 OK. So I'll speak about the wall crossing and geometry of infinity of beta-modular spaces, which I explained what is beta-modular space. It's named with Carlos, actually, I think, introduced to mathematics. It's character varieties, space of representation of fundamental group, or things like this. I don't know what is it. So my talk will have a lot of intersection with actually where is Andrew? OK, yeah. This Andy, when they talk, but I'll put kind of different perspective. Actually, how would you know? I never used Apple. OK, good, yeah. OK, so let's consider equations with, so I'll talk about equations with small parameter. And which Andy denoted by zeta, and I'm a mathematician denoted by h bar. Yeah, OK, so it's like constant, yeah. So it starts with kind of very simple case equation with regular singularities. So we have, let's say, several complex numbers and several matrices. And then we can write differential equations which in vector-valued functions, which has first-order poles. And to raise, you just say, give me these matrices. OK, so we get, and the most important thing, you put h bar in this sort of derivative. And now this equation has some monodromy. The question, how it depends on h bar. Yeah, what is the monodromy? You choose some base point, which is kind of a small white dot here. And then we get loops surrounding each puncture. Get some operators, which are invertible matrices. But we're interested in this thing, sub-to-conjugation. And this is beta-modular space, so collection of matrices, modular conjugation. There's actually a bit of a problem here because the conjugation acts, I think it's not really, but I will go to this point later. So if you're interested in what are functions on this quotient space, this is pretty complicated algebra, but it's finitely generated algebra. It's from a general result of algebraic geometry. We know it's finitely generated. We don't know any good class of generators. But maybe for small r and k, one can find them. And it's known that it generates traces of monodromes of all possible loops, which are conjugated to the classes. So you can forget about base point. But this is not a variety, this quotient space, yes. It's singular variety. Yeah, because the group is. No, the action is not free, but one can still consider functions invariant under the action of the group. And then identify some point. Some points, yeah. So with this. It's stable. That is way bad. Yeah, it's not a really good way to do this quotient. But when you go to this algebra functions, you identify several representations which from couple you identify kind of not semi-simple representation, which extension, but with a direct sum. Yeah, essentially, this spectrum of algebra functions, this variety will be equivalence class of semi-simple representations of the group. Yeah, OK. Now on this variety, you get plenty of functions, infinitely many functions. They form some finitely generated ring. And for example, for any loop, we get a function trace of loop. But now my equation depends on parameter h bar. And in fact, we get an entire function in inverse variable. When h bar goes to infinity, it means that consider constant connection. Put h bar from the left to the right, h inverse and you can have a well-defined limit. OK. So you get an entire function on complex plane. And we want to understand this function. Oops. Yeah. I will not speak more about regular case. It's actually more convenient to work in irregular case. So let's consider now Schrodinger operator with cubic potential. Yeah, m-thematician surfaces, I forget about i. Yeah, what would i h bar? But anyhow, cubic potential is not real. So it's not a question of Hilbert spaces. Yeah, so we get this differential equation. And of course, solutions are entire functions on plane. So it's no question about monodromia. There are two linear independent solutions. But as Andy mentioned, there is kind of a replacement of monodromia for this irregular case. It's called Stokes data. And this beta-modular space will be parameter space for this possible Stokes data. And in this case, it's algebraic surface. It has two complex coordinates. And what it parameterizes? It parameterizes configuration of five vectors in two-dimensional space, up to action of J2, with the conditions that which products in cyclic order are all non-zero and equal to each other. Yeah. Why is five? Yeah, you will see it in a minute. So one can, how to write coordinates on this modular space? First of all, the first vectors v1 and v2 consider the base vector. So they have coordinates 1, 0, and 0, 1. And then you write equations that which product v2 and v3 is equal to 1, then it fixes this number minus 1. But the next one is not known. And also, we have v5, so we get four unknown parameters. And if you look on the substitute to this equation, you see you get two non-trivial equations. You get some pretty simple algebraic surface given by two equations and four variables. Now, what it has to do with our differential equation? For a moment, I just simply put Planck constant to be equal to 1. So I have this differential equation and it has two-dimensional space of solutions because it's defined on one connected domain and given by value in derivative at any point. And in the space of solution, I want to have five vectors. And vector vi, it will be some function that is showing this equation, will be the unique solution which is you should draw appropriate sector in our plane z. And in this sector, it should have this leading term and this solution should decay. You see that if you take z to power square root of z in this expression, you have two square roots. And one solution will be big and another will be small in the sector. So it is which one of those sectors, number, where it indicates, so we have this sector. Yeah, you see that argument of z, if i is raised to the mod 5, like 1, 2, 3, 4, 5, yeah, then you draw a corresponding sector in a plane. And in this sector, you can write approximately solution of this form. And in this sector, one solution will be growth to infinity and another will go to 0. So here is the h. h is equal to 1, for simplicity. Yeah, one can put h bar after, but just for individual equation. But why do you think you choose this sector? Why this sector better? What are lines between sectors? Lines are called Stokes lines. Along the line, this exponent will be purely imaginary. So it will not go to infinity and it is to 0. It will stay the same. And you don't know which solution is larger. And in the sectors, we have one solution which is larger and another which is small. But the large solution is not really well defined. Because you can add to it small solution. It will be still large. But small solution is uniquely defined. So the small solution is uniquely defined? And this is a line. So it's only one solution in each sector which has a asymptotic. When I choose the branch of the thing which goes to such exponents, it goes to 0 in the sector. OK, so we get five vectors. And then it's easy to see that a wedge product with these two vectors is a constant. What is a wedge product? We have the equation, the monogram is SL2. And to make determinants, one should calculate a wrong scan. And wrong scan, if you have two solutions, one should calculate this wrong scan. And it falls from the equation that's a constant. It will be wedge product. And for these two small solutions, two nearby sectors, it's easy to see that the wrong scan