 okay thank you very much yeah so let's me remind you this was this example with sterling formula we have x and c and we have volume element dx yeah but one can write like this is df is equal to e x minus 1 dx yeah and if you denote by y is exponent of x then df is equal to y minus 1 dy y over 1 and volume is equal to dy over y yeah so denoted by alpha so you can see it's actually secretly is a whole thing is algebraic so we have more general situation we have let's say algebraic variety over c and we have closed one form and suppose we have volume element and again simplify my life I assume that my primitive for my form has more singularity so alpha has isolated zeros with multiplicity one zeroes j say is multiplicity one so it's equivalent to the fact that this local has more singularities and I should assume some extra conditions and kind of no zeros at infinity I can maybe tell later a little more precise what does it mean so and appropriate complication form has no zeros on strut at infinity then we can produce a bunch of series associated to each point and series something called ij formal and lift okay yeah okay yeah it's by definition it's will be series in each bar defined up to sign namely what you do you consider synthetic expansion of following integral take first to make this normalization and now I choose a function in the neighborhood of my points to have value zero sense will be uniquely characterized in the point and then they integrate exponent minus function divided by each bar and here I consider maybe truncated symbol not not not going to infinity it doesn't affect a synthetic expansion times volume element so I get this finite collection of series and and theorem is that this finite collection of series with certain analytic wall crossing structure will form a surgeon pack package so what does it yeah there is something which we can formulate in kind of classical terms of Borrelical bar calculation you you you make this series any of the series take Borrel transform yeah yes so make series ij formal will be some of CJK is bar to power k and we make Borrel transform some of CJK divided by pay k factorial and here you just it started zero start make series in zeta it's analytically it will be analytic in zeta and if you extend you'll have potentially countably many singularities and singularities will be logarithms multiplied by the same things in shifted variables and this will be generalization of this example with sterling formula here there are following troubles if you have like curves of maybe high genus like genus 2 even and have this one form and then the singularities will be periods of my one forms integrals for one form so one zero to another zero in certain domain it converges it's had some positive derivatives converges but then it should be extend to function in the starship domain and it will have infinitely many singularities the starship domain have infinitely many rays complex means not infinitely many cuts but if you go around the cut it will be logarithm of the same function times these things yeah so well defined function on the come on the on some cover on some cover yeah it's actually a bit hard to say which cover and you get you don't get a local system because eventually if you try to analyze these things in the whole totality you'll have certain lattice yeah maybe I'll say what is the lattice here as the lattice will be first homology group of your variety why and consider set of zeros which is set of this form points pi isn't it your coefficients you have this finite rank lattice and it maps to model torsion it maps to oh sorry mix mix letters maps by central charge to complex numbers by integration of my form alpha integrate over pass connect to points of closed loops and this lattice or at least first common drop five without zeros can have rank arbitrary let's rank and the image will be everywhere dense set of points yeah so it will be kind of a bit pathological situation you get infinite dense set of singularities of the function and when you try to reconstruct function by making a ray which avoid bad directions yeah and make Laplace transform I have really no idea where the integral will be convergent it's kind of it seems to be very deep delicate question in dynamical systems maybe depend on approximation of rational numbers were rational numbers yeah so so there's some subjects which we don't really want to dwell in yeah so it's good yeah at least yeah yeah yeah there's its convergence and and then good bye kind of a usual summation get kind of really ugly thing you get maybe maybe it's conversion but there are complements to count to play many race will be uncountably many rate uncountably many domains like if you remove all rational direction get and and then you get uncountably many functions yeah and I don't know I like I prefer countable something which you can write concretely is it this not deep which uncountables the race number of number if you can see the kind of components of race which do not go to things it will be like complement you're already your handles from comfortable set no no but these things are everywhere dense yeah and so this new language which has suggestions can overcomes this difficulties and yeah so endless and liquidation one can see in old language but how to make barrel transform and where one should really make money transform it will be answered better this language of its analytical crossing structure so that's kind of main advantage of this language now so that's that's the serum which I want to this and in fact here this analytical crossing structure all this kind of collection of stock synthesis this complicated thing it's it's it's not over not not only analytic but rational in certain sense and which I will explain you later so it's and rational it means that you get some rational a matrix valued functions is rational coefficients certain some variables sorry you said something with the uncomfortable