 And the first part will be dependence on parameters. Yes, so I recall that I'll consider the case of one function and function from a variety to a fine line, which is algebraic variety of C. And in general, I consider a divisor. But right now, just to simplify the notation, we'll know divisor and assume that f is proper just to simplify the notation. It will be easier to listen to this case. So you get kind of familiar of compact varieties as such a total space is smooth. And I recall that if you have such a thing, then you get space of critical values, some finite set, critical values. And then for each critical value, you get a local system on circle, which is kind of a set of theta in r mod 2 pi z. In the fiber of the local system, I denoted h bit from beta from my point. And theta in s1, the fiber will be homologous of the pair. I take pre-image of a small disk with center zi and radius r. The r is very small. And relative to the pre-image of zi plus exponent of 2 pi my theta times r, a point on the boundary of a circle. And take a mold with integer coefficient. Yeah, so I get this local system of a billion group of finite rank over circle. And then I had some gluing data, namely, if I have a line in the complex line of values of my function f, and get several d1, d2, dk, and have oriented line, which is parallel to some direction h bar, then I can see the stocks of my local systems corresponding to a point close to the ion length on this line bundle. And what I have, I get a bunch of linear operators between stocks, actually integers, between stocks of local systems. And this allows me to reconstruct total topology of all my picture. So what I have, one can say I have collection of operators from i is non-equal to j, which I map from h beta, let's say zi, certain c to ij, to h beta zj, theta ij, where c to ij is argument of zi minus zj. I have this data, but now I assume that everything now depends on parameters. So what I have, I have some projection pi from x to some space u. It could be topological space, or complex analytic space, algebraic space, and it will be pi will be locally trivial bundle, and fibers of pi will be complex algebraic varieties, and f will be a function from x to c, such that for any u, I get the function f u from, it will be just restriction of. And this should be like polynomial. No, no, no, it's not the same. x could be, or x could change as a variety. Local trivialist topological bundle. Yeah, smooth, yeah, in some sense, yeah. But also kind of really cool. And what I said, the map from u, maximum of absolute value of z, z, c, and s, u, remember, this map is because of the properness of the map, yeah. It's not clear. I will talk about it. In principle, they can disappear. So it is not clear. If it's not clear, it's spontaneous. Yes, yes, yeah. And do you have some properness in the family that is? Locally proper, yeah. Consider compact domain in u. Consider pullback of compact domain then it will get a proper map. Yeah, p restricted to y minus one of compact is proper. Yeah, so you get this nice family. But the number of critical values can change. Yeah, so let's assume that you have kind of big open domain, open domain when number of critical points stays the same. Stay the same. Then what will happen? So we have moving points. So z i, one can say that z i are functions on u, kind of moving points on plane. Critical values, yeah, critical values, yeah. No, set of open domain, number of critical values. At least in algebraic situations, it's clear. If you have this risk-open part when critical values stays the same, then you get complex numbers depending on parameter u and then over this u zero you get maybe some kind of universal set of critical points, fiber over u to SU. We have to assume more than the number is. Yeah, it's assuming more kind of move nicely. Move continuously as the two of them coincide. Yes, yes, yes. No, no, no, I don't assume that they coincide, yes. No, I assume that the set, the number of elements stays the same and points move continuously. And it's constant and points move continuously. Okay, then we get this universal set of critical points and this local systems, which we have before and we get kind of HB z i theta, it's conditional things. It's formed a local system on s one, this parameter theta times this universal set of critical points. But the operators which we have here, they will jump for some walls. Gij, jump along certain walls. And let's me kind of roughly describe what is going on. If points can move freely, generically, all these points, none three of them will lie on the same real line. I assume that this, if there's only points in line, of s u in a line in c, and if we move continuously, this operator will not jump. But suppose some point, three elements will stay on a line, on a real line, three critical values on a line, let's call z one, z two, z three, for example. Maybe I just continue the picture here because it's, yeah. Then if we move a little bit, it's for some, and it happens in real co-dimension, one wall in this u zero because it's one real condition that three points lie on the line. And if you move a little bit through this, so it gets some certain kind of hyper surface in u, and you get point, maybe u, maybe u zero call, like on the wall, and then you get two points in a parameter space which are on the left and right side of the wall. And if you move a little bit, you get a triangle looking in one direction, so u zero gets all three in the line, and u plus you get a triangle looking in opposite direction, and one really can even call the sides of the wall plus and minus according to two types of triangle. Yeah, same way, for example, I arrange this line also in one way. Then locally, because we have a local system, we can say that a billion groups do not change, and here we get some operators, t one two minus, t two three minus, t one three minus, which I mean operators coming from the corresponding stocks of local systems in this picture. Here I, and here we get two three plus, t one three plus. Yeah, so the claim what happens here, that it's operators which says on the shorter sides of triangles do not jump, but what sits on the larger side, it's jump, and here it depends on kind of conventions, how the normal separator is going from left to right to right to left, essentially to g one two minus. And what happens in the middle? In the middle, I think it will be left or right semi-continuous here, if I got it, just consider it is one of them. There's some kind of ambiguity on conventions, but I think it's, yeah. Yeah, so that's kind of, you get this nice behavior, behavior and how to understand it. Yeah, so there's some kind of very general language which I'll use on my next lecture. It's about wall crossing structures. So