is just constant equal to 2. Yeah, so we get exactly this picture. I don't know, so this function z must be one vector. Dependency of sine, sine prime. Sine, sine prime. Sorry? No, it's the equation of second order. No, this answer, you wrote vector equal psi i of z. Because it can see the space of solutions. In vectors, the function of z is thanks to the equation. But each sector, there are two solutions. Yeah, maybe just f psi i, function psi i and... Yeah, yeah, OK. This is the wrong scan in each domain. No, no, wrong scan, it's constant everywhere. If you have two solutions of the... For any two sectors. Any solution, wrong scan is constant everywhere. For any two sectors. Yeah, because you calculate it, see that it's derivative. It's two solutions which are different. Yeah, it's very easy. You calculate derivative of this thing, you see it's zero. I think they pass when we choose one big one so on each sector. Yeah, yeah, yeah. So use every sector. Yeah, yeah, all five sectors, yeah. I have all five sectors, I get false my solutions. OK, this i can fuse me. OK, this i of course can fuse me. Yeah, it's not square root of one. Oh, no, no, no, there's no square root of one at all. No, square root of one, I denote by square root of one. Yeah, it's not, yeah, OK, yes, right here. Yeah, yeah, this i is also not square root of one. Yeah, I'm sure. OK. OK, so what's the conclusion? So this stock's data, which is an analog of monodromia, and what's stock's data exactly, what's stock's data? It means that if you have equation of the second order with this property, we get five vectors in some two-dimensional vector space, yeah. And with this constraint, which products are cyclically the same, yeah. But now we can put also the small parameter h bar. And in what we get, we get four entire functions, but not invariable h inverse as I had before in the case of regular singularity. The entire function, what's taking where you were, which one? Complex valued functions? No, but each one, we take the Cb's or what? No, no, no, no, just go back. These elements in abstract vector space. But then these elements of vector space, one can encode four numbers, alpha, beta, gamma, and delta. Yeah, yeah. So it gets this algebraic surfaces, primetrazing this configuration of five vectors. Yeah, it means the vectors themselves will go, so if this space will be entire functions. No, but vectors in abstract space, we don't want to identify, you can see that. It's a space, it's just a space, yeah. Yeah, but now we can see the equation depending on parameters, they get different space of solutions, we don't want to identify them. You see the map from complex line to abstract space, right? It will be entire function is very near space. Yeah, it's will be entire functions is very sense-break variety. In this space, but a range of which will lie in this variety. Yes, yes, exactly, yeah, exactly, yeah, sure. So let's me go further, yeah. Yeah, so we get a holomorphic map from complex numbers to algebraic surface. Yeah, in fact, people in complex analysis are interested very much in these things, when, for which algebraic varieties, maybe non-compact one can have maps from complex lines. So the picture that's related to sign of curvature for kind of general type varieties should have little, very few curves. And here it's a very specific example of transcendental curve in algebraic variety. Yeah, that's also one important point here. There probably is a lot of line in this also, yeah? You must have something here. No, no, no, this variety has no rational line at all. It has no rational line? No, no, no, no, no, no, no, algebraics. No, no, there's no algebraic map from C to this variety. Yeah, in sense, I want to say that this variety is very similar to algebraic torus. That's right equation X, let's say XY is equal to one. You cannot solve with algebraic functions, but you can solve in transcendental functions. X will be exponent of t, y is exponent of minus t. But you cannot solve not really in algebraic functions. Yeah, so it's kind of similar behavior here. So it's not a line, but it will be line with sponges. No, no, no, with sponges, yes, but there can't be a rational kind of object. Yeah, no, no, but a really interesting line, yeah, C. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, so it's one important point, it's... But they need to know one punctured line, they need to know one punctured line? One punctured line, there will be plenty, yeah. Yeah, exactly, you can respond to the general, if you want to, yes, but... Yeah, so just one small point. No, but you see that in transcendental function, you'll be just uncovering many times of these different ones. No, no, no, no, this one is not covering, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. And A1 and A0 are fixed, for example. Yeah, are fixed, fixed, fixed parameters. So I can say the only parameter, it's a function of each bar. Of course, they can vary A1 and A0 as well, yeah, so they get familiar of such things. Yeah, so this one, important things here, in case of regular singularity, my functions, kind of one valued function, so they eventually functions in inverse of each bar. But here, if you vary H, this, all this sectors start to rotate and you get some kind of ambiguity. So you get a five valued function of each bar, or you can say it's a function of a fifth sort of each bar. It's actually an important point that in a regular case, you get some ambiguity rotating each bar. Yeah, all this thing, it's actually just beginning of, yeah, it was studied by many people, there's cubic singularity, but this can be generalized to high order polynomial differential equations. Yeah, first of all, the equation for not of order two, but of higher order and potential or coefficients will be polynomial source of high degree. You get kind of two parameters here, and then you get remarkable maps, you get hallomorphic maps from complex numbers to some generalization of this variety, which I explain, and to which I call Grasmanian cluster varieties and they prioritize completely similar story. You can see the cyclically ordered collections of vectors in a high dimensional vector space, like before we have five vectors in two dimensional space, but we have n plus m vectors in n dimensional space. The property is that all which product in the cyclic order are nonzero and coincide with each other. So it's kind of the basic example of cluster variety of which people study in very simple sentences, now it appears in amplitudes by Arkani-Hermet, and okay, so it's really the most basic cluster variety. I think it's more of the same, yeah. Yeah, I think it's more of the same, yeah. In this case, yeah, but it's kind of realizations, I'll just click on it here. Okay, so now I'll just touch a little bit the question of resurgence. Yeah, so this, what is the general framework? Suppose we get algebraic curve C and the hallomorphic, and we have some parameter space for this Planck constant, which will be just open disk, or some small radius epsilon, and then you can see the hallomorphic vector bundle on this product. And now suppose we get a connection along fibers when we fix H bar, which is meromorphic in both variables and has first order pole when H bar equal to zero. Yeah, so it means that we get really family of bundles, so this meromorphic connection. It's a curve there, how it positioned there. Sorry? It's pole not a point, it's a curve, right? It's a curve in the product