 Okay, so yeah, first, I guess I want to thank the organizers for giving me the opportunity to talk and for everyone sticking around for the very last talk and yeah, sorry, I couldn't be there in person. Yeah, everyone's had a great week and I look forward to watching some of the lectures that I was too asleep to attend on YouTube later. But okay yeah so let's get into the talk. Yeah, so today, I want to talk about some conjectures around resurgence associated to some asymptotic series that come from, on one hand, quantum invariance of three manifolds and sort of more generally proper Q hyper geometric functions. And so we'll start by sort of giving a little introduction to these series. And so one can sort of define these series from more generally than just from a three manifold but from just sort of the data of some interdress symplectic matrices. And so we'll give that sort of definition. And then I'll go on to some associated state integrals of that are associated to similar objects as these asymptotic series so these are sort of. Yeah, again, sort of a generalization of the constructions of Anderson and Kashaev to, again, sort of more general things than just the three manifold context. And then I'll talk about quantum modularity of all these objects, and then how that ties into resurgence. Finally, okay. So let's start off with the symplectic matrices and the asymptotic series. So yeah this all sort of starts off with Witten's interpretation of Jones's invariant of links so in the 80s Jones discovered this link invariant and Witten interpreted in terms of quantum field theory. And so interpreted this invariant as some kind of exponential integral. But over some infinite dimensional space of connections. So Witten would write an expression like this where we integrate am is sort of the space of SU two connections on a three manifold and G is the gauge group and so we integrate over connections modular gauge associated with this exponential measure here where we take the terms and Simons invariant of the connection divided by two pi h bar. Okay, so because the terms and Simons invariant is only defined modulo two pi squared Witten's invariant sort of quantizes this h bar I mean to make sense of this integral. And so Witten associated this integral with Jones's invariant at these specific sort of quantized values of h bar. Okay, and so, but if we take a step back and just look at this integral if this was a five dimensional integral then we'd expect to get nice asymptotic series associated to this when we send h bar to zero. And those should be sort of determined by the critical points of this Chern-Simons functional. And in this case the critical points are flat SL2C connections sort of complexify. And here the, yeah, this is now a five dimensional space so that's actually quite nice. And so we can really computationally get a handle on these critical points. And so the way that this is done in practice or the way that for given a three manifold you look at these SL2C connections is sort of using ideas of Thurston. Thurston would triangulate a hyperbolic manifold into sort of ideal tetrahedra. So say if we take a knot complement for example, we would triangulate this with ideal tetrahedra and sort of construct these connections locally on each tetrahedron and build it up into a connection on the three manifold. So yeah, so the hyperbolic tetrahedra we can describe very easily. Their modular space is very simple. It's just the upper half plane modulo Z-Mont3Z action. And so here we get some nice modular space of these tetrahedra. And then we want to understand sort of when we can glue these together. But yeah, for now, just the tetrahedra, you know, we'll look something like this. So for each of these parameters, this Z, Z prime, Z double prime. The geometry of this tetrahedron is determined by, yeah. So on one aspect, the angle along this edge of the tetrahedron will be determined by the argument of this Z. And then there'll be some sort of torsion associated to the length. And what's nice about this modular space is we can easily compute the volume of such a tetrahedra using the block of Wigner dialog rhythm. And so here, yeah, it's exactly just equal to the evaluation of this sort of modular Z of the block of Wigner. And so we can compute volumes or complexified volumes using some kind of dialog rhythm function. So the block of Wigner for specifically the volume associated to a triangulation and then more generally, complexified volumes for some other kind of dialog rhythm values. Okay, so that's sort of the local structure on each tetrahedron. And so we have this modular, we can describe them. But when we want to glue them together to make a hyperbolic three manifold, we need to understand sort of how these things will glue together geometrically. So because we're dealing with ideal tetrahedron topologically, we glue along faces and there's no issue here. And because for ideal tetrahedra, we remove the vertices. And this is the only thing that can cause us trouble when we're gluing tetrahedra together. And so therefore, we'll get a nice topological space and we just need to understand basically which modular can patch together to give us a geometric structure on the three manifold. And so the main constructions of thirsting is basically we look at so each of these ideal tetrahedron needs to glue them via an isometry of hyperbolic three space. And these isometry is exactly SL2C. And so then if you sort of choose your basis correctly so you send sort of one of the vertices or say so let's focus on an edge and we're trying to glue tetrahedra around that edge. Then we can sort of fix the two edges in a particular model at zero and say infinity for this ideal tetrahedron. And then this sort of turns the monodromy that we would like to glue. The monodromy would use to glue these tetrahedra together into sort of an abelian version so the upper triangular and also yes upper triangular side. And we can really see the monodromy around this edge which topologically is just contractible loop therefore it should vanish. And we can write that monodromy vanishing in terms of these modular as some simple equations like this. So here. Yeah, it's just this nice monomials in our parameters at J. I'm on July for each tetrahedra. If we have capital N tetrahedra and one minus ZJ. And then all of the gluing equations is determined by these integer matrices A and B and then some vector