 Thank you. First of all, I would like to say it's a great pleasure to be speaking here at this conference. Most of the new results which I'll speak about are joint work with Daisuke Yamakawa and Robert Paluba. So the basic story is about how to sort of one way to attach spaces to surfaces. There's a quite long and complicated talk, so I'll try to start with a precise statement before I sort of speed up in the rest of it. So suppose I have a smooth algebraic curve over C, then there's a well-known picture of how to attach a modular space of flat connections to this, the character varieties which were shown to be symplectic by a tier-bottom Goldman. So this is often looked at as modular spaces of flat connections, but there's a sort of a slightly stronger algebraic statement and you can taste the difference in algebraic structures on the spaces of representations of the fundamental group and the spaces of connections. So over here we look at algebraic connections on algebraic vector bundles on the curve which have this condition of having regular singularities at each puncture. So this is the statement that appears in Dilean's book in 1970 and maybe on curves had been looked at before, but there's debate about the precise history. Having phrased it in this way, it's clear that one should be able to extend this by loosening the condition of having regular poles. So I can look at the larger category of the connections on vector bundles that corresponds to the stokes and monogamy data and these also have nice symplectic structures as well. So this was looked at in my thesis and more algebraically based on and the statement was completed in this work with Daisuki from 2015. So we have a large class of spaces which extend these modular spaces of flat connections which had been looked at before and it's possible to get generalizations of the braid group actions which occur here in this generalized picture by enriching the notion of a Riemann surface by taking into account part of the boundary conditions which occur. So that's the basic picture. The original story was enriched by Hitchin who showed that these are not just complex symplectic but they actually have hypercalometrics as well and in a different complex structure they're isomorphic to Hitchin's systems and so in particular have Lagrangian torus vibrations. So if you think of that in the other complex structures these give a large class of examples of non-compact special Lagrangian vibrations. This story was extended with Bicard to show that even the wild spaces at the bottom also have this extra structure and so there's a sort of large class of rich spaces which occur. So this could perhaps be looked at as the state of the art in sort of moduli of linear ordinary differential equations but we actually got to this picture by trying to understand better certain classes of non-linear differential equations and so I want to back up a bit and explain the picture behind it and then ask questions about the classification of the spaces which occur. Okay so suppose you want to classify algebraic integrable systems you need to have a definition and so let's look at pairs M is a holomorphic or an algebraic symplectic manifold and I have a map chi to a vector space of half the dimension such that generic fibers are smooth Lagrangian abelian varieties. So that's an extremely sort of large and difficult problem so to make it easier I want to restrict to a special class those which are with a lax representation. So here I mean basically I want it to be a symplectic leaf of a meromorphic Hitching system so in the case where the base curve is genus 0 that's essentially the classical definition of a lax pair but it also includes the extension to arbitrary genus base curves as was first looked at by hitching in the case without poles. I've also put in the word good and so it's as if that there's a sort of a slightly naive algebraic perspective so it's not true if you look at Higgs bundles that have arbitrary order poles it's not true that every symplectic leaf is an algebraic integrable system in this definition here so you would like to pick out the symplectic leaves which are good such that they do have this structure and also I guess the main motivation was such that we do have hypercalo metrics on them and correspondences with modular space of connections as well and that's a description of these in terms of stokes and monogamy data and so I would like to at some point try to make slightly more precise what I mean by good but it's it's quite technical and so we would like to try to classify these up to isomorphism or perhaps up to deformation or isotogeny and then look at the different representations or realizations of each of these sort of abstract spaces or abstract is grubble systems can you take a second and say it's a curtail to a good variety or it's a symplectic leaf of a Meromorphic Hitchens system so in the moment I'll define Hitchens systems and so it's possible to do that with the poles and then you look at the symplectic leaf and some of those would be integrable and so that's the class of integrable systems I want to look at okay so a large class of examples looks at classically as isospectral deformations of rational matrices so here I have a matrix depending rationally on a parameter z and one looks at the coefficients of the characteristic polynomial and this gives the definition of the spectral curve and you can construct symplectic modular spaces of these by if you actually get a dz and fix the orbits of the polar parts at each pole of a then this amounts to be a symplectic quotient and this gives m star a a symplectic structure so I'm just fixing the the orbits of the principal parts at each pole and lots of examples of integrable systems can be put into this picture for instance going back to Jack of E and Garnier and others so the Hitchens systems look like this at least for GLN so Hitchens looks at the cotangent bundle of the monstrous base of stable bundles on the curve so this is pairs consisting of a stable vector bundle and a cotangent vector phi and so this is just a section of nv tens with only got one and then it's possible to relax slightly the fact that V is stable to have a stable pair so you get this slightly larger modularized space which the then has a proper map generalizing the characteristic polynomial that we looked at before this is the modularized space of stable Higgs bundles a Higgs pair is a pair like this where the pair is stable but that does not imply that V itself has to be stable so it's a partial compactification of the cotangent bundle of bungee so that's another large class of algebraic integrable systems H is a base of Hitchens yes H is a Hitchens base here it will be something else later on I think and so