Thank you very much and thank you to the organizers for organizing this very interesting workshop. I've certainly learned a lot and It's pretty clear lots of exciting things are happening in resurgence So I want to do several things in this talk first. I want to give you some motivation from physics Most people here are mathematicians, but I want to at least convey some reason of why physicists are so excited and enthralled with ideas in the world of resurgence and Then the latter part of the talk I want to report on some recent work and ongoing work with the video costume, which is much more mathematical flavor. So the physical motivation is actually extremely ambitious and Maybe the mathematicians don't realize just how ambitious this is We're really looking for some non-perturbative definition of quantum field theory that can address problems such as what's the difference between Minkowski quantum field theory and Euclidean quantum field theory. This is a very deep problem that people have thought about for a very long time It's extremely difficult. It's especially extremely difficult if you start to think about quantum gravity and Resurgence introduces some new ideas that can help us at least make new approaches There's a very famous problem in field theory called the sine problem related to finite density quantum field theory. I Would say it's one of the top ten problems in theoretical physics It has impact not just in particle physics nuclear physics atomic physics condensed metaphysics, but also in chemistry And it's a complete roadblock People have known this problem for a very long time and there's essentially no progress in 80 years Another similar level problem is non-equilibrium physics to really consider real-time time evolution in non-equilibrium systems without using just first-order perturbation theory to do it using pathenthrals there's also a really serious problem in many fields of physics and Resurgence has some new ideas about how do we even approach this this problem? So I want to talk a little bit in my motivation about phase transitions, which are related to both of those questions and the common thread which we've heard many times during the conference already Is to make sense of the idea of analytic continuation of path integrals? So to be a little bit more precise This is what physics gives us. This is the Feynman path integral the amplitude for a quantum process is a sum of phases Either the eye and action. That's what it is.  So for certain things like static thermodynamic processes even in QCD We can make this transformation to serious calculations and they agree with experiments It's an extremely successful program And the reason it's done is because on this side the problem makes a lot more sense Mathematically we have the Feynman-Katz formalism. We can do Monte Carlo and we can actually calculate things But this is the real world There are certain problems where you can't make this transformation in the Standard formal sense and that's why people are excited about resurgence we've already heard from Professor Konsevich that even in these finite dimensional exponential integrals There's a lot of interesting mathematics and topology related to making these analytic continuations and rotations and And so it at least raises the idea that since even in the finite dimensional context We need to think about complex Complexification and analytic continuation It certainly strongly motivates the idea of trying to think about doing the same thing for path integrals Okay, it doesn't solve the problem, but it at least tells you that it's a good idea to take this seriously And so this is the question that's Driving most of what I'm doing can resurgence and some form of infinite dimensional picket left-shits theory Be used to solve these really big problems in quantum field And a step along the way is to understand something about phase transitions So here's a sort of motivating thought That one of the important things coming from the general set of ideas behind resurgence from Professor E. Carl is The different critical points or special points in a problem can actually be related to one another in Quite surprising ways that people hadn't thought about before And so forget about all the details of how to do Borel summation Just the ideas in resurgence are important. They've actually raised Ideas of things to calculate that physicists hadn't even thought about calculating before Okay So this picture is supposed to be some sort of representation of these different critical points interacting with one another So let me be really specific This is a major unsolved problem in physics the quantum chromodynamic phase diagram is a function of temperature and can't vary on density We have techniques for calculating at zero density and very high temperature at zero density in very low temperature At zero temperature in low density and zero temperature in asymptotically high density We know how to do those calculations to some order But all the interesting physics lies in the middle And what we don't know how to do in any reliable way is to calculate in this region to study things like neutron stars from first principles and nuclear matter from first principles and So can we actually use some of these ideas to? Take our asymptotic methods out here and bring them into this interesting physical regime So those are the sorts of problems that