 It's a great pleasure to be speaking here So basically I want to talk about the simple case of how to define the stokes data of a linear connection on a curve So for most people here, this is probably a quite trivial thing to do But the problem is that lots of people think about the same objects in various different ways And I want to talk about how to how to pass between the different perspectives so Part of the motivation is because these you know spaces of stokes data give the Simplest description of modular spaces of connections on curves and modular spaces of connection on curves are important because They form the the fibers which occur in the isominerogamy and in particular in the panel of a equation So we want to understand these spaces extremely sort of properly In order to better understand questions about Isominerogamy and panel of a equations So in particular all of the basic examples occur in the theory of panel of a equations So the basic issue is that so then there was a paper of stokes in 1857 Which was the same year as the paper of Riemann where he looks at the the monogamy of the Gauss hyper geometric function And that perspective evolved and we now have the fundamental group and we can talk about the the space of Representations of the fundamental group and everyone knows what you mean Whereas if we talk about the space of stokes to data Various different people have different perspectives and most people don't know at all So one would like to try to upgrade the stokes perspective to be equally well known to The spaces of representations of fundamental groups that people know and love So let me describe this quite mathematically so we want to fix a curve So Sigma is a a smooth compact complex algebraic curve And I'll fix a finite collection of points a in this curve and have a look at the punctured curve Sigma o Which is Sigma minus a So we want to look at the category of connections on Vexa bundles on this punctured Curves says so we want to look at the category con So of algebraic connections on this open curve, and so we're looking at pairs e Nabla Where Nabla is a connection on E says so it's an offering operator that goes from e to e tensor I mean go one So from the sections of e to e tensor with the one forms and it has to obey Live-nits, which says that Nabla applied to a function f multiplied by a local section s is df s plus f Nabla of s and e Is an algebraic vector bundle Algebraic vector bundle Over our open curve So we want to look at the category for we have all of those and the the magical fact is that this category has a purely topological Description in fact has various different descriptions. I want to talk about three of them and compare them in this talk So the first piece of topological data to take is the local system of solutions and so we can Define a Functor to the category of local systems so the the Locally constant sheaves of finite dimensional complex vector to spaces and if we choose a base point This is equivalent to the category of representations of the fundamental group of the curve So this is quite simple and purely top logical so if I have an open subset you the sections of the local system on this open set you and sway have a Open set you that I just look at the the analytic solutions of this connection on this open sets which the kernel of The operator Nabla applied to the analytic vector bundle on this set you So Koshie tells us that if I have a small small enough disc I have a finite Dimensional complex vector space of dimension equal to the rank of the vector bundle E and basically the definition of local System is the object that these open these vector spaces V form Okay, so There's an equivalence between the subcategory con rs the tame or the regular singular Connections This is a subcategory and this is an equivalence So tame Is the focusing conditions so solutions have at most pulled denominal Growth or it's possible to choose an algebraic Extension of the bundle across the poles such that the connection has first-order poles So the basic question is you know, so what to put here such that I have an equivalence Between the local systems and the stokes local systems here So the basic point is that there's various different descriptions and we want to describe three So one particular did Description will be called the stokes so cool systems and others will be the stokes filtered local systems and the Stokes graded Systems so I have an inclusion here From the local systems into the stokes local systems. So There's a paper This is on the archive in March this year And the basic examples are in a paper I think on the archive in 15 Maybe 1501 This is basically explaining how to read a book of Sibuya From 1975 And it should probably talk more about the paper of Birkhoff, which I think I didn't mention there, but Lots of this is already in Birkhoff's paper in 1913 Okay, so the motivation is that we can then look at modular spaces of these So the basic picture if I fix the the rank at the top So if I look at the subgroupoid of local systems of Rank N I can look at the set of isomorphism classes there It's if I you map to the from the objects of this to the set of isomorphism classes then this becomes the the set of orbits of a reductive algebraic group on an affine Affine variety and so it ends up being HOM Pi 1 of the punctured curve Sigma O with respect to a base point B into glnc modulo