having this hypercaler metric means that the same space has a different algebraic structure it's naturally diphyomorphic to a modularized space of connections and then you have this other change in algebraic structure to go to the the better picture the space of representations of the fundamental group so the picture is more that you have this sort of abstract space with these three different algebraic structures and the various sort of diphyomorphisms but you know so yeah diphyomorphisms between the algebraic spaces which occur okay so over in the structure on the left here the dolbo structure we get this Hitchens system and so that's a certain class of non-linear differential equations when we look at it from the connection perspective here we get a different class of differential equations when you vary the modularity of the curve these are the isominojury equations or the non-Abelian Gauss-Mannin connection so now I have a family of curves all of which are smooth and we look at the corresponding Dirac modular spaces over that and so we get this bundle of Dirac spaces the basic statement is that that has a natural flat algebraic connection on it and if you write that explicitly in algebraic coordinates it gives non-linear differential equations with respect to the modularity of the curve so these are the isominojury equations so at the end of the day if we try to classify all of this structure we actually classes defying sort of two classes of differential equations the integral systems and the isominojury systems it's it's actually quite good to pass over to this isominojury picture because there's lots of very precise work on the modular spaces of low dimension in particular complex dimension two especially the work of Okamoto and we can use that to try to sort of build a theory of dinking graphs to try to classify the spaces of larger dimension so this leads to the definition of a non-belian hodge space which is a hypercalous space with these three preferred algebraic structures such that in one particular algebraic structure it is now an integrable system which is a symplectic leaf of a meromorphic kitchen system so that's sort of a vague version of the basic question I guess one point is that lots of this picture has been extended to higher dimensions but there's no new examples of such spaces which occur so it could be that all of the examples which occur occur for curves and so that's an open question which is not known as far as I know so let's go back to the sort of extension of the hitching picture to the case with poles we want to for instance put in the rational matrix picture into the sort of theory of meromorphic kitchen systems and this just appears by saying that a dz is a meromorphic higgs field on the trivial vector bundle or it could be looked at as a connection on a trivial vector bundle as well and then the statement is that the spaces M star we looked at before often have partial compactifications which have this full hypercaler picture so I've stated this quite vaguely this is the work of lots of different people and I've probably missed out several people the first algebraic part is to do with knit suree constructed modular space and showed that the meromorphic hitching map is proper both the chin and mark when looks at the algebraic plus on structures and showed that they were integrable systems in the plus on sense ignoring this question about which symplectic beads actually are integrable systems in the precise definition I have before it's possible to look at the sort of irregular extension of the atiyah bot symplectic structures and illicitly that was the infinite dimensional symplectic quotient structure that was up construction that was upgraded in this work with bicar to get these sort of generalizations of the hitching metrics and then which house and critch ever wanted to compute more explicitly the symplectic structures there and so there's papers of them in particular examples and this led to the quasi-Hamiltonian approach I'll speak about a bit here which started in 2002 and we completed the most general statement in 2015 even with the case with twisted formal structure at the poles and it's possible to also have twists in the interior of the curve as well so it's connections on on torsos for twisted group schemes over the curve but that's quite abstract and so I'll actually specialise to examples and explain those. What is Meromorphic Higgs Field and Meromorphic Connection? What is Higgs Field there? So a Higgs field was a section of NV tensor omega 1 and Meromorphic Higgs field is a section of NV tensor omega 1d for a positive divisor d so I'm a like to have poles and a connection is a operator which satisfies Leibniz rule. Okay so we have basically the same pictures before except we're allowed to put in the word wild and Meromorphic in various different places that you have these three algebraic structures you have the wild character for priority here and you have these Meromorphic Higgs binomodular spaces and Meromorphic Connections and you have this analytic or real analytic isomorphism between them which changes the complex structure. How old are you for? Riemann-Hilbert Burkov or if you like irregular Riemann-Hilbert there's this paper of Burkov from 1909 which was perhaps the first to suggest how to use the stokes data to classify connections with higher order poles and so people often speak of Riemann-Hilbert Burkov. Okay so the picture is something like this so we have a curve with some points and we do the real oriented blow-up to get these blue circles here and then we have this halo which is like a neighborhood of these boundary circles so the basic point is that there's a well-known and standard formal classification of irregular connections at the poles. Connections are formally classified by graded local systems on this circle or on this small annulus here and so no matter what else we do the formal classification gives us a graded local system and so you know that is the top logical thing which classifies irregular connections formally so the usual tame picture you just take the graded system to be all in one graded piece so one way to present what the stokes data is we want to glue the graded system here onto the local system of solutions in the interior of the curve and if you look at the multi-summation approach this gives us preferred analytic isomorphisms between the great graded system and the solutions of the curve here so it's possible to glue these together to give a local system on the whole curve except for the fact that these isomorphisms jump in certain singular or anti-stokes the directions and so we have to add these extra punches when we glue the graded system onto the local system of solutions in the interior in particular the