 Okay, maybe I start, yeah, so this will be a series of four lectures, following our lectures of, it was like a beer shaper and Masaki Kushevar, but it will be a slightly related subject. Yeah, so the rough plan, it will be four lectures, today we'll, today we'll speak about finite dimensional exponential integrals, then second lecture will be about one forms and wall crossing structures, and then certain force will be infinite dimensional story and applications to quantization, non-preturbative quantization. Yeah, so this, it will be quite different talks and I think, I think it will be quite entertaining to see the relations in the story. So I will start with kind of further integral, which everybody knows, which is Gaussian integral, which is square root of pi, yeah, and somehow I want to generalize this integral and calculate related things, or more general what we can integrate, let's say we integrate still in one variable, we have f of x, g of x dx, we integrate, such think we're, what, the f is maybe some constant times x to the power of d plus some lower terms is a polynomial, which is given, so it's, for example, this example is minus x square, and now g is also a polynomial, which is arbitrary, we write g, it can have any degree, here maybe I assume that d is at least 2, and I should integrate over some pass, yeah, so some pass, and what will be a good pass, pass will be, let's say, in C will be oriented semi-algebraic pass, semi-algebraic, I mean real semi-algebraic, like real line, or you can buy some real equation, polynomial equation, and such that real part of f is taken to pass, then we go to infinity, as we go to infinity in C, real part will go to minus infinity, so the, so the integral should be somehow convergent, so we have many such intervals, but claim that you have, one has to calculate if you vary both g and and pass, you have many integrals, but really interesting integrals are, you have only d minus 1 squared, kind of integrals to compute, and the rest will be given by some linear combinations, yeah, and this thing depends somehow on homology class of pass, and homology class of this form, in C, yeah, yeah, first of all about pass, the integral depends only on homology class, but pass is semi-algebraic, I said that pass is semi-algebraic, and that's, it's not semi-anal, then automatically the integral will be convergent, because in some local coordinates function will grow like a fractional power of small parameter, and the integral will be absolutely convergent, yeah, so the, so it depends on homology class of pass, maybe it's called gamma, which is homology class of the pass, and where it lies, it's an element of first homology group of the pair C, and I can see the subset f minus 1 of, maybe I'll just draw, what I draw, I draw the set of points in C, such that real part of these, I say less than minus t, and t is, I assume that t is very large real number, let me take these integral coefficients, yeah, this homology group is free group of rank d minus 1, with this degree of my polynomial, yeah, just for example, if my function is minus x square, as it's kind of basic example, then what we should draw, we have a C, when we consider the main, when minus x square has very negative real part, we get just two pieces, and we can connect them, this gives a homology class, yeah, this homology group stabilized in t is very large, so I mean just any one of them, and if for example, if degree of, if d is 5, you can, you should get the picture like this, you have in C, you get, depending on coefficients of your polynomial, you get maybe five domains where real part goes to infinity, and as the basis of homology, you can choose, let me choose some, put some orientation, yeah, this is an example of the basis of homology of the pair. Now, so you get essentially d minus 1 interesting pass, and what about g, this function g, let's consider, maybe I just put some blackboard, we consider twisted the RAM complex, which in this case is the following, we consider degree 0, it's just polynomial sin x, and goes by df to c of x, which dx, where df is differential plus which product is df, so if you look how it looks on some polynomial, consider some polynomial g, g goes to what, it goes to g prime, derivative of g plus f prime times g times dx, that's how this is the term differential acts, and why I wrote it, because if you multiply by exponent of f, it will be kind of isomorphism, and good here and here, you just consider expressions, which are polynomials for x times exponent, and polynomials for x times exponent times df, in one forms, this will be just usual differential, yeah, so it's just to write algebraically without referring to exponential function, and the integral again depends only on co-homology class of g dx, maybe, maybe g's wasn't, sorry, maybe we should put some another letter because it's not g, h, so g dx defined model as an image of this differential, and co-homology, so the co-homology of this complex are only in one degree, so here is a snow kernel, so that's only the kernel, so this class of g dx, and each one of this complex, of this twisted theorem complex, which is the same in this case, it's a co-kernel of df, so it's, I'll just, definition, and mod out by the image of this map, and this space is actually, again, finite dimension, it's the morphic to cd to power d minus one, and the basis given by representatives of one times dx, x times dx, x to power d minus two times dx, yeah, it's very easy to calculate this, yeah, for example, for d equal to, yeah, just write houses, houses twisted theorem complex looks like, looks like, so one goes to maybe, and function, maybe minus x square, minus two x, x goes to, yeah, so you get such formula, it's clear that you can kill top-degree terms and reduce to constant here, only constants you cannot get to the image of the differential, yeah, so that's the kind of basic example, generalizing this first thing, first story and now I'll go to general setup, so it will have several ingredients, so first I'll have algebraic, smooth algebraic variety over complex numbers, let's have certain dimension n, then I will have some extra ingredient which wasn't present before, I will have a some closed algebraic subset of smaller dimension, maybe singular, then I get a function, I can write it in various words, it's like a map from x to a fine line or complex numbers, it's kind of polynomial map or equivalently f is section of the structure shift on my variety, then I should have some volume form