glnc So this is gl NC and so I have a an affine variety Affine variety quotiented by a reductive group and so this is the So if I look at the stack quotient, you'll get the character stack And if you look at the affine GIT quotient, you get the character variety So this is often called the Betty space and noted MB Be for Betty Okay, so because we have the nice descriptions of the space at the bottom it's possible to do the same there The issue is that you need to fix more than just the rank in order to get nice spaces of finite dimensions and so we look at the Subgroupoid with fixed irregular class so password aim is to define what we mean by the irregular class So in particular that will fix the rank and then we'll look at the set of isomorphism Classes of these And it's possible to describe that again as the orbits of a reductive algebraic group on a affine variety all provided as home s Pi G modulo H and I'd like to explain. What is this? First of all perhaps it's best to explain this one slightly differently. It's it's also isomorphic to the homomorphisms from the groupoid Pi into G modulo G to the M so here I fixed Pi to be the fundamental groupoid of our curve sigma O and I want to choose a base point near each of the punches Well, I've got this set a which is like a one up to a M So I have a curve with some punches For instance, just the one a and I'm choosing a base point be near To there and so be visa is a collection of base points be one to be M and that's a good thing to do from the symplectic or the Poisson perspective in particular the action of G here basically corresponds to fixing a Framing at the base points and so The statement at the end of days that this is a quasi-Hamiltonian G to the M space and the moment map Is the map that takes the monogamy around these punches? So this is like new for the base point big B equals monogamy the local monogamy So that's a map from the space at the top here mu to G to DM So in particular this perspective implies that This space MB has an algebraic Poisson structure So I'll talk more about the background behind it on this picture is user Alex say of Malcolm and mine Benkin So once we phrase it in this perspective, it's closer to this And the same statement also holds in the wild picture The bottom and so this will be the wild character variety And it depends on the choice of this triple Sigma the points and the class Theta and it's natural to call that a wild Roman surface So it's the theory of group valued moment maps so you have a map which takes the values in the group And then you can do symplectic quotients and fusion there Maybe I'll give some Examples in a moment So in particular, it's the natural geometric structures which occur on on spaces of framed representations of fundamental groups so the space at the top here The space at the top here is a natural example of a quasi Hamiltonian G to the M space where M is the number of punches Okay Right, so let me draw a picture which did describes the three different perspectives that we want to get to in order to define this Okay, so I don't just want to list all of the definitions straight away, but let me do this in steps So I have a curve with a marked points I want to look at the the circle of directions and so this is the Circle which is the you know the boundary of the real oriented blood Blow up at the point a and so I take the tangent space to sigma at a and I Punch it and I quotient by r plus So if I have a point on the curve I get this circle So let's suppose I look at part of the circle So the first perspective is this perspective of these stokes filtered locals systems So this goes back to Deline I think it was conjectured by Deline and proves by Malgrange in the early 80s So there the picture is that you choose. Well, there's there's the class determines distinguished directions and so there's preferred points here I Will call these the stokes directions s So if you have a look at the picture in stokes is paper These are the directions on which the dominance ordering switches over So these are the directions which are important in the stokes filtration perspective and so you look at the the Filtrations given by the exponential growth rates of solutions and so on each of the sectors So I have the Dockall system V in each of the sectors between these stokes directions. There's a well-defined filtration Which I can call something like this F5 and It's possible to write down axioms of these if you Work out the topological meaning of the local analytic As in local asymptotic existence theorem That gives you axioms for how these filtrations are allowed to jump across the stokes Directions and Deline wrote Dynaxioms and Malgrange Proved that the category of these stokes filter the local systems is equivalent to the algebraic category of connections So it's an extension of the Riemann-Hilbert correspondence for these regular singular Connections which just involve taking the monogamy Okay, so another perspective is more sort of I don't know the resurgence or the Multisimation perspective. I'll draw that over here So the intrinsic statement is that it's possible to take the the associated Graded of these filtrations and this gives you a well-defined continuous Graded local system in a small punctured disc around the mark point and so on this picture that corresponds to a Halo or small Small annulus On which we naturally have a