no no if you kind of in original formulation if you want to construct function in each bar by making Laplace transform of this guy you should integrate over the ray but its function will define in some sector and it depends on the choice of race choose different race up to homotopy but there's no homotopy because it's everywhere dense you get really uncountably many things still get uncountably you get for each irrational number you should get a function in a certain sector that is assuming that really all the thing that it is happening the principle sheet isn't isn't there a principle sheet with slightly less long long thing all be all of them yeah yeah actually with Anton we're very very familiar with the story it's this is kind of some science in subscience of dynamic assist is called inter-exchange map or quadratic differentials and here's you typically have is people really study dynamics all things they have things dancing like billiards and you you get things really there is no principle branches no no no no no there's no principle branch at all yeah for one forms here for example yeah yeah so maybe I now see a little bit about return to the notion of analytical crossing structures I should have a lattice like here this lattice and I have some Lie algebra Lie algebra graded by this lattice again if you go this concrete example this Lie algebra will be direct sum over pairs of zeros and you can see the pi not of you can see the classes in set of classes in relative homology such that boundary of the things it's difference of these two points which I do not sounds like and this is definitely labeled by element of gamma yeah so so you consider that elements of my things will be roughly pass connecting these two point up to homotopy and it's a it's a source chief algebra you can compose to pass and get another pass and if it's you can compose a state zero and this is such chief algebra and take Lie algebra of this but this is such chief algebra is isomorphic not canonically to matrices of this of size set of my mark points zeros with coefficients in Laurent polynomials in the M is a rank of those homology it's it's really matrix valued functions with coefficients in the run series learn polynomials and I think associate Lie algebra yeah that's how things it looks in our example and we have this Lie algebra and analytical crossing structure it's something about this Lie algebra in grading letters so it's we have a map which is concrete weapon our example have additive map and so you can explain again so you have this puzzle algebra it's the same yeah because you can choose how motor can pass from each element to some base point isn't will be just choose of matrix element and big beauty of the second I'll take your water and that and thank for next speakers lose the water this over under here okay yeah so the story the following you get C plane with the values of my central charge we get some kind of high-dimensional space which maps here and I get a kernel of on the central charge make real and the Zanlik wall crossing structure was described by some can be specified non uniquely it depends up to use this some initial think you choose finite collection of rays and for each ray you choose certain corn maybe I just choose a corn which project properly the picture with the following it's calling some high-dimensional space and to the plane it projects to lay ray and something which is strictly less than 90 degrees on both side some angle sector yeah yeah so get this cons and this cons can overlap ups in high-dimensional space in the projections can also overlap downstairs and what you glue from these things you glue an analytic bundle over on on C on this H plane C by the following thing it's it's will be trivialized a bundle build rank number of of zeros maybe I do not something like rank you call it and before cold and yeah cold and yeah and will be number of zeros no m no m is rank of each one yeah and n will be n will be rank of number of zeros and so the rank of this group will be m plus m minus one yeah actually only disk yeah yeah it will be not on the whole plane but on a small disk so it will be trivialized outside of disks outside of this race it will be trivialized outside of trace and and the jump transformation will be element a whatever ak for each ray lk which is exponent of some it's logarithm and logarithm belong to some kind of completion of Lie algebra maybe associated to this cone sorry okay yeah it belong to to those the following six we have a cone and we have or maybe here we have a cone and had all integer points in sizes cone and consider part of my Lie algebra where a grade components lines strictly in sizes cone it will be it will be nil kind of you can make a filtration get pretty important completion and take element of this thing you get certain parental importantly group and take log of element but if you don't yeah so I consider certain element and the main property of this guy should be analytic so the sloggers the coefficients of this series should be grow no more than exponentially and this will imply for me a certain matrix valued function on on a defined on the neighborhood on some finite interval depending on radius of convergence for this lj does the completion of yeah probably important completion as a product of elements of Lie algebra maybe I'll take the following this local K kind of a priori for formal things it belongs to product of gamma non equal to 0 belong is to cone K K instead of direct some get into in some but in fact it belongs to kind of exponential gross part this race are not projections of elements of letters at all it's something which I choose at the wheel yeah like and then I said that it's it gives me some analytic structure and I said there is some equivalence relation can carton move race and which doesn't change something is choices actually sorry it's choices actually of the race choices of the race I can change the rate