which is a top-degree algebraic form, it's kind of top-degree algebraic form and I need something which will be something like denoted by c, chain of integration, I'll explain it in a second, but before going on I have to explain why I should introduce this set D, so it's kind of why have this extra set D, because there are other integrals which can similar to what we have, yeah for example one can have this guy and D will be something which contains the boundary of integration or I can have another integral which is, I think it's called something like E1 of T, it's not elementary function and it's, this is called exponential integral in fact, this guy is called this exponential integral, it's not elementary function and for example in this case what will be variety, in this case variety will be a fine line, this is D will be a device at just one point, T function is minus x and form is dx over x, yeah so it's a kind of integrate exponent of function times volume. Volume on what, on x? On x, yeah it's. But x is not compact, it's not compact, ah sorry x1 minus 0, sorry you're right, yeah x is, x is, and T maybe is positive number here. Okay so the idea that your pass sincerely is, ends either at infinity or. Yeah it's some some variety, yeah yeah it's, it's ends as at infinity or at some point T which is in this case algebraic sub-variety of C star. In general you have to integrate of an n-dimensional pass. Yeah and no it's, it's, it's a chain, it's not a pass. Yeah it's chain, yeah in this case it's, it's chain of integrations, it's will be n-dimensional. Yeah, yeah someone should be a bit careful about chain, what will be this chain? Now the chain will be, the following, it's will be some, to be kind of sum over n, n i c i, some finite sum. V c i is oriented n-dimensional semi-algebraic sub-manifold in. Sub-manifold you mean compact or it is. No no it has boundary, it's good. Ah it has boundary, is it smooth after the boundary or? No, no no it's, it's open part, it's, it's open kind of open, it's will be open. In general when you consider simplices, lots of simplices. I consider interior, I consider interior of simplices, like interior simplices, of simplices, yeah. Okay, I love to this. Yeah okay, real, real semi-algebraic sub-manifold and, and x with the property and such thing has, it's kind of rectifiable set, it has natural boundary and I assume that the boundary of support of the differential is in D. And, and the second thing which I assume is the following. If consider real part of f as a function as well as in real numbers and restrict to the, ah maybe in the closure of support and restrict to the that support of C which will be just union of this different sub-varities, then it's, it's a map from to R is has two properties, is bounded from above so it means that real part of f is less than some on the support is less than some constant and, and second this map is proper. Yeah, and if you get such a situation, yeah so one can integrate of a chain this expression and this expression, this integral will be absolutely convergent and that's generalization of this one-dimensional story. Maybe, yeah, yeah but it's, it doesn't matter. I will immediately, in one minute we'll go to some common older group so this exact chance will be relative, relative, yeah, yeah globally, yeah on some complication, yeah. So, so now I want to say what does it mean homology class of, homology class of, of the path belongs to what? Yeah, I want to describe real-life homology theory called gamma. So, first of all define certain set S sitting in, it's kind of set for f and the pair. This is the meaning of this set, it will be some finite subset mc, and it's finite. And what's the definition of this subset? It's a, in formal, it's a point of anthropology change of the fiber, more precisely define what is the complement of the set. So, I say that z is not, before bifurcation value is by definition is the following. There exists an open neighborhood of point z, you will just see which is part two. So, there exists an open neighborhood and plus such that over this, my neighborhood, my vibration is, this subset x to c is locally trivial. So, there is a homomorphism between the pullback and u tends to f minus one of z, compatible with projection to u. So, this diagram is commutative. So, it kind of trivialize topologically my bundle said that this things also trivialize restriction to the subset d, inducing homomorphism of what, of intersection with pre-image with d, with u intersecting, which is kind of subset here. Yeah, so it means that this things is locally trivial. Yeah, I claim that this happens for all points except finitely many. And why it's so? So, the explanation, one should compactify x and so the map will be expanded to the map from the complication to P1 and then to some algebraic variety containing x and then x prime minus x will be some vertical part, which is f minus one infinity union, some horizontal part, which is some another subvariety. Yeah, so you get some kind and also you have our things d. Yeah, so there are algebraic variety with many, many things inside, three devices inside and when you project, we know that topology of everything is locally constant except some finitely many points. Yeah, so you get this finite set of bad values and now what is homology group? One can consider the following various homology group, which will be all isomorphic to each other. First I can see the h, consider homology of x, this is subset d and union of f inverse, again the same story as before with integer coefficients, where minus t is less than real part of any zi, where s is the iso elements of this critical values. If I consider pre-image, it still doesn't depend on t, this is a natural map from one t to another and they are all isomorphism because we get all homotopic equivalences, kind of all isomorphic to each other. Also it's isomorphic to the same thing, but I take pre-image not of not of a half plane, but any point in the half plane, where real part of z is less than real part of the i for any. It's again, you just consider pre-image of one point because one point is not to be equivalent to half plane isomorphism and so I get many, many groups which are isomorphic to each other and call all of them just by one root hb, dot will be degree on homology, hb of pair x, d and f and b is for bt, it means it's kind of usual topology, not the risky topology, how do you know? And so the class of gamma, so the gamma which is class of c is belongs to h, maybe a variety