graded local system So at the end of the day the regular the class involves a particular cover a finite cover These are the exponents. I hopefully I'll Define that properly In a minute, but the fact is we naturally as an I graded Local system Which is just the associated graded of the stud stokes filtration there Now the stokes perspective which has evolved into this Multisimation perspective Says that there are a different distinguished directions So I'll draw these like this And I'll call these a the singular directions So this engine general is different to S. So in stokes's picture for airy these are the directions which are marked a b c here and One way to think about what stokes did Is that there's a natural way to glue this graded local system to the local system of solutions V that we had before So it's as if on this halo. We actually have two naturally defined local systems the Associated graded of the stokes Filtration and the restriction of V and stokes says that there's a preferred naturally defined way to glue This I graded local system Which I'll call V zero to V At least away from these singular directions so what we do we boldly add these extra punches the tangential Punches and restrict V outside the halo and then we know that there's a way to glue V to V Zero across each of the components between these ten Gentile punches So V zero glues To V. This is what Stokes says and this gives you a local system on this curve with these extra punches Taken out So this is what I'll call the stokes local system Blackboard bold V And it's got by gluing V zero and V together and then we can write Dianaxioms or we can read papers of Michelle Michelle and Grammice and others and see that there's preferred Stokes groups for each singled direction D and the Monogamy of this Stokes local system around these extra punches has to be restricted to be in these groups And we also have to add the condition which corresponds to having a graded local system in this halo And that gives you axioms for the local systems which occur, but at the end of the day It's just related to the representations of the fundamental group of this curve with these extra Drapunches and so we have sigma tilde is sigma o minus These extra punches e d for d a singular direction And then we can define pi to be the fundamental group weight of this curve with these extra punches Sigma tilde with some base points be one base points in each halo And so we get that the category of Stokes local systems embeds as the Stokes representations inside the space of all of the Representations of this groupoid pi into G and because we've chosen base points in the halo the local system is naturally Graded and so the thing that acts isn't all of the group G at each of the base points It's more this you know the the graded automorphisms of the Fiber so we get this group H Which is like the products of the graded automorphisms at each of the base points H 1 up to H M which is a subgroup of gene to the M and that naturally acts and the set of isomorphism classes of of Stokes local systems is the the quotient or these set of orbits of this group H on the space of Stokes representations here, so it's equally explicit to the Tame definition that we had before it's just that you need to add in these wild monogamies or Stokes or to Stokes or to morphisms around the boundary around these tangential punches at the boundary So that's just a way to phrase, you know what lots of people do in a nice intrinsic way Now the problem is that about half of the world works in this perspective and half of the world works in that perspective And they're kind of working in parallel. So what we would like to do is describe a picture in the middle Which kind of explains the bridge between the two perspectives And this will be the so here we've got the Stokes local systems here. We've got the Stokes filtered local systems. This is the Stokes graded local systems Or if you prefer the Stokes decompositions So these are very close to the Stokes local systems that we have before so I'll draw that there's just an epsilon difference It's just thinking about what happens there slightly differently So we have the same special directions. We again use the singular directions that we had before I'll try to draw those in the same place So these are again the single Directions a and now we So in Stokes's picture It's the ones where there's a maximal difference with if I'll have a e to the q1 there and e to the q2 here And you look at the The points where e to the q1 minus q2 has maximal decay the points of maximal decay over Poms all the apples if you like Right so I should point out that this is explained very carefully in the paper which took ages to write I'm just trying to present the picture to encourage people to look at the paper. It's it's much. Um, it's much more carefully explained there But the basic point is that these Filtrations are canonically split and we can describe it instead in terms of gradings And so here we have like gamma 1 gamma 2 gamma 3 gamma 4 gamma 5 So these are just the grading so if I start on this perspective I have a graded local system and I have an actual way to glue that to the local system V And so that in particular gives a grading on this sector of the local system V And that's the map from here to here just forgets the grading, you know, well where it's supposed to have this minodrome Yes, everything