it's in short. So what is the wall crossing structure? It will be wall crossing structure on what? The wall crossing structure I think will be on use zero times c star, maybe circle depending on plan constant or argument plan constant. Wall crossing structure is the following. You get certain topological space like this and you get local system of Lie algebras. For example, here, what will be Lie algebra? G, depending on u h bar, will be defined by its endomorphism of, you can see the endomorphisms of direct sum. You can see the sum of all critical points with this value of parameters. You can see the h, beta, z i, z i u and maybe argument of minus h bar. You can see the things and I take for example Lie algebra is rational coefficient, which also graded by some I billion, by a local system gain of lattices. Yeah, gamma u h bar, but in fact in our cases local system will depend only on u, it will not depend on h bar. And gamma u will be kernel of z to s u z. I take sum of all hyperplane given in coordinate space given by the question of sum is equal to zero, which is root lattice a, a minus one, isomorph to root lattice m is cardinality. So this root lattice contained special elements, contains elements e i j, i to power j, contains such vectors for the i non equal to z j belong to s u and graded components are the following. Why is this Lie algebra is graded by this lattice if gamma is equal to e i j? Then the graded component is, this grade components is home between two different spaces and if gamma is equal to zero, you get all diagonal terms. Yeah, so what does it mean in plain terms? You get finitely many lattices, so multiply back your, take rational vector spaces and if consider endomorphism of this direct sum, we get block matrices and you say that it's graded by root lattice of a n, kind of take diagonal terms and weight zero and the rest in the weight of a m. Yeah, so you get, sorry? If gamma equal to zero r. I take all diagonal terms. Yeah, I get all diagonal terms. Yeah, so the general story is that I have local system of lattices and local system of Lie algebra is graded by these lattices and what else I should have is continue the description of what is wall crossing structure and then I get for each point, I get a map for any grid bar in my space. I get a linear map for my grading lattice to C and the map is the following, kind of, I'll say there's a grade element, a base element a i j goes, a i j goes to j of u minus the i of u of h bar. So we get this thing just before to define what is wall crossing structure. By the way, this is kind of, one can kind of reformulate a little bit what is going on, we have gamma u grading plus the u h bar gives a derivation of my Lie algebra but with complex coefficient and derivation will be semi-simple. So it will be finite dimensional vector space and operators will be diagonalizable. So it's actually, there's a good question, can one generalize to non-semi-simple derivation instead of grading case? Sorry, but yeah, but it's kind of familiar of Lie algebra with semi-simple derivation and now what is the wall crossing structure? What is wall crossing structure? You can see that in this total space some real walls, real co-dimension one walls in u cross c star which consists of the following points. You can see the points u and h bar such that there exists non-trivial element in my lattice which is gamma such that z of u gamma is not, it belongs to, it's a strictly positive number and corresponding components, we'll denote u h bar, gamma is not zero. Yeah, so the real wall, in our case when it gets graded by this root lattice it means that z i z j such that z j minus z i divided in our case, it means that z j of u minus z i of u divided by h bar is strictly positive number so it means that we have two critical values on the real, on the line parallel to the real line and we should do it coincide. So there are real walls and wall crossing structures is the following thing. For any pass we can short pass intersecting wall. We should associate element of the algebra also there's some kind of wall and parameter space and we have some short pass and when we intersect the wall we should associate an element of what? Of the algebra I consider some of g u h bar gamma exactly this point of such gamma, such that z u h bar of gamma is strictly positive number. Yeah, generic it will be only one component. For the things I have an element, maybe called i a u h bar and then the condition is the following and constraint is the following. If you make again generic loop for small loop for small tractable loop which intersect walls in some several points if you take ordered product of exponent of the elements is identity in the corresponding group. Why it makes sense? Because one can see that it's if you make small loop as this all grading components will lie in some convex corner and you get nilpotently algebra. So it sits in some nilpotently algebra so it can speak about nilpotently group associated with nilpotently algebra. Yeah, only the example, yeah. In general, next lecture I'll speak about some infinite dimension example. One should put some more serious constraints on the story but that's basically the structure. And this property, it's essentially saying saying as that we have this wall crossing structure. Let me explain to you. It's kind of, oops, okay. Can you see the simple thing? I don't see the formula exactly. So you want some point of exponent equal to what was the? Identity. What was the exponent for? Exponent of elements of the algebra. And this is for a loop that intersects several walls? Several walls, yeah. It goes around some singularities of co-dimension too. So you take the order of those and the element is of the? Yeah, in this case it all makes sense because it's some finite dimensional group here. It is the direct sum of? No, no, my group is endomorphism of finite dimensional vector space. It's kind of group jail. But the given element for a given wall is you said it's the direct sum of? Of all gammas such as the gamma is strictly positive real number. I get a map of grading. What's this element A? It's a wall crossing structure. It's given by some collection of elements. For each wall, it is showing this constraint. Yeah, this is such activity constraint. Yeah, but okay, let me explain to you just some real simple cases. Kind of simplest case. Case when suppose all my singularities or singular points are isolated and actually holomorphic Morse. Then this vector spaces h are one dimensional. It's kind of this dimension of h. That was this b. If you h is one, monogram is plus minus one. Yeah, so it's essentially kind of have basis. And typically in co-dimension two, real co-dimension two in this u0 cross c, c star, you get two possible pictures. Like