 So today we'll explain what is the Riemann-Hebert correspondence in two examples for differential equations and Q difference equations and I'll better start with example before explain what are stocks data. So example is a formula. Suppose we have some polynomial in one variable and there's units of even degree and I write differential equation minus dx square plus v applied to some function psi equal to zero or the same as second derivative of psi. This differential equation gives you also d-module polynomial d-module on a fine line with coordinate x. In fact it will be ranked to bundle with flat connection. You consider this vector bundle as psi and psi prime and this is its first order differential equation. And space of solutions is in analytic functions is two-dimensional because it's c is contractible so you can specify solution to each point in derivative from the clinic solution. So it's c2. Okay now we look on formal solutions at infinity. So to write a formal solution you do this for the following things. We write a polynomial, assume the polynomial, you can rescale variable and can assume that polynomial start with coefficients one and then we can write some of two things. We will be w prime squared plus some rest where w is degree n plus one and degree of this correction term is at most n minus one and maybe you can also write leading coefficients will be kind of r n minus one x n minus one. Okay then one can solve this equation in expressions like this. So we can write psi plus minus it will be two formal solutions will be exponent plus minus w multiplied by some power of x and series will be series in x inverse. We lambda plus minus okay give you the exercise is equal to minus n plus minus this leading coefficient. Yeah so yeah so that's very easy to have two formal solutions and in general series of kind of last lecture, last lecture it means that if you work over, if you make a completion, go to this bundle sort of line to puncture disk, form puncture disk, then it belongs to module, belongs to the sum of two blocks corresponding to this E plus minus w times regular modules. Yeah so it will be sum of two one-dimensional orbital form power series. So in particular this for abstract isomorphism classes correction term r is irrelevant. All of them gives give the same formal module but they differ and differ by kind of stocks data namely array is called is stocks ray if for this equation if a real part of I think just x and plus one on the ray is equal to zero and what we get we get kind of we get two and plus two rays for case for n equals three and real axis is not on the stock ray. The complement called stock sectors, it's usually the stock sector the same as component of in this case connected component of C minus the set of stocks rays. In stock sectors the claim is there exists a unique solution which is alternatively asymptotic, which is asymptotic to say plans one is in sense of form of power series and and this is the smallest of two solutions because in the array one if you can if ignore this term real part of one expression will be big much bigger exponentially bigger than real part of another expression expression so be differently one smallest solution and formally smallest in the sector and what we get you get in two-dimensional space in C2 kind of abstract space of solutions you get a two and plus two vectors and you get C1, C2, C2 plus two which is equal to say zero and in the main conditions that say I which say I plus one is not zero for any I say from one zero to minus one plus one yeah so you get configuration of n non-zero vectors in a two-dimensional space such that each two two succincts form a basis this forms some interesting variety some more less space and that will be description of in Betty picture by room and here with correspondence of your equation so you say that it's encoded in this collection of vectors okay so that's example to keep in mind now I'll go to general story it's maybe a or b whatever this a x it's one of cluster there are various types of cluster varieties I think it's x cluster variety so I will speak about general classification of homing D modules on on a germ of punctured disc punctured disc so what does mean that I can see the module over the field you can see the inverse germs of meromorphic functions at zero it's will be funny dimensional space and plus connection to this link let me through yeah by the way here one can try to think that z is equal to x inverse is coordinated infinity in this example yeah so that's will be my next topic so recall formal classification so if we get over so we ignore convergence and strain then this any phonomic module will be M is canonically decomposed in the direct sum of certain finite set this thing this guy's belongs to different blocks so they don't talk to each other in the block will be of the following we consider for each alpha get certain number and alpha greater than one some integer number it's this ramification number and block will be the following we consider covering alpha to the coordinate on one disk covering to the alpha that's raise raise raise to and spar and block will be direct image of the following thing we consider exponent of some greater than integers see alpha names regular module over CV and here ck alpha it's some finite sum and greatest common divisors of of k said it's ck alpha is nonzero is one yeah and these things are defined up to action of roots of one okay yeah so we get so we get finite collection of this irregular this piece of polynomials because one just replace the alpha just to form right okay and get this finite collection of these polynomials and maybe one can say it's this formal classification in the following quay now for each this block appearing in the decomposition we consider a circle and take just pull back circle of ends