 So I just recall what we had yesterday, what we had talked about last week. So we have a complex algebraic variety, a smooth complex algebraic variety, and to get a function, just a polynomial function, and it has, and right now I make some small assumption which is not relevant. I assume that f is a proper map, and also last week I consider case of pair, and devices x is empty. I consider integration of chains with boundary on D, but I'll skip for a moment this thing, so it's a kind of minimal situation. And in this case we get finite set of critical values whose elements I denote something like zi, this is a critical values. And also what I have for any value of Planck constant, which is just known as zero complex number, I define the Ramm homology with, depending upon Planck constant of x of h is the same as the Ramm homology without Planck constant, divided by f of h bar. And it's hypercomology of x with the risk etapology, and we consider forms. And as a differential we can write as hd plus df, or d plus divided by h bar, and there's isomorphism in just the scale on forms between such homology. So we get this form, and this comparison isomorphism, that's this drum homology, isomorphic to some critical collate topologically, some BTI version of xf, depends on yc. We are this h BTI. Is there a relation between using notation h bar and Planck constant, is there a sum? Yes, yes, yeah, it's eventually related to quantum mechanics, yeah, and it will be a small parameter in quantum mechanics, yeah. Exactly, yeah, it's not accidental. It's a homology of pair x, and you can see the kind of topological homology. We rescaled my function by h bar, and take preimage of minus infinity, or you take a number with a sufficiently negative real part. Okay, that's what I'm talking about. So now I'll speak about some another comparison isomorphism, which is between BTI and topological homology, but in different kind of BTI global and BTI local. What do I mean here? So again, recall that for any, if you consider any disk in C, and a boundary point on the disk, and I assume that this boundary of my disk doesn't intersect a critical value set, I sorted maybe the graded group, adb, which was homology with all possible degrees of, let me see the preimage of the disk and preimage of the marked point on the boundary. Now, so I get such things. In particular, one can make very small disks around critical points, for any critical value, and for any argument, I have a disk, have a disk will be some disk with center that is the I, and radius is very small. Yeah, yeah, but let's take round disk in this example. This very small radius and point B will be the I plus exponent point this argument theta. Yeah, so I get this, the special d with the I, r, d with the I, r and beta depending on theta. And this thing I just denote by commulger of beta commulger of depending on point the I and theta. Yeah, it's just notation for the small disks. And what this is in other comparison isomorphism, so the claim is that if my Planck constant doesn't belong to the finite union of what's called Stokes race. In fact, the conditions are argument of h bar is not equal to arguments of difference between any two critical values. So, the I, z, j are in my critical values. Now, so we get typically some picture like this, get few friendly many rays. Then we get canonical isomorphism depending on h bar between what between global commulger and direct sum of all critical points of local commulger with the argument you take argument of h bar plus pi rotate by 180 degrees. So, what is the isomorphism with almost obvious. So, you get your various critical points. Now, for example, this will be direction of h bar slightly rotated. And now I take opposite points. I consider commulger of pair for this disk. This is this base point. And this is identified with global commulger of pair. Essentially, one can understand like this. To calculate the global commulger of pair, one can take very large disk containing everything can that take pre-image relative to pre-image of this part. And then one can contract it to union of such thing without changing commulger. So, it will be identification between one to another. Now, so the only thing which we needed that we don't have this all rays will be distinct. Then we have no problem at all. Okay. So, the question what happens if h bar varies continuously crosses stocks ray. So, in this h bar plane, we get many rays. And suppose it's h bar moves and start to cross stocks ray. Then we get isomorphism a little bit before the crossing, a little bit after the crossing. You get two isomorphisms kind of for h kind of if you change h bar a little bit, the spaces are canonical identified. This latest is typologically. So, you get two isomorphisms kind of h beta kind of global to h beta local. I mean this stuff. And so, you can combine to get automorphism of this direct sum. Now, how to understand this isomorphism. So, this automorphism it's very simple. Yeah. So, it's again I just have some direction of h bar. Now, h bar lies on the stock's rate means that there is a line on which parallel to h bar on which you have at least two critical values. Maybe you have several such lines. Maybe you have another parallel line. Maybe another get one. Yeah. So, I get sorry, don't really create roads very little. Yeah. So, imagine picture like this. So, it's clear that if you get this automorphism, it will act on contribution for each line separately. Yeah. So, the different parallel lines will not talk to each other. And the picture is a follow. If you rotate a little bit, h bar will get picture. This maybe choose more color. If you rotate h bar in one direction, that all this rate will became distinct in one way. Or if rotate to another direction, you get distinct in another way. So, what happens is here for every line which is parallel to h bar and containing at least two one is well critical values. You get certain isomorphism. So, the line which I draw and have this line. I just have several points staying here and called something like ze1, ze2, zeik. It's critical values which belong to one of this line. I get an isomorphism. So, direct sum from j equal one to k of this local thing. Yeah, the same theta will be argument of h bar plus pi. I get isomorphism with itself. I get automorphism. And the claim that this is automorphism is kind of preserve the filtration's identity on the filtration. So, it will be j equal one to j equal k. You get low diagonal matrix with the identity on the diagonal. Why it is so? It's finite. It's a finite dimensional lattices groups. And it's matrix is kind of integer coefficients. Yeah. Maybe your conventions are not, it depends on. Yeah. No, even your matrix also has integer coefficients. No, but the question is how the order is. So, when you draw the picture h. No, because I said that h bars ordered in this way. So, it means that I order, I have, it's oriented line, yeah. Because I have h bar and the line is parallel to h bar. So, it's automatically directed. Now, if I choose h bar on opposite direction, I order them in opposite way, yeah. Okay. So, you say the contribution of h point goes to the inverse, go to something which involves this plus correction in the lawyer. Yes, yes. Only in the lawyer's thing. And this is supposed to seem topologically but yes, yes, yeah. How things move around. Yeah. The picture is a