I can move race I can change a little bit cone yeah there was a several movements which do not change something interesting because according to my definition yeah yeah which I explained now here it's a social problem but in fact with young we consider differently algebra when can repeat definition namely you can see that if you have a lightest gamma okay Z to you know some L yeah lightest of finite rank then what we can consider we consider polynomial vector fields on on the dual torus yeah and it's automatically graded because you can see the commutation is shifts and and it's not such to father right so no it's a different with in fact in certain sense one consider this guy is a particle class of this guy because here what we do we can see the matrix valued function on our torus but matrix valued function will be the same as vector field on the total space of trivial bundle which are linear on the fiber so it will be particular class of vector fields on a total space yeah it's kind of a bit ugly in bedding but one can changing rank one can you bet one to another and this story is the definition we wrote a paper analysis here for in this language but it this is kind of young mills and gravitational type the algebras and is a peon what's called 4d wall crossing it's to do all crossing to explain how you get from your main examples this data sorry you have my example today yes yes yeah I will explain how you get this things it eventually yeah sorry no my my lee algebra it's kind of find a dimension each degree here it's actually one dimension for vector fields it's uniform with one dimension and that kind of uniform notion of norms and the conditions that element of lee algebra is because it's in the corner you choose choose a projection like distance along this ray you say that this coefficients grow no more than exponent times this distance yeah yeah in fact I have to say that it in vector field case you don't put condition on logarithm it's it's a wrong way to do things you put condition on a group element like if you have germ of different ophism analytic map identity plus higher the terms it's logarithm could be very well could be non-analytic it also part of this resurgent game called the Calvaronian theory which really fits into see yeah so it's in true research you put condition of exponential growth on ak itself not not on logarithm okay but in for matrix valid functions it's the same before exponential map it's locally by direction for this dimension groups yeah so get so that was this kind of abstract story if you have this you have this gluing and then it you have bundles which is also trivialized not only outside of rates it's also trivialized of a formal power disk for more power series neighborhood of H bar and you get invariant in kind of strange coset space which is jail and over a ring of analytic germs function of bar modular jail n of a formal power series sorry I mean left to right sorry sorry you are completely right yeah yeah yeah yeah yeah so the quotient of big group by small group I recall you what are possible transformations for example you have this description which you can just say that your data like a position of ray you just rotate it a little bit if it's strictly less than 90 degrees it'll be the same but you don't change the cyclic order of this race or you can increase one of this race or eventually can maybe cut these things in two pieces decompose and your element and product on one direction larger and separate a little bit race so the when you glue the six you do not change this invariant you don't you can if identify canonical this you have a holomorphic bundle trivialized in the formal neighborhood of a point which is what it's here says and that's that gives the element of this same element of this cassette which you want to keep the same now so there is this notion of analytical crossing structure there is a formal notion when I don't put condition of constraints but you move the same movement of formal closing structure and there is a kind of classification if you put in all this game what is going on you got just finitely many cons but each of them stay far away for the kernel and because I finally many they stay universally far away the corner currently you pick some maybe quadratic form or signature to something which will be positive definite here negative definite here and you say that all your corn stays in negative domain and if you fix this bound which come naturally from geometry of things on manifold at the end of the day this classification is a following again you choose your this axillary race for which you put for description in a way as you like you need at least few of them because you need maybe to have only few so what's what's will be this classification it's a bit hard to write in many words but what you do consider complement the possible support of this gammas which appear in in this expression it will be complement to this guy and you divide by several domains here by hyper planes which not do not pass through the vertical direction but close to it and copper planes which are rational yeah so the rational will have several meanings here if put several rational planes but because they cannot really aligned with this this kernel center church to some kind of very rational guy it's kind of pass through this when you project project the things you get a little bit overlapping overlapping angle sectors and what you do you this will be your call this will be my cons in between this direction and I put arbitrary analytic elements in concave and in order not to repeat my information twice I consider this kind of semi-close con to be closed on this guy said but open on another side so the integer vector slang on the boundary will belong to you'll belong to the kind of next cone so I never repeat my information so sort of free coefficients of my elements of the algebra will all distinct elements and the claim that it's really