 Thank you very much. Well, I was delighted and kind of surprised to be invited to speak at this conference, so it's a very great honour. Thank you. Obviously, I've learnt. Most of what I do is based on things I've learnt from Maxime's papers. Paul was... Then I kind of realised I was going to have to try and do some work so that I had something to talk about at the conference. So I did that, but I'm not sure it was effective. Actually, you were likening your progressive maths to assembling to Ikea furniture. I think, for me, a better analogy is one of these crazy people who makes buildings out of matchsticks, spends their entire life making a model of the Taj Mahal or something like that. Anyway, so just for a change, I'm talking about stability conditions. So stability conditions are some slightly strange subject which arose out of physics, out of Mike Douglas' work in string theory, on pi stability. So I think maybe the easiest thing to do, maybe this won't be so popular, is I'm just going to write up the definition, because then we know where we are. And you know we're in France, so everybody knows what a triangulated category is, so let me just start. So D, I should maybe start with the outside one. Whatever. It doesn't matter. So D is going to be some triangulated category, and I'm going to try and remember to call these things stability structures in Maxime's presence, because you don't like the term stability condition. It also seems to me that you've got variation of stability structures or structures. Right, so what's the stability structure on D? It's two pieces of data, it's a pair. Right, so first this is the easy bit to understand. A complex valued, so this is a group homomorphism from the Grotun D group to C, so this is called the central charge. And the other thing is some, so for each real number, a full subcategory, P phi in D, so these are called the semi-stables of those phi, satisfying some axioms. Let me write those up. I hope you remember what's up there. You can still see it. Satisfying four axioms. One is that if I take some non-zero object which is semi-stable of phase phi, then indeed this central charge should lie on the ray of phase phi in the complex numbers. OK, so you're supposed to be thinking about this one. Every object in my category gets mapped somewhere into the complex numbers and it should lie on the ray of phase phi. There's some compatibility between the shift or suspension function of my category and the grading like this, so it just shifts the grading. Then there are just two more. The first one is that if I take phi 1, if I take two real numbers, one of which is bigger than the other, and I take some objects which are semi-stable of these phases, then there are no maps. This is familiar hopefully from stable vector bundles or something. The last one is existence of hard and narrow seamen filtration. For all non-zero objects, there is a canonical. You don't need to assume unique, but it follows from the other axioms that it's unique filtration, such that the factors are semi-stable of descending phase. When I say factor and filtration here, I'm being a little bit cavalier, this is a triangulated category. What does a filtration really mean? It just means a bunch of objects with maps between them and the factor just means the cone on the map. That's the definition. The idea, the thing you should think about, is that you think about the fociar category. X is some compact carbonyl three-fold. This is just a heuristic. How do you get to these axioms of this whole business? You think about this category is only depending on the symplectic structure on this caliform, but you ask what does the complex structure. If we've actually got a complex structure on this three-fold as well, it gives us two things. It gives us the central charge, which if you have a Lagrangian, you can integrate the holomorphic three-fold against the Lagrangian. It also gives you the subcategory of special Lagrangians. That's somehow the idea that this category only depends on the symplectic structure, but as you vary the complex structure, you get different subcategories of special Lagrangians sitting inside there. I don't pretend to understand the fociar category. We would like to see the mirror of this, where we take the derived category of coherent sheaves, and then we want to see some situation where basically, as the scalar parameters vary, you see different classes of semi-stable objects sitting inside your derived class. If it's complex, we are completely irrelevant here. Sure, yeah, whatever. Yeah, absolutely. It's just the most familiar setting. Let's go over here. Now, I'm just going to assume the following thing. This greater and deep group is finite rank, a free-abilian group of finite rank. This is not really very essential, and it holds for the examples I'm going to discuss anyway. If it's not true, you just fix some math, k, r, o, d to such an abelian group like this, and then insist that the central charge, which was a map on here, factors via this thing. Anyway, so for now, I'm just going to assume this. Then I define stab of d to be the set of all stability structures on d, satisfying something called the support condition, which first appeared in paper of consavage and soil movement. Then the result you have is that firstly, there has a natural topology. Well, that's quite easy to see, but then the important fact is that this forgetful map, sending my stability structure just to the central charge, this is a local homeomorphism. Of course, this thing is just isomorphic to Cn, so you have some complex manifold here, something that's just much stronger than that. It's a complex manifold with a local homeomorphism to Cn. Already, you see