monogamy of the graded system does not need to be conjugates to the monogamy of the actual connection around the pole it makes no difference but yeah it's topologically equivalent and we just want to hear the top logical picture here i'll draw it on the real blood blow up but that just makes a difference about what's happening at the blue edge which is away from there um okay so you end up with an object i call a stokes local system which classifies the connections we had to start off with i'll come back with more precise examples in a moment so the basic picture is that there's a reduction of structure group to this graded system at the poles and so it's if you're breaking it into these individual pieces and you you have a picture like this at least in the untwisted case and the twisted picture is more like the picture i had at the start where the great graded pieces are twisted around as you go around the circle okay so it's actually graded by a cover of the circle i'll get to an example of the area equation where it's it's graded by the irregular type and that involves the z to the three over two and you'll see that that twists around later on um yeah i'm sketching a bit because there's lots of stuff here um there are papers which explain it all in great depth i'm just trying to give the picture of what's happening in this talk um so the basic aim would be to write out a list of examples just by having a look at papers and then try to work out which are isomorphic as abstract you know sort of non-abelian hodge spaces so for instance one might look at examples like this here you get the shift of arguments into kruble's system so i have one pole of order one and a pole of order two um the isomonodromy system there a special case of the jimbo miwa mori and sato system that was looked at in 1979 i think the corresponding stokes picture um well a slightly framed version of the stokes picture gives this holomorphic Poisson manifold g star which is the holomorphic Poisson manifold that you want to quantize to get the dreinfeld jimbo quantum group and so even quite simple examples of wild character spaces give you know extremely interesting Poisson varieties here um so the standard tamer picture on p1 would look like this you're looking at connections having first all the poles so logarithmic connections on the trivial vector bundle um the integral systems worked out by Garnier the isomonodromy systems were looked at by Schlesinger slightly earlier Garnier's work occurred by taking the autonomous limit of the Schlesinger system and the betty picture is the familiar space of representations of the fundamental group of a a punctured sphere into G um now the basic fact is that these two rows are isomorphic that there's algebraic isomorphisms between the full spaces in each column um so for the integral systems this goes back to Adam's Harnad-Herterby's um and then Harnad extended it to the isomonodromy picture over here and showed that the isomonodromy equations match up and one could understand it intrinsically as the Fourier Laplace transformation on d modules on the affine line or on sort of presentations of the d modules the matrices a b p and q which occur carry a particular presentation of d modules on the affine line um so basically it all goes back to the Fourier Laplace transformation um for example one of the simplest cases to have a look at is is gl2 on the four punctured sphere so if I take the matrices here to be two by two the corresponding isomonodromy deformations are equivalent to the panel of a six differential equation which is this sort of horrendous non-linear differential equation which can be viewed as a non-linear analog of the Gauss hyper geometric equation the corresponding betty space is this Frick Klein-Vocht cubic surface um so you've got constants a b c and d which match up with the four conjugacy classes of monodromy that you want to pick um and so this has this explicit description at least if we take the betty weights to be trivial otherwise it might not be affine okay and so we can then see how this appears from each of these descriptions from the standard two by two picture you just take four conjugacy classes in the group gl2 and you do this multiplicative or quasi-Hamiltonian reduction so you take the product to be equal to one and quotient by the group gl2 and you confuse it has dimension two um exactly the same space occurs in the irregular picture at the top here you start with sl3 so you just have one tame pole and regular pole and so you end up looking at a generic symplexic deep in the group sl3 star the dual Poisson v group these have the same dimension as color joins orbits of sl3 so a generic orbit as dimension six and then you want to do the symplexic quotient by the torus and so you subtract off two twice and again you get two and it's possible to compute explicitly what the space is and again get this Frick Klein cubic that these really are sort of different representations of the same abstract space and one can do the same with the lax pairs for the actual isominerogamy equations and for the integrable system which is really what hard add and Adam's hard hard add her to be did so you can look for different representations as well so for instance suppose you take the complex simple group g2 you can try to choose conjugacy classes in there such that you again get the dimension two so this is in the same picture so g2 has a nice six-dimensional semi-simple class so we take three of those at three of the poles and then take the fourth pole to be a generic class which has dimension 12 and so the multiplicative symplexic quotient has to the dimension three times six plus 12 minus twice the dimension of the group so you end up with dimension two and with paluba we computed explicitly what it is and we got the Frick-Klein-Vott surfaces in the case where they're symmetric so we have a equals b equals to c and so you just end up with two parameters which matches up with the eigenvalues in this generic class here and so there's even representations of these spaces if you go beyond general guinea groups as well okay you might look at a different example like panel of a2 here we have two by two matrices with one pole of order four this gives the panel of a2 differential equation the betty surface which occurs first appeared I think in the paper of flash renewal and looks like this it's like an irregular analog of the the Frick-Klein surface in yourself one parameter which corresponds to the monogamy of this graded local system around the pole of order two the formal monogamy so that's an explicit description of some space as a nice hypokalometric so one can carry on and draw this large table and then try to look at isomorphisms of them so I now want to discuss a bit about