of signs new. Okay. So these these are Thurston's gluing equations and the data of these gluing equations is often called no man's idea data and no man's idea showed that these matrices actually have a simple surprising symplectic property. So in particular. If you take a times B transposed this is a symmetric matrix. And if we sort of take a and B as an N by two N matrix and this is full rank of a queue. They actually show a little more it's full rank over sort of a half times Z in some sense so there's there's just a half that floats around but it's essentially it's almost full rank over Z but there's an extra half that's next in at some point. Okay. So yeah so if we go back to Witten's integral. This was an infinite dimensional integral over a space of connections and we said that it should be determined asymptotically at least by the critical points of this turn Simons functional which were the flat connections. So we can reduce this in some sense or it's believed that it can be reduced I don't think there's a sort of known physical argument to do this but in practice. People have had success in constructing something that could potentially be a reduction of this infinite dimensional integral. From work of Kashyap in the mid 90s. We expect this to be able to be reduced. And so basically, we would like to sort of perform Witten's integral over tetrahedron tetrahedra and then glue those integrals together because that Witten's invariant is local the Trans Simons functional is all local so we should be able to glue these sort of quantum invariants together to get the full thing. But so what's expected to happen is that when you do this integral sort of over a tetrahedron you get some kind of quantum dialog rhythm associated each tetrahedra. A tetrahedron. And so the quantum dialog rhythm was given by for Dave, and then studied by Kashyap and who used it to go on to construct three manifold invariance. So in the form that I like it's this quotient of two pocama symbols. So here. Do I give it no. And Q is either the two pi I B squared and Q tilde is either the two pi divide minus one divided by B squared. And so here there's sort of a modular flavor already that we can see. This is essentially a valuation of some kind of infinite pocama symbol at tau and then divided by its value minus one over town. Okay, so this is the quantum dialog rhythm it has many different formulas we'll see another one later. And this has a nice asymptotic series, which is sort of determined by just one of these pocama symbols so just give you the asymptotics of the pocama symbol here. And so you see that the leading order in this asymptotic series is given by the dialog rhythm and so this is maybe one reason called a quantum dialog rhythm. There's many more reasons but that's the sort of zero thought or reason. Okay, so then. Yeah, so we take these quantum dialog rhythms and we associate them to each tetrahedra we take a big product of them and then we integrate over some finite dimensional space and it's expected that this should be related to these sort of quantum invariance in some way so he came sort of initiated this study after Kashyav and then many others worked on this and it was sort of put on very firm footing by Anderson Kashyav who gave very precise contours of integration and proved invariance of. Yeah, of their construct of their invariance of these state integrals. Okay, we won't go into that directly we're still on the asymptotic series. So, yeah, the state integrals, we can roughly think a Gaussian measure at times a product of these for day of quantum dialog rhythms, and this Gaussian measure is determined, not by just the critical points themselves but the equations that determine the critical points. So in particular, if we have this known as a gear data A and B, then our Gaussian measure will be something like B inverse a, and then we will take a product of for day of each of the tetrahedra in the triangulation. And so this is the kind of invariant Anderson Kashyav give many more details but here I'm not sort of interested in constructing the three manifold invariance I'm just interested in the behavior of these the asymptotic series of these kind of integrals and the analytic properties. Okay. And so, how is this related back to the flat connections well so here we need to take these integrals and now we have a finite dimensional integral and we can try and apply something like stationary phase. And so this was started in detail by many people so one paper of note is the Moff de Gouk of Landau-Zagier, who gave many exact results. So these sort of perturbed expansions and use various I think three different kinds of methods to construct a certain asymptotic series that come from this stationary phase applied to the state integrals. And then the Moff de Gouk for leaders gave sort of formalized all of this and gave a precise definition in terms of the known as a gear data and sort of formal Gaussian integration. In particular they start with this known as a gear data which was this AB matrix that we saw before determined by these gluing equations and this sign vector new. And they start with a so called combinatorial flattening this sort of integral solution so f and f double prime are some integer vectors and they solve this sort of linear version of the gluing equations. So this will be a solution to the gluing equations that we saw before so a vector of not complex numbers say. And so when the determinant of B is not equal to zero, then they defined asymptotic series in this way so we take this product of these formal series size of h bar of X of Z so this was the asymptotics of the for day of quantum So this formal series here. So we take this series and we just take a product with this little exponential pre factor. So you can see here the dependence on f of this of this big Gaussian integral is just this h bar over eight term here. So it's quite, it's not a very strong dependence. And we perform the Gaussian integral around the sort of critical point Z. Okay, so then here. So recently, with the garrifolitis and thought so we actually showed that taking this definition of demoft and garrifolitis you can prove that these asymptotic series of invariance of three manifold for the special solution to the gluing equations that comes from the geometric solution. So if we have a hyperbolic not complement, then you can define the series. And we prove this is a topological