space, it should contain component H bar equal to zero, it should contain vertical line. It's a whole component, okay. Yeah, it could contain some other. It must contain this component. Yeah, and should have first order pole in this, and other components can have other high order poles. Yeah, in principle one can. This one might be reducible, this five fiber, it might be some other. No, no, no, I can see the constant family. I can see the product. We simply, in principle. What I'm saying, this particular pole. Yeah. So it's some device and it's anybody else coming through zero or not. Yeah, it should be some other things, yeah. It might be half-singulation. Yes, yes, yeah, you can have this curve, and you have this H bar, then you get this component pole here, but also you can have other component, yeah. Exactly. Yeah, in principle one also can allow the curve to vary, to depend on H bar, but just for simplicity, I don't put it in condition. Yeah, so you get a family of bundles with more of a connection, but they defined only for H bar and non-equal to zero. For H bar equal to zero we have, this is no connection anymore. And now for this bundles, this connection, we have this traces of monadoramia or the stux data, which you can imagine, again, completely analogous things. You can see the, some solutions decaying in some directions, and so on. So you get some algebraic variety. It's this Betty-Modell space, and... It's a break function where you find where, again, the function where. Ah, perimetrizing, now we have connections with maybe irregular singularities on curves, depending on H bar. And for the connections, singularity, we want to calculate some entity. So we say that... It's an F function of H, of H bar, F function of H. No, no, no, F is a function on a space of all local systems, say, or something like this. Yeah, yeah, yeah. But now, we have a particular local system, depending on H... Local system, I have different number of calls. Yeah, yeah, whatever, yeah. So you have a function, or something like that. Yeah, it's kind of... Yeah, it's a huge, complicated algebra. But what's the generalization of this variety, which I showed to you? But now we have, for each H bar, we have a particular local system, so we can evaluate it here and get function of small parameter H bar. And as I explained to you, it's in a regular case, it's actual function will be not on H bar, but on a universal cover. It depends on branch of logarithm. Okay, so we get a map from maybe universal cover of punctured disk to this beta-modular space. And then it's a complete reality to check that if you restrict this function to any array, so you go approach zero with a given phase, then this function will grow no more than exponentially. Exactly because this connection has first to the pull. Yeah, and there's a big conjecture in the subject, saying that this function is, it's called resurgent function. Yeah, I think that's all subject was essentially, it's the main example in all this conjecture. You can see the solution of equations, no parameters. Yeah, so this notion of resurgent function was introduced by Iqal. So what does it mean that I get a function on maybe universal covering of the disk and small H bar? And now I try to make Laplac transform, but in the inverse variable because you approach zero. And you integrate from some point to zero. And the, can... You integrate on which path? Yeah, you integrate from some path going to zero from some H bar zero to... Straight path. Straight path, yeah. But in reality it was function of one over H bar to the variable power. No, no, no, no, but in this case it's only first power. In this case it's first power. Is it regular? No, no, it's irregular singularity. The claim that this function, F-type of H bar, growing like exponent minus some constant over H bar roughly. It's no other powers here. Yeah, and yeah, so the function is called resurgent. So the function is called resurgent. If you make the Laplac transform and you get function which has endless analytic continuation, what does it mean? The Laplac transform is defined in some domain. You get some convergence by this bound. And the conjecture, there is a countable set which in reality it's everywhere dense. Said it to consider arbitrary path which doesn't cross the set, which should be generic path. The function will be analytically continued for this path. Yeah. So let me give you kind of the basic example which we have here. If I apply the whole formalism to the case of harmonic oscillator, one gets essentially the following function. You have, you consider this function and the claim that if you integrate, okay, denote by one over H bar is equal to X. And so if you integrate the amount of X plus one half, it's slightly diverged in zero, maybe talk about the point plus infinity. Consider this function. This Laplac transform, it's well defined in the right half plane with the claim it's analytic on universal cover of C minus two pi I integers. Yeah. Now, that's the basic example of resurgent function. But he is discreet. He is discreet, yeah. But in general, I think I use the covering to say minus rational point. No, no, no, no. This is exceptional set. You see that in this case, the discrete set is kind of a subgroup of rank one, a billion subgroup of rank one. In general, this exceptional set is subgroup of finite rank, but because it's, if at least rank three, it will be everywhere dense, yeah. Does he have good model of universal covering? No, no, there's no question about universal covering because points are dense. If you consider arbitrary branch, it will have no singularities nearby. But if you return back, you'll see again singularities. Interjection, the dense in projection. The dense in the projection, yeah. If you consider kind of Riemann surface, it's nice in my surface singularities are well separated from each other. But under projection, they're everywhere dense, yeah. I know, this actually in the Riemann surface where we got separated, yeah. Yeah, so it's a very beautiful object. And it was studied by Andrea Voros and Ekal, and I also get such a thing. And as far as I understand, I have to say that this is a really big effort to prove it, I think. But as far as I said, there are big mistakes made. At least mathematicians like Malgrange, Ramiz, tell me there's some kind of fatal mistake in the paper of FAM and so on. Yeah, so this really seems that the subject should be reworked together. So it's a big conjecture. So the only real example is this gamma function. I claim to prove that in my book in fact, for something with some hypotheses, I don't know if there's a big mistake in the book. Maybe, yeah, I don't know, yeah. Yeah, but for SL2, yeah, it's already, yeah. No, no, for SLR, yeah. But with some hypotheses on the entire function. Yeah. Some hypotheses on it. Yeah, yeah, yeah. No, it's a kind of very murky subject. And I think it's, sorry? Yeah, I think it's murky, yeah. Yeah, so I have also visited Sasha Getmanenko, who, yeah, you know, a little bit, but I think he went only to the first sheet, and not really far away, yeah. Yeah, and I think this wall crossing structure, it's kind of the framework how to attack, understand this whole topology and geometry for this resurgent. Yeah, but at least this one kind of looks like a modest achievement, but it's really important thing. This all wall crossing business gives a, for any function on beta model space and for any generic ray, you can