of some dimension n, middle dimensional. Okay, so we generalize the homology class of the path and now we should do something with the form and it's completely similar, so I use twisted Durham complex again, so I have df, d plus h product by df and it gives me a differential on, it gives me complex of shifts, one x df, but it's complex of shifts, now I should be very careful, I should do it in the risky topology, so I consider on the algebraic forms and covering by algebraic open sets and so on, so one can define twisted Durham homology of pair is the following, so now I have pair, just second I think I need one assumption, assume that d is divisor with normal crossing, yeah I said that it should be divisor with normal crossing, it's in general could be singular space, actually it's not a restriction at all, because you can blow up, yeah you can make blow up and replace by divisor with normal crossing, it doesn't change any commold which we consider at all, yeah no but commold of twisted Durham will, yeah no twisted Durham I defined only for normal crossing and it doesn't depend on resolution, oh maybe you can relative to d, yeah I can see the form vanishing on d, if d is singular I will be a bit unconfident, not logarithmic, forms vanishing on d, bond vanishing in which sense vanishing? Restriction to d is zero, yes yes yeah restriction to each component of d is zero here, yeah so I consider, yeah so it contains a sub complex, this is differential df, where omega is d are forms whose restriction and d is a union of some divisors, smooth components distinction to each d i is equal to zero, yeah and and this thing I think I defined as hypercromology in the risky topology of of this complex of shifts and I claim that the integral is actually can be the first of all if you get this my form which I have volume form, if I have n, in fact it's also belongs to because it's top degree form, it's vanish on the on the on divisor d automatically and also we have d of volume form is zero because there's no n plus one form it's top degree form and df which volume form is zero, yeah so it's killed by differential global section so it's definitely gives a class in this hypercromology and what I want to say is its exponential integral which we're interested, it's actually pairing, is a pairing between homology classes so I get and the pairing maps from this beta homology and df h derangue yes yeah there are dual over c, yeah there is a comparison isomorphism that's dual space kind of beta homology after confiscation isomorphic to derangue homology yeah and this comparison isomorphism can be so yeah so without reference to those chains so that's why it's I'm not sure yeah yeah it's it's so like derangue on a shift theoretic yeah one can prove it shifts heritically yeah it's following what what we listened last week's yeah it's also gives a proof yeah essentially it was I think it's actually it was proven maybe first by malgrange this comparison isomorphism I'm not sure yes no if consider homology in not in the risky topology you get just homology of a pair xf because you multiply by exponent of function you identify with usual forms yeah so it's it's completely wrong for if you put an analytic topology I will speak about how we calculate it using compactification all stuff it will take some time but now till the break I will now speak about kind of topological part on the beta side so for a moment we forget about this hypercomulge and this thing so maybe I'll even kind of hide it see too much yeah so first I think we should want to say it's some kind of panker duality yeah actually I go to this picture yeah so I assume I have the following station I get x prime some smooth compact variety compact algebraic variety over c which contains three three divisors horizontal vertical and some closure of my d this will be normal crossing divisor and I assume that this will be unit of some some different components those different three things can these things has no common components yeah I have such thing variety with three divisors and I assume that d bar intersecting between dh is empty and I have a map from I think to cp1 have just a map and a d vertical is pre-image of infinity that's theoretically yeah it's good to do some multiplicity which I don't care in the moment and and how it's related with original story x x x x prime my I remove divisor d horizontal and d vertical d will be d minus intersection of d vertical and f is a bar respective to x now so I get such a situation so let me draw the picture follow various colors yeah so I get cp1 I get point infinity and so some here it's on my x bar yeah so I have some devices normal crossing d vertical which goes to infinity then I get something will be I don't know d bar and have another thing which is completely this first one should be d horizontal now so what happens I remove this I think remove d horizontal take pair relative to no no no it's not a constraint this can be always achieved by making additional blobs yeah because my chain doesn't really touch d infinity if it intersect I can blow up the intersection and chain will not touch this and then you have to change the x yeah I change yeah I change a little bit the x yeah it's true no but to have kind of clean homology theory I really want to have such but the the previous comparison theorem I mean the setup in which you have this comparison theory between different models and this kind of the ramp and so this does not include this hypothesis I mean when you have a situation which this comparison is an opportunity it's not necessarily the case you can compactify it to it's true yeah but yeah yeah I think you are you are you are right yeah this seems yeah but one can at least numbers can reduce to I just I want to say that at least the situation or what is this point credibility I have this thing and and so I can define also beta co-omology by using co-omology of pairs and of rational numbers that do co-omology and the claim is that that if consider homology of original I think it's naturally isomorphic to co-omology after shift of some new variety minus f and maybe shift by two times dimension and what is the new variety you just exchange the role of these two devices you you remove one devices get pair relative to another or vice versa so x prime will be this is the complement but now I just remove instead of d union d vertical and d prime will be just now so you get different homology group and let me briefly say what the origin of this point credibility first