here is completely intrinsically defined at the start of the paper Sort of give a list of all of the things which are canonically attached to a connection And then these three different perspectives result from axiomatizing the various pieces of the data And so that's the the reason we get these different perspectives I'll try to get to it. Yeah So let me just so the map from here to here is easy Ah So So I have an isomorphism of local systems between V's 0 and V at each of the points between any consecutive pair of singular directions and Then you have to put in the conditions that correspond to the fact that the monogamy around here Has to be in the stokes group actually it's best to choose the base point there And then there's a well well well defined stokes group. It's if I have a projection of some roots Which give light to you the stokes arrows and then the stokes The stokes group is the group whose gli algebra is generated by these stokes arrows And these are you know the arrows which map to these points of maximal decay The stokes groups are carefully saying in this paper of Michelle Loddy Rishol from 1995 but it's also in the paper of Martin and Ramisse about the the wild fundamental group and It extends what lots of people do explicitly in the world of panel of eight equations as well so To relate the two pictures here. I just need to explain how to take the wild monogamy of a pair of consecutive Gradings and then the main work is how to pass between the stokes filtrations and the stokes gradings Let me just define the wild monogamy Since it's sort of an elementary fact, I'll just okay, so perhaps I'll just state the theorem at the end of the day Is that once you have these axioms Then there's a unique stokes graded local system For each stokes filtered local Systems so for each So it's stokes filtered local system of class. So let me call this V with the filtrations F of class theta So there exists a unique stokes grading on this local system which splits the filtrations at each points where both are defined There exists a unique Stokes graded local system so with the same V Grading so of class theta Such that you know gamma of D splits Fd for all D, which is not stokes or singular In fact, I think here's a kind of two-perseverance of both clockwise and to clockwise No, that's the Burkov perspective. That's different here. It's canonically to define. It's in the middle Yeah, I don't want to talk about the Burkov perspective, but but basically in the case of one Devil, yeah, so I haven't actually explained he did this and say Perhaps this perspective we should put the names of like stokes. I don't know Burkov And then perhaps jerk jerk at although his conventions are slightly different and then I don't know Rami's Lorde Richaud So that's the perspective of traditionally used in the past because it's much more explicit that we have these Presentations of course the way it's defined. I mean it's defined by taking the multisimation of a formalized morphism between the local system and the graded The graded local system Okay, so that's the statement one can prove at the end of the day It's not really clear what it means at the moment. I haven't defined everything what I do want to define So it's the wild monogamy of a pair of Compatible gradings so this is in order to define the map that passes from stokes graded the local systems to stokes local systems Grading So this is a very elementary fact when it's key to see sort of how the Wild monogamy or the stokes or to morphism that appears So we have a set I And we have V is a complex vector space I have a pair of I Grading so gamma one and gamma two I gradings of V So for instance, I would have V is the sum of sub spaces gamma one of I where it's indexed by I So in particular it's possible to have sub spaces of dimension zero And so I could be much bigger than the dimension of V And I would also have that for sub spaces Gamma two of I as well So if I have an order a total order Order on I Then I can talk about the associated filtration So Associated filtration Using that particular order so I get sort of F one Is the filtration associated to the grading gamma one with this particular order? So for instance, you would have like F one of I Is equal to the sum of smaller indices J Of these pieces gamma one of J and you would get F two as well Yes, but I'm saying that given an order I can define this this is a simple fact so We then say that gamma one and gamma two are compatible if there exists an order Such that F one is equal to F two So it's possible to have gradings which are distinct such that for a particular order their filtrations are the same So in our picture all of the consecutive gradings are compatible And then it's possible to define these preferred Automorphisms that take one grading into the other which just depend on the gradings So, you know, you might typically have a pair of bases and you could take the gradings that come From those and there's in general lots of automorphisms that take one grading into The other the point is if they're compatible that there's a preferred one Which in particular is unipotent, but all of the automorphisms That's not enough to Determine it and so what we do We take this graded vector space V gamma of one And that's isomorphic as a graded vector space to the Associated graded of the first filtration F one because this is the filtration