exactly say what is the right constant here. It could be, I think it's already on trivial, I think. Yeah, so now again, just kind of essentially repeat what Andy was talking about. So if you have equation depending on small parameter, and such it says first of the pole, then the limit object is not a bundle is connection, but it's, this time it remains, it's called Higgs bundle. So you get homomorphic bundle, but instead of connection, you get one form with coefficients and endomorphism. Connection's very close to one form. It's, and how we do it, if a connection depends on small parameters and local coordinates. We assume that it has first of the pole and H bar, and we consider this coefficients of H bar inverse, and it's well defined over the connection limit, which is Higgs field. So it's one form with endomorphism of the bundle, and then this Higgs field gives a spectral curve, which ended in by sigma, namely, this Higgs bundle can be understood, it's that if you have a tangent vector, then a pair with this one form, get operator on a vector space, and then calculate the spectrum. And if you rescale tangent vector, the spectral will be rescaled, so the whole thing, it's naturally understood as subset of a cotangent space in the curve. And when you now arise a point, a base point when you apply this tangent vector, you get some one-dimensional sub-variety of a cotangent space to the curve. And what does it mean in kind of real life example? Again, just, I will all speak about this cubic potential, tributary cubic potential, whereas equation H bar d z square is equal to something, and now we just replace H bar d u d z by variable w. Yeah, so that's how we get spectral curve. W is coordinate in the cotangent bundle. Yeah, so we get this static curve, and now my tech ability stops, so I just start to draw by hand. Yeah, so what is a Hitch and integral system? Again, we can see the Higgs bundles, which is a bundle with this Higgs field. We have to bundle this field on some curve, with some constraint in singular point. And then we can forget about, we have a social spectral curve. So we go, so if you have Higgs bundle, then I get a social spectral curve, and then get projection from a model space of Higgs bundle to something else, and then say this model space of spectral curves is always, it's not just very obvious, but it always has a natural structure of a fine vector space, a fine space, or complex numbers. Essentially, the spectral curve is a section of some line bundle, consider space of all possible sections on cotangent bundle. So it's linear system. Yeah, so we get a map from one manifold to another, and the most important point, how to understand it. It's very complicated stuff, but a generic point, generic Higgs bundle is a very simple object. It's just spectral curve and a line bundle on spectral curve. You can see the eigen space corresponding to the operator and the forms of line bundle. And now, what this projection means, it means that we forget line bundle. You can see the pair spectral curve plus line bundle, forget line bundle, get the spectral curve. The spectral curve can be very abstractly, just. Yeah, it's some kind of curve in cotangent bundle. Yeah, and in that situation, it's a smooth curve in cotangent bundle, but it will be ramified covering of the original curve. Yeah, and the fiber of the thing, it's something like Jacobian of my spectral curve because you should consider all possible line bundles on the spectral curve of some certain degree. Yeah, and the great thing, it's actually algebraic interval system. So the model space of Higgs bundles with appropriate conditions is algebraic symplectic and it's fiber, it's genetic fiber, it's abelian variety and actually it's Lagrangian abelian variety with respect to this algebraic symplectic structure. Yeah, so now I kind of just reformulate the things, what are equations, small parameter, why we have Higgs bundles at the limit in some geometric way. So it's Vistor family because like Andrew speaking about some special families of connections, but it can be applied to actually arbitrary family of equations depending on small parameter. So if you have family of equations depending on small parameter, then one can do the following. Then one can associate to some kind of universal question. Wait, one can define this defined base of Higgs, one can define this family. If you have an equation, if I want to understand some family of equations with small parameter, depending on small parameter, one can introduce kind of some ambient geometry to study it. Namely. But it's family has some gradient in zero, or what? Yeah, you have equation depending on, yeah. Equation, you can see the equation depending on small parameter, like of certain type, like which I can describe as regular singularity or some irregular singularities. And then some kind of the right geometric framework here is the following. So for each H, one can consider some family of modular spaces depending on parameter. So when parameter is non-zero, this modular space is modular spaces of algebraic differential equations or bundles with connections. And it's what's called DRAM modular space. It's almost the same as flat connections, but I want to see it's algebraic variety in this description as bundles with connections. And the limit H equal to zero can see the bundles, not this connection, but this one forms. It's kind of a nature of flat family of algebraic varieties. So the, so. Even if you start with the obituary family of algebraic varieties. No, no, one can, if you have an equation depending on small parameter. It's obituary, it's more than anything. Yeah, yeah, yeah. Yeah, but the equation- But it's not without any condition on zero. But for H equal to zero, just nothing special about H zero. It does have to have some singularity, whatever. No, no, no, it's not singular family of algebraic- It's not singular family of singularity of algebraic- Yes, exactly, yeah. Yeah, I get it, even in kind of nice situation, it will be family of algebraic varieties. Yeah. And to this, it's a shape, this Higgs bundles- No, no, no, no. No, no. Example of this conclusion, I'll calculate it. It's modular. No, no, the claims that one can construct some family of algebraic varieties depending on parameter H bar also say algebraically. Said that M zero is modular of Higgs bundles. Let's say without any connections. Without any singularities. And M H bar, it's modular of vector bundles with connections. So all fibers sub zero are the same, but the zeroes have different fiber. Yeah, in principle, yeah, so get this family of connections and, but now that when we describe our equations, this position of merit points and leading term singularities can move when we're So this algebraic varieties can little bit depend on H bar. Now, so we get this picture and equation is small parameter. Can when I said it's just a holomorphic section of this family, this red thing is a holomorphic section. For each H bar, we write some equation and for H bar equal to zero, we write not equation, but this connection. Now, so it's kind of geometric formulation. What is the equation of small parameter? It's just repeat what I said. But now there's this all interesting resurgent functions come from because it's complex analytic right, not algebraic right. Is the drama model space is the same as Betty model space, which is representation of fundamental group. And Betty model space is a rigid object. There's