following. It's kind of easy, easy to see, work with homology in this case. And in the homology, it's clear that you get direct sum decomposition. But it will have kind of natural filtration, which is preserved by this transformation because, for example, the law's term will be cycles which are supported in this domain. Yeah. And then it doesn't change. And so, you see preserved filtration also has identity on diagonal. But then it will be upper triangular matrix if we go to what happens, homology, get loaded, get opposite filtration on homology. And homology, get filtration in the opposite way. Okay. So, what is the conclusion of all this? So, given as this finite set of critical values and plus a bunch of local systems on circles. You see that this homology, local homology, depends on the argument. And they form a local system on circle, get kind of group with automorphisms. So, we get local systems on S1 which is just exponent of i theta. So, this local system is h beta zi theta for each i. And plus for any pair of i ordered pair of two indices, or maybe just two distinct points on S, singular set, we get a map of a billion groups. h, b, zi, n direction dependent on two points. Yeah, I do know this local system with same notation. Just i primus. So, that would be the conclusion. So, what we get? We get a collection of local systems and all this in bunch of non trivial matrix elements in this algebra. Yeah. So, eventually what you see is that you get how many non trivial elements it's will be for any pair of points. For any pair of ordered pair of points, you have unique directed lines which go through and then we get this bit of matches. So, we get this algebra data. No, but I don't know. So, you are going from zi. So, in the picture like you have zi1 to zi2. Yes, yes. You get some operation from zi2 and go from zi2 to zi1. It should take minus h bar. I want minus h bar is like minus h bar. This is their convention. Yeah, I think it's compatible. Yeah. So, I get the things, all this linear algebra data and the claim is it's the same as it's So, there are no constraints on this? There's no constraints whatsoever. Yeah. Last time I explained it's there was some kind of data for some constructible shift with arg of equal to zero. It's equivalent to this data. It's another description of the same data. It's a constructible shift on c with singularities in s with an arg of each component in fact of the shift is equal to zero. Yeah. So, that's kind of what's the apologies relevant for our consideration. Yeah. So, the main thing that's rather interesting operators, maybe I don't know what they call this operator. It will be usual shift. It will be both perverse shift and usual shift at the same time. So, this is the same as the perverse shift which I just talked about. Yes, yes. With the normalization of the perverse shift. Yeah, with some normalization. Yeah, maybe we should buy one from another. No. Yeah, but the main thing is what we describe is we describe kind of local system on circle send some maps and a lot of linear maps. What are these things? It's maybe called this operator something like tij, I don't know, tij prime. There's no constraints whatsoever on this, I think. Okay, let's try to understand what are these maps. It's just even kind of most special situation. I assume that my thing is proper, no divisor, but I assume that all singular points or critical points of f are isolated and moreover, as I assume, they're holomorphic Morse points. What does it mean, holomorphic points? Yeah, so it's maybe called p1, pi, sitting in x, and f of pi is equal to zi, sitting in c. What does it mean that points is holomorphic? So first, derivative at pi is vanish, and second derivative is non-degenerate quadratic form, and this means that there are some local coordinates, something like x1 to xn, n is the dimension of my variety, near points, which given by the coordinates vanish at this point, and function is sum of squares. Analytic, yeah, analytic, yeah. Yeah, I would do analytic. Yeah, so I make this thing, and in this situation, if you make this even special assumption, then this homology of pair will be, it's concentrated in degree one. Yeah, there are numbers, yeah. Ah, plus the i, yes, plus x maybe j, plus, yeah, right. It's constant plus sum of squares, yeah. So this homology is concentrated just in one degree, which is middle, and dimension is equal to the number of, maybe I assume, and I assume that all the i non-equal to zj, here, all critical values are distinct, kind of general situation. So the dimension of homology will be number of critical values. And this local homology, um, uh, sin theta, it will be just in middle degree, and it will be just one-dimensional space, one-dimensional lattice. So it has kind of canonical basis defined up to sin, because the size, is isomorphism is defined up to sin. So let me explain this. Uh, this very basic calculation, we have such a function. And, uh, if you look on the definition, it's some homology of pair, what will be this homology f b zi theta, uh, f beta, b is for beta, zi theta. It's the same as homology of a, it's the end of the day of a ball in sin. It's a coordinate sin. And relative to the pair, this coordinates x1, xn, which are complex coordinates, uh, modulus of, uh, hyper surface sum of x i square equal to epsilon times e i theta. And, um, and this is, uh, epsilon is very small. And, uh, this complex quadric, it's homotopy equivalent, equivalent to a sphere. And, uh, when all x i up to rotation are, let's see, real numbers, this is kind of square. And this is contractable, it's homotopy equivalent to the disk, to the end-dimensional disk. Uh, and, uh, yeah, so this whole story, it's isomorphic homology of pair, disk, and, and the sphere with integer coefficients, which is one-dimensional space in degree n. But identification with z depends on choice of orientation of, of the fundamental chain. So, there are really two choices. Choices, uh, uh, depends on orientation, this identification. And if you make a monodromia here, uh, this, uh, this argument c rotates, you get one parameter family of these things. The monodromia, is easy to calculate, is, uh, multiplication by minus one to power n. So, if n is odd, you cannot really choose the orientation in, uh, compatible ways, so it's, uh, it's essential data. And then, these operators, uh, whatever, maybe I remove them, tij, which I hit here, but, uh, these operators, tij are essentially integer numbers. It's one by one matrices, it's kind of one by one integer matrix. So, it may be called some n, some nij, i non equal to j, and defined up to sine. Uh, the space finds us to seek about orientation. Okay, so, so, the whole data in this case is given just a bunch of integer numbers for i non equal to j, how all these things glued together. And now I want to describe some meaning of nij. It's again, more convenient to think about homology instead of homology. Uh, uh, and, uh, the claim that it, for h bar, it doesn't belong to stocks race, we have a basis of, of, of homology, depending on h bar, uh, of my xf, uh, corresponding to critical, uh, defined up to h, basically defined up to sines, corresponding to, uh, critical, uh, points, critical values, and, uh, the basis can be represented by some sequence called, uh, left-shot symbols. Yeah, so, uh, so what are left-shot symbols? Yeah, I'll be, again, put more and more