something interesting and difficult about this, which is that I argued heuristically that every complex structure on x should give you one of these structures. This central charge is just the period of this complex structure. Of course, you don't at all expect the modular... It's just not true, the modular complex structure does not get mapped locally homeomorphically to a vector space by its period map, rather it cuts out some interesting transcendental submanifold of this Griffith transversality and this story. Somehow, what we would really like to be able to see is what I would like to be able to understand is some kind of extra structure on this space of stability conditions that maybe enables you to pick out these special geometric stability conditions in the case of a phrychiocat. I don't know how to do that. Let me go to discuss an example. Are there any questions while I've got this in my hand? This example is joint work with... I'll show you and Tom Sutherland. It's a very simple example, but it has the benefit that... To start with... This definition has been around for 10 years or so, and lots of examples of these spaces have been computed, but to start with, they tended to be lower dimension curves, derived categories of curves and CY2 situations, but now we're getting to the stage where we can work with the CY3 cases, which are the most interesting. In this case, I'm going to take Dd to be a CYD triangulated category, but a very simple one, which is the one associated to this A2 quiver. Just to show you what sort of thing is going to happen. While I'm at it, I'm also going to denote by D infinity the ordinary derived category of this quiver. Maybe it seems strange notation, but it makes sense, as I'll explain. The way you should think about this is that this is generated by two objects, and their X groups look like as follows, X star. These are spherical objects in this case. In the D infinity case, they're exceptional objects, because this isn't there at all. There's a unique extension in one direction, a corresponding dual thing. It's just about as simple as you can get an example of one of these categories. I'm going to describe what the space of stability conditions is. To give you the answer, let me introduce a little bit of notation. This is the Dinkin diagram for SU3, so you shouldn't be surprised when I write this. I want to write the carton and the regular part of that. What else do I want? The vial group acts on all this, which is, of course, S3. If I take this equation, of course, this is still just C2, with, say, coordinates A and B. Of course, the relation here is that if I write the polynomial like this, this is... That's just some notation. Then the theorem, which has two parts. Let's see. If I take stab of this Cyd version, and I mod out by a certain natural group of auto equivalences, then this is precisely H, right? A bit more interesting than that is what's the formula for the central charge. Under this isomorphism, what are the central charges of these two objects? You get them by taking integrals of this polynomial like this. Those are exactly the same integrals that are better on the subplastic geometry. Absolutely, yes. I mean, subplastic geometers should absolutely understand exactly what is going on. Sorry. This isn't on the right hand side. Well, there's a path here, a gamma-i. No, thanks. The picture that goes with this is this polynomial has some zeros, and these gamma-is are passed between them. Stab star is a component. Stab star is a component. There's always some small print with these theorems. Stab star is the only connected component I know about. This is the group generated by these Seidel-Thomas twists, but not quite in the full auto-equivalence group. Inside, well, the group which preserves this modulo those which act trivially. I'm not forbidding that there might be some auto-equivalences which just act trivially here, in which case I kill them. That one. Then what about stab d-infinity? Well, this is just isomorphic to h or w. Do you want to tell me the mapful? In this case, z of si is given by these oscillating integrals. E to the... The picture that goes with this, of course, when you evaluate one of these integrals, it only matters what direction you go to infinity. In the first part, do you spare down or equal to 2? Yes, that's important. Thank you. If you want to take d equals 1, I think the correct interpretation of this would then be the right category of an elliptic curve, in which case this result would still be true. In d equals 1, you say? I'm not claiming anything about d equals 1. It's just the ordinary derived category of this quiver. The motivation for the definition is that the sear dual to x1 appears in degree d minus 1 until eventually it doesn't appear at all. This is a very simple example, and as Paul and others pointed out, this is exactly what you would expect from mirror symmetry and the symplectic geometry and so on. Still, you have to prove this thing, and I'm not going to explain the proof. Actually, it's got a slightly 19th century feel to it. You find a fundamental domain for the action of this group, map it... You can always question everything by c star action, which is naturally around, so you end up with a domain sitting inside p1, and you have to work out which function maps that domain onto the upper half plane. These are the right functions to do that for you, so there's some kind of conformal mapping type argument that goes into this. What I want to take from this, and maybe people will disagree, is that this reminds me of CF, the Frobenius cytostructure on the unfolding space of the singularity x cubed. I guess that's exactly H mod w, and the discriminant is this thing. The things