no parameters here because it's given by some equation as integer coefficients, which I explained to you. Yeah, so it's locally the same space. So we see that we get a nonlinear connection on this bundle, which is a small, this planes which will be described a connection. So it gets, the connection actually can have no trivial monodrome in a regular case, which explains this ambiguity. Yeah, and then we kind of locally trivialize this connection and our section will be now a complicated resurgent function. Yeah, so we use kind of different algebraic structure, different connection and comparison gives the algebraic function, this resurgent functions. Okay, yeah, so now I will briefly review what is, what is the small crossing thing? And now then we'll turn back to Hitchin, to Hicks model space. Actually, we do it immediately, yeah. So as I told you, this base of the Hitchin integral system is a fine space. And I told you about kind of nice spectral curves which are smooth, but sometimes spectral curves are singular. And singular curves form some discriminant locus which is a hyper surface in the base of, in this affine vector space, one can write some explicit equation, and this hyper surface has completely horrible singularities, the worst possible singularities. And it has complicated fundamental group on the complement. Yeah, and on the complement, we have a, when spectral curve are smooth, we get a local system of lattices, this first homologous spectral curve. And the spectral curve could be not necessarily compact when the equation has poles, it goes to infinity. So it's, this lattice has an intersection pattern which could be degenerate. And then there's a Louisville one form on the Cartesian bundle, it gives you. It's smooth spectral curve. It's smooth spectral curve could spawn to point to you in the base. But because one open will not get a local integral vibration, we change infinity, you know? Sure. It's not a integral vibration, this graph, or they may change infinity, but they open. No, no, no, but I consider open part when the topology is the same. Of course, they also can change infinity as well. Yeah, as well, yeah. Yeah, I would consider. Formally, it's pro- What about my change? Yeah, no, I would consider appropriate part when H1 does stay the same, yeah. You're right. Yeah, so I get kind of this really generic smooth spectral curves. And then one can restrict the spectral curves is canonical one form on Cartesian bundle, get one form on a curve and it gives a functional on the lattice. You can integrate this form over cycle, so you get a linear function from the lattice to C. Yeah, oh, by the way, I want to say that the image of this map is will be exactly, it's analog of this 2 pi Z, yeah, that's it. Points with this Laplacian form. Can Jack say that or he's cool? But you say the statement, don't you? No, no, I think it's the moment where it should really abstain from exact statements. Yeah. Yeah. Yeah, yeah. Yeah, but now I want to say something about this, what is the whole crossing thing. And what Gaiotto Moronezki says is that there are some nice numbers associated with all this geometry, which are supposed to a point outside of discriminant and to point in this first homology. You should get some integer numbers, which jumps along some walls and satisfies some wall crossing formula. Yeah, so it's, yeah. So the walls actually also depends on also should introduce face, this thing, it's much better. Walls away, which space? Yeah, walls, where are the walls? Yeah, I kind of forgetting to write theta as a, ah, yeah, one can interpret this thing as a falling quay. Let's consider product of a circle and as a base. And here, now I can see the- The whole base will minus discriminant. Minus discriminant, yeah. Yeah. And here we get some real co-dimension one locus. Namely, we have special pairs, when we have point on the integral system and on the base and vector, and suppose it's this DT invariance, whatever omega is non-zero, something special happen, then we draw a point U and theta, the theta is argument of this complex number. Yeah, so we get some complicated co-dimension one thing, which can be sort of a collection of kind of stocks race in just an usual plane, depending continuously on points on the disk. Because if you fix a disk, you should get finite or countable subset of a circle. Sometimes everywhere dense. And you replace points on the circle by race and get- Yeah, it's good. Yeah, in real life, it's the main challenge to understand what is the dense domain. There definitely will be dense domains, yeah. Yeah, so it's, what I'm going to explain, it's a kind of a very nice example. We don't have this density. You get kind of locally finite picture, but- But this density comes from the stock's data, because- Yes, yes, it's, yeah. Yeah, I will explain what is this stock's data. And again, as next, and you explain that's what is all crossing structure can be under-setted. If you cross the wall, you should associate to crossing the wall some coordinate change transformation on some torus, or maybe in some domain in torus. And then we write this transformation in some kind of bizarre way. I will not really concentrate on this. It's kind of non-non-algebraic. It's very non-algebraic. It's actually acts on formal power series. Only, yeah, yeah. It's formal, it's completely formal, yeah. And the main constraint is that if you make a small loop and the composition is trivial, so it's, Andy, explained? Yeah, so this- It was explained, yeah. Yeah, yeah. And what are basic things to say about this wall crossing? First of all, this wall crossing constraints are so powerful. It's something like zero curvature equation of flat connection. And if you know these numbers for some point on the base and all vectors, so you associate kind of integer numbers to elements of the vectors, then by some kind of analytic continuation, you can recalculate it everywhere. It's like analytic functions. It's determined by Taylor expansion. So these things can be thought as analog of Taylor expansion at this point, and you recalculate it in different point. Yeah. Yeah, in fact, the problem is that there are too many numbers. We don't know what to do with them. Yeah, so it's, I'll say something about kind of nice way to see geometrically later. Yeah, but now the next important point that if consider this point on this product as one and the base, which is not lying on the wall, which could be kind of irrational point, things could be very, then you get a coordinate system on better modular space near some corner, in a certain sense. So you identify it locally, or maybe in some, we don't know really much about convergence with some domain in a torus. And when we cross the wall, this coordinate system changed by this wall crossing transformation, which I drew. So it explains associativity of the constraint because we reveal them. And then there's really nice things that in this product that domains, there's really no walls, kind of really big open domains when you don't have any walls. And this coordinate system should be algebraic. It's kind of a vibrational coordinate system. Yeah, essentially, in this case, it's you get embedding of the torus. You say that your beta modular space will be some, contain the risky open part, which is algebraic torus. So the like toric variety, or partially compacted by toric variety. And then algebra functions will be maps to algebra of Laurent polynomials. And so