constraints here. Uh, so, I assume x is a scalar, and, uh, and pick, kind of pick a scalar metric. Uh, the scalar metric gives two things. It gives a remaining metric, and it also gives a symplectic structure. And, uh, now we have, uh, of holomorphic function, we divide it by h bar. It also gives two, uh, real functions. It gives function real part of h minus one f, it's a holomorphic, and get imaginary part of h minus f. Uh, now, so you get, uh, two different things, and, uh, now consider gradient, uh, now let's use left function and left structure. Consider gradient flow for real part of h one minus one f. On the symplectic side, we can consider different story. We can see the Hamiltonian flow for imaginary part, one of f, and that's really just one second calculation shows it's just the same flow, same vector field. Uh, on, on your manifold, and that's really very remarkable. It's, it's, it implies that, so that some corollary, that, uh, if you consider trajectory of this gradient flow, uh, and map by my f, map f to c, trajectory will be some line in my manifold, it's to see what we get, we get, uh, horizontal line, kind of maps to, to positively oriented, oriented horizontal line. Uh, because, uh, it's Hamiltonian flow, it means that it preserves the imaginary part, so it means that it should go to horizontal line and real part will increase. Consider, yeah. So the real imaginary part was some corollary? No, no, no, no, no, the, the Poisson bracket is something. No, but you're saying that the gradient flow of the real part is the same as Hamiltonian? Ah, for the imaginary part of it. Yeah, yeah, yeah, it's imaginary part is preserved and real part increases, yeah. Yeah, so, um, so we define kind of left shift symbol depending on, uh, one critical points and, uh, direction theta is, this is left shift symbol is, uh, maybe it's called stable manifold for, uh, uh, for the gradient flow, uh, uh, uh, which is, uh, points which are attracted to point Pi. I'll read you, my, my theta is argument of H bar. So just this, this set of points in, in X, such that limit when time goes to plus infinity and I play this flow with time t of my point and the limit is Pi. Yeah, so the claim is that this thing is different morphic to Rn. Uh, Yn, uh, because a real part of H my, uh, it's my function for which I make gradient flow is a real Morse function and, uh, uh, it's Morse singularities and signature is n pluses and n minuses. It's two and real, two and dimension manifold. Oh, so this, uh, so it looks like kind of Rn which maps to, uh, C and if you look how it will be the projection to C by F, it's kind of C with values of F. Uh, you get, oh, maybe F divided by H bar. It's a Z bar divided by H bar and, uh, so the projection of the symbol is, is, uh, negative ray starting and is a critical value and, uh, and it's easy to see that the fibers are spheres except, uh, the tip of the fiber is a point. Uh, yeah, uh, why it, uh, why it happens because locally near this point, you can use local coordinates and see that it's kind of standard picture Morse theory. But then if you follow along the line, uh, one can do the following. The fibers of my map, uh, my map F are also symplectic manifolds, uh, just restrict color form. And because it's part of some hyper surface, pre-image of the ray, there's a connection. So they're canonically symplectomorphic to each other and I just, uh, transport by the symplectomorphism, I get fibers will be sphere, sphere everywhere. Yeah, so that's the picture how it, uh, uh, how represent basis elements of this local commulger and get, uh, get pony classes in, uh, uh, large commulger. Now again, assume that we are on a stock's line. We have only two, only two critical values on a real line in x parallel to h bar and h bar is on a stock's line in its own plane. Yeah, so you get two critical points like the i and the j. And, uh, again, I just repeat my picture what I had before. Uh, if I just move, rotate a little bit h bar, I get, uh, uh, uh, two, uh, left shift symbols and rotate another way, I get just another pair of left shift symbols. Uh, and the claims that, again, for generic calorimetric, n i j is, uh, exactly number of, uh, number counted with signs. I should take, take care about orientation of gradient lines of, uh, my function, uh, connecting, going from p i to p j. Uh, so how to, again, to understand it? Uh, yeah, if, if, uh, if h bar is on a stock's line, then after rotation I get, uh, z i and z j have the same imaginary part. And, uh, in principle, one can imagine there are such, the gradient lines static and z i can't end at z j. Uh, so how to understand it? If you, if h bar is not on stock's lines, then all imaginary parts are distinct. Uh, we don't have gradient lines and it's something which should, one should expect also from Morse theory because all, uh, uh, Morse indices are equal to n and generically Morse lines go when Morse index jump by one. So there should be no Morse lines at all and this really happens because imaginary parts are these things. But if we have one parameter family of functions, then, uh, its special values of this one parameter, you have this gradient lines which connecting two points with the same Morse index and that's exactly what happens. Uh, here, uh, sir, you don't actually need Kepler, right? This is back in your game. No, no, no, no. For this you need Kepler to, to, to say this gradient and Hamilton. But they have complex structure already. Yeah. Yeah. Yeah. So it's, it's essentially the picture what is n i j, but one can, uh, um, what did you, you said something about connecting Can you repeat what you said about connecting points of different or the same? Yeah. No, you can forget about complex geometry. You have real manifold which remain in metric in the function and with more function. Then it's, it's, it's generic. Then you have only, uh, could have only gradient lines connecting points when Morse index strictly decreases. Yeah. Yeah. We should be for the same Morse index which have no, uh, no gradient lines at all and isolated when it jumps by one and one parameter families for jumps by two and so on. Yeah. But now if you have one parameter family of functions, then for special values of parameters, this Morse lines will, will connect to points with the same Morse index. Yeah. And here's what's going on. We have all points with the same Morse index and have one parameter family because we rotate, uh, uh, variable h bar. Yeah, but it's not, yeah, it's, it's not really related to the story. Yeah. It's, it's kind of equivalent things because it's a little bit hard to decipher what is counting with science here. Uh, according description, I can just say is the following. You just gain two half, maybe say two, the I, Z, J. In fact, it works for many points. What you do, uh, you consider, uh, uh, this left shift symbol goes to one direction, which projects actually to some ray. But now from, from another point, you, uh, shut this, uh, left the table in kind of almost opposite direction. And then you get two, maybe non-compact manifolds, but we each intersect by compact in a compact part. And then we had to well-defined intersection number. So it's stable under deformation because, yeah, it's the same description, this intersection number is the same energy, which is, yeah, yeah, it's kind of more convenient to think in this way. It's, yeah, it's easy to prove