physicists are interested in And if resurgence can help us to solve any of those types of problems, then it's big news okay, so So more precisely, can we make some sort of mathematical physical and also computational sense of an expansion of a formal path integral into some sort of sum over saddles or thimbles Remember that a thimble is some sort of infinite dimensional version of a steepest descent contour So on a thimble the imaginary part of this thing in the exponent should be constant In which case you bring out a phase and What you're left with up to some Jacobian here is a well-defined integral that you can calculate in any number of ways You could do Monte Carlo. You could do other semi-classical techniques So if you could understand how to do this decomposition even approximately Not it not as an equality, but even approximately this would give a new way to do calculations that at the moment We don't know how to do so I've been kind of This is shorthand of some partition function depending on some coupling. We'll call it H bar But actually in realistic theories It will depend on more than just one coupling for example We've already heard many examples during these this week where there might be some another parameter Which I'll call it n in a matrix model It would be the size of the matrices in a gauge theory It might be the u n of the gauge group It might be the number of fermions in a theory but typically this partition function will depend on several parameters not just one and In phase transitions only occur the n could also be the volume of a system for example phase transitions technically only occur in infinite systems But of course we all know that's not quite right because we've all Played with magnets which are certainly not infinite. They're just very large Very large numbers of particles So we're interested in probing this end-to-infinity limit And we know we don't exactly have to go all the way to end to infinity But we have to have some understanding of what happens when n becomes large and We expect that phase transitions will happen in this limit But we should be able to see them as we approach this limit somehow and The interesting thing about and interesting thing about resurgence is that if we have some trans series description in one phase It will go through some weird metamorphosis and transmute into something very different looking in another phase so can we understand how that happens and Learn some physics from that transition Now there's something called liang zeros which I think Riccardo mentioned in his talk It's a very deep idea in physics is that in a finite system. There is no such thing as a phase transition But you can already see hints of a phase transition in a finite system by looking For zeros of the partition function, but in a finite system those zeros Will be off the real axis if the axis is a temperature or coupling or whatever is controlling the transition There'll be complex but as you approach this so-called thermodynamic limit these complex zeros will pinch the real axis at the location of the phase transitions is an extremely deep idea due to Lee and Yang and we'll see How that works in some cases For the mathematicians think of a partition function If you live in a pen of a world think of it as a tau function or as a path integral or a partition function If you're familiar with statistical mechanics think of it as a partition function Okay, so let me know run through a few examples and then I'll Turn to the more mathematical stuff So let's start with the Matthew equation. This is a quantum mechanical model I can think of it as a quantum mechanical model, but of course you can think of it as a classical system And I think everybody's familiar with the classical model where you have regions of stability and instability as that these two parameters q and a vary Okay, so there's this very intricate Spectrum if you like from the Matthew equation I Can actually rewrite this in quantum mechanics language I'll divide by q and call it 1 over h bar squared and a over q will be like an energy Okay, so I can rewrite it as a Schrodinger equation But of course I can also think of it as a quantum mechanical path integral and now here's the spectrum the energy you As a function of coupling h bar And you see that there are unfortunately doesn't show up very well. This should be these should be shaded these regions, okay? So there are these so-called bands and gaps in the spectrum corresponding to these stability and instability regions here They've just rescaled everything Now I could give an entire talk on just this picture. So let me point out a few things This mysterious end here doesn't appear in the equation In the equation. There's an h bar and there's an energy N is a monodermy parameter It comes from the boundary conditions you impose on the solutions and it corresponds to these bands The blue is the bottom of the band the red dash one is the top of the band and they're separated by gaps and You'll see that the spectrum is actually labeled not just by h bar, but also by N And there's some integer labeling the band zero one two three, etc These dashed lines here are the top and bottom of the potential one and minus one the cosine potential And so from physics we know from solid-state physics We know that this is some model of a crystal So if you're near the bottom of the potential Down here. It's as