and alpha roots of s1 covering sir abstract circle yeah and this one seat it's a it's our mode to play the z it's arguments of set circle of arguments of z and then what they would to say that we this regular demodal can be sort of the local system w alpha it's arguments of w alpha it's form the circle s1 alpha and this regular local system I said is the same as local system in a circle it's exactly on this circle so what we get we get a local system on disjoint union of circles that will be that will be what corresponds to this formal demodule my classification zero now I want to go to formal to analytic and here's stocks data which appears these expressions maybe I denote something like f alpha this multi-valued expression this collection of f alpha just find a collection of this beautiful level I will encode it to a stocks diagram what is the stock diagram you can maybe I'll start with first these examples and kind of if you have just regular singularity then the stocks diagram it's a circle surrounding zero and if it's for example exponent of one square root of z to power one half then this guy will be I will be more precise later yeah to stop the diagram maybe up to a certain isotope it will be not concrete subset of maybe in R2 minus zero for exponent z one half or two and for exponent say even have two blocks exponent one over z and exponent minus one over z what we draw two circles slightly eccentric and this correspond to exponent minus one over z what is the what is the meaning of this stock diagram yeah all this think this function f alpha we can consider as a one multi-valued function on c star I take all possible branches of my individual terms of alpha and also take all possible assets so get many many locally I get several analytic functions and because there are several alphas even some branches never go by monodromate to another so I get this multi-valued function and I consider generic generic ray if I restrict in generic ray I get just bunch of complex valued function of real argument and now I take absolute value of exponent bar this functions are I get positive valued functions valued functions and for generic ray for each two of them one will be up grow much faster exponentially faster than another if I approach zero the reason is the following because if I take two branches it's zero no no in infinity here was because my singlet was at infinity and coordinates was yeah but now it's all centers it's zero yeah yeah it's written it you don't you can't see it yeah it's exactly yeah sorry yeah yeah and the order by growth now I go to zero to infinity Nicholas and they're totally ordered now for example if consider like let's say here positive ray on positive ray if z is exposed exponent I zero times r just r which is positive number then exponent minus one over r it's very big oh sorry I think it's actually exponent plus one x point minus r it's much much less than one much much less than exponent of plus one over r but if r is positive negative then I get opposite order and this stocks diagram will be will do for the following if you can see the dust to draw stocks diagram generic stocks diagrams you do something like this you put you in hand your chalk and start to just make sinks like this in anything yeah I just any such things as stocks diagram and if you get a generic ray so intersection of generic ray is the stocks diagram maybe I do not something like big D with this diagram yeah the stock diagram will be effectively immersed stocks that will be immersed of this union of rays immersed in some way the intersection points will be all this all branches of my function multivariate functions and they totally ordered and here this kind of this the closest things to zero will represent the smallest solution the next solution so it will be initial order on the growth of functions but then if you get some kind of bad rays then you cannot decide when the orders jumps a little bit jumps when you move to the left to the right namely when for two kind of stocks rays are the following you get two branches two different branches of my multivariate functions it's called f first and f second which could be branches of one f alpha branches of two different f alphas such that f1 minus f2 it will be certain certain constant to be c times z to power where lambda zero is the most polar part of the expression it's a we'll write something like 1 plus 1 and and these things restricted to ray is takes values in imaginary pure imaginary numbers so the if you take exponent of this thing you get something of size one but if you move it right a little bit the way the ray then it will be in decrease yeah so get finite collection of stocks racing stocks racing this picture respond to points in several branches are immersed to each other now so that's the classical theorem by it's called delineated crunch and was reproven by Kashawar and Shepura in some yeah but it's classical stuff yeah so it says that the monomic demodule over this thing is given by the following it's first to fix kind of this demodule by because it can go to form the model gives a certain collection of use of polynomial and by this procedure bunch of the stocks diagram the claim is the following first it's a local system on s1 kind of original one which is set of arguments of Z or on c star this is a local system which will be solutions of my differential equation in the form naturally vector bundle is flat connection holomorphic bundles with connection but now and then stocks and the stocks data the stocks data says the following for any generic generic ray because if you can see the solutions along the ray it gets some finite dimensional space on gamma solutions long ray you get the filtration and filtration corresponding exactly to intersection of of the stocks ray is with the diagram and filtration by order of