here, yeah. This is to do with the table for the other type of forms, yeah, which are dual. Yeah, yes. Visuality and you'll use it. Yes, yeah. Yeah, maybe I just, uh, just to be even more concrete, I assume that X is just- The sign also. Sir, you have intersection number, it depends on the- Yeah, it's saying you should orient this because it's actually one-dimensional spaces. It's, it's, it's a basis up to a sign. You should orient the things. Yeah, yeah, that's a story. Yeah, no, I just want to be very concrete. Suppose X is just C and F is a generic polynomial of a degree D, at least two, and to take generic angle theta. Uh, so we get the basis up to sign of first homology because there's no other homology here, X, F, and how it looks like. So in C we just draw, for example, D is equal five here, draws something like uh, domains when real part of F is uh, negative, uh, in very far to each other. Then we get some, and then what we get, we get something like these left shift symbols. We'll be certain disjoint uh, uh, uh, lines will be, because in Cologne just will be real line embedded in the things and uh, passing through, through uh, critical points of F. So, so these are zeros, uh, these things are zeros of, of derivative of polynomial. And they form some, something dual to planar tree. Yeah. Okay, so that's uh, basic story, uh, yeah, just to get concrete feeling what is going on. In my first lecture I explained most general situation, we get a variety, maybe map is not compact and you get some divisor, so this, you can see homology of pairs. And the whole thing extends to this, uh, storage because it's really topology here, nothing about complex geometry. Okay, yeah, so the, uh, the conclusion is that there is a kind of typically left shift symbols and maybe some more complicated things if critical points are uh, more complicated. Uh, now I just want, um, yeah, so let's keep the same assumptions. Uh, it will be kind of, this F is proper and you get uh, divisors empty, so we don't have to speak about the boundary and, and all singularities are, uh, uh, holomorphic morphs, all critical values are distinct. So now pick some holomorphic, uh, picture algebraic volume form by volume and the nth dimension before. Then definitely, uh, I get a class in the RAMc homology. Uh, so, uh, d plus 1 over hdf is equal to 0 because both differential form is 0 because there's no n plus 1 formant. This is also to 0, so it defines the homology class in this RAM thing for nh bar. Uh, and it's comparison isomorphism which I talk about with beta, uh, the RAM beta. It's both sound just to calculating of some integrals. We get EI of h bar is equal to integral over my exponent of my polynomial divided by volume of n and integrate over left shift symbol plus pi. Uh, now, so the integral is convergent, so you get a bunch of, uh, uh, it's for each critical values you get a function, but this function is holomorphic outside of stock's rays. It's actually defined when h bar is not to stock's rays. Left shift symbol, and I go to the direction, negative direction, uh, on h bar belongs to C minus union of stock's sector, stock's rays. Yeah, in fact, uh, yeah, so you get, not, uh, in sense one, get not one function, but many functions because, because many sectors, but each functions admits analytic continuation, each kind of sector admits analytic continuation to, uh, to universal cover of C star. Uh, now the picture is something like this. You get some critical values, and if you want to calculate integral, you integrate over pullback or left shift symbols which, uh, which is the lift of this, of some ray, but now it starts to rotate h bar. What you do, you just eventually, uh, take, if you go some pass in the h bar plane, you just lift c, uh, uh, corresponding, uh, think, yeah, so it's clearly admits an analytic continuation. Um, uh, also let's me make some another assumption, uh, that, uh, there is no line connecting, uh, three point, at least three, uh, at least three points in s. Uh, so it means that equivalent that number of stocks rays is m times m minus one, where m is the number of critical values. So you, so you see you get a really huge collection and get something like m times m minus one functions. So as we assume that it is only one critical value? Yes, yes, yeah, yeah, I forgot to write it, yeah. Yeah, so get something like cubic, cubic will depend on number of functions because, uh, you have many, many sectors and functions have, functions have a syntax i. Now, so you have the number of stocks, not, not bigger than three, m minus one, so you have set this number of sectors, ah, okay, yeah, and then each of them has such and such number of, okay, yeah. So get plenty of kind of different functions, which I denote by one function defined in the joint union, but in real life it's many functions. And now if I have a, again, uh, this, uh, situation when the i goes to, let's say zj, uh, let's note like this. So, so you have the same picture. We see if you, if you cross the corresponding stocks raise, cross the ray, each bar goes to this direction, then the, the ij, which has a small integral, does not jump. It's, it's analytically continuous across the ray, does not jump. And, and i, i goes to i, i, again plus, uh, depending on the situation, it's replaced by, uh, uh, jumps by, um, this function on j, okay. Now define kind of modified functions of each bar. Yeah, I divide by what I expect for quadratic form, this is a, we can discriminate, determine this quadratic form times i, h bar, and you get some, uh, uh, functions. And the claim that it's asymptotic, yeah, asymptotic in each bar in this sector, yeah. So, you get some serious, uh, yeah, in principle it depends on the sector, but the claim it doesn't depend on the sector. You'll see it in a second. It's, uh, is it asymptotic in the, in the closed sub-sector or in the world sector? It's actually asymptotic in some, uh, uh, larger sector, uh, strictly larger sector. Because of the analytic. Yeah, yeah, yeah, yeah. Yeah, so, so we just modify, rescale the things. So, what we have, we have this jump of gj modified to zero, and jump of i, i modified is, uh, okay, this nij. Now we take minus the thing times i, i modified. This is, this i, i modified jumps by this, uh, by, by this expression. But, uh, this thing has a trivial asymptotic expansion. It's functions like x point one over x. It has trivial zero parallel expansion. Uh, so you see that, uh, this coefficients don't change. In fact, it's easy to see that one can calculate it's purely, uh, uh, algebraically using formal power expansion and kind of Feynman graphs. One can calculate them purely algebraically. What do you mean algebraically? Is there some topology also? No, no, no, there's no topology involved here to calculate this, this series. You can do a, use algebraic geometry if your variety defined over number filters will be serious. Okay, you don't need any, uh, any data coming from the complex topology. No, no, no, no, it's completely formal, uh, uh, thing. And the conclusion is that we have this, our integrals, uh, it gives a solution of some Riemann-Hilbert problem. Uh, explain what, what is this? This Riemann-Hilbert problem. Which, uh, the input for this Riemann-Hilbert problem is this, my collection of s of this zi and zn collection of, maybe just write z1, zm, like, like order them n, and then we get some integers. And I kind