this and like this. And what is the first picture? It means kind of typical point of the world. It means that you have two critical values divided by each bar. Same real line. So I get exactly at this point, what happens? You get some ze1, maybe they can same line as ze1 and somewhere else you get ze2, same real line as ze2. And on one of them you have real parts of the first two coincide and another two coincide. So these things completely do not talk to each other. And my operators, which I have here, operators PRJ are just numbers. Yeah, this is the co-dimension one. So this is a wall. But in the wall, they must be. The wall in your picture is just a sling of the line. No, no, no, no. Wall in my picture, it means that I have also each bar. It means that I have this equality. Each bar is also coordinated in my space. And these are just numbers up to sign, defined up to sign, defined up to two conventions. And here these things do not talk to each other. So the rule is the following. If you put some number a, some number b here, number b, numbers stay the same. And here is a, b, c. I claim that here numbers depend on parameters in the following way. So c, a and b plus ac. And what is, I propose in this picture, exactly this formula if you replace things by numbers. And what is the meaning of this formula is the following. You can see that in this parameter space, you go one point to another, or you go in this way. So across three, the total combination will be trivial. So it means that if you go one way or another way, it should get the same result. And you multiply three matrices. And matrices which one multiply, here I can easily make a mistake. Put plus minus. So you have this basic identity for three by three matrices, which tells you that if you go through these three walls and through these three walls, they sort of get the same result. So this explanation of this formula from wall crossing perspective. Yeah, so it's kind of pretty amusing structure, which you get. All this, it describes what happens when all critical values are still all distinct, but they also can match to each other. So now we can ask what happens on u minus this open part when eigenvalues match. And again, I assume isolated more, on u zero, isolated more singularities. And let's say it depends on parameters, this kind of holomorphic and kind of u analytic and projection is analytic. Then in complex co-dimension two, in complex co-dimension one, or the same as in real co-dimension two, what happens to critical values can merge? Yeah, there's also another possibility that these two critical values kind of coincide, but they do not talk to each other. But the interesting part when they really merge, so what is a typical example? Like dimension of my fiber is one and dimension of the base is also one. Let's call it coordinate u and coordinate x here and the function f will be something like x cubed over three minus ux. Yeah, you can imagine some other variables, but maybe it's not necessary one here. f coordinates x1, xn. And I write only formal near singular points. You should imagine that it's compactified some way. Yeah, so I get this stuff. So what are critical points? f of s of u. You can see the derivative is equal to zero. So it says that x one of u squared is equal to u. So it's plus minus square root of u. And critical values. You substitute here square root of u. You get minus two-third u to the power of three over two. Now, when you consider this full crossing picture, for example, I just put h bar equal to one. I restrict to small sub-manifold. Then I get volts. Volts, it means that I have just two critical values and the difference should be, so z1 and z2 of u are two critical values. So the difference should be real and positive. So it means that imaginary part of u to the power of three-half is equal to zero. And so it means that u to the power of three-half is a positive real number. And what do we get? So you see that locally these three volts are three rays. And here we get 120 degrees everywhere. And this u-plane. And what I claim is that my numbers, which I have here, my numbers are all equal to one. Because in the simple case of space of one dimension, these matrices are just one by one matrices, just numbers. And claim equal to one. And this is something really nice here. It's related to the folic identity. It's kind of similar to this identity. What is going on here? Why write these things? I want to go along this path. And I claim that the total monogram which I constructed is again trivial. So it's like wall crossing associative condition for wall crossing condition, but in a point when I don't have any wall crossing structure anymore. It's kind of generalization of associativity constraint. Why multiply such things? Yeah, 1, 1, 0, 1, it's a matrix which I associate to crossing the wall. It's upper triangle matrix with element 1. But why multiply by this guy? Because it's a matrix actually in two dimensional space, which I have two critical points. But if I move from here to here, these two critical points interchange the role. So it shouldn't interchange them. And it also should care about orientation. Of course I said that it's a one dimensional, but base element you know up to sign. One should take care of orientation. It turns out that one of them should be minus one. And then you get this remarkable thing. So it's cube equal to zero. So there is something which happens on Eucrose U0. One can say that it's kind of generalized associativity constraint. If you take product over some small loop in U0 surrounding this divisor, again composition or the product is trivial. And in what all this will give us? Because composition for any loop is trivial. So the result maybe in again in Eucrose C star. And the conclusion that we get local system on U cross C star. We get kind of wrong local system on U0, which don't accept places when critical values collide. And we modify it along the whole source. It will be extend to local system everywhere. And this local system is what is it? It's global H beta H bar of it's fiber at U H bar is equal to global beta comology of X U, which come comology of pair when I put comology of pair of X U and the main when F U divided by H bar tends to minus infinity. So that's rough description of what happens topologically. And in fact there's something else which I want to describe when critical values collide in some kind of stability effect for merging critical values. This will be something very general. It kind of depends on the half algebraic variety and so on. So first speak about isolated singularities. Suppose we have a germ of function, of analytic function. Again this is my function F from a ball in Cn to C. Germ at zero. So restriction of function of ball containing zero. And this