another important thing that many of these numbers are vanishing. There is some inequality, which was suggested based on some kind of temperature of three-dimensional caliber. You're a completely different think it. That you see that what we're going to get a lattice. Let's suppose it's kind of this rate maps to C. Yeah, so you get this kind of integer points in three-dimensional space maps to a plane. And then there's a kernel here. And then the claim that omegas are equal to zero in some neighborhood of the kernel. That's a rough picture. And it's related to kind of areas of convergence or how you even understand power series in kind of added convergence and all this business. And finally, that's the kind of most beautiful part here. There are some absolutely remarkable set of functions on this beta modular space, which form a basis of the algebra functions as a billion group. It's variety of integers and it has some nice basis like Chebyshev polynomials, roughly speaking. The generalization of Chebyshev polynomials in very simple case, and which is co-variant transaction of generalized mapping class group. But this set of basis is labeled by some abstract set. And now if you have again point out sort of all this abstract set, it will be identified with the lattice. Yeah, and then it's really nice for asymptotics because if you restrict for this to the family of equations by monodrama map, then this function will have a naive growth, which kind of you expect from WKB asymptotics. And what is nice, if you kind of generate point, all different points will have different growth. They never coincide because you get a map from lattice to real numbers, say. And generically it will be inclusion. There will be linear independent of integers. And it means that if you consider linear combinations of this four-quantia of basis and want to study the order of growth, so only one kind of term of this linear combination will dominate the system. You can see the two small ray in each bar. Yeah, along the ray, yeah. It depends on the ray, yeah. Yeah, yeah, so why is this? Yes, yes, yes, exactly this stuff, yeah. Yeah, yeah, and then one can see kind of intrinsically without even understanding what is this omega, so what is it, what is the meaning of walls? It's pairs, yeah, if we have point on the base, then I can see that you get a lot of resurgent functions and the resurgent functions have stocks rays because along certain rays, the main terms of synthetic will jump from one contribution to another. And all possible stocks rays for all possible functions on beta form walls, yeah. So it's completely intrinsic definition of at least the picture. No, no, without bases, you can see the all possible functions. It's a claim that you can see the linear combinations. How all possible solutions equation? Yeah, now all possible combinations of these bases, you can see all expression traces, yeah, you don't have to go to this one. But you can see abstractly with the spaces, you know, you can describe it with something. Not really, yeah. In origin, you can describe it with anything. No, no, yeah, no, but without going to understanding what is for going to your basis and so on. Kind of solving on computer differential equations, you can immediately see where are walls, yeah. You see the certain expressions jump their behavior. Yeah, but now the things, it's more the space. I told you that it's kind of similar to algebraic torus. Yeah, it's called, it's low-caliberal varieties. It has some volume element. And, yeah, in fact, it's for this Mbeta and descent. But it's got some element somewhere, it generates somewhere, or everywhere points to. Not everywhere, not zero, yeah. Yeah, no, there isn't the form, like it's... Because it's a lot, so it has some behavior. Yeah, yeah, no, because varieties is not compact, yeah. Yeah, it has compactification. In fact, there are many configurations, none of them is better than another. So, volume form has first-order poly along all components infinity, yeah. Yeah, so it's really analog of algebraic torus and the compactification we see gets analog of toric compactification. Toric, if you have a torus, you can compactify it in different ways by different polygons. Yeah, so there's a similar story here. So we have toric compactification. And now, why we need this compactification? Before we get a holomorphic map from a punctured disk or maybe a universal cover, let's ignore this universal cover, from punctured disk to my algebraic variety, transcendental map. I can see the small circle, very, very small circle of radius r, so I get a curve. And I want to understand how the curve look like, but if a radius compact goes to infinity, when I compactify, I kind of have a tool to... Because you don't get covering it goes to infinity. Yeah, myself don't go to infinity compact, yeah. No, no, but it depends on radius of the... When I shrink it. When I shrink, yeah. That kind of small circle which depends on the parameter and eventually will travel somewhere around infinity, yeah. So now with this compactification, you can really ask reasonable question, how it behave like because space now compact. And the claim that this loop, so we can see the very small radius, we can see the this in each bar parameter, very small loop. And now in compactification, I claim that this loop travels only near one dimensional strata. On a compactification, I get devices to infinity. And... It converges particular one dimensional strata. To chain of one dimensional strata. It depends on compactification, yeah. Yeah, so what I draw here, it's kind of on the left is a real picture, on the right is kind of mixture of complex and real picture. What is drawn by blue is suppose it's kind of three dimensional space. It's compactified by some two dimensional surfaces, but they're complex two dimensional surfaces. Intersected along some curves, and then there's some eventually points. So you get kind of compactification of polyhedron. And then this, this my circle goes to a circle, but why I draw them on different colors? In fact, they're drawn the same color. If you look very carefully, so this circle I draw them by kind of gray and there's tiny kind of red pieces of red here, yeah? Yeah, but now this gray pieces go to very tiny pieces here, but red pieces will be very long, along, yeah, so it's kind of, small things became large and large things became small. Yeah, yeah. And now near zero dimensional strata, when this gray things is very small, we can kind of make again, write different coordinates to see properly in different coordinates because we take logarithm of complex coordinates. And consider the logarithms of complex coordinates, the claim is that we get ellipse. Yeah, so it will be really straight map. Where comes the ellipse? We have a linear map from lattice to C, C is R2, and when we draw lines, we get the map from R2 to a vector space, and the image of a circle will be certain ellipse. Yeah, but now, you see that when we get this question of some parameter, we get a curve which travels around several one-dimensional strata. And you get kind of, the limiting things will be degenerative elliptic curve. It will be union of Cp1s. Yeah, so you get several Cp1s, and at infinity, and my circle approaches the chain of Cp1s, it will be degenerative elliptic curve. And now what we can do with this degenerative elliptic curve? Let's consider a small neighborhood of this union of Cp1s. We get some chain of one-dimensional