invariant of the not. Okay. So, these are the kind of series that I'll be interested in studying their, their research and properties, and their asymptotics. And so just for a quick example, very easy example. So if we take the figure eight not for one, then, then you can find some no man's idea data given as follows. So you just have this collection of integral matrices and you can choose a solution to these equations. Oh, sorry, well there is the geometric solution to these equations is Z one and Z two equals the sixth route of unity. And so if we just start with that formula and apply a particular identity to the quantum dialog rhythm we can reduce this to a one dimensional Gaussian integral. So here this is extremely simple every time we see an x we just replace it by an x to the n we just replace it by an n double factorial with a square root of minus three to some power. If it's odd and otherwise, if it's even it vanishes. Oh, sorry the other way around. Um, okay. And, yeah, in this example you can compute, you know, hundreds of coefficients and so you're just the first 10, wherever moves sort of the, all the pre factors so the series starts with one. Okay, and so this is sort of a typical example of these things and so you can really compute a lot of these terms, or even the order of hundreds. Um, but yeah as I said at the start there's no need to take a particular three manifold so for one. The norm is a gear data of a three manifold these a and b matrices will be quite sparse in general they'll be the entries will be at minimum minus two, and a maximum two and often they'll be zero. But we can define these series for more general a and b any upper half symplectic matrix will do and we can construct some asymptotic series for this. So really we're just starting with this combinatorial data, and then this algebraic solution so we need a little more. And then we can construct these asymptotic series and all the proofs that we gave with Garafelletis and Storza they'll go through for these these data so if you apply. There's sort of combinatorial equivalences that come from certain equivalences between triangulations of three manifold so in particular a two three pack in a move so if you have a stack of two tetrahedra and then you split it into three. This will give you a different expression for these gluing equations and you can track that and so that gives you some combinatorial move between the this data. And all of these moves, the series are invariant under these moves so even if we don't start with the three manifold we still have sort of a bunch of equivalence relations between this combinatorial data plus a little algebraic and all of our asymptotic series will invariant under these. But so here to define the Gaussian integration, you do need non degenerate quadratic form. So for the geometric connection so for that theorem we had a hyperbolic manifolds. It has a geometric connection and for this example of combinatorial data and algebraic solution. The quadratic form that would appear is in the Gaussian integral is non degenerate and so therefore you can really construct this series but in general. This might not be the case and you have to work a little harder to even define the asymptotic series. But in that case there is a little work that I've done with Garra for leaders. This is a period. This is a paper that appeared the end of last year. And it's the Meromorphic 3D index and try bureau invariant. And so the in these examples. So in the previous examples where we have a non degenerate quadratic form. These all give rise to sort of asymptotic series with algebraic coefficients. But more generally when we have some positive dimensional. Focus of critical points in examples then we start to see periods of these loci appearing in as coefficients in the asymptotic series and so in the examples of this paper we had. Yeah, we had basically the critical points associated to the objects we were looking at. They were located at the A polynomial of an associated not. And so then the asymptotic series we're seeing we're seeing periods of this a polynomial appear. But so in more generally these are the kind of things that would appear. If you're not in the ideal situation where you have a non degenerate critical point. But okay. So to introduce you to the series give you a little flavor for where they show up in three manifold topology and their invariance. Okay, so now let's dive into the state integrals in a little more detail and how they relate to q hyper geometric sums. Okay, so the state integrals we saw before they have a natural. Log for q hyper geometric sums either as elements of the her bureau ring or as Q series. So in particular these are two different objects that will have asymptotics determined or the same kind of asymptotics as these formal Gaussian integrals that we gave at the start. And so to prove these kind of asymptotic statements, you can do this in practice for examples and small enough examples you can do it so basically the. The main idea is that you need to take these sums and you want to apply some summation method but then you need to justify that all the boundary terms are small enough. And then your summation method will turn these into some kind of integral and then you want to justify use of the saddle point method so in examples. So these kind of asymptotics can be done but in general, you know just depends on the geometry of sort of the critical points and to do that in general is a little harder but in an example you can sort of prove that these kind of sums will have the asymptotics that we have the star just sort of by formally hoping that we could apply stationary phase to a sum. These kind of sums are actually quite general so here it looks very specific you know we have a product. So these pocama symbols here that have appeared they're a product of. You know n pocama symbols so these are vector of these case of vectors so you know they're a product of n pocama symbols but all very just straight pocamas is nothing crazy going on no linear forms appearing in the arguments of the pocama symbols. So the sums are actually quite general because we can always take a pocama symbol and turn it into one of the sums of these forms by using the q by no meal theorem so in particular if you have a pocama symbol in the numerator of a q have a geometric sum we can always put it into the denominator it's the clean pocama L here with you know