growth so it says that element belongs to each term some terms filtration is a certain equivalent to exponent of the things times function with polynomial growth by intersection ray intersected this big diagram and some condition along stocks ray yeah so it's doesn't the filtration locally doesn't change if you rotate a bit your angle but if you go to stocks rays you get two filtrations on the same space and they're satisfying some rules actually what are rules so what we have new one stocks ray so we get plain we get stocks ray and we get the permutation is n where n is he'll be number of intersection of diagram is generic ray so from here to be something like 8 we get 1 2 3 4 5 6 7 8 and if you and you order them in one way and then you get different order if you go through stocks ray so the question now I get two filtrations if you can see the space of solutions near near near stocks rates it's contractable domains certain vector space on space of the stocks ray get two filtrations and they should interact in certain ways are not arbitrary filtration on the left and the right let's consider the simplest case and which appears kind of if you have put generic coefficients in your equation simplest case when you cross the stocks rays you get just a standard generator of symmetric group in the solution of two new terms yes so suppose we had some vector space whatever we will be space of solutions near ray and then we get two filtrations we get maybe zero zero and then you get another filtration prime how they interact so there's something drop just in one place and the pitch is it these two filtrations coincide everywhere except one term so you get some click f i minus one is equal to f minus one prime and all the all the previous also coincide it contains f i contains f i plus one all this can set but this in general not do not coincide so it means that what happens if you make quotient of this term by this term it's the same for both filtration and we get two subspaces and the kind of condition in this case is that this spaces are complementary complementary in this quotient on this quotient space now by the way if you have such a picture when you when you go to associated graded they will be permuted exactly by this permutation because this if the spaces are complementary then this quotient will be equal to this quotient and canonically yeah yeah and in general if you get these stocks daggers when all intersections are only like this then you immediately get the condition for the for the whole picture get this flex related in the special way intersection points and we get identified and in particular we get local systems on this kind of normalization of this picture we put here disjoint union of two intervals instead of intersecting automatically we get local system on the disjoint union of s one alphas and it will be the same and kind of uh fact it's the same it's this data coming from formal classification identified if I associate associated graded spaces get a local system on the disjoint union of these circles what to do in general in general if you have some kind of more complicated permutation you write as a product of simple generators in the shortest form they're not unique way if you get for example deform like this yeah so it will be some part of my picture or you can deform in different way and and try to describe the same reduce to the case of uh simple generators and the claim you get the same category it doesn't depend on choice of shortest decomposition for a couple here this local picture you analyze either in this way or in this way and you get that this my space of solutions is split in the direct sum of three spaces and they're just permutant in opposite order yeah so so this is a story and here's some kind of really uh things which has no uh importance whatsoever but I found it pretty amusing that not all permutations can appear in crossing stocks ray for example what is impossible from if you have like n equal 4 in the first case in some permutation cannot appear the permutation 3 1 4 2 and 2 4 1 3 of uh never appears so you get 22 which is possible and 2 are impossible and if n is very large the number of possible permutation grows uh number of possible permutations it's something called large Schroeder number I don't know what is mean but it grows something like 4 to power n not not like in factorial yeah yeah it's uh yeah uh it does it really play any role at all yeah one can formulate for arbitrary permutation but in fact the it gets very special permutations in real life uh and and it's kind of a general result about and uh following thank you suppose you get n real analytic functions germs of real analytic functions in one variable and variable like maybe I can imagine like ceta is this slope and and suppose uh my functions are not equal to each other from i equal to j and all functions of zero takes very zero uh then uh yeah why ordered my functions because I forget several functions uh I uh because I differ they differ uh they're not identical to each other it means they really differ for let's say small negative the values of x I say that if you want of x then n of x for x less than zero and very close to zero but then if I go for positive numbers I get some other or not x it's called theta for theta so I get just bunch of I get certain permutations and these permutations are exactly those which appear in uh uh general situations why so because these expressions are kind of real analytic functions of angles and and permutations how to maybe I just draw uh this diagram of permutation in uh some please yeah you decompose your n elements in some groups yeah so this will be from one to n and now I decompose by several groups and each group flips the order to the opposite no this permutation is like for one two three goes to three two one yeah yeah