you consider kind of common open part you remove from from x bar all three all three devices consider the complement you get some open manifold and then because it's you have devices normal crossing one can add some boundary with corners can add a real boundary and get manifold with corners using polar coordinates and things which clear yeah so get essentially closed manifold and if I analyze all this pairs one can do something on the boundary and eventually uh maybe I'll just draw the picture what happens you have some kind of compactification kind of real manifold real compact manifold with boundary into also these corners and then the boundary of this guy which is again manifold with corners one can decompose one can put some negative part plus maybe some something middle part plus positive part and roughly the picture like this you get some two disjoint open domains in the boundary and the middle you get something like color so it will be the cylinder and then you get a usual point credibility so commulger of pair x0 and d minus x0 is dual to commulger of x0 d plus 0 yeah so one can do analysis like this but there is some interesting point why I replace function by minus function because when I do this uh domains in in in a boundary what I do roughly I can imagine a kind of complexified series sphere at infinity a circle at infinity with the values of my function and instead of minus infinity plus infinity I take two uh angular domains and take them back and you see that it's really looks like two domains which you connect by by a color and the same goes to high dimension why so you said the boundary is a union of d minus d plus and other things and other things which is which is something which if you add to d minus which will be contracts both to be something like um hemomorphic to the to the product of of whatever and minus two dimensions some manifold small dimension times the interval and this comes from the devices from different components yes yes one can yeah make yeah of course one can do it in different languages using shifts and so on so um same story yeah yeah no so the main thing it's uh this objects allow duality that's one thing which I want to say but now uh before the break I want to spend some time some very basic topology question about topology of of the map uh from a pair get x in pair d and and maps to c because it's with topology I think c will just think it's a real plane and we forget about complication story just do something very basic uh so suppose I get a map from a pair to r2 and I define before occasion set as before is a complement to the points when it's locally trivial topological local trivial and assume it's fine but now x and d will be for me very general for x and d will be let's say just topological spaces I ignore all details of this story and uh then I fix integer number and uh uh I claim this case I can construct a constructible shift on c and what is the fiber of the shift uh fiber of x the axis of point z with these any point and c is defined uh stock at point c defines this case topology of the pair x take d and take union of of the pre-image not of the point z but uh small disc with simple z integer coefficients where epsilon z is set of z prime and c open disc and this uh by this uh finiteness condition we see that it stabilizes as epsilon goes to zero so we identify all of them now so the claim it's it's a constructible shift uh so obviously if we outside the formification point then it's local system because we have locally trivial bundle outside we get structures local system and what is the structure of a shift at sorry I don't assume anything it could be in our case it's finite but in general it's yeah it's very very general story it's the main point that it's this is finiteness that's uh what plays for me essential role nothing else it's a constructible shift uh and what's the structure of shift just defined just a collection of a billion groups and one should kind of make the translation suppose I have a critical kind of critical this bad point when I have formification then what should I do what should I do so let's draw kind of a take point z i take pre-image of this disc union with some set and take a mold of pair but now if I have point very close to z prime I can have kind of smallest thing this you see that d epsilon z i what is whatever u z i epsilon uh minus point z i can contain some u some another prime maybe z some epsilon prime z is not the prime and epsilon is smaller than it's much smaller then I can have a restriction map from commode of pair to commode of pair and that gives this shift structure so I have for a for a section for for the stock of my shift at this point I have stocks in nearby points it's exactly definition what is a constructible shift yeah so I get constructible shift which is smooth outside of s and kind of the theorem which is not obvious at all that this shift has a zero co homology there's no global sections no first homology in fact there's no higher group or what can say this r gamma and here it's a very very bizarre story goes on you will see it in a moment it's doesn't really homo it's not really part of homological algebra as I speak about homology so what is explanation of this fact it goes through some different formalism uh let's consider following pairs we have pair d and b consider pair d and b where d is let's say uh in c is uh closed subset and which is homomorphic to closed disk the closed disk this curly d will be some uh maybe not homomorphic I'll say maybe piecewise whatever you want linear an elite analytics infinity this could be maybe some corners and d is a subset of c uh b is a point on the boundary of d and we have only one constraint constraint that there's no uh bad point from the seat of my set on the boundary okay so this bad points which I have will be some inside some outside and here I get a point b my disc you avoid some some spikes or some yeah it's it's doesn't really matter one can use semi-algebra semi-analytic semi-algebraic stuff yeah it's just to uh with horrible singularity and for any such thing I can associate a billion group which denotes something like by a of this pair of d and b which is a case homologous of the pair the formula can take pre-image of disk and and take again you uh d intersecting with pre-image of disk union uh f minus one of b and take homologous integer coefficients I have this uh homologous pair yeah so this uh and they they form the following thing this a if I associate this uh association of the