determined by that for a particular order And by definition that space be equal to the graded of V of F two and that's isomorphic to V of gamma of two So we end up with this stokes automorphism or wild minodramy Which is a graded automorphism from V gamma one to V gamma two. So g gamma one gamma two So this in particular is in GLV. So there's a preferred Automorphism of the vector space that takes one grading in To the other so so that's the basic trick to pass from that Picture in the middle to the picture on the right and that's how to define the graded local system around the boundary if you just given the the Stokes gradings on the sexes Okay, so it's then possible to prove that these two perspectives are equivalent Right, so maybe I should define what is in the regular class to try to help Define what we've done Sorry in the theorem with uniqueness. Is there a condition on s GLS or is it any? So I'll have some axioms, and there'll be a unique one that obeys those axioms Okay, there's axioms for the s GLS right so it's the axioms that it has to be graded around the boundary It has to be in these stokes groups Okay, so let's get back to this picture We have a a Curve with a marked point a and I can define this circle. So the basic fact is that there exists a canonically defined I can't spell canonical Covering space This is curly eye over this circle of directions. This is the exponential local system Or the local system of exponents of irregular connections So The local sections so it's a huge infinite thing The local sections are functions Of the form so it has an intrinsic perspective and here is the perspective that depends on a Coordinate the local sections Functions of the form Q Is the sum? so a i x to the minus i over r so Summing for i from 1 to k for a particular k and r is an integer which is allowed to vary for different q So each of these Functions defined on a certain sector so here Maybe for later use I want to call this z so z is a coordinate that is equal to zero at the marked point So a such a function q would have a gal what all bits in particular it determines a circle Which is a finite cover So if I take r to be minimal it will be a finite cover of degree r So this thing i is a huge infinite union of all of these circles and There's a basic fact theorem Which is due to huck a hurrah Turritin and Levelle's and this is the version of it that the phrasing of it due to delinia Is that the formal connections on a vector bundles over the puncture disc are equivalent to the top logical objects of just i graded local systems on this circle at the bottom so formal connections So connections on the formal puncture disc Connections on the formal puncture disc This is equivalent to the category of i graded local systems systems of vector spaces So this is a rephrasing of this classical Results, but it helps to explain what's happening in the picture on the right that any connection has a natural form Formalization and in particular we automatically get this graded Graded local system on the boundary and then the stokes business is about how to glue that graded local system To the local system of solutions in the interior of the curve And so the true picture of what's happening is something like this where you go close enough to the boundary You know the boundary does does break up into these graded pieces And stokes explains us how to glue that back into the local system of solutions in the Interior of the curve we're breaking the structure group from some group G for instance GL to to for instance a maximum Torus or a block diagonal subgroup on The boundary this picture also ought to have the tangential Punches, but it was hard to draw and so They're not there, but that is the basic picture of how you know connections behave at the boundary Okay, so let me talk slightly more about this quasi-Hamiltonian perspective It's basically always a fifth think about the the explicit presentations of the spaces which occur Which zero So if you have a point on the curve you get an exponential local system And then that is what you use to grade the local systems in order to Yeah, I Blame in the paper that a local system is a covering and then you have special cases of local systems of vector Spaces where the clutching maps are linear So let me talk briefly about the present tations And how this leads into this quasi-Hamiltonian approach So basically everyone knows what is the fundamental group of a punctured curve So I can just write down that you get presentations with relations something like the product of the group commutators a I be I And then the product of MI so from one to G and from one to N is equal to one So if you look in Poincare's paper He does something slightly different where he chooses a base point near to each of the punches like we did there And this is then written differently. So this part is then changed to be like C inverse I H I see I like this So we have a connection matrix from a fixed base point to the base point near the puncher the the the the local The local monogamy and then the the path back So this is basically the way it was presented by Poincare and it also fits into the extended modularized space Perspective that was looked at by Lisa Jeffrey at least in the case of compact groups So immediately this starts to look like a group-valued Simplexity reduction So the basic fact you need to prove These are due to