and put several this total flips one on top of another then I again these groups decompose in several groups and put them again in some order example here can do like this and now what I do just maybe just again this big groups again flip one over and other yeah so hope that it's kind of clear then we get one of the special permutations yeah but it's really irrelevant fact I have to say but it's kind of amusing that you get the story and where it comes comes from all this uh this second permutation comes from the fact that you don't have a full unipotent group but when you consider the small permutations it's not the full wide group no it's subset of wide group it's certain subset but it's kind of stating about collection of real analytic functions and uh the reason is the fork if you get real analytic functions if you want to compare which one is bigger for small negative and small positive values of function you take the difference and this is still non-zero analytic functions who has some leading coefficients will be even or odd power of monomial and then if you analyze a little bit this story then you get this strange statement and sinfinity function of obviously one can put arbitrary order yeah so that's classical picture of stokes filtration and and it had some kind of new life uh uh recently maybe I'll say just stop off yeah so this application it's will be kind of a bit description description of the following thing you're supposed to have x y there's some compact complex curve y bar and minus finite subset and what I interested in algebraic bundles with flat connect with connections on algebraic bundle plus connection on this complement uh so this was picture four uh one for the puncture disc but in general the picture is a fork you if you have your curve whatever certain genes you remove several points and then around each point you draw like peaceful atom diagram yeah I just remind my question because your picture was abstract it's abstract but uh for this it was up to isotope because I said what is the order of it was uh maybe I say with the following I get immersion of s one of this union of s ones which intersect transversely race and the self-intersection points correspond to stokes race now so these things are dependent maybe on local coordinates and not terribly canonical and it explains that one can really move the picture and get the same equivalent descriptions it's different okay yeah so so draw several diagrams and what you want to have have local system outside of these diagrams outside these diagrams embedded in your surface and then if you consider the neighborhood of each point you you do the say you get local system on the homotopically circle and you replace the same story as filtration plus iterations satisfying this constraints near each point near each point yeah so it's nice but those uh something which uh arise kind of pretty recently and from various directions uh there is a very clean way to formulate it in terms of constructible shifts and microlocal supports yeah suppose if you have m and get some c infinity manifold just real manifold and to get a shift of essentially anything on m then I get something like ss of f which is a cotangent bundle and it's kind it's conical invariant closed subset and if l in a cotangent bundle will be singular closed conical subset single closed and in fact in our case will be a singular Lagrangian closed subset I'm not terribly clear about what kind of singular Lagrangian let's say sub subanalytic yeah but it's it's it's one it's a good choice sub subanalytic then I get a category maybe triangulated category in this our case will be of uh shifts whose support is contained in now conical sorry conical sorry this uh yeah yeah suppose if l is zero section then this shifts a singular support and single support of f is in l is equivalent to the statement that f is the local system is is locally constant and yeah let's do case just one variable m is equal to r so the cotangent bundle to m is cotangent fiber yeah so so in this case when l is zero we get local system but suppose now l is zero section union of cotangent fiber at zero so l will be coordinate cross then this thing you get uh you get a construct shifts constructible with uh shifts on r constructible with respect to stratification kind of negative numbers in zero positive numbers get three strata and uh then I get so it will be local system on two rays because a constructible contractible grid just to say vector spaces you get so you get three vector spaces and f zero maps to f minus and f plus because kind of if you get germ of section point zero it will give a germ point with some on the right hand side yeah so you get um so the category of shifts will be just three vector spaces since two maps but now if you do such guy this means to get positive positively very co-oriented point then it means uh that for example this map from f zero to f minus is isomorphism in what you get you get just two vector spaces because it's a coincide in one maps to another this category representation of quiver of quiver e2 not e3 now now suppose we have a collection suppose my manifold is surface topological surface it should be a complex curve eventually and I have a collection of immersed co-oriented real curves then I take union of the positive conormal bundles and zero section like L maybe zero section of positive conormal bundles and you get certain condition on shifts um let's uh try uh yeah suppose uh we have locally just one curve and it's co-oriented so it's boundary of some like line boundary of some hyperplane then what we get we get fiber let's say left fiber right of my shift and uh essentially because it's it add kind of one dummy variable to one to one dimensional case and you get a map uh the the category of shifts with a macro local support and this local