if you're just solving a harmonic oscillator problem And you just have discrete levels Okay, but because of all the other potentials in the tunneling they're actually Widened into bands which are exponentially narrow near the bottom of the well But as you go up towards the top of the well, they're broadened into wider bands okay, if Your Energy is way above the top of the potential very high up than the potential is essentially irrelevant So it's essentially a free particle But it's still periodic so the energies You have these narrow gaps and they go like n squared because it's a periodic problem And so in physics we learn how to calculate the exponentially small splitting And that's what I would call non-perturbed of semi classical physics of one instant on approximation But you see that as you approach this what's going to be a honest phase transition here at the top of the barrier That one instant on approximation is hopeless and you need to sum all of these non-perturbed of effects to get the broadening of the band and Going to a region where the bands are broad and the gaps are now very narrow One thing we've learned from resurgence is that the formal series that you just do by getting Get by just doing perturbation theory in this well for example For the nth level in this well you get a formal series in H bar for each end Those series of course are divergent They develop into a trans series using resurgence And we've learned that in fact that formal series encodes all the information About all orders of the trans series. It's really remarkable I mean it's something that could have been done a hundred years ago, but nobody even thought of doing this calculation But if you look at the problem through the eyes of resurgence, it's an obvious calculation to do So now let me talk about this as a phase transition if you now look near the top of the potential You'll notice unfortunately. This is really not showing up very well. This is a band. This is a band This is a gap you'll notice here that the width of this band is equal to the width of that gap and The width of that band is equal to the width of that gap as you make the transition The bands have broadened and if you're coming from above the gaps have broadened and at the transition point they're equal Okay, and it's a it's a real phase transition and the parameter that's driving the phase transition is what physicists call a tuft coupling It's not h bar. It's not n. It's n times h bar Which is the classical action and that's of order one at the top Yeah, it's in fact eight over pi and the this normalization and it's a transition from the physics of Isolated atoms in a crystal Where they're separate So these are narrow bands, so that's called the tight binding model in solid-state physics It's a transition to the free electron model where you've got conduction electrons flowing in a crystal Okay, so even the language we use to describe the physics here and here is completely different and Correspondingly the trans series that we use to describe the spectrum is completely different here and here And it's interestingly different here also And so one thing I want to be able to understand is how this Trans series structure here, which is divergent Changes as you tune this tuft parameter and these things are no longer exponentially small And you need to sum all instantons and you now go into a new regime Where the physics is completely different your language is completely different the expressions are completely different and even The expressions here are divergent series the expressions up here are convergent series So how does that happen? How does a divergent trend series turn into a convergent? Expansion as you vary this tuft parameter and the physics of this is this phase transition is is driven by what we call instanton condensation where The one instanton approximation is no good. You have to think of a gas of instantons condensing The other thing that's very interesting. There's this beautiful old very short paper in the solar state literature by I don't even know how to pronounce his name correctly picnic maybe That The way to compute these very narrow widths of gaps high up in the spectrum is to look at complex instantons Because if you think in language of wkb You would want the turning points when the energy is way above the potential But there are obviously no real turning points there but there are complex turning points and if you find the classical solutions that interpolate between those complex turning points and Evaluate the action you get exactly the widths of these gaps Okay, so you can also think of this transition as a transition from a dominant saddles That are real instantons to a region where the dominant saddles are complex instantons And all of this can be worked out in complete detail in this simple model Now the simple model is not so simple because actually There's an exact mapping of this to a famous n equals two supersymmetric quantum field theory The cross-off at al and this is made very explicit in some nice work by Miranoff at al and many other people So even though this may sound very simple because I'm just talking about a quantum mechanical model There's actually some very interesting quantum field theory buried in this problem also All right, so let me move on. Let me give you another example And it's a pleasure to be talking about that this here because