now one can do the following. If you have this L, which is kind of polyhedral, whatever, whatever stuff in polyhedral Lagrangian cone, it's a singular Lagrangian variety, kind of singular Lagrangian variety. It's not smooth. It's kind of maybe, here's a bit less linear. And there was a general idea that if you have such variety, you get canonical kind of combinatorial type of categories of Foucaille type category associated to L. In fact, L will be automatically, L will be automatically exact Lagrangian cone because it's conical. If you have form, PDQ, you can integrate along race. Yes, it's even better than just Lagrangian variety. But anyhow, we have canonical Foucaille type category which is described. You take some two-wheeler neighborhood, make some flows. But thanks to kind of progress with Guillemot, Shapira and Kishavara, we know that it can be described as category of constructible shifts or maybe on some domain on a C infinity manifold. So that singular support in microlocal support is belongs to some conical, Legendre and subset. And so there's really no good way to go through this conical in symplectic case, the conical in cotangent space case. One can, for example, add one variable because Lagrangian manifold is exact one variable and then produce eventually something in n plus one dimension, certain conical in cotangent direction, direction subspace. And then you get something very, very concrete. And in the case of this collection of race, you get A n categories, A n number of spikes minus one. And the description will be the following. The conjecture is the following. This category of holonomic modules, holonomic Aq to power n modules in support is contained in given L. So it will be category which is much like we specify irregularity and gratis should be equivalent to the following things. You can see it will be full subcategory in dB of n's power of elliptic curve, multiply it, make tensor product, it makes sense for this category of this constructible shifts are sorted to L. Satisfying some property. And the property will be completely analogous to what I said that for each ray, I have a semi-stable object over the certain slope. And the property is the following. When you consider objects on this guy, you get, just forgetting about dB of elliptic curve of a billion variety, you get a smooth locus of this singular grand set will be just union of some cones. And then on each interior of this, each simplicial cone, you will have a complex and then we'll be some related in one way to some very intersection points. In particular, you get a function from this category to dB of vector space for each smooth point. Strictly speaking, also it's fun to define up to shift. You should choose argument of restriction of volume element. And also if you have a rational Lagrangian variety, a rational Lagrangian subspace. It's kind of Lagrangian subspace in Q2N. You have a canonical subcategory in dB of a billion variety. It's defined the following way. On this gadget, there's section of some, again, universal, some central extension of Cp2Nz. But for real, okay, transform. Yeah, yeah, so a Cartesian product, yeah. You can see the n-dimensional of a billion variety and then free MOC transform gives the action of simplecative group of extension of, central extension of simplecative group here. And the same group acts on the space of Lagrangian subspace, it's transitively, for example, you have kind of horizontal subspace. And for horizontal subspace, QN, we are sourced to category dB, consider category of coherent shifts with zero-dimensional support on e to power n. And then, then they put the constraint that if you apply each factor, lent to the corresponding category, it's everything's well-defined. It's completely mutation of the story which I explained for n-cal-1. And the conjecture, that's its right-hand side answer for the deli, Riemann-Hilbert correspondence. Yeah, so there are many reasons to believe in it. I don't have a pro, but I think it should be really easy. What sounds is the analogous to what you said for n-cal-1, because there had some... For n-cal-1, sorry? Yeah, you have coherent shifts on elliptic current in here. Yes, yes, no, I said that this thing that I put on each array, it's a copy of coherent shifts, but they apply some element of fsl to z, yeah. And what does it mean, support is contained in L? Support contained of d-modules contained in L. It means that if you, it has certain, it should put maybe certain filtration, new filtration on algebra, and you can imitate notion if you have coherent shift on Lagrange variety, when Lagrange variety goes in a certain direction, you imitate the definition, and you do sense it was quantum algebra. Yeah, in fact, the conjecture, it's all works. If you replace C by non-archimedean field, and you're still between less than one, you can just, you can do the same story, but for non-archimedean field, or any characteristic, yeah, it will be, again, this kind of Riemann-Hilbert correspondence, and it's actually a very nice example, because like in usual Riemann-Hilbert correspondence, we get kind of two algebraic variety, two complex varieties of the same algebraic structures, but it doesn't work for Q difference equation. No, for differential equation, for non-archimedean field, because they don't have monodromy, but here for Q difference, I