in one, these functions are called... What are they called? Twisted periods with parameter. So there's some parameter here, which is d minus 2 over 2. And in two, these things are deformed flat coordinates. And one other coincidence that's worth mentioning is that the Euler vector field, and then this Frobenius manifold has an Euler vector field, E, and this corresponds exactly to Z. I mean, the tangent space to the space of stability conditions of course is just this vector space, and at each point you have a particular element of this tangent space. So you have a vector field given by Z, and that corresponds to the Euler vector field. There's a factor there, but anyway. Okay, so my dream, I feel you should be able to get this Frobenius structure out of the space of... By thinking of categorically, you should be able to construct this Frobenius structure. I don't know how to do that yet, but that's somehow what I'm wanting to do. I think Maxime disagrees with the whole project. You're completely disagreeing. I think just volume elements or something. I love the default of one fault here. Yeah, well, okay, you can say that, but I mean... I'm powerless to disagree, because I can't prove the difference, but anyway. So... But yes, anyway, we'll go on. So how to see. I should say that... So I've only treated this quite simple example, but this is part of a... So let's focus on the CY3 case for the rest. So this is part of some much bigger family of examples. So... So this space parameterises. You can think of this as the set of... quadratic differentials, somewhat bizarrely, on P1 with pole of order 7 at infinity. I mean, and the correspondence is not so earth-shattering. It's just that you consider this thing. Okay? With simple zeros. It sounds like a slightly mad thing to say, except that basically there is a generalisation of this to every space of Meromorphic quadratic differentials on Riemann surfaces. So this was joint work with Ivan Smith, and this very much came out of work of Gato, Maureen Nijski, which is that for any g-basin equal to 0 and non-empty collection of pole multiplicities, there exists some category D, such that you have a similar result. Stab of D is isomorphic to the space of quadratic differential on a surface of genus G with these numbers of poles. So these are simple zeros as well. No, so that's a good point. So this is roughly speaking fibring over MGK. So you first take a surface with K marked points. You choose a complex... You take a Riemann surface with K marked points and then you consider the quadratic differentials on that surface with fixed poles at those marked points. This space is gotten by varying all that data. You vary your complex structure and the other thing. Can you get the pole of D or C? And D is always... D is CY3. So that's some big story that takes a whole seminar to talk about. Does it take a whole seminar to tell us what the map said is in that example? No, it's still the square root of the different... Whenever you have a quadratic differential you can take its square root. But don't tempt me to talk about that. I didn't want to. So now I want to talk a little bit about the structure you have on this space which is due to concevich and soymann this wall crossing formula. I realize this will be very abstract and unpleasant for many of you. But for now, if you're lost about space and stability conditions just think that in this particular example you have this very concrete space which is just H-reg mod this complement of the discriminant locus and you have this local system over it given by these twisted periods. So what structure do you have on this space? Let me see. So now we're in the CY-3 case. I guess we have no idea what happens in higher globial dimensions. And again, I'm assuming that this thing is free of alien group and I'll probably call it gamma sometimes. So there is a plason torus algebraic torus knocking around which is invariantly it's just C still at the end. And what's the plason structure? So the characters of this torus are indexed by the elements of my lattice and because this is a growth and deat group it has an Euler form on it. So this is the Euler form which exists on any kind of finite type triangulated category but because this is CY-3 this is skew symmetric. If we give me some stability condition now we have the following data. So at each point of this complex manifold parameterising your stability conditions you have the following data. So a collection of what I would call active rays so I mean this is going to be a picture we've sort of already had this picture so L, Li these are the for E semi-stable so they're the ray I mean as I said every object gets mapped somewhere into C by its central charge and I just take the rays through the semi-stable guys so that's potentially a countable number of rays so I mean I guess it's also by definition it's the same as this for all the p5 which are non-empty I mean we had these subcategories of semi-stables objects of phase 5 but most of them are going to be empty. Right, okay so far not very interesting but then for each ray a Poisson I'm going to put automorphism in inverted commas explain this in a minute but to first order you should think of this as an automorphism SL so a Poisson automorphism of this torus and how's that defined well it's pullback on functions if I pull back a character I get sorry what do I get I get the exponential of the Poisson brackets of a generator okay I can see I'm going to have to explain this okay so what's all this mean so I've fixed a ray here I'm trying to give you a Poisson automorphism for this ray and on this ray by definition because it's active there are semi-stable objects living on this ray so we can count