isolated critical point. Then for such thing you have a Milner number. Some invariant. Milner number. Milner of function and maybe point P zero. Just called mu. It just, it can be described in two ways. Consider homology of the ball. And consider pre-image. And assume that F of P zero is some zero. Zero plus epsilon. Where epsilon very small. So this homology will stabilize and you'll get, I think, and this Milner number is the same as rank of, consider function of the ball. And mod out by ideal generated by derivatives. Yeah, in fact it's better to consider it's actually a rank of top degree homology of complex of omega of the ball with differential cap with dF. The homology will be sits on it only top degree. And if you realize, kind of identify with volume element you get top degree homology here. Yeah, so it gets Milner number which is just some, for non-trivial singularity, non-negative number. It's mu equal one. It's equivalent to get more singularity. And for X cubed mu equal two and so on. Yeah, so it's some characterization of singularity. And this number behaves kind of stable way. Suppose you move a little bit, deform a little bit function. So it will be some function depending on parameter. And then in this ball this critical point will be replaced by several critical points. Maybe P0, you get something like collection of Pi of U. You get collection of critical points. And Milner number is all Milner number of my function and the point F0 and P0 is equal to sum over I and a few P0. No, that could be the several critical points above it even. Yes, yes. No, no, no. I can see the critical points. It's for each critical point. Yeah, so it means that the X cubed can decompose to two more singularities. Milner number is number of more singularities on which the critical points can decompose. So I get this kind of positive number. I decompose some of another positive number. You get this conservation law. And this conservation law has a generalization. There's some nice generalization. Suppose you get some holomorphic function on some global variety. And let's consider a set of critical points of F. So the set of all points in X, so that the F restricting to tangent space X is 0. And decomposing the union of connected components, some union, and suppose it's one of them is compact. Now, I will speak about only this connected component. Then one can assort with this the following contribution for this component, depending again on angle. B for betas is for the direction F. And it's defined in a similar way to the first definition of Milner number. Namely, you do the following. You consider, yeah, first of all, you pick some remaining metric. And don't assume this thing is scalar. So the remaining metric on X. And what I take? I take delta neighborhood of my component. And take F minus 1. Or maybe just call it 0. Smalls is 0. Smalls is 0 is f of this large component because on every component of critical set, the function is constant. So it will be just some complex number. And I take pre-image of this complex number plus epsilon exponent i theta is integer coefficient. And I take limit, I take double limit. First epsilon will go to 0. And then delta will go to 0. There are some natural naps. And this thing stabilizes. So limit is well-defined. So it's a definition of generalization of this commolder of pair. And I define kind of Milner polynomial. Of course, from my component is sum from i to minus n to plus n. n is the dimension of x, as usual. And the rank of h a plus n does depend on theta of this 0f times t to power e. And it's a polynomial in Laurent polynomial variable t. It is non-negative coefficients. Maybe call this kind of mu, my f. And this sort of critical points can see the connected component of critical points. I get this polynomial. And claim, it also satisfies the same property. If you deform a little bit, so you get very kind of complicated critical set. I deform it, can be decomposed by several pieces, maybe of smaller dimension. I get the same equality. This polynomial will be equal to sum of Poincare polynomials of deformed thing. And in fact, one can construct isomorphism similar to what I explained to you before, not just equality of polynomials. And get similar property. Actually, how to write it more scientifically? This thing, when you can write as this HBT signal, is the same as hypercromology of this theta. And consider shift of finite functions from maybe constant shift. You get a more scientific notation, if you like. There is something very bizarre. For isolithic singularities, Milner number, it's always positive. But for non-isolithic singularities, this Milner polynomial could be zero. So the phantoms, things which cannot really guarantee that they appear or disappear. And there exist the phantoms. I said that this Milner, this thing is actually zero. Sir? You don't attack thousands of coefficients. But on a global variety, yeah. It's not clear. Maybe it will be not surprising that this situation even put any local system on this critical local set to get zero hypercromology. At least with constant coefficients, at least I can explain it. Yeah, so not only the rank, but the group themselves. Groups themselves are zero, yeah. Okay. Comological, it kind of doesn't exist, yeah. And example is very simple. Your variety, you take product of elliptic curve. You don't know if the component can disappear somehow. Sorry? Yeah, it could disappear in principle, yeah. Yeah, maybe there's some other, in fact, I don't believe that this thing can really disappear. But yeah, it looks hard to believe, but at least chemologically I cannot see that they cannot disappear. So let me show this simple example of phantom. So you get point x and t, say. And you get function equal to t square. It just depends on second variable, but you're more doubt by involution. x goes to x plus x zero, t goes to minus t. And x zero, it's two torsion point on elliptic curve. Then on a quotient to get critical points that will be elliptic curve divided by this shift. But local system will be a rank one local system. It's non-trivial monodromia, and it will kill all comology. Yeah, so it's, this doesn't disappear, yeah. Because it can put homology with non-trivial local system, you can twist by local system, then it will not disappear. But I don't know, maybe one can cook out, because there are some people constructed examples in, I think one connected examples when comology are zero of these phantoms. Yeah, so, yeah, so these things. Disappears and just maybe before break, I'll just say one couple of words about hot theory. First of all, this Milner polynomial is symmetric with reflection t to go to the inverse. It's follows from Poincare duality, but in kind of good case. So in fact, it's maybe put a conjecture. I assume