kind of one-to-one correspondence all this equivalence classes are given by finite collection of analytic transformations but with the precise conditions that this month that monomials which appear in this transformation belong to specific subsets and the kind of drawback of this the things which is kind of a bit unpleasant I can prove the series only if I use this rational hyperplanes I cannot use hyperplanes which compatible with irrational hyperplanes compatible with correction because it's leads to very unpleasant questions which may be true but I don't want to answer them and and then there was a kind of very again I recall why this rational hyperplanes plays so important role in in this business it's because of the following kind of basic lemma which I remember if you have element in gl let's say n there is some lemma just in two variables if I have matrix if you have a matrix valued function depending on two parameters q1 and q2 and suppose have matrix values two parameters which and it's convergent for small q1 q2 you can uniquely decompose in the program to guys but here a priority can put form of our series and to make it really unique you maybe should ask that Taylor monomials which appear in can log of a minus don't have dag diagonal terms so it will be really unique decomposition and the series that this actually canonical factors are also analytic so what is this a q and q1 q2 are just the variables I see three symbols I don't understand what it's it's analytic functions in product of q1 q2 and oh oh it's a product it's a product yeah no come yeah yeah so it's so those are the constant matrices no no no but I assume for example I assume that logarithm of a log of a minus doesn't contain terms which is doesn't contain q1 q2 to some positive degrees and constant matrices I assume that matrices equal to both equal to identity at zero yeah so that's the basic lemma which has kind of a bit long history and again I remind you briefly how how one can prove it the idea is the following you take a second now consider some kind of double cassette space like matrix valid function model to trivialization depending on q1 q2 that's one can understand geometrically is a set of vector bundles on the following things you consider such domain city can p1 cross p1 you consider either points where q1 equal to zero and q2 runs up to infinity or q2 equal to zero can formally a neighborhood of two projective lines and consider bundles on the skies which are trivial on vertical and horizontal things of p1 then the same is the same as double cassette space it's because you can trivialize it on these things it's ambiguous depends on series and q1 can't relate to nothing and doesn't get overlap identification so that's exactly what you say in this double cassette language yeah so it described geometrically set of bundles with some open condition and now how to see this decomposition it's with the function p1 cross p1 maybe make a little bit even blow up at two angle points then you get in this domain get three families of three one parameter families of p1's and consider bundles which are trivial on each three of them and if you consider one comparison get one double cassette get another comparison we get essentially a plus and other essentially minus yeah so it's that was a basic argument and what was historical origin this was wall crossing this polynomial vector fields originally this was more started with young with more complicated wall crossing called for 40 wall crossing for some sickly simplectomorphisms and then you have this DT invariance story and then one make calculation see that you get bunch of transformation which are algebraic yeah for example if you have instead of JLN you do the same story in more complicated setting with different or fisms and you get kind of like rational maps in decomposed to things you get algebraic maps yeah and in this DT invariance you get plenty of complicated algebraic functions but what goes on here in a case of matrix valued functions if you look on this argument you see that there's something really remarkable goes on if you get this series belong to rational function it's it's it's not only convergent in zero but it's a rational function this a plus and minus will be gained rational functions but rational not in said not in the sense of rational hyperplanes but in sense of rational functions for many variables yeah so the rational functions if you center for posthumously obviously analytic and this series the analyticity follows from the fact it's rational functions and so it's kind of free kind of finite amount of information you cut it will this domain and your finitely many domains with rational hyperplanes you get kind of finitely many kind of integer coefficients and then I got rational valued matrix function which in fact in this case we have integer coefficients yeah so get some kind of some finite amount of information how did to encode this wall crossing structure for this case of one form and this kind of manageable part of this huge story and as we'll see later for what we see in chair assignments we get not rational functions but modular forms but still it's kind of the whole things in court and some nice algebraic data and this one forms kind of intermediate example between just finitely many singularities and more complicated world to have complicated tricky picture yeah so now probably I have to go explain a little bit how one proves the theorem just before going on I want to say that to check this criteria you see that it's something very unwieldy you divide things by not by different kind of slightly tilted hyperplanes if central charge is rational in substance in the sense that the of gamma belongs to let's say Gaussian integers and real real pattern imaginary part will be interesting one can there's no such troubles one can do the following things to choose kind of rational race in our tool choose a rational race then they get canonical