them for each class in the gradient group which gets mapped onto this ray I can count the number of semi-stable objects where counting means take the DT invariant of the modular space of these things assemble this into a generating function so this looks like a function on this torus except that it's probably an infinite power series so we have to be worried about that then we take the Hamiltonian flow at time 1 of this function and that's supposed to give me some automorphism of this torus okay and I come back to what sense it's an automorphism and then there's this very beautiful fact which is the wall crossing formula which is that if I take some a convex sector it doesn't really matter what in C star okay so now the picture is fixed on the sector like this and and I vary my stability condition so that no so that no active ray crosses the boundary of this region then if I take the product over all rays living in this sector of these automorphisms then this should be constant so so here's the picture so here that the active rays and I take the product over these in that order okay and that looks a little bit ill defined and so on but in fact it's absolutely fine because because we're in a convex sector we can choose a half plane so that all these all the classes appearing here are all positive combinations of some basis if you like so in other words this makes sense in some automorphisms plus on automorphisms of some ring that looks like okay I mean before we were talking about a ring that looks like this okay so the ring of functions on this tourist but because these are all positive all these coefficients appearing here in this generating function are all positive in terms of this basis then then everything makes sense in this formal power series ring so this is a perfectly rigorous and decent statement and where does it come from it basically comes from the existence and uniqueness of hard and narrow semen filtrations although it's hard to see by now it's been disguised but basically there is a formula like this holds in the hall algebra almost for free just because I mean that just if you think of a formula like this in the hall algebra it just expresses the fact that every object has a unique filtration by semi-stable objects and the really kind of amazing thing here is that there's a ring on a homomorphism from the hall algebra to this well to the quantum tourist and so you can transfer your identity to the quantum tourist and then you can cook up this thing which I guess is the semi-classical limit and so let's just say what it says in this example so again we're taking the CY3 drive category of this thing then there are just kind of two chambers in this space of stability conditions you've either got two stable objects or you've got three stable objects and of course you have this as well but let's not put the negatives on and the rays are right so let's let's take a basis the obvious basis consisting of given by the simple at here and the simple here so what would I write here and I would probably write SR for one and here I would write SR for two SR for one where SR of any alpha is this cluster style transformation and then there is a non-trivial so this is now I'm thinking of this as now a birational automorphism of the torus so one thing I explained here is you can just expand all these things as power series but in nice situations and it's not true in general in nice situations this thing really does define a birational automorphism of your torus and this is the case here so for each of these so as I say there are kind of two parts of this complex manifold in one part there are just two stable objects up to shift and in the other part there are three stable objects up to shift so you get these two different regions and so you end up this wall crossing formula becomes this identity that SR for one SR for two equals SR for two SR for one plus SR for two SR for one which is kind of this pentagon identity which comes up in cluster algebras so it's related to the pentagon identity for the quantum dialogue so that's just the simplest example and I should say that in these spaces I mentioned here these much more general things this is also true these are always birational automorphisms there may be countably many rays but all the automorphisms are birational any questions? I know this looks somewhat abstract but at the end of the day there's something very concrete here we just have some complex manifold you have some rays in C which are varying with your point around in this complex manifold these system of rays is moving to each such ray you have a birational automorphism of a torus and these things have the magic property that when two rays collide the automorphism associated to the two things turns into the product of these two things so that's kind of the structure you naturally have so that's exactly what the structure of the automorphism is the pentagon identity just follows from this general machinery of I mean it follows from this wall crossing identity it's the simple example of this wall crossing identity I don't know my question is what's your idea of the automorphism probably so structure of the automorphism has a true architecture not that I know of right so where are we at the time I brought that board down so I should remark that at least this stuff is where's my I should find my placement yeah so there is at least here a very strong analogy with the Frobenius structure which of course is living on the same space I mean no one can deny that there is a Frobenius structure on this space and what that means is that so over so we have a have a family of or rather the points of this Frobenius manifold parameterised connections gln connections