Alex say of Malcolm mine Mine Wrenken is that it's possible to define a space So the internally fused double is G cross G and this has a moment map given by the group come Utata, so mu is a map from D to G cross G To G and so a B Maps to a B a inverse B reverse which I denoted the brackets a B above So they have this and they prove that this is a quasi-Hamiltonian G space So I don't want to list all of the axioms, but the basic I did here is that the group G acts on this space D D by diagonal conjugation And it has the analog of a symplectic form such that it's possible to take the products of these such that the Moment map for the product the fusion Product is the product of the moment maps that you had before and it's possible to do a multiplicative Symplexic reduction mu inverse of one modulo G will be Symplexic or if you just do the straight quotient modulo G is will be plus on and then there's the double D which is The double this is again G cross G and this has the moment map mu CH is like in this part here, so H inverse is the monogamy around the boundary and then the other one will be C inverse H C So the space we get here is the fusion of G copies of the The double like that and then the M copies it of this and then you want to look at the multi the multiplicative Symplexic quotient at the value one of the moment Maps so that's isomorphic to your character Righty, um, yeah end of the day So this still has an action of G to the M So which fixes the H is and if I quotient by that that's also I will get the ones on which we had at the start So the basic idea is that then Birkov extended this Birkov 1913 So basically you want to put the stokes data in here. So you want to have a space so the C inverse H C Goes to C inverse H product of stokes data multiplied by C And all of the discussion about these stokes groups is a way to make precise Exactly what is the product of stokes data there? He looks it's a particular generic case where each of the stoke groups is a dimensional one But now we understand what happens in general So what you want to prove There this was in the archive in 2002 Yes, this is by me Is that I can define a space a to be G cross H Cross the product of the stokes groups To do D. This is a quasi Hamiltonian G cross H space So it's a generalization of the space which is here except we have this reduction of structure group to the block diagonal group H in G And we have these stokes groups as well So this is a cosy Hamiltonian G P cross H space with moment map With moment map so as for H inverse and then the product that occurs here see it inverse H the product of the states data C So this is in H cross G. So automatically we get algebraic symplectic and Poisson structures on the spaces which occur So this was only in the generic picture and then there's an evolution I guess from 2002 In ended in this paper with Yamakawa In 2015 which does the most general twisting case So this is the general case So we have this nice algebraic construction of the symplectic and Poisson's Drugs is up on all of these spaces So at the end of the day you have to fuse you know a copy of this for each of the marked points a Okay, so I haven't really talked about the motivation that much perhaps I'll go back and try to do that now So basically I wanted to describe, you know, how one can think intrinsically about the various facts that were proved by other people So the basic project has been how to extend lots of statements about spaces of representation of fundamental group to These wild character spaces, so I at least ought to try to list Some statements that have been proved Even if the basic aim of the talk was not to do that, but I think So this is like the fundamental theorems of character varieties So there's lots of things that have been proved in the case of compact curves So for instance, I think the Poisson and symplectic structures are first looked at by a tier of bots Poisson and the symplectic structures So this goes back to a tier and bot And then in 1982 and there's a more algebraic perspective user Goldman in 1984 or so And the symplectic structures which occur here were upgraded by hitching to hypercaler metrics so we have these Hypercaler metrics and the correspondence between modular space of connections and modular spaces of Higgs bundles This is I guess hitching for the metrics 1987 And to get the correspondence it involves work of Colette, Donaldson and Simpson as well And then I got interested in this because I was interested in what happens if you vary the choice of the curve with the mark Point so we vary sigma and a and in the irregular picture It's natural to vary the irregular class also, which I didn't define no one mentioned this. Okay So the irregular class So if I have an I graded locals System it's only graded by a finite number of the circles and for each of those it picks out a particular, you know Subspace of each of the fibers so in the regular the class is just a choice of finite number of circles So QI plus a multiplicity for each so This is an integer great greater than equal to one So equivalently it's a map Theta from this Covering I I have this huge number of circles From here to n Which is equal to zero and all but a five finite number of components And is constant on it each of the circles and so that's the the basic Dead data basically picking as a finite number of exponential factors So in Stokes's