pitch yeah and what happens if you have two co-oriented then you have something like four different uh uh stocks of shifts of smooth points in four quadrants and you get a map then then you get a map now the condition that uh at this point you don't add anything else to your singular support means the following that you get commutative diagram uh plus the following condition if consider kind of three-step complex when you put uh like I know s1 s2 s3 s4 and you change sign somewhere s1 plus s2 and maybe minus s3 plus s4 uh that this complex is a cyclic so the equivalent of the condition that you had complementary vector space yes yes exactly yeah yes yes yeah so the typical situation we get something like c2 power a c2 power a plus b c2 power a plus b plus c and c2 power a plus b plus c and it's called cartesian and co-cartesian diagram and like this yeah yeah so the so this uh this also even local story about classifications of this uh stocks filtration can be said the following kind of local uh classification these things uh it's it's it's the following can I I deform my diagram to have on the trans transverseful intersection again so I draw it's not really realistic in case I draw these things and orient everything outwards and consider uh constructible shifts with microlocal support in this thing which are equal to zero inside so I remove part of zero section which is the neighborhood of zero remove part of zero section section which will be kind of innermost part of the diagram component of let's say r2 minus diagram around zero so here will be no shifts will be zero and what will happen we get here exactly pieces of stocks filtration describing some kind of if you intersect with array then we increase our space this space became larger and larger and I thought it's great to get local system over as you know first once uh so I need to uh region it's a constant shift okay from a locally closed set or on open set on local close up because on the boundary it's extent extent yeah yeah yeah yeah but on outside it will get maybe non-trivial monodrama yeah so it's uh that's essentially the whole given in description and uh next lecture I will explain that it's kind of natural from point of your fukai categories it's just reformulation of uh dealing one great story but uh so this constructible shifts which wasn't set at time um yeah so maybe I can a little bit return to this original example in this this point zero will be point infinity in my for my equation or in general we can write differential equation of some order some minus minus dx to power n plus plus potential start plus x to power n if you write such differential equation uh for example what what it will what the question of linear algebra it will appear from this description n will be exactly number of uh this I think you get a picture which it's a bit hard to draw it's kind of I imagine that you draw something like this it will be a kind of periodic diagram uh yeah for example in this original case you get something like for question second order you get such story and uh in this case uh the local system outside will have trivial monodromy because the outside will be c it will be contractable yeah so what you get uh you you get uh n dimensional vector space kind of solutions of your equations and then you get uh vectors say one say n plus m could say zero a kind of cyclically ordered um I consider psi i where i belongs to z mod n plus m collection of cyclically ordered vectors there's a condition that psi i plus one which which psi i plus n is not zero for any i so it means that each n of them in cyclic order form a basis it's kind of famous cluster variety for grass mania in which people pusing now on many many places and that's its concrete example of all this uh stocks data geometry so it'll take break about maybe five minutes yeah yeah so the second part will be uh something which also can be called Riemann Hilbert correspondence I'll explain next time why it's natural it's about q difference equations so what is this suppose q is a complex number and I assume that it's not a unit circle let's say that's the one uh and I denote by a q the the following algebra it's c of x one hat for sinners x two hat it's in two invertible elements kind of free group of group wrinkle of free group model is a relation that x two hat x one hat x two hat and this is algebra it's it admits the filtration as I explained on previous lectures and you can speak about holonomic modules and why it's q difference equation because one can try to think it's x one maybe call it's z then x two hat will be exponent of log q z or d z it will be a shift operator a multiplicative shift z goes to q z instead of derivative uh so you get uh one can ask about holonomic modules and the story hits remarkably parallel to differential equation story first of all uh like in uh differential equations we go to formal uh disk so you can see the completion in text two maybe denote by t this operator shift shift and now I can see the finite sum of powers of shift operators coefficients in uh lauren series first so it's it's it's completely it completely parallel of differential operators there is a notion of regular modules in this case yeah uh first of all if consider uh modules of this guys will be modules over these things it will be automatically finite dimensional spaces over the field of lauren series equivalent under dilation this is action of dilation plus action of t instead of connection and definition so there's no analog of delta function here in this case so the definition that this module is just the action is called uh regular uh if uh it contains uh if it contains a sub module m zero which is m zero uh free or finite rank and again uh and connection preserves t preserves t zero belongs it's preserved under this shift operator yeah so it's a in