the catch of introducing an extra sum and similarly in the denominator. So therefore, these kind of sums, if we consider proper q have a geometric arms they'll sort of always be able to be reduced to something of this form that I considered on the previous page here. So in particular these q series definitely we can always reduce to something of this form. So then. Yeah okay so we have these. We have the sums we have the state integrals and sort of the question is how they're related. So these state integrals can often be factorized into sort of two objects so we saw that the for day of dialogue or them itself was factorized into sort of. Into two pieces one that depended on q, if q is either the two pi i tau, and one that depended on q tilde e to the minus one over town. And this factorization also happens for the state integrals in general. And so this was sort of known in the physics literature for example there's this work of being democracy. And this was explicitly proved for, you know, a large class of examples by Garry for leadership and shy of in two papers. And the reason that there were two papers was that there's sort of two cases that you need to deal with, and it sort of behaves differently in either case so in particular, we can either have. You know this argument tau b squared, if we're thinking in terms of the for day of that I gave previously, this can either be in the complex numbers anywhere but the real so the upper and lower half planes, or it can be at a rational point. And so these two cases correspond to these two papers that I just mentioned. Okay, and so if we go into the upper half plane. Then, basically the way that we would factorize a state interval is simply by using you know sort of undergraduate complex analysis and collect residues we just have a contour and we push it to infinity and collect residues. And so then the main point that we need to understand is just keep track of the poles and zeros of the for day of quantum algorithm. So with this formula that I gave as a definition. This is extremely easy so here it's a product formula we can see exactly where all the poles and zeros are. And so here, you know we're going to have poles in this upper half plane and zeros in the lower. And then we have these sort of currents out here where nothing is happening. And if we have a contour along the reels we've sort of often push it to the positive infinity and just catch these poles and this will give us a double sum, because it's a sum over a two dimensional coin. Okay, so then let's just look at some examples. So this is a state integral for the figure eight not so if you remember back to the asymptotic series that I gave we had a formal gas integral of square of this asymptotic series associated to the for day of. And then it was, you know, with some Gaussian form. And so here this is exactly the same. You know the for day of quantum dialogue to them squared appearing is for exactly the same reason. And so here, this one on the left is the Anderson-Cashire state integral or at least a reduction of it. And then the one on the right is a state integral that we introduced with Marcos, Gia and Stavros in a paper from two years ago, I guess now. And so these can be factorized again using this sort of residue calculus. The first one factorizes this nice bilinear product of two series G1 and G0. And the second factorizes in a similar way with a G2 and then you have the G1 and G0 and then some additional functions appearing. So these functions here are all nice Q hyper geometric functions. This G1, G2, G0, you can these higher order terms sort of appear. You can think if you're familiar with differential equations then or Q difference equations, the basically they're sort of some kind of Frobenius deformation of the G0 series. So if we solve a certain Q difference equation associated to G0, then G1 and G2 would sort of be like the logarithmic solutions. So we were talking about differential equations, but here it's Q difference. So we see these sort of Eisenstein series appearing here. Okay, so, yeah, so you can completely do this it's all very clear for simple enough examples here we just have one dimensional integral. So you can explicitly calculate it and yeah factorize the same tools as these sort of bilinear product of Q and Q tilde objects. Okay, so that's in a case where we can push the contour to infinity but this is not always possible. So they might of course if you want to push the contour to infinity you want it to vanish at infinity this contour, or at least this integral sorry. But this might not be the case for certain state intervals that we might want to consider that could be convergent but just not convergent adding the direction at infinity that we want. And so like colloquially we call these integrals trapped and so here. It might seem that we're stuck but in fact we can go a little bit further so yeah it's an unpublished work of Garafalidas and Kashyav. These can be dealt with with a so-called untrapping procedure so here. Yeah basically we apply a Fourier transform identity to the for day of quantum dialog rhythm. And this will decouple this Gaussian form from the for day of and then we just have a Gaussian integral so then we can just perform the Gaussian integral. And we're left with some other for day of and some other interval left and so here in this example if you applied an untrapping procedure. And then you would reduce down to an integral of this form so now we've turned out to into a three divided by eight. And so in this if we have a product if we just have one for day of times a Gaussian basically we want the number in front of the x squared to be between minus one and zero. That's the case. Then sorry minus half and zero. And if that's the case then all the untrapping so here we've achieved that. Okay. And so there's the analogous I mean with the state intervals, they're really very analogous to the Q series themselves or these proper Q hyper geometric series. And so if you see something happening today integral there should be some analogous being happening for the Q series, and there is. And so the other Fourier, the analog of the Fourier transform identity for the for day of is the cubanomial theorem. And so we can apply a similar a similar method to this Q series here. And then we would write instead of getting a Gaussian integral, we would find theta functions appearing, which again makes sense as an actual