that the zero, there exists an embedding the zero to a Keller manifold. Not my manifold, but some another Keller manifold of different dimension. I would say that's kind of singular space is Keller. The zero is one of components of my critical. I have just complex manifold, I have a function, and I assume we have So we just did a given value of the parameter. Yeah, yeah, yeah. I mean, speak about parameters now. I just speak about taken with the reduced structure. Even with reduced structure, I think, yeah. I think it's Z reduced could be embedded to killer manifold. Then this could be some kind of good hot theory. Then you can see the hypercomology of again neighborhood of the zero with form, with a kind of analytic forms with differential HD plus DF. That's this homology. This rank is doesn't jump as each bar goes to zero. And yeah, it's clear for each bar not equal to zero that it's coincide with this HBT. Automatically without any assumptions equal to, for each bundle equal to zero get completed a RAM comparison. Oh, sorry. Sorry, I'm just writing nonsense. I think this is things. So it means it's kind of vector bundle or formal line. This guy can be compared with vanishing cycles. Yeah, so it gives you some kind of hot theory and moreover, you get kind of left shift decomposition. So it means that if consider even an odd part of this polynomial and consider what is the graph of k-efficient you get kind of bell-like curve. So it will be increasing then decrease this k-efficient. So it will be some primitive homology. It's a little bit more than what one can extract from literature but I think that's a really sufficient condition that reduce part of critical point set is itself a scalar and the rest is automatic. Okay, so now I'll make a break for maybe five, seven minutes. Yeah, so it will be kind of a very kind of vague idea then I will go to concrete example. Yeah, suppose we get infinite dimensional kind of x and kind of infinite dimensional complex analytics manifold. Yeah, I don't know what is it but kind of at least should have tangent spaces which are complex vector spaces. And I get some function from these things you get a holomorphic function and assume that set of critical points kind of critical points is say finite union the joint union of first it's compact and finite union of a complex analytic space or unusual sense kind of finite type. Yeah, finite dimension. Maybe very singular one. Why it's kind of natural assumption because you more or less should fall from the conditions that consider second derivative of this function is operator from tangent space. This should be some kind of friend column operator so it should have only finite dimensional kernel because these two spaces are more of the same dimension, infinite dimension. Yeah, so it's a difficult situation and in this case we can replace f near each component each the alpha by finite dimensional model. In general it's not what one should do. You get some kind of compact finite dimensional set the alpha in some of this infinite dimensional space and then one should choose a sub bundle in tangent space to x infinity restricted to the alpha of finite co-dimension. Finite co-dimension and kind of it will be kind of t-transversal and t-transversal intersecting with maybe skin theoretic tangent space at each point which is finite dimensional is zero. So it means that you can extend it to a kind of vibration a fullation of maybe finite co-dimension of finite co-dimension near alpha and one can think about it's really kind of vibration or fullation. So you get some neighborhood of the alpha an open neighborhood. It maps to some finite dimensional space and this condition essentially means that function restricted to fibers has get some kind of projection maybe you find it and function restricted to fibers of any point you find it has only one isolated more singular point for this projection. You have neighborhood what essentially I want to say that in all variables except finitely many one can think that my function is Morse function sum of squares. Your function is kind of a finite dimensional function finite dimensional manifold. Here I get function infinite dimension manifold. But the infinite dimensional part is quadratic. And then I can make a new function. Is there any topological obstruction for this? No. You have to find a family of such things. No, no, no. That's not topological obstruction. It's analytic. It's analytic. It takes some finite complex co-dimension so this chain classes will be not obstruction. And the fact that the forming the quotient Yeah, because it's in general quotient thing is not Hausdorff. Yeah. Yeah. But I think here there's no trouble. Yeah. And you get new function on on you find it kind of a finite which is defined the following. It's valid point U will be critical value unique critical value of F restricting to the fiber. So here there are no coincidences when you're going up because of the assumption. And we just do the story so we get some finite dimensional replacement. And then you see that you get a constructable sheaf of finishing cycle in derived category Phi of F finite of the U is indeed be constructable of this the alpha. The alpha will be the same as critical points of F F finite. Okay. So you get a sheaf of finishing cycles but there is a trouble here. If you choose different finite dimensional reduction to get a different sheaf it's it's some ambiguity. Ambiguity by tensoring by rank one local system with monodromy plus minus one. And also yeah so one could should make this choice why it happens. Now kind of emergence is some finite dimensional thing with some finite is a finite and make multiply make kind of Tom Sebastianis sum kind of add function additional variable then consider vector bundle complex vector bundle even finite dimensional vector bundle E with quadratic form Q. Yeah and then quadratic form it gives you change your sheaf of finishing cycle by one dimensional space and it depends on rotation. But can you relate any two finite dimensional reductions by going through a third one such a jump? Yes, but it still get ambiguity. You have kind of many in infinite dimensional case I think it's you don't have canonical sheaf of finishing cycles. You get sheaf of finishing cycles defined up to multiplication by this plus minus one. Yeah. It looks like this. Yes, yes, yes, exactly, yeah. That one can do. Yeah, so we need some do this choice of this local system and that's it. And this something I'll call orientation choice which something which have to be done and now and then what? Then it will be a very nice thing. What happens if you make this choice? Then you get sheaf of finishing cycles so you get some set of critical values kind of the I the alpha which will be function of the alpha