meromorphic on p1 so these things look like d minus u over t squared plus v over t okay so so this defines some you know meromorphic connection on a cn bundle over projective line with some irregular singularity at the origin and some regular singularity at infinity and here u so u is some diagonal matrix so this is multiplication by the Euler vector field so you're on a Frobenius manifold you can multiply tangent vectors together and v is some skew symmetric matrix and v is the gradient of this Euler vector field and so what people do or what the Brovin explained is that you can well I guess this is not quite true there's a diagonal bit is that you can consider the kind of irregular monogamy of this connection at the origin so that's you know stokes data and you get a picture like this very similar to what we've had before whereas now the rays these are the stokes rays of this connection so this is spanned by the differences of the eigenvalues of this diagonal matrix and you have stokes factors so SL but instead of being automorphisms of a tourist or anything like this these are just in GLN but you have exactly this same you know this is now the kind of isostokes property I mean somehow the big the important thing here is that this is a nice monodromic family of connections so you have exactly this same property SL equals constant okay so again as you move I mean these UIs are the canonical coordinates I mean these servers are a coordinate system on your Frobenius manifold as you vary around in your Frobenius manifold these rays can collide but when they do these stokes factors form a product okay so there's some and and so and sort of given U I mean very roughly V is uniquely determined by a set of this stokes data I mean at least locally so I mean can I just explain that this is the identity here so this is the identity so this is a identity matrix and that's the gradient of the oil of Oxford and D is the dimension of the Frobenius manifold which is some number okay so maybe I'm just a sucker for analogies but to me these things look very similar okay and as I emphasize they're living on exactly the same space it's a way different kind of group but I mean well let me say a little bit more so there's one other thing here that you have in this Frobenius manifold picture is that if you go to the negative stokes ray so these come in you can go to the opposite stokes ray this is kind of the same matrix it's inverse transpose you have this property and this is kind of the same this is the same thing as saying that V is skew symmetric I mean in other words you could consider other connections where V wasn't skew symmetric you would get stokes factors and they wouldn't satisfy this but in this quantum cosmology picture you always have this kind of happy thing going on so let's just speculate what we might have on the left-ab of D should look at some family of connections like this on P1 where now this group is this much bigger thing okay so let's just think about the Lie algebra of this group so that's going to be symplectic vector fields on my torus so this has a bit you can think of as the cartan this is you know rotations infinitesimal just rotations and then you have functions I mean Hamiltonian vector fields so this is either given by functions on this torus either way or the group algebra of this lattice what's F? F is going to be an F right so this is an element of the Lie algebra of my Lie group this is going to be in the cartan bit just like that was in the diagonal bit so this is living in just the rotations of the torus this is going to be a function on my I mean well I think of it as a Hamiltonian vector field but it's I guess I have to mod out my constants here and then there's one other coincidence just like this one which is that you have this property that DT alpha is DT minus alpha okay so that's very much like this so what that tells you is that this automorphism if you go to the negative ray it's related to the automorphism of the torus for the other ray by conjugating by the inverse map on the torus so what does that tell you about F you know which is the analogy of this V being skew symmetric then F is um well if we write it as I mean if I think of it as a function on the torus and expand it like this it just tells me that F alpha equals F minus alpha um so I mean I'm not saying you can actually solve this I mean so in the GLN case we know how to get from Stokes matrices to this connection by inverting the irregular Riemann Hilbert map well I mean it's a delicate thing but we could we sort of expand that but this is an infinite dimensional group we don't really know whether we can do this but let's just pretend we can we will have this condition um and if you think about it what that tells you is that this function is invariant under this inverse map so it's corresponding Hamiltonian vector field fixes the identity um and therefore induces an automorphism of the tangent space to the identity so this implies that flowing um fixes well fixes the identity torus plus induces automorphism of the tangent space endomorphism so this was Dominic Joyce's idea so so I mean so what he then writes down is that you can say well I mean if you write down this now I mean you can write down what is the induced connection on the tangent space you see I mean the tangent space to the identity of this torus is the same as the tangent space to stability conditions at that point so you can write this where P alpha is just this reflection okay and you can prove that this is a well I mean formally this is a flat connection so the so the conjecture that I want to sort of try and take seriously is that this is actually and you know you can you can assemble various bits of supporting