picture here. He would have something like x to the three over two As his his you know, and it's a cover of order two Okay, so it's natural to vary this triple And this is what we call in regular the curve and so this comes from the isominogeomy story And in particular if you once know intrinsically what are the times in the panel of a equations? Those are coordinates on a moduli of irregular the curve So for the panel of a six picture you're looking at the cross ratio of a fourth couple of points Which is the case when the irregular? The class is equal to zero But in the general picture for panel of a one to five your time is a coordinates on perhaps the cover of the the modular space of irregular the curves which occur So lots of people still, you know, just restricts looking at curves with mark points, but it's it's equally Natural to look at these Moduli of triples in particular if you want one to stand the panel of a equations Equations And the irregular isominogeomy story is basically the same as what people call wall crossing the walls occurred in Burkov's paper and it was understood how to cross those In this work of Jimbo meet me Rino Eno in 1980 And then there was a story about you know the counting of BPS states and the TT star equations and Chikotium baffin Dubrov in Embedded that story in the isominogeomy story So basically I got to this picture by just wanting to understand how the braiding works in general And not the particular case is having poles of order two Which occurred in the TT star and the Frobenius story that Dubrov in had looked at Okay, so we're able to extend each of these pictures So this I guess is due to Goldman, too This is what Goldman calls the symplectic nature of the fundamental group Fundamental group and so we wanted it to understand in particular. What is the symplectic nature of the wild fundamental group? Which basically comes down to the fact that you get When you vary this triple you get a local system of Poisson varieties whose fibers are the wild character spaces Which is the exact analog of the statement that Goldman has here so each of these gets extended so there should be a tame Column also, but let me skip over that So this the irregularity a bot picture was in a paper I wrote in 2001 So there's basically a straightening trick at each poles to get this a tier bot integral perspective to work This was upgraded to give complete hypercalo metrics in this paper with big card In Yes, it was published in 2004 So these are expected to be the same metrics which Gaiotto more and nights you have a look at and so the conjecture seems to be that all of these series that Jan and Maxine look at Should to be the clutching maps for the twister spaces of these metrics But I don't think there's a single example in which that's known to be true, but that seems to be the most hopeful, you know completely geometric interpretation of what these Series mean, please correct me if I'm wrong And then yeah, so Right, there's various papers that look at this also so Yeah, so this starts out in this paper for the generic Picture and then goes on to the general picture in some paper in published in 2014 And then you need to look at the twisted picture also, but that works Works the same Maybe I'll just end with a particular example So of the two descriptions of the modernized spaces which occur So if I look at the example of Panel of a to we can describe the space of stokes filtrations and the space of wild monogamy Representations and you get you know two equivalents, but different looking algebraic descriptions of the Same space that first appeared in a paper of flash cut in Newall in 1981 So this is for panel of a to And it's the same type of irregular the classes occurs in this chaotic oscillator Which may be better known To people here So the stokes filtered perspective I would have six directions and the local system would be trivial so I have a Rank two connection with just one pole at the infinite point on the sphere I do the real oriented glow up to get this disk So I have a rank two local system of solutions And you have a the sub-dominant solutions in each of the sectors So you end up with a six tuple of points P1 to P6 So that their Filtrations of this vector space of dimension two so you get six points of the sphere And the stokes conditions across the poles says that PI is distinct from PI plus one So this much of space was looked at by Sibuya in his book in 1975 And I went over it in this archive paper in 1501 and so perhaps that's a better place to have a look than here So you get So the symplectic spaces are got by fixing the formal monogamy around the boundary and that corresponds to fixing the multi ratio of these points so we fix Q which is the multi ratio Of these six points P1 to P6 and that corresponds to fixing the formal monogamy So the statement is that for Generic Q. I think perhaps Q not equal to one So this is a complete Complete hypercaler manifold of complex dimension two So it's a consequence. This is a special case of this result with Bicard and then the other Perspective so we need to look at the singular directions that here will interlace the six Stokes directions so that's something like this And then we have to have these tangential punches And we get a graded local system by a trivial cover. So it's as if I So it's it's got two Q something like x cubed sort of plus minus x cubed