differential case it was invariance under zero dz now to invariance under this story and you want m zero to span yeah and m zero span yeah sorry and m is equal to c z now such guys uh parallel to usual d models and classification of regular d modules is uh uh regular difference models is uh similar yeah yeah you can take for example z to power mu z kind of morally you can take this guy uh and uh it will be simple regular modules and uh how is the operator tx operator tx and the generator z mu goes to q to power mu so you should say what is q to power mu in the sense uh uh so this model m mu and mu mu one is equal to isomorph to m mu two if mu one minus mu two belongs to z is to power z q or one can what you essentially you get uh things correspond to points of elliptic curve in fact this elliptic curve it's c start by q to power z uh you have a generator and you say that it's a generated vectors by action by action of t multiplied by some constant so it gets a certain invertible constant but if you make a different gene between particular power z then it is constant multiplied by power of q so you get quotient the simple models correspond to this guys and and the whole category of regular modules a billion category of no q is has no less than one less than one yeah one can do it also for algebraically for not to uh category of irregular modules difference modules over this thing is the same as coherent shifts with finite support on elliptic curve yeah that's one story now what to do is for irregular uh things kind of like irregular terms or blocks uh in this case are in one-to-one correspondence to rational numbers yeah uh so in uh in case of differential equations blocks as this feels this piece of polynomials and they have continuous parameters here's no continuous parameter whatsoever and um what is the block supposed to get number t which is a over b and usual whatever this b greater than one a in z and iq prime write rational number in niquet then i write you can see the c of v and um and this as endomorphism keep prime w goes to q to one over and here we choose b's root of q if you choose b's root of q then we have a spectrum of this ring maps to the spectrum of ring it's it's uh uh map compatible with with automorphism so we get this projection and what we do we consider push forward of uh regular c of v module densoring by c of v by some standard analog of exponential of uh um uh module uh module e alpha it's kind of and e alpha is c of w it's meant by some generator e e a it's some generative e a it's one dimensional bundle and the action of t e a goes to w to power b times e a bar bar to power a sorry c w the map is c w to c z that the spectrum spectrum spectrum yeah w goes to z equal to w to power b it's b full cover yeah yeah yeah so for any integer number you get this standard one dimensional module multiplied by generated by monomial and take push forward by discovering map and the theorem that you get again direct some decomposition so it's analog of uh the higher level to retain decomposition the category of uh the holonomic it directs some or all rational numbers of and all categories are eventually the same categories like coherent shifts with zero dimensional support on elliptic curve uh what is geometry of the set of irregular terms by rational numbers with parallel with d models and difference models i remind you what was the geometric interpretation of this uh use of polynomials in demodal case we consider the following guy we consider uh spectrum so here the algebra was like this yeah and here you get algebra or c of z maybe z and here we get c of z and this get Laurent polynomials and here polynomials and differential so here uh we we do the following we consider uh we consider formal disc contains zero and consider logarithmic cotangent bundle and compactify fiber viscid infinity and we get certain Poisson manifold which projects to puncture disc or disc and my Poisson form has zeros of order one along vertical devices and order two along horizontal devices and now we start to make kind of various blow ups we can increase sort of pole and eventually we will have some new devices when the order of poles is one get new logarithmic devices for uh to form and all the set of all possible logarithmic devices on universal blow up it's exactly log devices in the universal it's called the risky space kind of blow up i in one to one correspondence to uh blocks in in the compositions yeah so it was a geometry in as a classical case here we do uh instead of cotangent bundle we get kind of c star bundle we can see the cp1 multiplied by spec of c of z and here we can small piece of toric variety so again projects to spec c of z and here my uh symplectic form has poles of order one at uh one big device and two kind of formal germs of devices and now we can make blow ups but if you make blow up again we get a logarithmic devices if you make blow up you get again here get blow up again here and then the set of all kind of big devices except with the small ones the set of all kind of big log devices it's not a formal but actual cp1 because these devices are kind of very short pieces of curves uh i in one to one correspond with the rational numbers t namely if you introduce coordinate z1 and z2 here you can see the curve z2 equal to z1 to power a a b and get certain curve if a or b is zero you get curve which intersect vertical fiber or if it's negative you go up if it's kind of q negative it's t equal to zero and get q positive goes down and uh and then this device appeared from kind of very decomposition of uh fraction for rational numbers okay yeah so it's so the fundamental result here it's theorem it's analog of the linear manganese malgrange classification it's a theorem by uh zloy ramiz and zhank about a right yes ramiz yeah maybe about five six years ago it's a pretty