of put under, under the re-execution about orientation. So it's up to same issues. Now, if you get just collection of complex numbers and integers, then we get a halomorphic m dimensional bundle, vector bundle on c. On c, uh, coordinates I denote by h bar. And, uh, the, by definitions will be the following. The sections of this bundle, global sections of the collections of, of functions psi i of h bar, uh, h bar belongs to c minus stock sectors, stocks raise, kind of like my integrals, admit a synthetic, uh, which admits a synthetic expansion. Yes, h bar goes to zero. And, uh, jump is given by the same formula. In this situation, kind of, in this situation, if you explain here, uh, jump of, if you go across this, uh, stocks line, so psi j goes to psi j and psi i goes to psi i plus n i j psi j, uh, multiplied by this exponential factor. Okay. Yeah, so this is some question which has, you can forget about your variety, about, uh, things, you just have this data, uh, complex numbers and integer numbers, and you get constructive vector bundle, halomorphic vector bundle. And there is explicit trivialization of this bundle. And I'll write your formula in a second on an open disk, uh, where h bar is, uh, and this r, uh, of radius r, where r is, will be very small. And this formula I learned from, I described what is the section of the bundle? Global section. Global section, all can take neighborhood. Is it algebraic bundle or halomorphic? It's halomorphic vector bundle. I describe, I describe what is a halomorphic, I describe the bundle in terms of what is this model of its sections. Yeah, I put. Yeah, so let's say on every open section, see the, the, you mean the section on this? Yes, yes, yeah, you can restrict the question. Yeah, the mean, you describe this, this, this bundle outside of zero by this jump formula. But what is the, the, what kind of a synthetic expansion you know? No, I need, I need this, it wasn't a bit sloppy. I said that it's, it's extend to open, uh, uh, the functions are in stock sectors, but actually they extended to some, uh, larger sectors and admitted synthetic expansions, uh, within this, uh, of this form. Oh, the extension also. Yes. So it's clear what the bundle maybe is outside zero, but it's zero. Yeah, it's, yeah, that's exactly this whole, yeah, it's not clear what this bundle is zero, yeah. What is the fiber at zero? But is it clear that it is really a bundle? Yeah, it's a bit, uh, yeah, it's not totally clear, but, uh, let's miskeep it, yeah. Uh, because functions are really, the, the thing that this functions has zero Taylor expansion. It's, it could be anything finty functions with zero Taylor expansion. One can look out that you get still holomorphic bundle. Yeah. Maybe is it related to the theory of, there is a theory of, of classifying homomorphic connections. Is it related to? It's related, yeah, but it's more general. Yeah, yeah, yeah. Yeah. Yeah. Yeah. So those, now I'll write, now write a formula and then I think your questions will be answered. Yeah. Yeah. So the formula, what is the trivialization of this bundle? And I learned it from, some paper of Gaiotto Muranetski. Uh, um, so what they propose, they propose a base of sections, which is called a psi i for i for one to m. Uh, so, so what, what, what means the section of bundles? So it means that you shift, uh, some functions phi ij of h bar, where h bar belongs to c minus m times m plus one raise. You have many, many, uh, holomorphic functions. So it is thanks to this jump property and now, now write the formula. The formula it's, maybe there's some small mistakes with science, but basically it's correct. I want to write the value of functions of a formula. First of all, it says delta ij. So, so, if there's no jumps, this will be just standard basis and plus sum over all n greater than one and takes some over all sequences of indices such no, no, no, no, no two subsequent sequences are distinct, but they can repeat it after some intervals. Now I take n i 0 and 1 and i 1, i 2. Okay, I'm going to take it. Sorry, maybe let's call it k. n i k minus 1 i k divided by 2 pi n multiplied by certain integral, and the integral is like this. You take, yeah, and now I should. Sorry? How does it depend on h bar? Well, the last guy. And where is the integration? That means integration is a following. This h l belongs to z i l plus 1 minus z i l multiplied by positive numbers, maybe strictly positive numbers, and h l less than r. So the form h l runs to some interval. Some interval of lengths r. And if you, yeah, first the formula, it's very easy. If I get convergence conditions that it satisfies the jump formula because you integrate, if you jump the h, the h you get integrated a little bit, one side and another side. So it means that you substitute h bar h k, h equal to h k, and you go to the previous expression, yeah. So it's definitely satisfies the formula. And when the r is very small, these things in the numerator will be so small that it will be a guarantee convergence of the whole sum. Yeah, one can make. You sum over h in this? No, no, it's integral of h l. Yeah, h l runs through some interval. And the integral converges in the usual sense? Yeah, because, yeah, because what happens here, you get exponent minus positive number, positive real number. And in the numerator, you get h bar, but different h bar lies in different rays. And to get a little bit of small denominator, but this guy in the numerator, it's, you get polynomial growth from denominator, but the numerator is exponentially decay. Decay, so. So essentially it's real, well, yeah, here it's a real number, real positive numbers and very, very small than h bar close to zero. And the poles in the numerator get something big only if it all points close to zero. Yeah, one can estimate the norms here. It's some work here, of course, but. Yes, you need r small enough for this, because. No, some of, for some of okay, I think you need r small enough, because otherwise it will like geometric progression, which you do not control. Yeah, maybe make a small break, something like five, between five and ten minutes and then continue. Yeah, maybe just slowly continue. Yeah. So we get this bunch of sections here, or maybe I just exchange whatever, and annotation. Yeah, so we get many things which are sections of my bundle and define m as, I just put as a column vector say one, say two, and so on, say m. So what does it, it's glmc valued function, or again on h bar, which it doesn't on, of h bar, which is c minus sectors, on sectors, and the limit of this function is its identity matrix, because, yeah, the leading term is delta and the rest will be exponentially small. And when you consider jump, then m goes to up or low triangle or depends on, oh, it's jump by matrix, which is, sorry, yeah, yeah, sorry, I just started. I repeat, so I have my explicit sections in a small disk, and I form them together as the columns of some, together form a square matrix, and to the glm valued matrix is limit equal to one. And when I consider jump, if I cross the ray, it means that I multiply matrix, I think maybe on the left, I hope, and it ij place, what I put here, it ij place, I put this n ij, this exponentially small correction, which I... And it has an asymptotic expansion. It also has a symptotic expansion, and so, yeah, I'll just see it in a second. And I kind of have i is, it will be just the