this was Edward Brazzan's PhD thesis He was a student of it's Xen and in his thesis he studied the transition from What is effectively real instantons to complex instantons in the problem of vacuum pair production The vacuum is unstable to the application of an electric field because the electric field can accelerate apart electron positron pairs and create real electrons and positrons and if you do this problem with a Time-dependent electric field with one frequency omega It's an extremely interesting WKB problem and you get an expression for the rate of this pair production Which looks exponentially small And it has some pre-factor Involving the electric field the mass of the electron speed of light e h bar these fundamental constants and some function Which is some arc tan or something and can't even remember what it is But it's some explicit during a metric function of this Adiabaticity parameter which basically tells you if gamma is large. It's a very rapidly varying field and if gamma is small It's a roughly constant field So there's a very interesting transition This function here goes to one when gamma goes to zero and you just get this exponentially small thing Which looks like this sort of one instanton approximation here Whereas if you go to the other limit where this thing is rapidly oscillating It develops a logarithmic behavior and what you thought was exponentially small is in fact a power So it's actually a power of the perturbation Epsilon And that's the perturbed of limit and this one expression interpolates between these This is this was what presented in his PhD thesis building on earlier work of Keldish in the world of ionization And this expression is extremely interesting because from the exponent here the power We recognize this as the number of photons for doing this Pair production because this the barrier to production is 2 mc squared That's the rest mass energy of the electron positron pair It's four because it's a probability. So we've squared it and it's that divided by the photon energy So that's the number of photons needed to not tunnel, but to go over the barrier Okay So this is a also a phase transition between this non-perturbative type expression to a perturbed of expression As tunneling versus multi photon ionization or pair production in this case extremely beautiful calculation and This can be interpreted also in the language of this transition from real to complex saddles and instantons So this is now more complicated than the quantum mechanics Problem because this is real quantum filter a quantum electrodynamics But it's still not quite full quantum filter. So let me go to a more complicated system So this is the essentially a relativistic cousin of the example that Marcus talked about the other day And his beautiful talk about these superconducting systems There's a famous model in quantum field theory due to David Grosse and under in the vote From the mid 1970s when they were struggling to understand what asymptotic freedom meant in Qcd so asymptotic freedom is this weird property that in certain Systems the strength of the interaction actually gets weaker as Things get closer together, which is completely counterintuitive if you think about it and this simple model of two-dimensional quantum field theory with a four fermion interaction Has the property of being asymptotically free having Dynamical mass generation and chiral symmetry So it's a very nice model that captures some of the features of Qcd while being much simpler to analyze than Qcd It has a chiral symmetry breaking phase transition in the large number of flavors number of fermions limit and The physics of it is a relativistic version of something. That's also well known in the condensed metal literature Coder piles instability that in one dimensions With these four fermion interactions things tend to clump and dimerize and form kink anti-kink crystals So the phase diagram of this could be studied at finite temperature and finite density think back to that problem I mentioned at the beginning of the Qcd phase diagram This is a baby version of that problem and in this case you can actually solve it in detail Unfortunately people first solved it in detail. We've got the wrong answer So in textbooks and review articles, there's this phase diagram, which is actually completely incorrect The true phase diagram is the thing on the right That in fact as a function of temperature and chemical potential think of that roughly as a measure of density There's a region where you develop crystals It's a periodic structures in space and As you vary temperature and chemical potential you can go from a phase where this M is the expectation value of psi bar psi is Either zero a non-zero constant or an x-dependent crystal and The breakthrough in this came from work by Michel Thies who solved this problem using Hartree-Fock originally numerically and then analytically and then later it was understood in the language of saddles Solving this so-called gap equation, which is the sort of saddle equation in these fermionic systems There's a very very tricky computation You have to choose some here's a Dirac operator with some potential Sigma of x that's the expectation value of psi bar psi. You have to evaluate these determinants Such that the variation with respect to sigma is equal to sigma It's an extremely non-linear non-trivial problem and you need to solve