think it's really analytic problems are kind of the same, both in complex numbers and non-archimedean. Field, and maybe the last very small remark, it's about another SL2Z, which goes here. When you write Q is equal to exponent of two pi I tau, and you take maybe tau prime is equal to one one tau, get Q prime, you get the same category, how it acts on, let's say, let's say in one variable, how it acts from one equation to another, and here I get the following suggestion, it's similar through the following story. Like for differential equations, one can consider space of Schwarz functions, dual space to Schwarz functions of C. It's a module over Cx over dx, but also it's a model of a complex conjugate. It's a bimodule. And it was Kashiwara conjecture proven by Michizuki that if you have a holonomic module, consider Holmes through this guy, you get a holonomic module over dual guy, and it's anti-equivalence of categories. And here it seems to be the same story, but without holomorphic complex conjugation goes on. How go from tau to one over tau? Namely, you can see there's a following thing, you can see that Schwarz space is dual to, distributions are dual to Schwarz space. And analog of Schwarz space will be the following. It will be a holomorphic function on C, some coordinate Z, with the following property. You get one tau and consider basic trees. Yeah, so now a tau will be not a negative real number. It's actually works even for real numbers. And you can see the functions said that along these directions have exponential decay. Yeah, so you get this set of vector space of functions. And there is kind of a natural, at least some class of dual expressions. Namely, dual space will have functions which has poles in the neighborhood of two sectors. And you integrate over this, over the path. And now you have exponential growth along these lines. Yeah, so it's some kind of hyper function story. And also you get analog of S prime. And there are two commuting algebras by shift, Z goes to Z plus one. And function multiplied by exponent of two pi i tau Z. You get one algebra, or also Z goes to Z plus one over tau. And F goes to exponent of two pi i Z. It's kind of Fadeev proposed two commuting quantum torus acting here and contractures it the same story holds. That if you go home, you get space of distributions. And if you have harmonic demos, you want to consider solutions. So you get dual harmonic demos on dual torus. And this will be eventually be exactly this model transformation on logarithm of tau, okay. Thank you. Thank you very much. Mike seems to have a question, so, three more, yeah? Is there an analog of regular? No, no, no, nothing, yeah, it's all irregular. In one dimensional case, there is, but. Yeah, so that's what we're talking about. Yeah, because there are only two singular points. It's a very, very stupid story for regular models. Nothing, yeah, yeah? Yeah, for example, you're supposed to be informed in terms of the queue. Yeah. And if you change the queue, what do you do? I have separate queues, yeah. Then the category will be much, much smaller, yeah. If the logarithm is linear, it will be no object, I think it's interesting. Oh, yeah, yeah. And similarly, in algebraic situation, you'll have no logarithm variety if you take generic, simplecative form, yeah. Oh, yeah, yeah. Yeah. Was that a question? Yeah. So in the conjecture, do you start from autonomic modules or DB? No, autonomic ones, abelian category. Abelian category, actually. And then you have a DB of. Yeah, but consider object in this category with the property, when you have restriction, you go to appropriate t-structure in, yeah. Yeah, one can formulate it pretty exactly, yeah. More questions? At the beginning, when you went through your, I think, different equations and different equations, you had C star on one side and the other on the other side. Was that C star really interior of a nodal curve, or? C star, I mean, I was wondering if it was like a limit of elliptic curves, or a limit. Yeah, no, no, there is something called elliptic algebras. I have to say, you can see the CP2 minus smooth cubic, you can remove, kind of, instead of triangle, which we can see the C star square, you can see the smooth cubic, and there's some algebra, but here's the whole story, it's kind of really trivial, get only one divisor, you can do, and I think this whole series have only one block, but still get Riemann-Hilbert's correspondence by Eric Reins, I think, make some Hilbert, Riemann-Hilbert's correspondence for this sync with itself, with different parameters. Any other questions? Cluster, mutations? Yeah, no, no, no, in principle, all the C star to part two, and it's only approximation to cluster variety. Yeah, it should be some bigger story, I would tell you. But you don't know what would be the other side? Yeah, I don't know, yeah. That's what I meant, yeah. Instead of this AQ, the Ns take these deformations of the cluster out, but you expect some. No, in fact, I expect the right-hand set will be Foucaille's category of the same variety. With the symplectic form, it will be real part of symplectic form, B-field imaginary part of symplectic form. That's a general Riemann-Hilbert's correspondence which you have as Yann for arbitrary symplectic manifolds. Okay, if there are no other questions, we start in five minutes and we thank Maxime again.