m-valued function on h bar, going to union of sectors. And the jump of i bar, it's essentially the same, so it's the same jump formula. So the conclusion is that if I take inverse matrix multiplied by i, it has no jumps, and then this means it will be halomorphic c to power m-valued function. Sorry? On the disk, yeah. E-modified, yeah, it's modified, yeah, sorry, yeah. Yeah, so falsity is on the function of this, but then you can use the expansion. Because it has trivial, it has asymptotic expansion, so definitely. Yeah, but now what happens is, let's look on asymptotic, on asymptotic expansions. More carefully, you get some formal power series. Again, this expansion doesn't depend on the rate, so it's, which belongs to Jlm, it's a formal power series, and you get i formal, which belongs to Cm. Yeah, these things can be calculated algebraically using my variety, and this is some universal question which should be studied using the same integrals. Then because, then we see that m minus asymptotic expansion of m minus inverse i is m minus inverse formal times i formal. And so if you calculate these formal things, we see that because it's analytic, it should be convergent things. So it's automatically convergent for h less than r. And so if you, so without calculating integrals, we can calculate it just because we calculate the series and take the sum. And now we see that what is ai at h bar, it's m of h bar, it's something, it depends only on this data, stock's data on z i and n ij, multiplied by convergent sum of a formal power series which we calculate for algebraically. Yeah, so it's, yeah, so it's pretty remarkable, and one can, presumably you can make a kind of computer program to calculate integrals without calculating integrals in a sense. So we, what about the m, you said that the m formal is asymptotic. Yeah, because, yeah, this is, there's some certain m. Also this is algebraic. No, no, no, no, it's, it needs to calculate these integrals if you consider the terms of asymptotic inspection h bar, it will be some complicated iterated integrals. Okay, so this is not, yeah, but it's not related to original variety in the function, it's related only to the points and z i n ij. Yeah, yeah, so in a sense one can be very, very optimistic and hope that it's, can be used in a quantum field theory. Not just it's, I will maybe go to, if you consider some Feynman integral, if you don't know what is going on, it's infinite dimensional integral, what are cycles, but we can calculate what are critical values of the action on very complex effect things, solving classical equation. This n ij will be some number of instantons. That tends to be a very good theory, because usually the expansion every term will be a generally quantum field theory. No, of course, no, you get also renormalization for formal power series and so on, yeah, but the whole story is very general, it can generalize to everything, to arbitrary case and maybe not proper and so on, the whole business and the only difference that I modified will be not serious in age, but will be in general, maybe have Lorentz series, maybe in fractional power of h of certain whatever k and polynomials in a logarithm of h bar. Yeah, so that's only, in general case, you get just a little bit more tricky. Story. Oh, I will be pretty brief. Do you like what is ii mod? This modified things, which I didn't even have known as a little, yeah, I consider pairing between the Raman beta and then I get. And what do you have in your thing? What do you have the k and the low k? No, k for some k, some integer k, some ramification, yeah, some monodrama could be quasi-unimportant and so on, yeah. Do you usually, quantum fields are not convergent subs? No, no, no, formal series are not convergent. Now, what I explained to you, you get divergent series, but this universal way, you made them convergent. Right, so what I'm saying is that as far as I understand, you generate quantum fields or you cannot make it to work? No, at least what I explained in my lectures, one can do it in series up to three dimensions. Yeah, definitely, yeah. So, we can think of this as flat sections of some collection. And this is in monodromes, the transformations in monodromes of some collection? I don't know. Yeah, so half an hour time, yeah. That's part of my lecture today, will be generalization to closed one forms. Yeah, so my original situation was that you have smooth variety, divisor is normal crossing and some function. And I will generalize it to this setup when we get variety, divisor and one form, algebraic one form on x, which is closed. Yeah, so think, if you get a function, just take differential function. Yeah, so it's, yeah, it's all the same story. So, for h bar non-equal to zero, one can make the Ramco model, depending on h bar, d alpha will be, I just repeat the definition, because in the definition I use omega xd, d plus one h bar alpha. Okay, and what is omega xd? I use this notation last time. It's a shift of forms whose restriction to any component of divisor d is equal to zero, the kind of nature of forms we use to come out of pairs. And there's some kind of funny curious equality that omega xd, in fact, one doesn't have to introduce this new notation, but one can use standard notation of log forms and you can, and you twist it by functions vanishing in divisor d. Yeah, it's, yeah, just one can, maybe to understand what's going on, as I mentioned, nx is kind of c, this coordinate x contains d, this is the point, x equal to zero. Then you can see the omega xd, global section of omega xd, you get polynomials vanishing in zero and one forms. And if you consider log forms, you get just polynomials on x and cx multiplied dx over x. And if you multiply by x, you get the same shifts, yeah. Okay, so the claim it's easy exercise in resolution is singularity, that there exists algebraic compactification x bar containing x, and tangent x bar has normal crossing divisor whose components are divided into four groups, divisor I will note. So I borrow part of notation from my previous lecture, I get kind of horizontal and vertical part when I get a function. So what is this relation? So there are four different devices all formed together in normal crossing, x is x bar minus d horizontal union d logarithmic union d vertical and d is x intersect with d bar d. It's inset and water conditions that's constrained, it's locally in analytic topology, not in the risk topology. One can have the following. Form alpha is sum of three terms, where a regular is a regular form, has no poles, all forms are closed. So a logarithmic is locally can be written as a sum c i d log z i, where c i are nonzero complex numbers and z i equal to zero equations for logarithmic part of the divisor. So that's why it's denote by log this part of divisor. And what is infinite? It's locally differential of some function, and function is some constant, non equal to zero divided by product over some other indices, j is zj to some maybe some nj times one plus o one, where nj are strictly bigger than zero, and the product of zj equal to zero is locally equation for d vertical. If one form is differential of function, this part is absent, and it's essentially the composition which I used in my last talk. And formal locally cannot be necessary exactly, because if you add a comparatory by divisors normal crossing, it can have some non-trivial periods