disk and point on the boundary is is a local system on uh space of disk is boundary such a boundary disk do not touch so I allow to move my thing but boundary should should cross this red point this s yeah okay yeah it's some infinite dimensional space but uh this should form a local system then we have such thing then we get restriction maps if one disk contains another and a b is equal to b prime is the same point then we get a map from a of db to a of d prime b and it's again locally constant map if we deform both things yeah so what happens here I get uh this d I get d prime and some bad points stay in one some both of them some outside and because I have just a map of pairs I just get restriction to the smaller space it's clear and uh and then it satisfies some kind of additivity axiom namely uh imagine the situation like this I have d and maybe I just make a little bit smaller d contain d prime and union d double prime uh and there's no point no black black points no bad points in the middle so there are some points here here outside so d is a big uh uh disk contains two small disks yeah all points will be mark points will be the same then the additivity property says the following that's the condition about this thing in the winter no no no no I said that d and d minus d prime intersecting with s is equal to put this condition and and d is kind of something some disc containing is not the contains the union yeah it's not yeah and there may be disjoint here point b the touch at one point then then what I have in this station by restriction maps I get a map from ad b maps to ad prime b plus ad double prime d you take d minus the union or d minus the intersection sorry d minus the intersection the uh the union sorry sorry yeah you're right yeah by restriction maps I get a map from uh to one group to another group I get a map to direct sum and this thing says isomorphism yeah so I get this uh different types of structure which I have in this situation so what is the relation of this to my shift to constructible shift f uh the relations uh is the following if z is in c minus s that's the stock of my shift is can be written as one of the spaces namely when you have the following picture uh here b will be z and d contains s there are many such discs you choose any one of them and uh you define this like this it's not really definition because then you should say how you identify one thing to another it's it's a pretty complicated story but uh and a similar story if z belongs to s one of these special points uh then um then you should draw the following picture you should make uh you should just exclude point z i that's where the point b and all the rest will be inside yeah that's the definition uh what f of z i is a of such I think so what yeah if you look on on uh this uh axiomatics that I get uh groups depending on discs and some restriction maps and additivity axioms it works uh so axiomatics works if we replace the commologo of pairs with some integer coefficients with some given for given in any key by any contra variant functor uh called the h from what from uh topological spaces or pair or pairs because um pairs can be replaced by topological spaces with base point by contracting close six to zero topological spaces with base point two groups or even to any i billion category it doesn't have to be groups it's contra variant functor it's up to homotopy but it doesn't shouldn't be some homologous theory it has only one properties if you have two spaces I don't know some space uh y one and y two with base point y two then have the union where you identify you can book a you identify the base point of course it contains both y one and y two and then then we have a map and this should be should be isomorphism yeah so it's definitely any homologous theory like k-serial bordiasms what the commologous aesthetic efficiency works but maybe some there's some kind of bizarre I don't know function functors on comotopy types which do not come from homologous theory because I just sit in only in one degree yeah so I don't do things like suspension yeah then one can repeat the same procedure and get a constructable shift is argum equal to zero yeah yeah that should be what is right there should be isomorphism yeah make maybe in 10 minutes ah it's it's not clear at all yet it kind of this redid uses from a homological algebra just give you some kind of theorem which we proved with katzarkov and punty a few years ago that says kind of fix finite set c to q naught two or c that's really meta then the following things are equivalent this category of constructable shifts of let's say five billion groups which are local system on the complement to s and cited the snow h zero and h one one condition then it's equivalent to the following data to the following things it's equivalent to collection of the sabilian groups adb boundary b the second piece s is empty which form a local system on on the space of parameters dnb plus restriction maps as I explained before here satisfies the utility axiom and so I think that's something very explicit so this third description uh choose some uh topological data priori choose some kind of di bi uh sorted di interior of di contains only zi and no other yeah so for each for each point in my set yeah so for each point I choose a small uh disk containing the input choose a base point on it and and also choose for any for any ordered pair of indices which are different uh choose a simple path uh and i'll draw the picture and the simple path say it should go from start from some point uh so I have map from interval to r2 so zero goes to point of my first disk but not point bi one goes to point bj and the interval goes to r2 minus union of all all disks decay for my of all points it doesn't intersect all so I draw this path outside of my points so I just make this uh choose but not everything it's uh for example it should be something like sorry this guy disjoint yeah I see that also I think this this collection of paths uh it could be not arbitrary I don't yeah for example I can choose of something called Gabrielle of this type uh what does it mean I kind of imagine to have some points the infinity which is very far from my point zi and kind of first connect by disjoint path to all zi there's no no disk yet but now what I what I do I just uh I have this bi di I have I just draw a small little disk and have first intersection points to the points bi but now how would I connect them it's uh let me just yeah there are two possibilities I go