analog for the Gaussian integral course. And so then we can write these series as sort of combinations of again simpler series here. Now okay I had to introduce an eye and that's because of the three of eight but so some functions like this so basically that I could be, you know one minus one I minus I, all of those will come into play. And then we'll take these series and they'll be a product of these with a bunch of theta functions and that will sort of reproduce our first series. And the point now is that these simpler series down here, sort of the analog of being untrapped for the Q series is that we have a Q hyper geometric sum that makes sense, both when Q is less than one and when Q is greater than one. The other way you could think about it is if I send Q to Q inverse. I want that Q inverse series to be a convergent Q series. And so we can apply basically if there's some untrapping that can happen at the state integral there's a similar thing that will happen for these Q series. And so then it's quite good because we sort of can, there's natural ways to extend these theta functions from Q being less than one. So, we have a bit more freedom with these data functions and we can sort of make sense of an extension of these things. And so then, yes, so then once we've done this sort of untrapping, we can apply the method that I mentioned before basically pushing the contour to infinity for these state integrals. And you will reproduce these kind of series, these simplified series down here, and then you can go back. So you sort of do the untrapping at the level of the state integral and then you undo it at the level of the Q series. And so you can rewrite your state integral in terms of the original Q series without all of these sort of messy theta functions and stuff floating around. Okay. So that's sort of the structures that go into factorizing the state integrals into Q series. So then the other case that I mentioned was factorizing them at rationals. And so this uses the sort of following nice lemma. So if you have a function on some domain that has some translational invariance, then if we consider this function shifted with this translation we're interested in divided by the original. If this quotient is invariant under the shift, then we can rewrite this integral in the following way. So we take the integral, this contour, some contour here, f of z, and we can rewrite it as the difference between two contours of the same form, f of z is z, but divided by one minus g of z. Okay. So this is quite nice because now our contour is sort of, if we think, okay, so the contours that we're going to take for state integrals, this gamma is going to be the reals, and the shift is going to be i times some integer. And so then we've sort of trapped a region between two contours. However, we've introduced some potential poles with this one minus g of z. And actually, this is quite nice in the examples of the state integrals because exactly, you know, if we take the higher dimensional analog of this lemma, these one minus g is that will appear and could give rise to poles. The equation g equals g of z equals one is exactly in examples will be exactly the gluing equations that we had before. What we see is when we try and factorize these state integrals at rational points, at the rational numbers. Then we get a sum over essentially the critical points of the, yeah, the original series just using this sort of residue theorem. So this is quite nice. This is an example that I gave before. So this integral, the state integral that we introduced with my first year in Stavros. You can apply the fundamental lemma to can or this lemma that is gave to compute the state integral at rationales. And so here you find quite nice results. So you see in a similar way, if you remember the factorization for q series we saw this g2 function appeared out the front of this factorization. And so here the g2 is replaced by this J. And this J is the Kashyv invariant of the figure eight not. And then we get some sort of homogeneous corrections to this. And we see some sort of volumes figure eight not appearing. So these v1 and v2 and sort of volume and the minus volume of the figure eight. And anyway, so then we can rewrite this in terms of these series. And so here these other terms are given by similar kind of expressions to the Kashyv invariant. However, defined over a number field. And so here these, they define over this square root of minus three field or some kuma extensions of this. So this. And so we can completely calculate this integral in terms of these exact formulae. And this is quite nice because then these exact formulae if we sort of pen together we can learn things about this bilinear combination. Just from studying the original integral itself. Okay, so that's sort of the state intervals, how you factorize them as q series and it rationals. And so now let's get into the quantum modularity. So just to remind us, I'm sure there's, I feel like there's been many talks about similar things. So that I'll just be very brief. So just what our modular forms or how could we think about them in the context of this so we'll be interested. I mean, these quantum modular forms will appear for these kind of examples. They'll all be vector valued. So let's just think, let's just recall what a vector valued modular form would be. So in particular, if we have a representation of SL2Z into GLNC, for example, then a function from the upper half plane to CDN is a modular form for this representation of weight k if it satisfies, you know, the following equation. So if we take the, you know, the action of SL2Z on the upper half plane by Mobius transformations, then if this ABCD matrix is in SL2Z, then we should expect the function to transform by just the product of this vector times the representation. And with some weight factor appearing in the CTL plus dk. And more generally that weight factor could be just an automorphi factor. So we could have a little factor of its own, but anyway. Another example of this is the theta functions. So here if we take this triple of theta functions, then we can understand the action of the modular group on these vectors very easily. So if we shift, so if these are the two, these two equations I'll give here are basically the generators of SL2Z, T and S matrix or the 1, 1, 0, 1 and 1, 0, sorry, 0, 1, minus 1, 0, or the other sign, as you like. And we can completely describe