the dinking graphs that one can try to classify certain examples of these with are there any questions at the moment before I sort of switch over to graphs so they're all plastic I don't know you need to ask the expert so I I think it's more sort of reconstructed these first and then they came up with some spaces so it's more their job to sort of prove if it's isomorphic or not as far as I can tell so in this work of Okamoto in the 80s he computed the affine vial symmetry groups of the panel of a equations so for instance for panel of a6 he found the affine d4 vial group and for panel of a2 he had affine a1 so you would like to think that you know affine d4 is something to do with rank two connections on a four-punched sphere so you draw a picture of that and then you draw the picture of the affine dinking graph here and you think well there's four legs and so I'd have these four punches and maybe the two the dimension vector that you pick there matches up with the two over here so you know that looks reasonably clear and that's been extended to arbitrary sort of tame connections on p1 you can prove slightly more if you look at the the quiver variety of type d4 this is on the nose isomorphic this open part of the modular space where the bum is holomorphically trivial and so there's a slightly different precise statement that this open part is naturally attached to affine d4 it's the corresponding al e space so one would like to extend this to panel of a2 for example that there should be some direct variation between the affine a1 graph with dimensions one and one and rank two bundles having a pole of order four so the singular directions would look like this and it's not completely clear I mean this looks different that but the story is that one can relate the stokes data here to this and one can also look at the open parts as well the open parts is reasonably easy but perhaps not obvious so there's an open part to the modularized space this m star and that is isomorphic to the a1 al e space or the iguchi Hansen space the first example of a a non flat hypercaler metric so in one complex structure it's just the cotangent bundle of p1 and in a different complex structure it's a generic orbit a generic coad joint orbit of sl2c okay so let me recall briefly this story of Nakajima quiver spaces so there's a well established story of how to attach a space to a graph so suppose the graph is this we put a vector space at each node and we look at the space of maps in both directions along each edge so we have maps a and b in each direction like this and if you choose an orientation of the graph this is a complex symplectic vector space and the graded automorphisms acts in a Hamiltonian way so here the moment map for the action of this group h the product of the groups of the nodes is given by something like this a b and one component and minus b a in the other and the aim is to just look at the the complex symplectic quotient of this large symplectic vector space by the graded group h at a central or a scalar value of the moment maps we choose a complex scalar at each node and we look at mu inverse of lambda quotient by h so we have a space attached to the graph with this list of dimensions at each node plus the choice of these complex scalars there's a slightly richer version where you look at stability conditions but let's ignore that here right so the picture extends to an arbitrary graph easily just by repeating on each edge so i guess this is first looked at in dimension two examples in kronheim as thesis and he looks at the al e spaces for arbitrary a d e subgroups of s u 2 or ad dink in graphs affine graphs and the point is here that if you take the vector of dimensions to be the minimal null roots you end up with something of complex dimension two so in particular for these two examples that we looked at before you do indeed get space of complex dimension two which fits in with the fact that these these panel of a equations are second order that they're geometrically a a connection on a fiber bundle with dimension two fibers okay this has a multiplicative version that was looked at by crony booby and shore and then extended by vandenberg to get the symplectic structures as well so we look at the multiplicative version of before in the sense that you look at the open parts of the representations of the graph what could be called the invertible representations by putting in this condition that one plus ab is invertible so this is a well-known condition that occurs in the local classification of regular holonomic d modules on a disc taking a and b to be the var and can maps but for us the crucial point perhaps is this statement proved by vandenberg that it has a natural quasi-homotonian or multiplicative Hamiltonian structure with the group as before but the moment map is given by one plus ab and one plus ba the the monogamy and the micro monogamy that occurred in the previous talk with slightly different conventions on the signs so one can now take an arbitrary graph and fuse together this construction on each edge and you get a holomorphic symplectic space by doing the reduction as before and this is good in the tame case because it's possible to check that the multiplicative quiver variety attached to the affine d4 star with dimensions two one one one one is indeed isomorphic algebraically to this Frick Klein-Vott surface so let's call this space bv1 v2 the vandenberg space and the question would be well suppose I have a double edge well the classical theory of multiplicative quivers would say to take two of these edges and to fuse together and so this means you take the product of the moment maps that you had before so you get something like one plus ab times one plus cd for the moment map there and you can look at the reductions of this and no one knows how to put a complete hypercalometric on these uh the point is that through this stokes business taking the wild character space associated to panel of a2 we get a space we do know how to get a metric on that it involves slightly different quasi-Hamiltonian spaces and we can repeat this for triple edges and lots of other types of graphs as well so the question is perhaps what is rep star and can we put a a quasi-Hamiltonian structure on it so the point is perhaps that it's it's not quite the right statement to put these edges together okay so to understand dan what to do you can go back to the paper of 1764 of Euler and if you read the third page of this there's a list of the Euler Continuant polynomial the second of which is the one plus ab that we got from this var and can picture before and the fourth of which is this degree four polynomial which is very close to one plus ab times one plus cd but it has this extra term this ad which is here so