recent uh result uh which is a complete analogy of uh case of differential equations you can see the holonomic modules q difference modules and now the fact that's q less than one it's became really essential this here it could be just not root of one in all this picture but how much q difference modules over now I take germs of meromorphic functions and again take polynomials in this q difference curator uh it will be vector band uh the theorem is the following it's the same as holomorphic vector bundle on elliptic curve bundle very cold plus the filtration by rational numbers is finally many step filtration labeled by q by sub bundles such that associated graded this is kind of f lambda is semi-stable with slope you can make mistake lambda or minus lambda lambda and uh what is the slope and what is semi-stable uh if you have a shift uh you have a vector bundle on elliptic curve it's not zero then it has um it's kind of slope will be degree of the bundle divided by rank it's some rational number and a definition f it is slope of f f is semi-stable with given slope t if this thing is equal to t and for any sub bundle which is strictly between f and prime this degree of f prime divided by rank of f prime is uh that's equal to t um then for every bundle get canonical it's called hardon-narsimum filtration the degree you get a sub bundle with uh largest possible ratio and uh and iterate the procedure uh so get kind of filtration and uh and this degree over rank will decrease strictly decrease yeah here i just want to see what is important here you get here also get a filtration and associate degree it will be semi-stable but degree or rank will increase not decrease yeah so it's uh so here what i get is called something like anti-hardon-narsimum filtration so it's not unique object it's it's some additional not trivial additional data but then you should have a direct sound decomposition not direct sound no no it's filtration by some numbers but it's filtration by some different numbers it's it's filtration by certain collection of rational numbers yeah so this will be analog over the stock's data if you take some of associate gradient uh just uh one important uh uh fact it's kind of theorem by a tier uh say it as a following uh he he proved that uh any vector bundles is actually direct sum of semi of semi-stable bundles and semi-stable bundles is given are the same as coherence in this category are all equivalent to each other for example if consider the slope zero it will you make free mokai transform we get shift with zero dimensional support and all categories are equivalent to coherent shifts with zero dimensional support on a elliptic curve which was exactly our all our blocks in in in this picture yeah yeah so if you believe this is kind of black box then the correspondence between a formal and non-formal classification go to associate gradient and you get uh this formal classification how this correspondence grows goes you consider your meromorphic bundle consider point consider point z then you get something like whatever qz then get q square z q q z you get some kind of spiral uh if you iterate uh application you get maybe spiral which goes to zero and um suppose you're interested in some solution maybe along the spiral uh analytic solution along the spiral uh now there are those basic models which i explained to you just this one generator it was upstairs for this kind of basic uh q difference equation if you want to write what is the solution you get something like f of qz is equal to z to para a b which is t to power f of z so f of q square z will be qz to power t f of z f of qz and so on and then when you see that if you consider very far point of this far f of q to power n to power z it will be z to power n t times q to power n times n minus one over two t f of z so you see it's so you see that the norm of f q and z goes like this it's q to power n square times t plus capital of n so it's it's analog of this exponential growth so it's in power it goes like exponent of square of logarithm and from this uh i think you you immediately see that you can also filter solutions by various slope and what what it what it gives us first of all if you have this q difference equation you can forget about manifold structure zero and because of its equivalence under this q difference operator you get a holomorphic bundle on elliptic curve you identify fibers at fibers of your bundle here here and here they all identify so you get holomorphic bundle on the elliptic curve then at each point of this elliptic curve for each vector you can see in which terms of filtration it goes you just put a put a vector here and then transfer it some meromorphic trivialization of bundle along this spiral and see how fast it's growth and this is so it gives a filtration and the fact it's what they prove that it's these three people that you get filtration by sub bundles and associated gradients are semi-stable and it's one-to-one correspondence and uh finally i can go to oh global case now i have really this my algebra of q which is q difference operators the picture is the following first i'll draw uh kind of picture with devices i consider c cross z one cross c cross z two and compactify by cp one the one cross cp one z two you consider classical torus first with uh symplectic form d z one over the two and here in this classification i get four devices uh when my two form has polo four to one yeah uh this kind of previous local picture was somewhere here yeah and is equal to zero and now i i can start to make blow ups at all four points make kind of toric blow ups and the set of all kind of limit all set of all the set of all this log devices and have log pol log pol divisor it'll be a two fold copy of qp one let me consider uh double cover rp one and consider pullbacks of qp one uh so what will