them like this. And so this is how we could think of a modular form. There should be some growth condition and infinity, but okay, it should satisfy some simple equations like this. Okay, so then how, like what, so how do we go from this to a quantum modular form? So Zagie introduced this notion of quantum modular forms around 2010, because there were certain behaviors that kept cropping up in various examples of sort of special functions. And so Zagie's article didn't quite give a definition, but I gave a list of interesting examples. And we've already seen one of them, but we'll see it again. But so the main idea is that instead of insisting equality for these modular transformations, we sort of insist something a bit weaker. So the original version of this quantum modularity started with a function from the rational numbers to the complex numbers same. And this is called a quantum modular form of weight k if we take, when we take the difference of the function and minus seed, tau plus d k times the original function. This is a quantum modular form if it's better behaved than the original function. So the original function just could be a complete mess, completely discontinuous, no rhyme or reason. But when we take this difference, we want this to be somehow better. So, you know, in an ideal situation, maybe it's analytic on R minus a bunch of points. But we could insist, we could ask for less and the further, you know, so the most interesting example of Zagie's article did ask for less. And so this was the Kashyav invariant of the figure eight, which we saw before, which came from the factorization of that state integral. So if we take the logarithm of this Kashyav invariant, then we can look at these plots of these are plots from Zagie's article from 2010. So if we plot this logarithm, this, by the way, this J of X is real valued. Just out of luck about this not has extra symmetries. But okay, so this is a graph of what the function looks like from Zagie's article and you can see it's, you know, you can see some structures, but mostly it's a bit of a mess. And so then, if we take now the difference, or here we take, you know, let's take a quotient and then take a log of difference as you like. We don't know if this is better for branching issues. But so now if we take the quotient of these two functions, one evaluated X and one evaluated one minus X, take the log. Then we see this is a much nicer looking graph, it's still discontinuous at each rational but if you approach any rational number from the left, then you will see sort of a full asymptotic expansion. So one way to say this is that it's see infinity from the left and right at each rational point, but discontinuous at the rational point itself. Okay. So this example led Zagie to make this quantum modularity conjecture which stated that for hyperbolic knots, their K'shive invariance should be quantum modular forms in this sense. And so then, yeah, these asymptotic series I was mentioning is approached from the left and right at each rational. These are sort of versions of this series introduced originally this Phi m of h bar. And so this series, I sort of introduced just one special example of this series. But, and this is essentially the asymptotics of this invariant as we send roots of unity to one. But you have asymptotic series when you send roots of unity to any other root of unity of this K'shive invariant. And so you get a whole family of these asymptotic series and so at each of these different rationales you sort of get series associated to these other series. But they're all sort of determined by each other in a sense. And those functions that I gave before, let's see. That's kind of further back. Yeah. So these J sub i functions i equal one and two. And these are sort of the constant terms of these asymptotic series are the roots of unity. But so you can extend that to a full asymptotic series, but they're just the sort of constants. But anyway, so yeah, we get these nice asymptotic series everywhere. But then the question is saw these state integrals could be factorized into sort of combinations of these functions. And we know that they should all have asymptotics and so how they will relate to each other. So this we'll see in a little bit. But yeah, first I want to talk, I want to show how you could prove this kind of quantum modularity in many examples. And so just before doing that. So the example I'll actually do is the Q series not one of these functions at roots of unity like a K'shive invariant. And so here one way you could define a quantum modular form for Q series is you look at sort of horizontal asymptotic. So instead of if we take tau in the upper half plane, instead of sending tau to i infinity and taking, you know, the function evaluated at minus one on tau so sending sort of this q tilde to one. You could send it horizontally to infinity so sort of along a horror sphere. And then the Q corrections are these E of tau corrections they'll all be sort of roughly of the same order. And so then you can really make sense of a full series corrections asymptotically. So in particular be nice that maybe if you take the quotient of you know the Q series evaluated q tilde. And the Q and the Q series itself. Then if you take these horizontal asymptotics. If this has a nice asymptotic series and you could define this as a quantum modular form in a similar way to these functions evaluated at the rationals. But anyway, okay so then how would you prove this so the main tools you need to prove this is of course studying the properties of the pocama symbol or the day of quantum dialogue rhythm. And so the first so yeah the figure eight not in this example because I had invariant this was proved by Garfield leader since I get using sort of positivity of the sum. As I mentioned it was real valued and get positive. And so this was then generalized by betting and Drapau, like 10 or so hyperbolic knots. And the sort of main tool as I said is to study the modularity properties of the pocama symbol itself, which basically just the properties of the for day of quantum dialogue rhythm. And so here the main tool you need to use is this formula of one bits, which rewrite so this infinite product these two pocama symbols on the left, this is related to the for day of quantum dialogue rhythm. And then on the right we have this nice integral expression here that we can rewrite the left hand side with and so