it's different and this is the one that occurs in the stokes story um the general rule for these continuant polynomials is that you take the product of the monomials which occur the first monomial is the product of the symbols which occur and then you delete all possible consecutive pairs so for instance for abc i have the product of all it and then i would delete ab and then i would delete bc and that's it so you get this extra term here by deleting bc in the middle there yeah and you can check its difference to one plus ab times one plus cd and as i said if you learn about the stokes data it's possible to see that these actually are cosy Hamiltonian moment maps and i want to sort of explain how this occurs in this particular example there are various different ways to describe what the stokes data is we use this multisimation approach that goes back to ecal and gramese and others and this is actually closest to what occurs in stokes is paper actually you know where he has his discon tenuities are the things that we now call the singular or the anti-stokes directions even though he did not have borrell summation uh so the or another other way to describe the stokes data is just to look at the the flags of exponential growth rates of solutions so i have a connection on a disk and you go towards the pole at the infinite point and you just look at the flags or the filtrations which occur by looking at the the exponential growth rates of solutions on the sectors and these switch around as you pass these stokes directions so over here we have the anti-stokes directions but the the flag switched around on the the the stokes directions this was first looked at i think at least as far as i know for rank two by sebuya where in rank two your flag just amounts to one particular way of solutions which were the sub-dominant solutions that he looked at and it was extended to arbitrary gln connections in the the lessons of deline in 1978 and i think that this was proved by malgrange um so in particular in this example you have a slightly different description of the flash your surface in terms of the six tuples of points which occur by taking these rays so the formal monogamy the constant which occurs turns out to be the multiracial of the six points taking the rays of the sub-dominant solutions on on the sectors okay so these are equivalent pictures it's possible to pass between the two pictures algebraically but it's easier to classify what the stokes local systems are it is a special type of local um it is a special type of local system on this disk having extra punches and so it's it's classified by its monogamy um so here's a picture of how to pass between the two pictures you basically take the associated graded and um all of the maps here isomorphism perhaps i'll skip over it this picture is out of paper um but yeah that explains what to do um okay so the general picture looks like this um the framework for most of the complex symplectic things we do so over here you have this sort of standard sort of finite dimensional symplectic picture and you have the Hamiltonian picture having actions of the groups here and lots of these additive spaces occur as simplex equations of products of co-adjoints orbits using the standard you know symplectic equation or you know the complex symplectic quotient at the top is this infinite dimensional a teabot picture um where you have this nice fact that the curvature is the moment map for the action of the gauge group on the space of all smooth connections on a fixed underlying top project bundle and so when you look at the symplectic quotient here you end up with the modulator of flat connections and that that explains nicely why the character spaces are symplectic um so this has a an algebraic approach where you just go up a bit you frame just at one point on the boundary and you get to this quasi-hamiltonian picture um first looks at by alexa of malkin and mine brenken as a sort of a reinterpretation of work just before by lisa jeffrey and hubschmann and perhaps others as well so the point is by lifting up a bit it's possible to work in the world of smooth affine varieties um you don't need to go up to this analytic perspective at least to get the complex symplectic structure to get the metric it seems you still need to go back up to the top um and then the multiplicative symplectic quotient is the way to forget the framing and the moment map turns out to be the monogamy around the boundary in most of the examples and so you end up with a nice algebraic construction of the spaces at the bottom um so for instance you might look at spaces of tame connections like this and interpret this symplectic quotient as a a space of connections having first-order poles and then you have the riemann-hillbert map the map which is a transcendental map between these two algebraic symplectic spaces um like an analog of yeah the exponential um this this matches up the complex symplectic structures but both spaces actually do have hypercalo metrics as well the metric here is more algebraic and is different to the metric which occurs from this this hitching type picture yes yeah i'm always looking at things like gln or arbitrary complex reductive group uh and then we can um extend the picture so in my thesis i looked at the picture at the top and extended the a tier bot construction and then later on after this work of woodhouse i extended the quasi-hamiltonian picture as well to get both an analytic and an algebraic construction of this symplectic structures on these wild character varieties at the bottom and this was completed in this work with daisuki in 2015 um so we have a sort of enrichment of the spaces that people have looked at before um okay so now i want to run through quite quickly the sort of the definition of these spaces what's happening i guess we have the picture um so i'll be quite quick the usual picture is just that you take a surface you look at the space of representations of the fundamental group so e g g g g g m c um and then you have the ream and hillbott correspondence to the connections um and then you can look at the tamed picture if you have some marked points as well um you can look at the space of representations of the punctured curve um then you have like a naive space here or a set of isomorphism classes to do a modular space properly you need to look at extensions across the punches and i'll mention something about that in a moment but the basic picture is that you have these two different algebraic structures here um if you just look at all of the connections not the ones was important to have these punctures or not you can have no puncture you can but that would be the picture that was looked at before that's usual hitch in a tear bot picture um uh so if you