happen we get kind of like curves goes to this intersection to the center we get one bunch of rational numbers another bunch of rational numbers but also we can go here we can go here we get uh two copies of infinity and this double cover and what is this set it's a set it's the same set of kind of rational arrays in r2 or primitive vectors um so this uh the general claim is a following any holonomic aq model produce a finite subset in either description in z2 primitive uh so it's just kind of find a collection of some um uh arrays this will be analog of something like singular support for regular holonomic models demodels and usual station or this more irregular form of classification coming in irregular case and uh the whole category will be described in the following way uh I claim that it's kind of little abandon then kind of addition to this really hard analytic result uh that's category of holonomic modules modules is equivalent to the following thing you can see the coherent shift instead of vector bundle plus two anti-harden or single filtration on this coherent shifts coherent shifts are slightly more general than vector bundles you consider in case of curves you can see just torsion shifts plus vector bundle and torsion shift one can say that it belongs to uh confuse maybe to plus its slope is either plus or minus maybe plus infinity yeah yeah so yeah yeah so so for for coherent shifts how we do make hardness and filtration you get um canonical torsion part it's will be canonical sub-object here yeah so it means it's something like degree of rank I think it's plus infinity in these things if you make quotient to get it's will be the same as because the object is the direct sum of torsion plus vector bundle get this residual vector bundle and then with this vector bundle you do you can see the usual hardness and filtration and here we do kind of for coherent shifts we do anti-harden or single filtration and um that's kind of essential for example we can make equation something like fc of say z is equal to z minus one and say z that's a typical equation which produce naturally some coherent shift of filtration steps which are coherent shift in one of one filtration we have coherent shifts torsion part in the filter and the reason is that this this equation you get a morphism for fiber from z to qz but it's not isomorphism when z equal to one when z goes to zero infinity it's isomorphism so you get this vector bundle story but you get a little correction and one can figure out and you get again some line bundle and one of filtration will be one step filtration another filtration will be two step filtration with torsion part yeah this two filtrations coming from two limits you can z goes to zero z goes to infinity yeah yeah so it's uh yeah at least kind of formula it looks now have complete analogy of the classification for differential equations and next time i'll explain how all the things naturally come from focac category considerations maybe just one little thing to add here SL2z in semi-direct probability starts square acts by automorphisms of this algebraic q and because this algebraic q one can describe in kind of SL2z covariant way so it's essentially by manual transformations the question how it acts on this description yeah so it's it's kind of baby it's a q-analog of Fourier transform for example Fourier transform acts on demodules and in principle we know how to calculate this is beta description how to calculate for beta description Fourier transform and here it's it's part of the following general statement suppose we have a c a category with region stability condition structure stability it's triangulated category subregions stability structure means that you get mf from t0 group to r2 kind of oriented r2 and you get some notion of semi-stable object for uh in this category sit for some maximatics what i claim that this thing produce canonical abelian category this abelian category uh it's basically objects in the heart of t structure because you get stability structure you get a heart of t structure the heart of t structure plus two anti-hardness infiltrations how to understand this thing if you have an object usually it is structure called something like kappa half plane or right half plane and if you have the social degree which are semi-stable objects they have certain slopes which are ordered in some anti-clockwise order and you get another filtration or same object but you consider part of this diagram below half plane you maybe get something like n raise and this thing you can analyze is the same as the representation of quiver function for quiver a n minus 1 to your category uh satisfying that some the property that some specific n object in this representation of quiver goes to this semi-stable object for example if you have picture like this n equal 3 get quiver a 2 and say we get three semi-stable objects form exact triangle uh and and this notion it's uh from this notion it's particularly as it's the all invariant if you start to rotate your stability structure or deform by sl2r and then the fact is it's for elliptic curve you get essentially unique up to universal cover f sl2r stability structure on on category of on db of coherent shifts uh yeah so what is it's important at this construction abelian category at this construction invariant under action of universal cover of sl2r this action certain action of universal cover cell thermal space stability structures so it's preserved here yeah so it means that for elliptic curve we get some abelian category which will be this this guy but now um automorphism group of this category start to act Fourier mocha transform start to act on this thing in this Fourier mocha transforms we'll produce a sl2z action on uh this description yeah so it's um pretty symmetric picture and i think i better stop now