this left hand side only makes sense when towers in the upper half plane. But this right hand side makes sense more generally so we can use this to study the pocama symbol at roots of unity. And so the way we use this is we start with our start without you have geometric sum. And then we look at our pocama symbol and we rewrite it as follows so we pull out basically these integral expressions appearing here. And the dialog rhythm the interval of the logarithm we put them inside this five function here and we pull them out and we replace the case pocama by this pocama symbol evaluated at sort of the floor of K divided the real part of one of the tower. Okay. And then we just keep rewriting so we start splitting this sum up into pieces so we we take our original summer now we break it into basically the parts determined by this floor in the argument of the pocama symbol. And then we start pulling out factors of Q that appear and rewriting this Q tilde in terms of some sort of small parameter X and the sum now we've sort of if we keep if we pull out all of the two dependence. Then we're just left with the sum over a compact sort of integral with as we send tower to infinity it gets sort of the number of terms in that compact interval is getting more and more and more. And so then we can reduce the problem now to understanding the asymptotics of the sum that is left. Okay, so then the sort of last thing to note is that when we go from the second last equation to the last, we've basically introduced this J parameter here so if we take some integer J. So that means that we shift sort of some factor of Q at the front but it also shifts this sum that we have inside here over this sort of compact interval. And that's important because when we try and apply a summation method to the second sum and then try and apply stationary phase, the particular J that we choose will be important for actually being able to apply stationary phase approximation. So this is quite nice because being able to apply this stationary phase approximation sort of picks out a J for us. And this sort of determines essentially uniquely the sort of correct correction in these Q series here on the left. And so there yeah in these class of examples you can prove that this these series where a is some integer you can prove that these are quantum modular forms in the sense of we take. If we evaluate this Q series at minus one on tower and send tower horizontally to infinity, then it will have a Q series correction given by the same function. We can multiply it by a nice asymptotic series. And so this is kind of nice but we'd actually like a little more because as I as we saw previously these quantum modular forms although they're getting better. They still sort of discontinuous at each rational which is not sort of the nicest kind of functions better but sort of not so not so nice. And so to get this better we need to refine the original modularity conjecture. And so this was sort of done over the last 10 years in work of Garafletis and Zacchia. And so basically the main issue with the original version of the quantum modularity conjecture was that we should have been taking these sort of bilinear combinations but we just took the largest term and divided by some other term. On the asymptotics and so therefore we essentially just ignored all exponentially small corrections. And so, in a sense, I mean we'll see in a bit but the, you know, we were dividing entries of matrices when we should have been dividing matrices. And so to get that improved version we should work a little harder. And so this is of course best done with grow re summation but in Garafletis and Zacchia originally used sort of some smooth version of optimal truncation. So to do this you need a bit more information about the series I mean for bro re summation of course we know the kind of structures that you need on the asymptotic series. And you need the same sort of things for this application. And so if you use this kind of re summation, then when you look at the asymptotics instead of just dividing the left hand side by this J. To see the asymptotic series fire we take a difference where we replace fire now by the smooth optimal truncation or later the bro re summation. And then we see new asymptotics. And so, as I said this sort of indicates this J should be lifted from just a number it should be part of either a vector or a matrix. And so the matrix perspective is the nicest of course because you can't divide vectors by vectors but you can divide matrices by matrices. And so the what happens in practice is you start with this function J or you know any of these sort of geometric functions you want to realize this as part of matrix of similar objects. The matrix will be indexed by some sort of basis of an associated Q difference equation all of these hyper geometric functions have nice Q difference equations that they solve or they're in families that solve Q difference equations. And the so this is one index of the matrix, and then another index is given by sort of objects associated to the critical points of these functions. So the sort of gluing equations that we saw originally. So in exactly what this index should be in general it's not quite known so if you have gluing equations that have that is zero dimensional so a zero dimensional set of solutions sorry. Then this is just indexed by these points. But more generally, as I said is probably related to something like the number of periods or something but there's a sort of some ongoing work at the moment so at least for a zero dimensional set of solutions this is index clearly by these. And then the this matrix this matrix now should satisfy a sort of matrix version of our modularity where now this before we just had a representation row. But now we have this function omega and these omega should satisfy better properties for example be analytic on a cut plane so extend analytically and this is much nicer than just having seen infinity from the left and right at each. Rational point. And so this is sort of a nice better structure. And this, these kind of statements we can really prove using state intervals. And yes so for the example of the figure eight you know we had these functions from before to this cashier invariant and then some similar kind of functions here. And we can put these into a matrix as follows. And this would be the kind of matrix that would be considering for these