just delete the condition you'll get some infinite dimensional Poisson scheme here and so we that has a Poisson structure you want to look at the symplectic leaves which turn out to be finite the dimensional and so we need to fix extra structure at the poles which gives us the finite dimensional spaces so we need to fix the irregular type now part of the wisdom that comes from isom monogamy is that the extra data that you fix at the poles behaves exactly like the modular of the curve um so this is not at all clear to start off with but there's particular examples with the Laplace transformation where the irregular type does match up with the positions of the the poles and so this this you genuinely is an extension of the notion of the underlying beam and surface so at the end of the day we'll define a wild surface or an irregular curve to be in a curve with the marked points plus an irregular type and we can braid the irregular types just like you might braid the points as well but we could take a different structure group and so immediately we start to get the g-brae groups occur and cabled versions of them okay so there are notions of the twisted irregular types where you fix a carton tub algebra these can always be straightened out by taking a root of the coordinate so let's start out with a straightened carton tub algebra and we use the principal part of this to fix the irregular type of the connections we put in the condition that the connection in some trivialization is isomorphic to dq the irregular part plus something having poles of at most one something logarithmic and so that fixes you the structure at the pole at least the irregular part and then we get nice spaces that have symplectic structures and so the picture before was that you had a surface that gave you a space now the data you want to fix is a wild surface which is this triple and at the end of the day you want to look at the deformations of that which gives the wild mapping class group but i'll generally just fix the curve today so that's the picture as i mentioned before that to do this properly you really want to look at extensions across the punches and you also look at the parabolic version of that and then in general for arbitrary reductive groups you want to look at the parahoric extension and then you need to put in a condition about how the connection is compatible with the parahoric extension so there's a paper which explains carefully what a logahoric connection is but the statement is this you want to be able to pass to a cover such should it becomes very good so basically you fix a weight and this gives a filtration of your loop algebra and you want to look at connections which go one step beyond the positive part of that and so that's the notion of a thesis of a thesis logahoric connection um so you want connect connections which have any regular parts plus something which is thesis logahoric and that's the natural extension of what we did before here and then you want to look at the ones which are twisted and so you only put in the condition that after a finite cyclic cover it becomes very good in this in this sense um so it seems to me that these are the connections for which we have a Riemann-Hillbert-Berkel correspondence and we have a correspondence with Higgs Bundels as well and it seems to me that the other ones we don't know what to do um there's also the question of whether or not these are the good ones in the sense that is the corresponding hitching system and integrable system in the definition which i had is he okay is should i stop the talk um so there is the question about whether or not good here also matches up with the fact that the hitching systems are integrable or not i've not looked at it but there are some work of baraglia and camgarpour that they have a weaker conjecture and that is compatible with this general statement here but that would be good to know um okay so let's go back to our example to get the continuance um so you have a disc just with one marked point and an irregular type having a polar border k um you just have one term in the irregular type this a here so you have a disc and the irregular type determines various data like the centralised with the maximal torus the singular the directions um so solutions of these connections involve the term e to the q so this might look like this and the terms e to the q1 and q2 they switch they switch order of growth in different directions and so you can draw this stokes diagram of where it switches over so these will be the stokes directions here and the singular directions of the ones which have the biggest difference um and then we put these extra punches in in the singular directions and we get this halo and these extra punches um so h is now the halo if we're worried about h um and you have a stokes group um so here you just get these unipotent groups u plus and u minus and you can define a stokes local system to be a g locals system on this curve with the extra punches and you have this flat reduction where grading in the halo and then you put in the conditions at the monotony around the extra punches has to be in the corresponding stokes group if i start in the halo it's it's graded and so that condition makes sense um so this defines a category which is equivalent to the category of connections with the fixed irregular types so it's a it's a clean way to describe what the stokes data is but as i explained it's just the the data that the multi summation picture gives you um so if you start out with the filtration picture you want to classify those you basically pass over to to this picture and then say hey it's classified by the stokes matrices which are the monogamy around the extra punches but here's a a base point independent description okay so if we did want to classify these we choose base points and we look at the fundamental group points we look at paths between these and loops as well and so we end up with this framed space of stokes local systems the stokes representations of this group point in the group g and so it's just the representations such that the monogamy around the halo is in this graded group H and the monogamy around the extra punches is in the stokes groups so i proved in 2002 that these have quasi-homotonian structures as well this is probably the first large class of examples which don't appear from modular spaces of flat connections on curves directly this was actually proved for arbitrary complex reductive groups in this paper here the space is quite explicit i can choose certain paths and i just get that the space is a product of g cross u plus cross u minus to the k cross h so we can play with this to get to the continuance let's define the space to be a and call this the fission space because it's