strata. You say we get abstractly, how do we write? I'll do it right, and some Cp1s. But you see some, yeah. Alderic variety has some Poisson structure. These modular spaces all have Poisson structure. So I get a formal skin, formal variety, which is kind of high-dimensional, but it's kind of... Phytosome neighborhood. Phytosome neighborhood of union of Cp1. With this Poisson structure. It's this Poisson structure, which vanishes and some stratification, so on. And near each point, it looks like a neighborhood, formal neighborhood of a one-dimensional strata and an entoric variety with constant Poisson structure. So it's locally completely standard object. And then with Jan Söbelmann, we proved the following theorem, that if you consider abstract, this formal Poisson variety, formal Poisson variety, which is formal neighborhood of chain of Cp1 with Poisson structure, then the modular space of such things is unobstructed. Actually, it's infinite-dimensional affine space. But how do you... Because there are these discrete elements, but they're in useful components. Yeah, no, I mean each component, yeah. Each component, yeah. I can see the deformation of kind of toric. Why are they in useful component or in this shape? No, no, no. No, I can see the kind of this formal neighborhood of the things. I can see the algebraic formal varieties, which are formal neighborhood of chain of Cp1. But you take deformation if you allow to... No, no, you don't change the Cp1s, Cp1s has no parameters, but kind of when you can see the high-tail coefficients, how this thing glue it, you get some freedom. How they glue it, I'll explain in detail. I end of course speaking. Yeah, yeah, that's okay. But they transfer some tension. Do we tension or... No, no, they transfer some. Any kind of normal crossings, like coordinate... It's like coordinate lines. Like two coordinate lines. You don't allow tension. No, no, no, they don't see the different line. Yeah, like a local, it looks like two lines and two coordinate lines in the neighborhood. Yeah. Yeah, so the claim that it's a model of space. Yeah, kind of in first order you have no parameters and have at more and more terms and tail coefficients, you get more and more parameters and no constraints whatsoever. You get some infinitely many parameters. And these parameters could respond actually to some domain in the lattice. And then the... It seems you can find many parameters when you break it. No, no, no, but consider tail or coefficients. It's, for example, consider kind of stainless thing, but glue by some form of power series transformation. Yeah. And this transformation has infinitely many tail or coefficients, which are three parameters, yeah. Yeah. And the space of this is a model space of such a person's skin. It's the same as model space of wall crossing solutions when you, in this picture product of S1 times, better consider neighborhood of one S1. Yeah, I told you this is kind of this post property of analytic continuation. If you know numbers, omega, gamma, for all gamma in the lattice, then you can canonically calculate nearby for other things. And the claim that all this control, all this can be explained in different ways that you get the same formal scheme, which is not in general algebraic. Yeah. Now, so it's some kind of purely formal reformulation what is wall crossing. And now... Pause, I'm confused. What is the whole portion in case of... No, the reason is the following. Because when you consider automorphism of Poisson variety, they are given by Poisson bracket with a function. And this algebra has the same size as the functions. Otherwise you'll get vector fields, you'll get larger. Smaller. Yeah, smaller than all vector fields. Yeah, one can do similar things with vector fields, but it will be not wall crossing, because this is in some kind of more general sense. Yeah. And... Why Poisson in case of wall crossing? No, no, no, no. The reason is because this all model spaces are Poisson varieties. It's nothing deep in it. Yeah. In principle, the whole story can be generalized to without Poisson structure. Yeah. And now the story is the following. There's some kind of purely algebraic game how to extract numbers, omegas from this formal scheme. And now we get this nice algebraic varities. This take formal neighborhoods and extract these numbers. Yeah, in principle, this can be done so we completely computerized. You can look at this one, this scheme for this Poisson structure and number actually. Yeah, from the six, you can extract omegas and come completely formal way. If you get arbitrarily, it's kind of telecoefficient. You essentially cut by two pieces, you get no parameters and there'll be transformation how we glue to ends. The telecoefficient are free parameters. Yeah, so in principle, it's one can make it, from it's a computer program. It should be very effective. And it will completely different way to calculate this omegas, which Andrew talked about spectrum metrics and drew some complicated pictures on the curve and so on. Here we don't do anything. We consider some algebraic varities and do calculations with form power series should get the same thing. Yeah, strictly speaking with Yan, we didn't prove that it's, did we, yeah, no, it's some kind of all bunch of contracts how whole things are related. Yeah, that's this procedure with spectrum vectors should be the same as the spiral algebraic procedure. Yeah, but now, as I told you, it would be really nice to have domains when you have no walls. Yeah, and so this will be last topic, which I'll describe. There's a big class source kind of, at least hypothetical of such thing. So I claim that in the Hitching base, there is one specific point, but just should be a little bit more precise. So the Hitching system depends on plenty of parameters. And how many parameters it's some kind of random number not related to geometry, not related, not universal, so it can change complex structure on the curve, punctures, can have change in regular terms. And the most important things from when we have first order poles of the Hicks field, you can change eigenvalues of the residues of the Hicks filtered punctures. Yeah, so the kind of some, some parameters are more important than another. And one can vary this parameter freely and all this one can vary, arbitrarily residues, I recall that spectral curve has one form, and it's not compact, so it has some punctures and one form has the residues. This residues are important parameters, kind of central parameters, casemiers, and then I also vary them arbitrarily. The only condition is sum of residues is equal to zero. Yeah, now let's assume that our Hitching system says that all residues are real. Yeah, so it means that integrals of the one forms are not two pi i times zero, number, they are purely imaginary. And the claim is that for such a Hitching system, there will be some absolutely remarkable unique point said that not only integrals of one form about small loops or punctures are imaginary, but all periods are purely imaginary. Yeah, yeah, so this I called a generalized table differential. I'll explain you in a minute why I call it like this. So in fact, we guess the explicit weight, how to determine this point. Yeah, if you assume that all residues are real, then outside of the discriminant, we get a map to, we have real part of the central charge is the element of some vector space, and this identifies locally outside of discriminant my base with an open