this problem at finite temperature and finite density Okay, fortunately it can be done by some magic. This is an integrable model So you can think of the thermodynamic potential the the partition function roughly speaking. It's some integral over the density of states with these Fermi factor and The Ginzburg-Lando expansion says you expand this in powers and powers of derivatives of this condensate field sigma This is gap that Marcos was talking about So it turns out there's an amazing well, maybe not amazing that there's a useful Relation between this Ginzburg-Lando expansion and the MKDV hierarchy turns out these quantities here when you expand this Are exactly the conserved quantities of the MKDV hierarchy? You can now use all the machinery of the integrable systems to actually solve this gap equation at any temperature in any chemical potential and So here's an illustration of what happens if you were only able to do that to a certain number of orders of this expansion Because this Ginzburg-Lando expansion itself is divergent Expansion so if you just go to the first non-trivial order, you get a crystal phase In shown in red if you go to the next order, it's like this the next order is like this This is in the vicinity of this trichritical points. You see this little wedge here Yes, yes, they are exactly the conserved charges So this is a divergent expansion and you fill out this full crystal region as you go to higher and higher orders of this expansion And you see from this picture already Unfortunately, this is not showing up very well This phase transition point here where you go from this massive phase Homogeneous phase to a crystal this point is the most difficult point to access using this Ginzburg-Lando expansion So now you can go and look at the sort of standard ways of doing this calculation a low density expansion a high density expansion And if we look for simplicity on the zero-temperature axis You see that the high density expansion row is the density is a nice convergent series The low density expansion It's convergent because there are only two terms But it turns out that there are exponentially small terms That are not you don't see them in the normal low density expansion But since we have the full solution We can see that those terms are there and the appearance of these terms are what is responsible for this phase transition here So this is an extremely non-trivial Extremely is maybe extreme It's a non-trivial quantum field theory. It's integrable. So it's not extremely non-trivial, but it's an honest interesting Quantum field theory in which we can see some very interesting phase transition behavior Triggered by some trans series structure in something that we usually just calculate using perturbation theory There are many other models. I'm going to have to zip through some of these just Let me flash them 2d Yang-Mills is a famous example That's very much like the matrix models examples that we've heard about in several talks This has two ways of doing the calculation of the partition function So there are two parameters one is the area of the sphere and one is the n of the UN or SUN or whatever you want to do the gauge group You can either think of it in the Hamiltonian language as a sum over the spectrum So this is sum of the representations and this is the chasm here Or you can think of it as a saddle expansion the sum over saddles of the partition function their monopoles And that gives you a different type of expansion and you notice that this one has a over n and this has n over a and The relation between them is exactly a possum duality Transformation between them and the phase transition the third-order phase transition discovered by Douglas and Kazegorff is Exactly at the sort of self-dual point where the critical area is pi squared in this normalization So this type of transition happens in many many systems. This is a particularly well studied and elegant model But it's representative of many other such transitions It's impossible to talk about phase transitions without mentioning the icing model It's sort of the most studied phase transition in all of physics. So the two-dimensionalizing model has a well-known Kramers-Vanier duality relating high temperature and low temperature it has a phase transition at some critical temperature somewhere in between zero and infinity and The expansions around zero temperature and around infinite temperature are actually related to one another because of this duality But they're both convergent So one of the points I want to make is that even in systems where there's convergence of expansions You can still use some of the ideas of resurgence for example If you expand around zero temperature just learning the radius of convergence of that tells you the critical temperature So you've learned something if you now look at the large order behavior of those convergent Coefficients around zero you learn the nature of that phase transition. You learn that it's logarithmic The fact that there's this duality means you can do the same thing starting at infinite temperature And you do that and you learn that it's not just log of t minus tc. It's log of the modulus of t minus tc okay So even in a situation like this where the expansions Convergent you can use some of these ideas of relating expansions in one region to determine information about expansions in other regions It's actually