along divisors, so it means I should correct it by this part. And those things are uniquely characterized in some way? Yeah. Given the complication. No, no, no, one can find it's not every complication. No, no, but suppose you have a complication which is good. Yeah, of course the d bar is the closure of what you have. But the other, the decomposition of the other start this. Yeah, the h is something which you throw away, but it's the form has no zero, no poles at all. Ah, a form, its form is, sorry, I forgot to say that form is, where is the form? It is on, I mean one closed. You can see the x bar minus d log and d vertical. Okay, so you look locally at alpha, and now it bears analytically, and you construct your stuff. Yeah. In fact, in my last talk, last week, I said that it's d bar and d, d horizontal should not intersect. It was a unnecessary condition, so everything works without any constraints. No, in my last talk, I just assumed that these two things do not intersect, and it's, it was, it was too cautious. It's everything works fine, it responds credibly to, in all case. And when you write your, in the last line on the blackboard, alpha is in omega 1, what is the substrate? No, on the middle, on the middle, go down. What is the substrate of omega 1? Closed. Sorry, yeah, I'm back to writing. Yeah, this, then this, in order to calculate Rammkommulger, what should we do? We can use this compactification. We have inclusion of x 2x, 2x check, and maybe a little bit, yeah, you can consider, you can calculate Kommulger using direct image on x bar, and what we get here, we get omega x bar. Yeah, this is instead of writing, it's omega x bar d. It's kind of this finite notation, and then I allow poles of arbitrary order on d horizontal plus d log plus d vertical. Yeah, this star I remind, it means it's poles of arbitrary order, finite order. And it contains a sub-shift, sub-shift, which I, in order to, I just denote by omega tilde dot, and depends on all these things, but what is it? It's this, locally it's considered forms on x bar, but we, which are logarithmic everywhere, minus d bar, so it says it's actually not logarithmic forms, but vanishing one, and such that if you multiply by alpha, this form, it belongs to the same thing. Yeah, so it's some complex which I proposed about a year ago in case of functions, but it works the same for forms. So I get sub-shift, and in fact it is a vector bundle, which is not terribly obvious. It's clear that it's torsion free-shift, sub-shift of vector bundle, but in fact it's a vector bundle, and so there is a serum, just you can take it locally, this omega tilde, with differential d plus alpha h bar, maps to this star, with differential d plus alpha h bar, induces a, induces a quasi-isomorphism. One should check some circuit infinities that get some acyclic complexes, so it means it can replace by some something much smaller, and it's actually, it's quasi-isomorphism, but here's a really funny constraint here, for h bar, which doesn't belong to finite union over i, which is a component of the log of what I consider in this h bar plane should remove finitely many kind of inverse arithmetic progressions going to zero, strictly positive. Yeah, it's really a local calculation, you can check its quasi-isomorphism, who checked some associated graded, and then we see that sometimes it's first. Yeah, so we get this reduction to some nice finite dimensional problem, except this finitely many arithmetic progression in h inverse. The union of i is in the component? Are components of the logarithmic, for each component I get some number, which will be residue of my one form along this component. See, i is integral of alpha of some loop surrounding corresponding components, it's a residue. Now, so it's really a small local calculation, yeah, in fact I can just show where this condition looks for, kind of sample calculation is the following. So, imagine that you get x equal to zero, it's just in one variable, it's equation for this logarithmic guy. So, you can see the c star, it's my x, and more or less, and c is a compactification. Along a small loop surrounding or residue of one form, close one form on the divisor. Yeah, so when you can see the differential compactified story, we get something like this, and xn to power n goes to n plus c divided by h bar, xn dx over x. But now we can extend, it's something like i star, and this is something like omega tilde. We get Laurent polynomials, the same differential, so it means that we allow n, now, arbitrary integer, and you see that you get multiplication by zero, only if c divided by h bar is positive integer. Yeah, so that's basic calculation. So, one should have the quasi-izomorphism, some exceptions. Now, what about comparison theorems between, what does it mean we take integrals and so on? Kind of, there's a global comparison theorem between bet and, between bet and deram, it is the following. So, this h deram, h bar xd alpha, is canonical isomorphic to h beta, h bar is the alpha, which is, I'll kind of write very roughly what is going on. We have this compactification with four types of divisors, and we can see the x0 will be x bar minus all four devices which we have here, rd h, deloc and d vertical. It's some open, the risk opens some variety and we add real boundary to it. Real boundary using real blobs, get manifold with corners, and this h beta can be identified with the cumulogy of, you add boundary to this open variety, it's kind of real nullity construction, and gets certain sub, in certain domain, certain domain in the boundary. And, but, but not yet, and you take the risk efficiency not in z, but in local system associated to alpha divided by h bar, and what does this, what is this local system? On x, on x0, you can see the shift of solutions of the equation of this thing, you get closed transform. So, you get a local system which depends on h bar, or, right, or, all right, or rank one, but, x0, it's homotopic feelings to, to this real compactification, so you just consider, extend to the local system to x0. And, the domain is something similar to what I described you last time, just look on this divisors and you add appropriate boundaries. where your chain should have a boundary. Sorry? What is y as a function? No, it's all cosets, it's considered a sheaf of functions, locally don't have solutions, so get, represent rank one local system. You consider this flat connection on rank one local system. Yeah. It should be kind of topological. It's topological, but here you get some periods of one form alpha. And you can see the local system is monodromed given by exponent of periods. Say again, what is the fiber of the local system? It's solutions of this, no, I can see the connection. I can see the connection, I can see the flat connection. Flat connection on trivial bundle. So you say you add real what? Real boundary? Yeah, add real boundary. Yeah, X zero will be open manifold and add to its real boundary without changing homotopy type. Yeah, and then take some appropriate dimension. It's appeared, but instead of fine, now we get the periods of connection. Yes, yes, yes, yeah. You need only integrals of this one form, that's it, I think. Yeah, so it's really easy. Statement, yeah, it's. Sorry? What was the relation between the one CX but CX over X over X, what was good? Ah, no, no, no, I just explained why it needs these exceptions, why