contribution for this each angle sectors and this canonical contribution should be rational as rational function of several variables so that rationally use in three different ways yeah it's rational matrix valid function okay yeah so eventually want to have a central charge to be rational and it needs certain amount of work yeah so now return to my situation you get function and variety and one form yeah get variety or complex numbers and closed one form now let's assume for simplicity that's wise compact this condition of infinity maybe I'll skip it in my lecture because it's a bit long story so I have one form have compact then wage direction we take we take a closed real one form real closed one form and it has again isolated zeros and as a real it's derivative for real function a real function has more in this more syntax it's kind of the same as dimension complex dimension of x it's kind of middle middle much more syntax the same number of positive and negative right is why oh sorry why yeah and then you see that generically should be no gradient lines because they have the same more syntax is kind of no differential in most complex and then there was very long time ago it was there was something like more snorkels theory if you have in general close of real one form with like more close one form with more singularities but maybe it's arbitrary more synthesis you can start to count gradient lines and get certain complex here get no no differential and what I claim is that so what does what does this more snorkels theory gives you it gives us get obvious complex or more snorkels complex was over some field which I like fractional powers real infinite series is real powers of some variable and which gradient line from one point to another point your your associate a term t to the interval of alpha it will be positive number and you get some complex of this ring and what is calculated by this complex it's actually we look on literature it's not wasn't really properly stated what does it calculate the claim it's quasi isomorphic to homology of y with coefficients in the local system given by monodromy along with shrewd is equal to exponent of integral of this your alpha is alpha set up right into alpha teta yeah alpha teta yeah consider local system over this manifold with coefficients the snorkel field or of rank one and it calculates its its homology yeah that's a general story so now what goes on in this complex case example in this here get kind of in a complex case because we don't really integrate form alpha not alpha teta yeah so instead of a consider not a field but a ring we can see the complex powers of my variable but so we can see the sums over c j t to lambda j we are real part of lambda j goes to plus infinity things like this you can see the such infinite sums set of such infinite sums it's a local system of the snorkel field the monodromy around this loop it's a power of variable this dummy variable t which is integral of one form along the slope yeah yeah so you should consider maybe this ring and but you cannot multiply sorry you can multiply sure something goes to plus infinity yeah it's a ring and so you identify so you got this identification and one can maybe make a little bit more so you get this canonical we get canonical a local system el fita with monodromy along this loop is equal to t exponent minus theta integral of complex my form of Bowser's loop and get this canonical a local system and in a complex case it's it will be the same and this okay maybe what I want to say that for generic ceta for non-stock ceta not equal to there is not equal to argument of gamma for gamma in my algebra for non stocks direction I claim this will be no no most differential and the claim that we have canonical isomorphism of a commulger of my of my manifold why this coefficients is sigma l ceta this direct sum of one-dimensional space and basis vector corresponding to my points pj and why it's the isomorphism is canonical we have a complex manifold and in definition of most complex we choose some remaining metric but now we choose Hermitian metrics not even Keller metric choose any Hermitian metric to draw most complex and then we get gradient lines stable unstable manifolds and the claim that on this for any Hermitian metric if ceta is not stocks this left should stimbles which I start to draw from different points they never touch each other there's no gradient lines the story the phone can go to universal cover then my function will be the global holomorphic function and then imagine a part of minus e ct alpha will be constant along along this thing and so under projection the project to really straight lines and because we choose this not doing things that they never intersect yeah and and this space of Hermitian metric is contractible so we have this contractible set of identification so this really canonical identification after changing the ring the rings are different right local system with this ring but but you chose a smaller ring when you take a theta no no no no no I just some dummy variable have just this big one you don't restrict to the ones that are alpha theta periods yeah of course one can make little yeah one can do a bit more my my accurate things but you get this canonical isomorphism and if you do slightly different direction get different isomorphism and the contribution of sector between them is different ratio of two canonical isomorphisms yeah it's all so that's this description what is going on but in fact here's a little bit more or one can say like this it's also one can this local system L theta I can I said what small and wrong in things but actually one can say that's local system trivialized at each point set it as it's colon colon it's in t to power integral of one form from past one point to another actually it's local system L theta is trivialized at every point of way of way yes no no no way sorry or you can trivialize it zero it any minute