you can see this critical values then you get a local systems over S1 by taking homology with sheaf of finishing cycles of the alpha and this kind of sheaf of regular sheaf of finishing cycles of my function divided minus critical value and divided by each bar of constant sheaf in some finite dimensional model. Yeah, also sheaf dimension. Yeah, there is something interesting also going on. When I go to S1 shift when I go different reductions you get a shift in degree homology you should remove dimension of your space. Like in this Milner polynomial I shift the grading by dimension complex dimension. And then it will be well defined so you get local systems and then one kind of then one should have this operator from the i to the j from homology from the alpha i to the alpha j and if you try to think should have some notion of gradient floor and so on in infinite dimension and so do you consider only a finite function what? for finite function for finite function then a shift by dimension of finite dimensional guy you find it and you stabilize it this is a choice which you should make how you organize the things then it should count tij it will be something like generalizing of number of gradient lines and this should be kind of elliptic problem again space of solution should be of infinite dimensional maybe real semi-analytic space gradient lines for some gradient floor because you remember in finite dimensional for more singularities this operators in case when you get more things which are integer numbers counting some gradient lines and then in infinite dimension we should be able to write some notion of gradient floor and to rephrase what you are talking about here what is the elliptic problem what do you refer to to write a gradient line from one critical point to another it should be reduced to some elliptic solution of non-linear differential question with elliptic symbol you will see an example right now by the way the infinite dimensional spaces are they seen like finite analytic spaces no the idea it should not matter at all in concrete examples you should make sense of individual stuff of water gradient lines without going to foundations and then define homology through kind of backdoor without looking for the infinite dimension homology and so on so what is concrete example which I look as a following suppose I get infinite dimension manifold with the function I start with holomorphic symplectic manifold so it is a complex analytic variety with a holomorphic symplectic form and dimension is now 2n for some n and I pick two let's say closed again complex analytic, holomorphic, Lagrangian subvarities submanifolds and what this will be my space it will be space of c infinity maps one can probably put some Banach k maps maps from what from interval 0 1 to m such that f0 belongs to f0 1 belongs to f1 so I get the space carries a canonical I think it's kind of complex manifold it's kind of like product of all points of intervals from manifold so you are consulting quantum mechanics yes quantum mechanics but this infinite dimension complex manifold dungeon space is complex if you look to this and it has canonical one form closed one form what is one form called alpha it's two form over the interval so what precise definition and take pullback of omega by evaluation map which maps from x infinity and 0 1 to m yeah it's one form but it's not exact it's to be exact so you want to write this differential some function some assumption here so we assume that omega is symplectic manifold is exact in the sense that omega is written as differential one form eta eta is kind of one form m and so eta is chosen and also both Lagrangian manifolds are exact in the sense that eta restricted to li is written as differential f i where f i is holomorphic map from li to c so it makes choice of eta and f 0 of 1 ok then the function f then f infinity such that alpha is differential of infinity it's very easy to write it's function on a path given by integral of the path of pullback of my form eta let me begin plus minus sense it's very easy to get confused something plus f 1 at one end of the interval minus f 0 to another end of the interval yeah it's easy to check that differential gives two form ok what are critical points no no no nothing that's it yeah just two Lagrangians PDQ and that's it yeah exactly what are critical points it's very easy to write what a critical just see when derivative is 0 in terms of its kind of constant maps from 0 to some point m city can intersection so the set of critical points is either morphic to the intersection and let's assume that intersection is compact ok there is a kind of question which I explained in general situation there is kind of ambiguity you could kind of it's well defined up to plus minus one and here critical point is a manifold sorry this are not the points no no set of critical points intersection of two sub-manifolds could be very bad non-transversal these are not the points they don't intersect in some maybe very singular spaces yeah the intersection is not transversal in general and so locally one can do the for like one locally can identify mm near some points in L0 one can identify m with cotangent bundle to L0 I just put some kind of transversal Lagrangian fallation and also one can put transversal Lagrangian fallation such it will be also transversal to L1 and then L1 will be a graph of function maybe some function f L0 graph of differential function f L0 is some function in L0 it's 0 sorry sorry it's the mixing languages here sorry this is only in only locally and then you see that locally it looks like a critical point of function finite dimensional variable so get this finite dimensional reduction and then I get shift of vanishing cycles and assume that f L0 on intersection, on component of intersection is 0 near my points I get shift of vanishing cycles but the problem is so locally it's well defined but again I've defined optimal multiplication by plus minus one dimensional space if you analyze carefully if it changes splitting globally it's not well defined so one should do something and in order to get orientation choice it's something which several people analyzed recently I don't understand the ambiguity because locally because of the symplectic form you can there is not much but you can still choose this identification with the tangent bundle in different way and then if I follow no but the differential is the same all those ways are connected it's connected but it's not it will be not simply connected space of choices will be not simply connected at the end of the day because I think that it's the fact that the symplectic form goes over to the standard form yes and this I think is you get some eventually you get some not simply connected parameter space even