from up or to down and one should choose pass uh something like uh point in my set so you want something which is topologically equivalent to such a thing or what yeah topological equivalent to such a thing here you get just collection of things you get collection of invertible maps so it exists in yours from each guy to itself and collection of maps tij from ui to j for i non-equal to j so to think no further constraints what you do there is not I didn't understand the principle so because you you have different choices it's a and could be j maybe first and be the j second i you start from one i and yeah we want to reach another disk but it could be above or below in this picture and if you have more uh no no it's it's many of them it's maybe some intermediate here yeah yeah I have many many a disk but they're from order set and if I go from one to another from which I start could be below to the result or above and according to this picture you use different yes yes yeah it's for it's ordered pair ordered pair yeah so this is just yeah it's it's really a long story how it goes I just want to explain you and tell explain it's in different terms in my letter for the lectures what is tii what is space ui ui is exactly this a di bi this is definition of spaces ui then what is tii you consider if you have a disk okay yeah you can see the one parameter family of pairs disk is marked points you just travel your point around the boundary of the disk and you make by because all constraints it's kind of local system get automorphism of this space ui and and what is tij just a second yeah so get for i and get something for j and uh what I just finished in one minute you have a pass uh and now now again you can see the one parameter family of disk with marked points which will start with the following thing it will be the mark points will be always bi I start with this then intermediate step will be something like this and the final step will be again get one parameter family of disk with uh uh marked points and and what happens here uh so I get some kind of again disk depending on uh oops sorry disk depending on parameter theta uh some angle along this variable and so what what I get I get ad0 maybe di by uh monodromia along the pass or holonomy along the pass it's identified as d maybe 2 pi the same di then both of them mapped by restriction map because both contains original disks adi bi and also also both of them isomorphic by ax axon to the direct sum also the morphic to direct sum and uh one can check that the matrix when I have so I get isomorphism the space to itself in the matrix will be identity identity zero here because this one is quotient object and sum operator which will be the definition tij yeah that's uh that's the definition of operator tij um yeah yeah so it's a really long story but um if I choose um yeah and if I get this explicit data one can explicitly construct a shift which is argama is equal to zero and the whole thing it doesn't depends on description and yeah so eventually we see that's argama is equal to zero it's a kind of calculational result it's not follows from abstract homological algebra reasons um yeah so uh one can choose different passes in fact if no three points right on the same line one can choose kind of straight passes between disks and sort of maybe rotate a little bit these things yeah so in general what kind of collection of passes you can choose to have this equivalence I don't know it's uh it's kind of interesting question of topology about braid groups so now we have small break for maybe 10 minutes half so before I talk to you a lot about this topological how topology looks like so eventually it's very simple data but depends on this drawing and now about hodge theory so all this thing which you can see the x d and f is it is a generalization of a just smooth variety and divisor say with normal crossing but variety it's not compact and in what sense the generalization we just consider usual rate geometry it's apart when f is equal to zero it's uh and uh of course then then we consider h deram x d f uh it's kind of generalization of deram homology of pair when there's no function we don't have this correction to differential uh and uh so if you try to think it's uh about hodge theory this is commolder of pair it's typical example of mixed hodge structure uh it actually it's not enough considered just open varieties you really can see the pairs to get interesting hodge mixed hodge structures and pure case in uh respond to usually that then d is empty and also x is compact now and uh what is uh analogy of pure case in the case of function uh kind of analogy analogous to pure okay situation is when d is empty but f is proper uh in fact it's very common situation at uh that is it's proper um often when d is empty but f is not proper not proper uh there exists a compactification or partial compactification x tilde contains x uh and f tilde also uh c is proper uh such that compliments say again uh is kind of d horizontal is uh divisible is normal crossing and f f tilde restricted to any and let's say it's union of some d alpha of any intersection is vibration so topology at infinity doesn't uh change uh yeah that's kind of really nice situation in this situation first of all see that critical points of f is the same as critical points of f tilde it's it's automatically belongs to x and um then whatever commolder of pair for x and f is the same as commolder of x tilde f tilde and the same with the RAM in comparison isomorphism um yeah so one can uh kind of get a non-proper station can replace by proper without changing anything and yeah it's very concrete example I can just say the following suppose x is c star square with coordinates let's say x one x two and the map is those two x some low run polynomial one c star square uh then the generic fiber is elliptic curve minus three points and even singular fiber singularity curve minus three points you just add three points to the fiber and compactify the fiber so x tilde will be x and you use three copies of of c computerized somehow and this this things became now proper map and without changing topology yeah and um and such things are really very convenient just instead of uh this proper map you can consider these things which are equivalent to proper uh you have the same co-homology and uh yeah for example we can make kind of tame polynomials even for example take x is c and f is x square x square it's you can yeah no here's a term yeah in the case of one dimension