yeah no you can you can define the following you consider I just want to consider trivial local system I want to say what is parallel transport from one point to another yeah along the path I take it multiplication by t to power integral of the path it's kind of like give the connection in trivial bundle and so it's actually trivialized but in particular trivialized at zero yeah and you can also multiply by connection local system trivialized at zeros and local system is kind of like small monodrome monodrome will be just let's say complex numbers not powers of t of rank one yeah rank one one quantity I said zeros and if you can see the local system trivialized at zeros you get to catch a dual torus to this gamma yeah and what is going on we get in certain domain in certain a priori non-archimedean domain in the torus home gamma to 2gm we get a bundle we trivialize this common of why this coefficients in local system so local system will be kind of small deformation of this things is very big monodrome actually let me tell you she wrote how to say what is what this means it's the main and just a question from the chairman to the organizers one hour yeah it's yeah okay I'll finish in one minute maybe okay yeah yeah now I just want to say that this what is really goes on you can see this is this torus yeah T is home from gamma 2gm it using this collection of cones which you can extract from geometry we can construct a toric variety this collection of cones yeah a priori which want to play this yeah it's yeah yes yes you have this cyclical order sequence of convex cones so get cones of top dimension of co dimension one and here it goes to in mirror picture it goes to points in and projective lines because top dimension calls goes to point in toric variety and what here goes on in the formal neighborhood of the things you trivialize your bundle of co-mology and my co-mology some algebraic story you consider universal rank one more local systems give you since point and realize as long things get some trivialization yeah so it's still things of a formal power series now what goes on if we are happy if the central charge is rational then something very remarkable will go on if the central charge will be rational then what I will have periods of real imaginary part of my water form will be integer number I will get the map from from my from my variety to two-dimensional torus is it because the periods will be defined up to integers so class of alpha real part of alpha and imaginary part of alpha and belong to co-mology of these integer coefficients and maybe I assume also this mark I also assume the relative to awesome zeros let's consider all not only close periods but all zeros are integral coefficients it maps to the torus and all my single all my critical points maps to the point zero in what I get I get a local system on a torus minus a pump minus one point I get so it's a story will be vibrational on a two-dimensional torus minus a outside of a point and because everything is kind of CW complex of finite type I see the co-mology I get a co-molder of fiber and fundamental group of torus minus a point free group is to generate you can actually consist of things and then in this situation one can calculate everything in terms of spectral sequence downstairs and you get some kind of in some elementary six you get contribution for positive octant is a rational map yeah so so we can as you that you get rational map in this special situation one can write some simple form in terms of two operators in finite dimensional space and what is going on now the question how one achieves this position which is have kind of complex manifold with complex variables and here's also kind of final idea in all these topological arguments you don't does really need holomorph holomorph 50 tall what one needs it real part of one form and the major part of one form are linearly independent outside of zeros independent at any point why which is not the set of zeros it's very big condition it's you don't need any holomorph 50 tall it just need complex complex value to one form with this condition under you very easy to change something with some small support outside of zeros is very small form it doesn't it's more factored doesn't change this linear independence and then one can move a little bit in and achieve the situation with the central chances rational and in this non-holomorphic situation but the whole things works and then one get this proof of rationality yeah I'll try to I try to calculate very simple examples like a curve of genus 2 with one form and get this rational that will it matrix matrix function with are so huge already in this example I don't want to frighten you but the huge but fine it okay yeah okay okay thank you maybe maybe I'll maybe I'll just write you leave little formula you have the stores and essentially get vector space with two operators yeah some kind of like this monodromia and this monodromia I don't know operator t1 and t2 you get two automorphisms maybe get some some special fiber and a commode which you calculate you make cell the composition of the things and you calculate some sick like this complex but this complex you also may multiply by some parameters like q1 q2 q2 q1 you should be very small so you get this bunch of isomorphism but then but then you identify is this common all Javis looking about sorry ah no you do like this you can see this such complex we goes to these things and you can identify this complex with V in two different ways you say it's contained you take this this quotient will be a cyclic complex of this type or a cycle complex of this type essentially what you you do this identification of isomorphisms will be something like this it's always a rational functions of q1 q2 and if you want you to depends on some parameters it's also a rational function yeah essentially you used to simple calculus like this and then it's it yeah yeah maybe it's a good idea