locally yes so the story which was analyzed maybe by Joyce and Braf and some other people so you need definitely certain choices and the choice is the following I can define you should fix some kind of class H2 of manifold in Z mode 2 and in fact one needs some kind of representative of this class plus representative and convenient choice will be the following you choose some line bundle LM line bundle on M just apological said that beta will be the first chain class of the things mod 2 and then on each Li in 0 and 1 you should identify restriction of beta should identify it's again some choice with first chain class of canonical class of Li it's kind of top degree form mod 2 yeah so for example if you make these things what you do you choose square root of the line bundle which is LM restricted to Li tethering by canonical class of Li and it's in real life it's really very essential thing even in the case of Cartesian bundle this beta kind of if beta if M is Cartesian bundle to X then beta is this LM will be tangent pullback of canonical class to X it's definitely not trivial things to choose yeah so you need this nasty plus minus one choice so then we get well defined shift of finishing cycles just it's pure topological yeah it's completely topological data yeah and what are gradient lines gradient lines pass in X infinity yeah now for example if M is scalar if you choose some scalar metric then on the space of X infinity also one can choose scalar metric so you choose scalar form 1 1 what will be scalar form 1 X infinity if you get some let's say pass and consider tangent vector so you get two sections of a tangent bundle kind of you get some pass phi and you get 5.1 and 5.2 are tangent vectors in the space of pass then the things the scalar form will be kind of integral from 0 to 1 pairing in tangent space my manifold multiplied by dt so we use volume form dt on interval one can choose different volume form get different scalar metric yeah so yeah so it's example of graded line and if you write what is a pass one such it will be maps from what from 0 1 times r kind of time parameter along the pass and this is so it's a get the map from a strip to M which is pseudo holomorphic well that means the point where did see everything was kind of mechanical with one dimensional yes yes passing through suddenly you get two dimensional yes now because I should consider in general gradient lines in my space so consider pass in space of pass yeah so automatically get right equation what is the gradient line get pseudo holomorphic curves for some pseudo for some almost complex structure which is not the original structure but something completely different on M yeah so get the thing and and the claim that this is a series of kind of pseudo holomorphic curves this boundary and this gives some this integer number so in general operators between homology which I talked about and then one can play the same game consider manifolds depending on parameters with functions so start to move maybe Lagrangian manifold start to move and then you get flat connections so what will be kind of concrete example and then I just want to show that it gives some explicit formulas so the M original manifold yeah complex complex structure it's determined by killer metric and holomorphic form kind of point wise you use some kind of hyper if it's hyper killer you can use one of the integrable yeah it's yeah it's something which in principle you don't want to do but claim is the following suppose MS cotangent bundle to some manifold X and I will have L0 in M will be exactly Lagrangian sub manifold yeah here this two form is kind of DPDQ is differential of neutral form eta this canonical one form of cotangent bundle and then we can speak about exactly Lagrangian so it means that L0 is DF0 for certain no no no I just want the main point I take an manifold with Lagrangian and also assume that L0 intersects with projection of L0 to X this projection is proper map so it has only fibers don't go to infinity yeah so there are plenty of such things such story I claim that you have a such canonical bundle with flat connection on X maybe depending on parameter H bar you can introduce parameter H bar it's always in my game how do we do this how define this bundle with connection so I get fibers depending on point on X and H bar the definition is the following again kind of for generic X and H bar you do the following you consider my manifold related to pair space of pass from L0 to L1 depending on point X which is cotangents point and point X so I have this will be my variety L0 this will be my variety L1 to point X I have two varieties and as I explained X is a long X right X capital X no capital X it's here X is here X point on my X L1 is base constant section L1 is not constant it's fiber it's just momentum just fiber direction then I get a family of my infinite dimension manifolds with function depending on parameter this pass space I get a family of my infinite dimension manifolds depending on parameter as I explained to you that should be some walls some isomorphisms and so on and then we should get a local system on parameter space multiplied by C star that's something which one can do very concretely ah it's kind of transcendental construction but I think this construction will solve this I have this bill of canal many years ago conjecture that automorphism of very algebra is the same as polynomial simplectomorphism of vector space and then towards the generalization if we get kind of simply connected Lagrangian sub manifold should give some canonical demodule but simply connected is automatically exact because this form can have any trivial periods so it's kind of concrete story here so you've got another some construction which is analytic instead of using characteristic yes, yes instead of using characteristic I don't know how it's related here but ah so as far as I understand so the calculation should be that you are in language of quantum mechanics you are calculating you have in the first in L1 you have way functions as functions of one moment in PDQ they are PDQ maybe you don't have very little time maybe I'll just finish the discussion yeah, yeah for example there are some kind of tricky very concrete example let's consider one can construct some embedding like of C to C square algebraic embedding in the image it will be cotangent bundle to X which is X as C Lagrangian which will be M Lagrangian L0 one can construct very tricky exact Lagrangians so for example we've got rational curve C to C2 yeah I just want to give you some example like point T C to C goes to point coordinates T plus whatever T to power 4 T square yeah it's kind of not obvious but one can check that it's embedding different points goes to different points because in fact if you