there's really no problem but what is advantage uh such things uh uh uh call x e kind of is a singular to infinity if there exists x tilde satisfying all this property and uh and if you get these things one is either singular to infinity and you have another thing is either singular to infinity you can make a tensor product and here will be some kind of tom sivastiani some sort of the kind of preimage of one plus preimage of f2 and just take these things is again is a singular yeah so in sense it's proper kind of kind of pure hodge whatever unlike of pure hodge structures we can really multiply and just take product yeah it's um there's some sedge mark that's it's very common yeah there is something called tame polynomial for example people sometimes consider x just c in and you get f is some polynomial and for good polynomials you can compactify fibers uniformly without changing topology and also reduce to this pure case yeah yeah but now uh I assume that it's assume d is empty and f proper uh I can do the following I consider some h bar will be parameter plant constant for any plant constant I associate h d run depending on the constant on x and f there's no device I skip it from notation and just by definition to be hypercomology of x is the risk topology you take forms with differential h bar d plus df yeah so the claim claim is the following so this uh this this form this collection of h around h so this forms an algebraic vector bundle on plain c this h bar coordinates and uh for each degree in doubt this uh it has a meromorphic connection natural meromorphic connection uh this pole at h bar equal to zero and in irregular singularity in general here and regular singularity at infinity no other singular points uh and and and the so we get a connection and the connection has second order pole it's irregular has second order pole at h bar equal to zero yes yes uh kind of remark yeah that's kind of yes yes it's kind of first fact it's its vector bundle essentially says it's rank of h h bar equal to zero it's this it doesn't jump it's uh our old result with Sergei Baranikov which was reproved several times and I will explain the proof in the next lecture of more general result yeah yeah that's it's analog of degeneration of hodge to degram spectral sequence and this allows to speak is about uh fact that the connection has second order pole because otherwise you don't have canonical notion of trivialization yeah so the actually what's the origin of this connection uh this uh for for h bar non equal to zero one get conveniently constant into dual lattice uh in in this homology kind of gamma h sitting to h and in fact it's to describe connections better describe this lattice uh the the reason is uh is the following you see that this thing is isomorphic to the homology of d plus df divided by h bar namely isomorphisms you multiply h times degree of form you add you kind of rescale complex and you replace by this uh complex and and then here by comparison isomorphism so it's h deram of x and f divided by h bar and then it's isomorphic to h beta x f divided by h bar turns around c and the image of this things will be a lattice gamma h gives the lattice and lattice gives you a connection there'll be a unique connection preserving this lattice uh yeah so the non trivial thing to check its lattice this has this thing has a real second order pole uh first thing that's if you consider connection over uh go to loran series in variable h bar uh then this uh and to allow poles then it will be isomorphic to the following thing uh to direct some overall critical values of my function uh you take exponent of the i divided by h bar as a kind of generator of demodule in h bar variable uh changing some regular uh holonomic uh demodule over this uh formal uh loran series and uh this part if consider regular holonomic demodule over uh uh loran series is the same as vector space with plus automorphism and in fact what is this vector space in this automorphism we can see the uh homology of kind of beta homology of uh uh neighborhood of maybe uh f minus one of the i this coefficients this shift of vanishing functions of vanishing cycles of function in shift by the i so it'll be value zero and this is a billion group with automorphism and it gives the regular holon demodules and and this is I think it's maybe slightly and let's survive the answer it was actually my question a few years ago and it was the answer that it's uh this is holonomic demodule it's canonicalism of to this guy okay then there was things which I probably will not explain right now the stocks filtration uh at h bar equal to zero in any direction in any direction uh is compatible with lattice structure yeah so there are several statements here if you also go to form power series but without inverting variable h bar uh then it's canonical isomorphic to direct sum of all critical points and you take homology of formal or maybe analytic neighborhood or analytic algebraic neighborhood of what um you consider these things and intersect with this critical set of f called it's maybe kind of capital z i of z i it could be very singular subset an algebraic subset of my x but take arbitrary neighborhood and in neighborhood I consider uh forms now add formal parameter and take again hd plus df the same differential yeah so I can calculate it uh calculate these things uh locally and uh what else I want to say maybe a couple of points here is that fiber at zero is uh it's natural isomorphic sum over i homology again of neighborhood of z i is complex of forms with differential adjustment application by df and it's uh and it's isomorphic homology of x with the risky topology with df and isomorphic homology of x with analytic topology with the same thing so formal you don't think that the right formal completion yes yes and the analytic you don't you do you mean complex analytic complex analytic but then you have to take uh for you take formal or convergent power series no no still formal yeah in each part of the formal power series yeah in fact so when you take analytic is a fixed sum of the limit of what i was for any given neighborhood yeah yeah yeah it was yeah it was pretty complicated story here yeah but I think it's all of them are isomorphic whatever you do and maybe what the last point here get uh and also get kind of non-degenerate pairing it's essentially the spawn carrier pairing which I explained to you before between h and the ram h bar xf is isomorphic to h maybe take do minus h bar xf because it's take a for h