domain in real vector space. Then on a real vector space, we get earlier field which contracts all vectors. And so we get vector fields outside of discriminant and conjectures, it's extent to continuous flow everywhere in singularities. So it will be not differentiable. It will be, yeah, it's a really bad question, harmonic analysis appear here, but there is some, for example, function which is convex outside of discriminant and it will be gradient flow through this function. So it's very plausible conjecture that this thing exists. And in the case of SL2 system, this quadratic differential, the spectral curve is the same as quadratic differential. And this very nice point, the existence of this very nice point is exactly another theorem of Schrebel which Andrew didn't mention, but this kind of most famous result of Schrebel, that if you have any curve with punctures and some positive real numbers, then there exists a unique quadratic differential with such that my curve will be union of half-tubes with given perimeters. Yeah, so it's, my conjecture, it's kind of straight generalization of Schrebel result, but with a different approach, yeah, so even in this case. And kind of, but even for quadratic differential, it's something which is not, you can find in Schrebel book and it doesn't follow from his result. Again, for this cubic Schredinger, you get very simple things. For any number A1, which can be sort of this parameter of interval system, A0 can be sort of the point on the base. For any A1, there exists A0 such all periods are real or pre-imaginary. Yeah, so it's very simple things which I can prove, but... So it's a statement. That's a statement, yeah? If you take both additional... No, no, no, no, any complex number A1. And the gamma waves, gamma... Gamma will be all contours where this integral can be on the double cover, which is when the integral is defined, yeah? For any A... You said that this period will be pre-imaginary. All periods will be pre-imaginary, yeah? And the space of parameter C1 will decompose again. How could be? Is there a guarantee assumption? No, no, reality assumption. Ah, the story is the following. Because here the curve is elliptic curve is one puncture and the result is automatically zero. Yeah, it's a really stupid case because we have just one puncture. It's... And similar one can have higher order equation. It still will get considered odd degree polynomial and it will have still one puncture. In the same story. No, in this case, reality conditions kind of empty. You get automatically this thing. But in general, you need some reality conditions. Yeah, and then if... Yeah, the claim that if you're... Suppose that all the stable differential is kind of nice. So we don't have point on discriminant, which is general position when we have at least one puncture. So it's corresponding exactly to what and the considered quadratic differential, stable differential is simple zeros. Yeah, yeah, and then we get a point when we have this domain in very interesting involves. And there's really very little walls here because exactly at my point is zero, all the map from my lattice will be to imaginary numbers. And if you move a little bit, it will be open a little bit. Yeah, so you see, you get this still left and right domain where there's no walls. Yeah, so you automatically get two nice domains when you get this in general stable differential. And each domain should give some rational puncturation to get some birational map from one in coordinates, from one coordinate system to another. Yeah, so it will be some birational map. And, but the claim that this is even some kind of more basic object, this will be some sort of quiver. Because in the previous picture, when you have this transformation, when you expand this transformation, you get various vectors in a lattice. And the claim is that they lie in certain octant, coordinate, so it will be this vector in a lattice which appeared in the composition. We lie in the octant and it will be generators of this octant. So we get some nice collection of some basis of the lattice and then from skew symmetric form I can construct a quiver. Yeah, yeah, and the claim that all this quiver kind of controls everything. And this even by rational maps, everything is controlled by this quiver. The quiver is something else, it's transom quiver. Quiver, it's the same as... It varies in what category. No, no, no, quiver, abstract combinatorial quiver is the same as skew symmetric into geometrics. Because you wrote positive quiver, still the number of arrows. Yeah, yeah, so it's quiver is just way to draw skew symmetric into geometries. And now I just finished with two basic examples. So we can see the cubic, Schrodinger operator with cubic potential, which was an example, the quiver is just A2 quivers. A2 quiver. And by rational transformation is some of this kind of typical cluster, guy. But now the next example is really interesting. You can see the elliptic collodron mozer. Then this quiver has three vertices and two loops. And the rational transformation, I finally after many years of attempts calculated, it's such a monstrous, rational transformation which controls all secrets of this all-over-crossing structure. Okay, thank you. Okay, okay. Can you, is there a way of seeing, why is it that when you have a region which is free of walls upstairs, why does that allow you to get a bi-rational transformation, rather bi-rational coordinate system, rather than just a formal one? No, the reason it's looks that you get embedding of, in this case, you just get the risky embedding from C star to parental. Why is that related to not having walls in here? Yeah, it's a long story. Yeah, in this region, I think it's, one can see that when functions on your embedding model space will restrict to Laurent polynomials, not, there will be some kind of finiteness. I actually don't really understand it very well, I have to say. Yeah, mostly it's all kind of half experimental, what I explained to you. It's quite far from established. That's on this slide. So the user, the quantum model is a many parts for quantum. Yeah, it's kind of the simplest one. Yeah, 150. Have you tried to decompose this one into the elementary invariance? No, no, the story is the folk. In this case, one cannot decompose, or it can decompose. Yeah, it definitely would be not a finite composition of. No, no, I didn't try here. This means that it seems to be equal to take. Oh, X1, in indices one to three. So it's, yeah, just permute one to three cyclically. And it's clear it's actually invertible, the other transformation. So there are squares there, yeah? Yeah, actually it's a vibrational map. And, it's not clear, yeah, no, no, I calculated, it's a computer fine for me, the formula. No, we don't create all Y's invertible. No, no, no, no. No, for formulas, nothing is. It does no formula that look good. No, but it is, yeah. No, the thing is the folk, if we compose this antiportal map, XI goes to X inverse, then it will be in evolution. Yes, it's, yes, it's special property. And this has to do with the Hitchin system on the Taurus with one puncture? One puncture, yeah, quadratic differential with a tube, yeah, kind of second-order pole, yeah, that's it. Essentially, the stable differential is a very simple one. You can see the hexagonal lattice, kind of, and glue to tube to this lattice, and mod out by shifts. So you get a tube glued to itself, which will be punctured Taurus. Sorry.