this isomorphism breaks. No, no, here, it used to some finite dimensional problem. It's kind of unrelated. Okay. Ah, decoil there is a morphism. Yeah, breaks when each bar is in this finitely many arithmetic progressions. So, I do understand how for the exact form you get this topology from theory we see break the team box and others. No, it's another story. I didn't explain it, you are explaining some global theorem, Chris. So, then the exact form is not a special case of flow. No, exact form is special. I use the idea when you don't get the theory. Sorry? No, you get the previous definition because you got trivial local system. If alpha is there, you get the same definition because the system will be canonical trivialized. Sorry? No, integral will be pairing between, this isomorphism, how you look on a basis it's all the integrals. Yeah, one can define. Yeah, no, I didn't define it yet here. Yeah, you just said that it's like previous case. It's completely similar, yeah, but yeah, but just maybe. So, you don't need to invert, I mean, you don't need to locally write it as a defile. Yeah, of course you can locally write this defile, yeah. And then you repeat the previous part. Yeah, so yeah, sure, yeah. No, but this is really interesting. And maybe it's a bit, don't have much time to say it's properly, maybe I'll next lecture say it's properly, but our rough picture is the following. Again, assume, no, just to get feeling what will go on. Assume that alpha equal to zero are isolated points. As I let it at Morse points, because in a point we can write as differential as let Morse points, and you fix generic theta, you fix theta and take alpha theta and it's function, it's again one form on X and then go to universal cover. No, no, by H bar, theta will be maybe, what I really mean, H bar will be R times E theta, something like this. Yeah, you go to the universal cover. On the universal cover, this alpha H bar will be differential of a function. But now if you fix a point, now you can draw left-hand symbols. Yeah, it's actually, it's not clear where this thing is really well defined because it's non-compact variety. And if you project to original, where to get everywhere dance, maybe a thing, and how to write integrals. Yeah, you're asking for example, how to write integrals. You can see the gain top degree form on X and then you get pull back the form on the symbol and function, you normalize such that function is equal to zero at the tip of the symbol. This unique normalization. And then if you consider exponent of F divided by H bar, it will be real numbers which tends to minus infinity very fast. And then I can try to integrate the same things over symbols. Yeah, so this really question why these integrals are convergent and why it doesn't depends on choice of calorimetric when you define symbols, yeah. So there are really host of questions, but and here I think that actually things will be diverged if H bars is sufficiently large by some kind of entropy reasons for some, for the flow. Because when you make this symbols, you kind of make a flow on your manifold using calorimetric in one form. And the spheres, if you apply the flow, the volume can grow exponentially, but one can bound the order of exponent and then for each bar small enough, you get convergent stuff, yeah. But yeah, actually I have properly started in one week here. This. But then compare some, the theorem is not the theorem. It's conjecture. It's comparison conjecture. It's some kind of conjecture that this global beta-comology is equal to direct sum of local beta-comology. Can you announce a little more about next lecture, what it will be there? Yeah, and then I will go to infinite dimensional integrals. Still zero forms and one forms, or it will. No, one forms in infinite dimensional spaces. Like. In the loop space? Yeah, loop spaces, yeah, I'll get. Quantum mechanics. Quantum mechanics, yeah, go to quantum mechanics and yeah. Yeah, yeah. Quantum integrals. Yeah. I don't know. It will be quantum mechanics, yeah, and proper. Instead of alpha, what's the PDQ or something? PDQ, yeah, yeah. No, integral of PDQ, no, I can see the DP integral of two forms of a loop, of a loop. Two forms of the greater of a loop, I get one form of space of loops, yeah, yeah. Okay, so I think I should stop. So the thing that you had about comparison of loop or logical of the random or the vertical are you first headed it for a function? Yeah. Do you say there is a similar thing for alpha or? Yes, yes, for alpha, yeah, global homologer when you use something at infinity, homologer of pair at infinity. But you said the previous one was proved and the one for alpha is more difficult. It's not difficult as well, yeah, it's very easy. It's very easy, yeah, yeah. Okay. Yeah, yeah, it's like for functions, yeah. So what is easier? No, no, to compare the random homologer and global beta homologer, you use this stuff, I think it's, you use this shift which is kind of, and here you can use an analytic topology and because it's finite. Something is conjectured? No, no, conjecture, it's other stuff, it's relation like left, it's symbols. I have in my, one hour ago I spoke about isomorphism, beta homologer with some of things of critical values and for one form, one can formulate the same state, statement but it's analytically more chilling here, I think it's. Actually, it'll stop, take me next time, yeah. I'm a little upset that a bit earlier that you seem that you have a manipulation and you obtain convergent H bar expansion. Ah, yeah, yeah. If it's a non-trivial expansion, I'm upset because I don't see how you can obtain non-trivial expansions in such calculations. No, no, because there are the iterated integrals and they lead to some kind of form of power series which is not related to original details. Okay, and I see that, you're sure that your series will not be trivial? No, no, no, no, no, definitely. This algebraic is computable, still? No, no, no, this is iterated integrals, I don't know how to calculate them, even numerically, yeah, that's it, yeah, yeah. No, it's not algebraic story, unfortunately, yeah, so it's, yeah, so it's transcendental number, yeah. Suppose you take the simplest integral, the integral that defines the error function. Yes, yes. Can you do all this explicitly or not? No, no, this is iterated integrals, it's something which should be calculated once for all, but it's not. But you cannot do, even for the error case, you cannot. No, no, no, sir? No, no, no, but still one should, it will be infinite sum over iterated integrals, yeah. No, no, no. How does the homology define the point for this one problem? Yeah, no, essentially, model of the slog are different questions, you locally represent as differential of a function, and you do the same pair as for the case of function. But f minus one still makes sense? F, what is f minus one? Inverse. Inverse. Yeah, of course, locally, yeah, but you add it to the boundary, when we consider f minus, at real boundary, you can see the, f minus infinity, it's subset of the boundary, yes. So we do under universal cover? No, no, no, no, no, it's not a universal cover, you don't write it on X bar, yeah. But you have to define a part of the boundary and you don't have the function, you just have it for all. Yeah, just part of the boundary, you don't need to function at all, yeah. Okay, no, I think I should leave now, sir.