Those are the exponential for factors which occur and so I have a grading there and I have these six stokes matrices So you end up with something like h s1 s6 is equal to one where h is in a Taurus and these are in sort of u plus cross u minus cubed so they have to They have to alternate between being upper and lower triangular And this ends up being isomorphic has an explicit description so We get this space be the reduced fission space so it's just So I can fix the h and the s1 and s2 as a function of the the others and So I just end up with these four basically independent Entries, which I'll call a b cd the off diagonal matrix entries which occur there And the fact you want h to be invertible means that the degree for Euler Continuance is not equal to zero and that's mu is a b cd plus a b plus a D plus cd is not equal to zero So this is a quasi-Hamiltonian t t space And then the space are interested in is the multiplicative symplectic reduction by this action of the Taurus acting by conjugations with ziff, you know a will be scaled up and b is scaled Down and so this space MQ Sibuya is isomorphic to be simplex equation tip by the Taurus. I can reduce to c star And you can compute explicitly what it is and you get this surface written down by a flash cut in your also This is flash cut in your They actually had a different lax pair for panel of a tube it's known to be equivalent is x y z plus x plus Y plus z is equal to a constant Let me call that D Um No, let me call it b minus b inverse and then I know that b is minus q squared to match up between this picture of the Continuance to the picture of the multi ratio over yes, but has the this complete completely Explicit description So this then fits into a story of of multiplicative quivers So this basically ends up being the invertible representations of the affine a1 River on c2 So there's a large story of Nakajima quiv fear spaces And the Nakajima spaces occur. There's the additive modulite spaces of connections which occur here Here we're looking at the stokes picture the multiplicative version But basically the maps a b c d can be used as maps in both directions Along each edge So it's natural to think of this as the multiplicative version of The Iguchi Hansen's space Which was the first known non-flat complete hypercaler metric Which is in one complex structure, it's the cotangent bundle of p1 the if you like the affine a one River space. So there's lots of other aspects of this. I've not talked about maybe I should mention at the end that You know, we now have these three different perspectives in which one can think about the stokes Danger in particular it's possible to think about it as stokes gradings and when you look at the non in the air picture You want to define what is a stokes grading there? Will you take the pro torus? Whose character to lattice is the fibers of ice you get this local system of pro Tori and you want to have an action of that So you would expect to get, you know, Tori at the boundary of your spaces And if you look look at the asymptotics of panel of eight Equations, I mean there's papers that have Tori at the boundary and so that should be the the stokes Gradings which occur So in particular that there's a paper of kit I have which has you know an explicit Description of the the the stokes Tori for panel of a one Maybe I should stop there. Thank you I have no question, but the remark because this workshop is about resurgence Where is the resurgence in the picture if you look at the model on the right you have these talks You take the logarithm of one of this music here, and then you will you will be read by the exponential thoughts The conjugation of the external so that is the risk-a-ling of exponential and then you perform the for analysis and the alien deletion Right the alien derivations of the matrix entries of the Lie algebra or the The logarithm of the stokes matrix, I mean I read this in your papers, but I guess it comes with a cowl or something Yeah, it's good to have a better connection to the subject of the conference, but This is your problem. It's had to classify the value of the equations Um, I don't know So it's not really clear precisely what is known. I mean we sort of think there would not be any others, but Has that actually been proved? Yeah, I Don't know I wouldn't like to say Well, you know are there any other complete hypercalous spaces of dimension two of complex Don't mention to it would be a you know, we had this business I had a students and I we looked at you know There's certain character varieties for G2 which have a modular space of complex dimension two And we worked hard and proved that you get the Frick Klein Vot surfaces for you know It's it's isomorphic to the case of panel of a Dicks and so you know there's an interesting lax pair for the symmetric panel of a six equations which has new structure group G2 But there's Lots of others and one would like each of those I mean, maybe it's known how to do it for GLN, but even I'm not completely clear. There's a complete Proof of that, but once you start to look at other groups, also, there's lots of groups out there And I've looked at certain examples, but you would like to have a proof You know sort of what the panel of a equation is there and that it's equivalent to the one Which is here. I think you know one would like to have a cleverer perspective Yeah, there's lots of problems