breaking the group from g to h and so if we call elements of it c s and h um so here i had a disc now i want to work in the case of a sphere just having the one mark point so i want to glue a disc on the back of the sphere um so this is what the quasi-homotonian reduction by g does here um so you end up with this space which we call b um and explicitly we just write down what the moment map is and compute it that we we've proved that this space is now a quasi-homotonian h space because we're reduced by g um so that's the statement we've got to at the moment what time am i supposed to stop okay very good um so this leads easy to the continuance by computing what if you use the Gauss factorization you get to this and so that means that the one one matrix entry is not zero so if you look at the case of k equals two you get the one plus ab that vandenberg looked at that cares in regular holonomic d modules um and you can prove actually that the quasi-homotonian structure matches up as well with the same moment and that's the h so h is a function of the matrix entries which are left over is indeed this one plus ab that we had before and then we look at these arbitrary sort of deeper products and we do do indeed get the Euler continuance and so we see that those are moment maps as well and we can define our open part the answer to the question what is rep star to be the space of representations of the graph where the Euler continuance is not zero as an extension of this fact that one plus ab is not not zero um so of course i got to this by trying to work out you know the quasi-homotonian space is in stokes data for arbitrary reductive groups if you look at particular examples you get to sort of explicit statements like this so this is a very simple example and one can ask you what is the sort of more general picture that occurs and in particular how much of that picture is related to to quivers or graphs so first of all we could just so here we had dimensions one and one first of all we could take an arbitrary vector space at each node so h then becomes non-abelian um that's been done and then we can replace the irregular type by this very simple one that we had before by an arbitrary gln irregular type and the class of graphs that replace the graph which are here um uh easy to describe explicit the i call them fission graphs and so there's lots of examples which occur for instance if i have a queue like this you sort of work down and look at how the eigenspaces break up so first of all i would look at the eigenspace of the first term i might have two here and then it will break up into other pieces i look at the joint eigenspaces of consecutive terms so you get this fission tree um and then there's a way to attach a graph to a fission tree um so you get a quiver attached to an irregular type and this quiver plays the role of the quiver that we had before um that it defines in a space of invertible representations and a multiplicative quiver propriety description of the wild character space okay for instance if you look at try to look for the ones of dimension two you will get the affine d4 and you will get the square this occurred at orco motos will work for um panel of a five uh the triangle occurs for panel a four and the affine a one for panel of a two as we discussed before um but you don't get the pentagon because it isn't a complete k-partite graph um and it does not occur at all so that's the type of explanation of why the list of bases stops at to square um and you don't get affine a four and then we need to look at the reduction so we need to fix the graded graded monogamy in terms of graphs that corresponds to gluing on extra legs and so you end up with a class of graphs so called supernova graphs which um like the fission graphs put with extra edges and then we can look at the reductions and look at the corresponding multiplicative quiver spaces and indeed we do get the panel of a two space as we expected um and you can go back and look at the additive picture and prove that the the Nakajima quiver space of that graph is indeed um simplexically isomorphic to the open part that occurs as a modular space of connections on the trivial bundle um like that and so you have a picture that the Riemann-Hilbert-Berkov map takes the additive the standard quiver variety picture it's a holomorphic symplectic map to the wild character space that has a description as a multiplicative quiver variety like this with this new definition of multiplicative quivers um you can look at the spaces of complex dimension two these are sort of a non-compact analog of the k3 surfaces um which I now call h3 surfaces after Higgs, Hitchin and Hodge I mean we have these three h's there and so I think it's perhaps the correct name um for these non-compact hypercaler but but but um hypercaler manifolds um so most of these occur in the panel of A's Dory um like this for panel of A6 down to panel of A1 and 3 and things um if you look at what the open parts are so these become simpler hypercaler for manifolds um the ones on the left to differ your morphic to the ALE spaces with this um the corresponding symbol and the ones at the top right of the ALF spaces for instance the T-Hitchin space here um at the bottom right the open part is this C2 with the flat metric um okay so in the remaining time I'll just go over what we did and and sort of see how to understand the fan part of it slightly differently um so we had this story where we had an edge and then we looked at a double edge and we said we could put the those together and the moment map is the products of what we did before but there's a better thing we can do that involves this degree four continuance such that we know how to put hypercaler metrics on the quotient um and then we play with these a bit they have factorizations um you can either factorize on the left or the right like this and if you think about what that means it's actually an algebraic inclusion of the spaces either the left inclusion or the right inclusion and it's possible to check directly that the quasi Hamiltonian structures match up um and then you try to count all of the inclusions which to carry if you factorize all of the way down um so for this we get 14 and in general it's possible to prove that you get the Catalan number so this does starts look sort of familiar um you can deduce this because the operations L and R form something so each of which is associative and then together they commute in one way but not the other um this is the free duplication algebra and it's known that the pieces of that have dimension equal to the Catalan numbers and so that explains that but also the Catalan numbers occur in terms of triangulations of an n plus two gone um of course here i'm now describing it in terms of