bar not equal to zero we get up to shift we get this pairing which I explained before it extends uh to uh non-degenerate pairing pairing at h bar equal to zero yeah so get this picture I will not really yeah it's uh yeah no this is this kind of analog of hot theory and let's explain in a second why it's analog that's going to be sort of the analog of hot theory now yeah so what here really goes on for h bar not equal to zero we already get this notion and no no trouble at all uh what happens if we remove zero we can we just uh uh consider remove zero obtain get uh irregular holonomic demodule on c star with coordinate h bar which has irregular singularity at h bar equal to zero and regular at h to zero regular h bar goes to infinity but now one can make the following thing let's introduce inverse variable call it t just uh so we get demodule on c star t to get vector bundle this connection algebraic connection uh but now it has irregular singularity as t goes to infinity and irregular as t goes to zero and demodule of this thing it means that we have an action of partition t inverse and do dt since it's like usual commutation relations but this demodule we can interpret as demodule over c just forget that t is invertible and the same is demodule on c this variable t uh such that t gives invert multiplication by t gives is invertible operator it does the same story uh plus some regularity condition yeah again the same regularity condition yeah which you have here plus the same same regularity condition and such thing yeah so it's uh and then this thing gives by Fourier transform so Fourier transform it means that t we goes to do d z with z some dual variables and do dt goes to minus z okay and we get demodule on c with z coordinates such a do d z is invertible plus some regularity yeah but this demodule will be regular will have regular singularity everywhere uh and uh then what everywhere including infinity yeah and uh this demodule has regular singularity infinity then it gives a by Riemann Hilbert correspondence it gives a perverse shift on c such that our gammas of the shift is equal to zero because how we calculate the ramp homodule we get a module kind of d uh demodule over this invariable z and when we calculate the ramp homodule you just consider m tethering o by o m tethering o by forms and uh to get the ramp differential you just act by do d z and consider kernel and kernel of these things here so it's the drama homologous kernel kernel of do d z acting on this total space model and because it's invertible it means exactly means that it's our gamma of corresponding constructible shift is zero uh and this is exactly this shift which I explained to you before or or it's functions functions and polynomial functions in the invariable yeah so you see that it's um yeah so it's eventually it's equivalent to this topological data which I have before uh to have these things and uh so if you just throw away zero you get the same topological data just rephrase in different way and extension to h bar equal to zero it's kind of analog of hodge filtration that's where hodge structure uh appears uh one can treat the very simple case when f equal to zero yeah and if you follow the the line then h deram h equal to zero of x f equal to zero you will be direct sum of hp of x it's a bit uh direct sum of commulger forms yeah and for h non-cozy will get the ramp commulger and how we glue one thing to another it's using hodge filtration yeah so because it's well known for things like filtered spaces are the same as c stark variant bundles and you get essentially this uh this picture uh and uh what I want to say is it yet it it looks it's all this uh things can be extend to general case if x bar is contains this d vertical just using d you want to plus the d bar I have the three things and uh I really want this property uh this is before and I get a bar from my extension to p1 then what one should do one should consider on x bar minus d vertical we can see the things like this we can see the forms which vanish whose restriction to d bar is zero and our logarithmic forms with respect to d horizontal and we have this shift and and we end up end out with differential h d plus df take our our gamma and uh this again form a at least a hop form a vector bundle as h bar goes to zero and the same story works uh yeah so just minus d vertical yeah sorry yeah sorry you're right you can see the forms vanish it's easy to see the vector bundle outside yeah because it's the same you have topological description topol and homologous pair don't jump yeah but it's bar to zero it I think it's it's the spectral sequence again degenerates but um but in principle should be part of more general things about so this is a generalization of the previous yes yes yeah yeah yeah I think it's should work even for general mixed hodge models so let's me formulate the general story there's some kind of notion of things called mixed hodge models uh it's by by seto uh yeah yeah so it's uh if you have uh but these mixed hodge models could and very interesting mixed hodge model on a fine line see this algebraic all right you're not an analytic one what is it it's a constructible shift it's a perverse shift of the modules which correspond to some things with regular singularities uh in corresponding demodule with regular singularities and to treat care something like hodge filtration this will be vector bundle this connection to some delta functions yeah so there's some kind of notion here everywhere including infinity yeah and um so we get some some category of mixed hodge modules and contains spot uh uh which do not by something like this those such that r gamma of constructible shift is zero and uh is it a q shift or z shift z shift say yeah yeah or maybe q shift maybe a different perverse yeah yeah maybe to q shift yeah forget about torsion it's also it's not terribly good yeah and this is a rigid tensor category where you use convolution additive convolution to make a tensor product yeah actually this perverse shifts will be automatically constructible shifts shift sitting in degree one is follows from this condition yeah there's some simple result that in case of unveritable it's automatically just sits in one degree shift and and there is some kind of weight filtration here yeah so this one can make analog of weight filtration story which we developed with the unsuible one for some auxiliary reasons and um yeah so this can complete parallel to usual story and maybe I stop here and continue next week