evidence for this I would claim that this is is the Levy-Civita connection on the Frobenius manifold okay so you know I mean we could speculate about this in various degrees of generality but for now let's just do it in this A2 case right so there is a particularly concrete Is this not a German connection it's a connection with the values on the hugely algebra? No because we got rid of that because this connection it had values in the Le algebra so this F business had values in the Le algebra of vector fields which are invariant which vanish at the identity of my torus so you take the tangent so if you think so I mean F alphas are just numbers F alphas are just some number depending on my stability condition so this is just some linear I mean this is an infinite so this is unfortunately an infinite sum of elements of N by N matrices aha sorry because you can see the contribution of each rate of vector fields that's right but that's just like this I mean it's exactly the same relationship as going on as is going on well you know I mean this relationship is a bit is a bit strange so again on the level of stokes matrices you get this relationship which is not saying that anything vanishes at the identity it's saying that this matrix is invariant under conjugation by the inverse map what that tells you for this function F in just the same way that you get a function which is invariant under the the inverse map so if it's infinite dimension of the algebra what's the dimension and it's a connection out of a thing of what dimension dimension N I mean the tangent is to this building before you go on to the special case can you tell us whether you know whether this is torsion free yes it is torsion free you can play a whole load of formal games you know I'm not claiming that this is some major breakthrough I mean for one thing it's Dominic's idea completely it appears in a paper of his from four or five years ago now and for the other I mean we just have no idea whether these things converge right so I mean it could just be a formal game of nonsense but inside that formal game of nonsense you can define a metric you can say this is the levitivity connection for the metric it's torsion free and so on so how long have I got two minutes five minutes so actually the thing I'm sort of you know I mean as I say this is a somewhat old story what one really wants to be able to do is actually calculate these F alphas in some examples to see whether this makes any sense okay but I mean formally it makes sense so let's not do a Kevin did I just put that one up so I mean again going to the going back to the analogy of the Frobenius manifold I mean somehow the whole Frobenius manifold structure is encoded in this stokes data and how do you do that how do you reconstruct the Frobenius manifold you solve some Riemann-Hilbert problem okay so I'm just going to write down the same thing here so you write down a Riemann-Hilbert problem which is for maps well I'm just going to write it into the torus which should have a couple of properties which is that I mean this is very much GMN I'll turn more nightsky Snipe story did I ever write Dominic's name he'll kill me if I didn't sorry Dominic so again where's our picture so we have a bunch of rays and let's think about why don't we just do this example then there's just two rays say and what we're asking for is a function from C star into my torus which is discontinuous across these rays and jumps precisely by this automorphism and also has some asymptotics at the origin I mean in coordinates this is just saying that X i H e to the z i over H tends to 1 if you write your element of your torus in coordinates like that okay and the point is why is such a thing a sort of relevant thing to write down of course you now want to I mean this is for a single stability condition you write this problem down but now you want to think about this thing varying with z as well with my stability with my central charge and why does that make sense well precisely because these discontinuities match up when one of these when two of these rays collide the discontinuity is the kind of product of the discontinuities I mean that's exactly what the wall crossing formula tells you that this is a sensible sensible thing to write down and it turns out that well there's a recent paper by Iwaki and Nakanishi and this is called EXAC WKB analysis and you know I haven't processed this entirely yet so but I want to give you the idea because I think it's very beautiful there's also a paper by Massive Arrow which exactly deals with this very exactly this example and what we're doing here is some kind of conformal limit of what Gaiotomor and Niteski do okay so there's my last sheet so what do we do we consider a family of projective structures on P1 minus infinity so this sounds a little grand it's going to be something very concrete in a minute some fixed one Q of X is this thing that keeps coming up this is this by a projective structure I mean a family of an atlas of maps to Cp1 whose transition functions are in amobious transformations right so on any ream and surface you can consider a set of these projective structures so I'm fixing some basic one which is the standard one on P1 there's an obvious projective structure on P1 and I'm perturbing it by this quadratic differential one thing you know about projective structures is that they form a torso for quadratic differentials an affine space for quadratic differential so I should say all this applies perfectly well to all these other examples but quadratic differential and what does this actually mean I mean so it means you should consider the following familiar equation okay I mean that's how you get I mean that's what this affine