get two points goes to the same points then you see that if you get to equation then it implies that T1 to power 4 is equal to T2 to power 4 by squaring this equation then you get this T1 equal to T2 so it's really embedding so it's yeah this square here yeah it's easy to check that this guy is embedding it goes to some tricky demodule and how I construct this demodule I just follow all lines and construct wall crossing structure I write it my form Q and P like standard coordinates going to this boundary sorry yeah it's the boundary but it will have also some irregular structure in this case it's not interesting to speak about it will be bundle on C but also with some stocks filtration to infinity algebraic yeah I get algebraic construction is it algebraic no no no construction it's not algebraic it will be completely transcendental but and so it doesn't have an algebraic structure no no no it's just all of it but also one have kind of stocks filtration to infinity so one can construct by classification of regular singularity algebraic bundle in addition to those yeah so you write form PDQ we stick into the things you write this differential of function F0 you should have this function of zero you get some function of zero which is I don't know TQ 3 plus 236 you just substitute to get some polynomial one variable and what together and all together it means that you get multi-valued function in one variable algebraic kind of algebraic function which has course with maybe called kind of G G of Q is equal to F0 it's equal to F0 of T VT is a solution equation which is a solution of question T plus T04 is equal to Q yeah it's kind of what is the motivation for this power for F0 just because from this I restrict my one form to Lagrangian manifold yeah so eventually what I get after all these things I get some algebraic function in one variable G of Q or Q of maybe X it's point you get algebraic function on one variable and at times this is really funny game with algebraic function which you can do then one can try to we do it on a computer whatever we get a function which has kind of have four valued function because my equation has for solutions solution of equations substitute to these things now now I should write wall crossing structure I put H bar equal to 1 and I have this walls and there are walls walls where we call it GIT from 1 to 4 maybe X maybe 1 to 4 and I wrote in C namely also imaginary part of G of X is equal to imaginary part of G of X for I non equal to J yeah so I get this I think and if I analyze what happens you get something like 3 ramification points so one get like this you get 3 critical points and then you should kind of continue you get some picture you get some finite graph yeah I change Q is equal to X yeah I get some finite graph but the graph will be oriented because along edges real part on edges on edges you get two branches have the same imaginary part orientation then read G of X minus read G of X it's positive and increases I just first take order and automatically can order which one is first which one is second by real part direction in which direction difference between real part increases so I get some kind of positive increasing function on my graph and now I should put some now I told you that for wall crossing structure I should put some integer numbers because I is a critical point the numbers I know I should put one here near each thing and then I have one rule and I have another rule which uniquely by induction kind of by growing to this flow reconstruct my numbers so I don't have to solve this equations for holomorphic curves at all yeah so it's automatically I can do it so it's canonically defined wall crossing structure is canonically defined by initial data and what you get you get local system on C which is not very interesting rank 4 rank 4 because projections here get rank 4 local system on C just vector space of dimension 4 but what goes on so I get this picture but then at infinity I can go I can do some other lines so it will be blue lines at infinity and as x goes to infinity given very different equation but when real part of gix is equal to real part of gjx it's no those are not walls but these are things but these are things responsible for stocks for stocks filtration when consider corresponding differential equations we should have filtration solutions labeled by real parts of these things and my vector bundle will be will have kind of basis for elements solution of everywhere outside of the walls and I glue them together and I know how the order outside of the walls and what I get I get kind of stocks data for these things so I get kind of transcendental description of differential because my local system has canonical basis corresponding to solution of my equations and then outside of this blue lines I have the order of them given by real parts so I get filtration of this story and then the service case works marvel also so I get filtration different stock sectors these things are all the semantics of essentially what goes on see it infinity decomposed by several sectors and in your vector space you get a complete flag in each sector which change if you go through the stock's direction and this we can read from all this picture in completely explicit way so one can really run computer program and get the story Is it some additional structure to this wall cross? No, no it just follows from this stock 6 and infinity it's something which you should I didn't tell you but it's kind of naturally follows from this all the game but that's how eventually it leads to some absolutely elementary story and this is very mysterious in fact because I conducted this canonical demodule and this demodule should be exponential motivic so eventually it should expect that should have some something like on X cross a fine line you get motivic demodule then multiply by exponential function take projection and here it's not clear to why it's getting motivic so I get only better realisation What did you say now I didn't catch the last phrase I said it should be some demodule associated to this exact Lagrangian manifold not arbitrary it should be what's called exponential motivic which means that it should have motivic demodule on the product of manifold with a fine line again some sub quotient of some Gauss-Mannian connection Where is that fine line here It's it's something very general it's really to irregular singularities in general on complex algebraic varieties one can see the exponential motivic demodules defined in the following way consider product of your variety with universal a fine line take motivic demodule there multiply by exponential function in the last variable and take push forward to your manifold kind of familiar Fourier transforms I think I now have to stop here