modularity properties. And so in this example of the figure eight not. Yeah, this we proved a satisfied these sort of matrix valued versions of this quantum modularity in this very strict sense, which is called holomorphic quantum modularity and so this was proved in a bunch of works with. Yeah, Garfield is good because I have Marineo myself and zag yeah. And so as I said you basically prove this using the state integrals. And a certain duality associated with the Q difference equations that are floating around for these intervals basically you need to rewrite a certain inverse matrix in terms of some other functions so the sort of L. The L functions that appeared before you the LJ functions if you recall. These appear as some inverse or entries of the inverse matrix of this J function here. And so you need this and then you use analytic properties of this for day of so this for day of quantum dialog rhythm is analytic on a cut plane. So basically use that to show that the the state integral itself is analytic and then when you rewrite this omega in terms of the state integrals then this gives you the proof. And so another example is this trap state integral that we had before. So these are also holomorphic quantum modular forms. And the proof is essentially the same I mean you have to do this untrapping procedure but again you just rewrite them in terms of state integrals and then you're done. Okay, so then we're interested in how this relates to resurgence and so I'm pretty much almost out of my time so. Let me just sort of briefly give you the conjectures and then give you a quick burst through an example. Okay, so how this all relates to resurgence was given or was sort of understood in work of Garaflet as good Marineo. And then they basically showed that or they conjectured that these so they had numerical evidence to show their conjectures that these state integrals appear to be the Borel resumption of the these asymptotic series that were appearing from these gluing equations. And so certain combinations of this dating for so certain combinations is sort of the hard bit that's where you have to sit down and do the numerical computations and work out what these combinations are. But in practice this can seemingly always be done. So we can rewrite the Borel resumption of these series in terms of these state integrals. So the fact that these are Borel resum was conjectured by Stavros, you know, 10 years prior. And so the structure of the Borel resumability of these series is also very well understood again, appeared in Stavros's paper where he conjectured this sort of resurgence for these asymptotic series. And so the branch cuts of these series appear in these peacock patterns basically have these towers, which appear as differences is sort of the values of these critical, these critical values of the asymptotic series. And if you package these all up. We also they also give garrifolitis goon marina also give conjectures about sort of the structure of the Stokes phenomenon. So at each of these Stokes lines. They expect that this jumping that should appear as we sort of perform the Borel transform of the Laplace transform across these branch cuts should just be given by sums of a sort of nice packet of asymptotic series where we have sort of a Q correction and then just some integral coefficients. And so this is this conjecture is very strong because in practice we can really use it to compute. And so if we look at this example of specific example from that family that I had before. So we look at this Q series. Then we can look at the asymptotic series that it has so here it has some asymptotics of this form. And then we have this exponential singularity with this dialog rhythm value. You know we have this free factor here this one a square root of delta it's in this number field this quartic number field. And then the first few coefficients you can compute as follows just by doing sort of stand gas integral. So we can start with these functions we can really just evaluate them at points and then divide by we can do a numerical Borel re summation of this asymptotic series. The asymptotic series we expect to appear there. We can divide by this and then we can just start peeling off Q's corrections Q corrections to the asymptotic series and so we start keep going and going and then we can recognize these corrections as sort of a value of our original function. And then we can keep going and we just start pulling off more series so that we can completely pull out one of the asymptotic series out of the four that come from this quartic field. And we can keep doing this for the others as well and we find these Q series corrections to each of those asymptotic series. And this leads us to a sort of conjectural identity of this form, where we have the original function of q tilde equals you know our Borel re summation times these Q series corrections. And so we can do that just above the positive reels and we can do it just above the negative reels and we find two different expressions. And so then we can sort of compute the difference between those Q corrections. Using this sort of the formula just above the reels and just above sorry just above the positive reels and just above the negative reels. And we just basically take a quotient of those two terms, where we use the original conjecture to say these Borel re summations should just be the state integrals that we had before, and we can factorize the state integrals so we can rewrite the function of this form. And in the end reduce it just to a Q series. So in particular this F matrix is just you know the first column is given by these little f functions that we saw before but then you extend this to a full matrix. And then the end you write this nice. You can write this nice formula for a generating series of stokes constants and so in this example you know the first few coefficients given by this. And yeah so this is super powerful you can do it for many Q hyper geometric functions. And you can apply this to many many like all of the quantum invariance associated to SL to a sort of these proper Q hyper geometric expressions as you can always apply these methods to not just for knots you can also do with the closed manifolds and so my thesis I did this for a particular closed manifolds relating WITNs and hat invariance. But yeah, so sorry for going over, you know, bon courage for all of you did very well. Thanks for coming.