graphs but i started out with gluing surfaces and so we can go back to the picture to start that you know our edge is a surface and our doubled edge is a surface with these six singular directions um the picture in the middle is the fusion of two edges so i take the two edges and i fuse the fusion corresponds as usual to gluing into two of the three holes the three holes here um so the inclusions say that this space is an open part of that in two different ways and for instance the left product might look like this where that gets glued like that and that describes an open part of this but we want to get to triangulation so we need to have a space attached to a triangle um at the moment we've got spaces attached to squares and hex hexagons and so we then open Stokes's paper so i can attest to the fact it's possible to think about Stokes data for sort of 10 or 15 years without actually reading Stokes's paper but if you do you'll see that there is this picture in it um and this is the Stokes diagram of the airy function which has a q which is involves w to the three over two where w is one over z which is the coordinate i had before so the pole is at the infinite point and then you can repeat what we did before and define twisted Stokes local systems and prove that these have twisted quasi-hungle tonic structures and you indeed have a space attached to the triangle and then you can fuse those together just like we did before um this completes this project of understanding the symplectic nature of the wild fundamental group as an analog of goldman's work on the symplectic nature of the fundamental group of a surface so we now have these irregular poles and now we get the odd continuance as well um in particular b3 occurs for the case for the pentagon that we we had in the picture at the start of the talk um so this explains that we are actually triangulating a polygon by fusing these pictures these pieces together um of course if you take the dimensions to be equal to one this is familiar from the complex w k b picture if you open for us his paper there are pictures like this this is actually the same type of example which occurs for panel of a two so you have these hexagons and these really are pictures of triangulations of a hexagon so let's look at the picture at the top um so you have these stokes lines and the cuts if you delete the cuts you get this and then um you replace these by triangles like that and so his picture does mean that you have a triangulation of the hexagon and if you look in quantum mechanics textbooks then they really do approximate what's happening at the turning point so the zero of the quadratic differential which occurs there is called the turning point they they do approximate in the asymptotics what's happening there by the airy picture and so that does match up with gluing to get together these airy triangles to get a triangulated hexagon so I will stop there thank you this worst picture you know there was paper by gaiota more in nature sure yeah all right we generalize it highly and they also give some they're saying something to try triangles so yeah I'm sure it must be great if I don't know if you know I'd like to draw a precise statement out of their paper they have all of these pictures and there's the stuff that was known before by takei and others in Japan it's very hard to work out I think if I go back to this 1983 paper it's clear that that's before most of what was looked up before and the picture is here and so we we have this this fusion picture which says how to glue pieces of the surface and you know it's not yeah any other question is there a understanding how the time spaces degenerate the wild space as well so when singularities merge together my not really goes to infinity how space space continues to degenerate there's a paper of Garnier from 1920 or something that started to look at this my perspective is a bit that we just want to understand what the spaces are and the structures on them first and then one can ask questions like that that afters I think maybe one can now start having a look at it I think it's hard to make precise statements I mean one would like to get the wild metric from the tame ones or vice versa I think there are stories that are analytically trying to glue the pieces together to approximate the hitching metric by the wild pieces and so perhaps one wants to do that backwards that it's actually simpler to work with the wild spaces and with the tames spaces but there's lots of different things that people want to do I mean it's a quite rich story my question is kind of the same sort of for more etheric reasons well now I'd rather sing something but the these things should have some nice compactification almost no like normal crossing compactification essentially unique do you know any single one of them that has this nice like the one thing you had was this cubic surface right you know x y z dimension two then yeah you get I wonder if you know a single example where you have the compactification and you know it kind of in your world as some kind of you know some kind of space of things with connection the odd spaces in rank two I mean you basically get this list of the odd continuance and so it's all quite explicit there so I think there were some examples that Simpson was having a look at I was thinking that these might be simpler to look at first rather he was looking at sort of five points on p1 really I've not looked at compactifications I mean the metric and the simplex structure does not extend okay and then real quick back to the very beginning when you had said the things you wanted to classify you had integral system that looked like a reasonable thing and then the second thing was this lax condition and that looked to just you know very quickly look to me like said my thing is in some way a Hitchin system right but what about some kind of more is it possible to give a sort of more abstract criterion something that would just sort of be in the purely in the integral system world or purely in symplectic geometry or like which symplectic manifolds are we looking at or yeah I mean people in the integral systems have looked at this I mean are their integral bullet point you know does every integral bullet equation have a lax pair there are ones that don't have these like multiplicative lax pairs and this gets into you know sort of a multiplicative Hitchin systems it could be that there are integral systems which don't fit into either of those that there are the ones that are more additive that have these exponentials that don't have a billion fibers but C stars yeah I think it's difficult I would first of all I'd like to get the simplest question and then okay thank you any other question okay if not let's thank Philip again