space thing is it's this well let me just say it like this you consider a ratio of two linearly independent solutions to this equation that will give you a map to P1 from your complex surface of course it will go wrong at infinity but you know we don't care about that that will give me some projective structure but okay and a projective structure in particular well so I draw the following big diagram and then I'll sit down so so this what I've just described is a way of getting yourself to a meromorphic projective structures on P1 and then I want to take just the monodromy of this thing so this is there's a map to the wild character variety of you know I'm thinking here of the work of Phil Bolch or you could also think in terms of these I hope framed local systems and the thing is on here you have some cluster coordinates I mean what what does it really mean it means again you take the stokes you take the stokes matrices of this equation at infinity those actually those agree with these these Fogontroff coordinates and for these these depend on a triangulation but of course anyway that's another story but this quadratic differential gives you a triangulation I realise there are lots of things I'm not really having time to explain but okay so I claim that this map gives you the solution to this Riemann-Hilbert problem okay and I think well this is I mean these guys will probably kill me because I've rephrased what they did in possibly wrong possibly wrong way but anyway there is this very precise and you know 80 or 90 page paper explaining the connection between cluster varieties and the WKB analysis of this of this Schrodinger equation and you know it's kind of a beautiful story relating to voros symbols and so on but I think you know in a nutshell this is what's going on and I think this will provide you a way to understand this structure so the story is not finished I'm hoping to use this to somehow compute these F-alphas and see if they match up but until I manage to do that I think Maxime will not believe me so anyway I'll stop there Question Maybe I just confused in the case of these manipular structures in our A-model it should depend only on these structures not on these structures so kind of why this stop being to be parameterising complex structure Yes I mean I think this is isn't this just the mirror I mean roughly speaking stability conditions I mean there are kind of two categories we could think about the Fakaya category of X and the derived category of X and somehow stability conditions on one so say stability conditions on the derived category of X that's like Kehla parameters that's the same as defamations of the Fakaya category of X and it's on that defamations space you usually get all this kind of non commutive hodge structure the thing that they studied but mirror symmetry kind of says that these spaces should be the same or very closely related but this is some completely different story not related directly to Gromof Witton invariant so this from this one structure is not one which can come from Gromof Witton no it's kind of somehow correspond to the Gromof Witton on the mirror Yeah exactly so I mean what you should be thinking of is if I take stab of dx how am I going to get a Frobenius structure on this well I'm telling you that it should be something to do with dt invariance of X but of course that's kind of the same as Gromof Witton of X and this should be something like defamations of the mirror well this should be like symplectic structures on the mirror No I mean this is the dt Gromof Witton correspondence I mean it stays on the same space There is this Gromof Witton but corresponding to some players and actually you can try to hook that up to give you some heuristic I mean I have some very heuristic argument as to why this why this leverage of a connection is the right thing involving this dt Gromof Witton correspondence but that's certainly not in a state I can present But you agree that there is a Frobenius manifold structure in these examples and there is also the fact that you don't believe me is a huge motivation to try to try to work out the details I wish I could have done it in time I also know for those two characters they don't pay them Thank you very much absolutely I mean yes thank you that's a very important remark so I mean that's a very important remark so I mean this first theorem I stated I was in too much of a rush I mean various parts of this had been proved before by various people but Richard Thomas who's even here and Alistair King and then there are these but I mean the absolutely most important thing is that this guy Ikea who scooped us he not only proved this theorem before us well part A part one but he did much better because he did not just A2 but all A n so he absolutely blew us out of the water it's kind of criminal to forget so yeah Question do Froben studies the powers of the Euler vector field because they satisfy this nice I mean obviously in the simple case it's obvious what they do they just like vector fields on the line I'm just curious if the one could hope to find formulas for these powers I don't know you're going to have to explain that explain that to me later, sorry Alistair The question is part varus they seem to be yes are these varus symbols which you also mentioned in your paper so these people have studied the Schrodinger equation obviously people have studied these kinds of equations I mean this is a Schrodinger equation in a cubic potential I mean people have studied this since the 30's I guess and there are many results but in particular this varus there's a paper of varus from the 80's which introduced these things which turned out I think to be the same as these cluster coordinates