much deeper than that because we can now look at Correlators between two spins on this ising lattice And if you go along the diagonal of a square ising lattice these correlators are tau functions for penal base six Penal base six with a special choice of parameters. There's an n here Which is the number of diagonal steps. So n is sort of an integer Right and the boundary conditions are actually special boundary conditions. So they're they're what Jean-Pierre would call non-generic right these are very special And that's what's responsible for this logarithmic type behavior in the in the ising system these these correlators have simple topless determinant representations in terms of hyper geometric functions so For these special versions of Penal Bay six the problem actually linearizes There's this very clever You can think of it as a change of function change of variables which completely linearizes the problem And that's not true in general of course in Penal Bay six You can study the scaling limit where this correlation length goes to infinity and you approach the critical temperature It's a little bit like a tough Limit and that was the big discovery back in the 60s of McCoy and Wu and Tracy and Baruch that in doing so the Penal Bay six reduces to Penal Bay three. I also want to make the point that these expansions here have convergent conformal block expansions at both high and low temperature and This was first pointed out by Jimbo back in the early 70s and Recently there's been a lot of activity from the field theory side showing that these There are these explicit conformal block expansions of tau functions And I want to just point out that this is an expression of resurgence Because the expression for the tau function is an instant on some Again, this is along the lines of what Riccardo talked about the other day. It's not just instantons It's also anti-instantons. So it's a sum from minus infinity to infinity This is the instant on counting transduce parameter and there are some coefficients which are explicitly expressed in terms of barn's functions and then there are the temperature depend temperature variable dependent stuff Which is expressed as sums over young tableau young diagrams Think of it as sums over partitions And the key property is that there's a Prefactor t to the sigma squared and sigma is the other trans series parameter So there are two trans series parameters because it's a second-order problem and this is quadratic in sigma That's the key feature and why is that important? because this instant on some says that in The different instanton sectors all you have to do is shift sigma by n So this is the origin of that n-squared type behavior that Riccardo was talking about the other day And it just comes from this conformal block expansion. And so for Penelope 6. There's a completely explicit expression combinatorial expression for all the coefficients in this expansion and By reduction you can do the same thing for any Penelope system Okay So this is yes Quick questions. What is s? s it e to the i theta So it's an instant on counting parameter, but counting instantons and anti-instantons So, you know, there are details here. I have to talk about which lines I'm talking on about for this to make sense I don't want to go into that Simple way to see why or the origin of this conformal block expansion Yeah, but I don't have time to talk about it now. I can tell you about it later Okay, I just want to make the point that this is by now very well understood in Penelope world But it's an explicit expression of resurgence because it says that All you need to know is this thing this expressions when n equals zero and Then any instanton number you just shift sigma by n Okay, completely explicit expressions. This is very very beautiful work Essentially upgrading this idea of Jimbo to all orders I Think I'm gonna have to skip through this stuff very quickly So there's another example along these lines where you can do things in complete Detail, there's a famous matrix model called the gross Witten-Wadier Unitary matrix model Integral over unitary n by n matrices with a coupling one of a g squared It has this a third-order phase transition just like in the 2d Yang-Mills on a sphere This is driven by pain of a 3 at finite n and pain of a 2 at The strict the double scaling limit near the transition and this transition this double scaling limit Is this Tracy Witten transition which occurs in I don't know I could make a list of 15 different problems where this is the transition and the reason it's so universal is it's the nonlinear version of The single turning point transition in quantum mechanics, which is described by the airy function This is basically the nonlinear airy function. And so this is why it shows up in all these problems This also has this topless determinant representation As n goes to infinity what gross and Witten pointed out was there was a kink in the specific heat That's a signal of the phase transition But I think I'm going to have to skip through this quickly. You can map it exactly to a pain of a 3 guy you can check in weak coupling, there's a large n expansion the former one of ran expansion has Exponential corrections, so there's a full trans series here. You can check that the low-order Behavior, sorry the high-order behavior of these perturbed of coefficients for any t has this characteristic low-order high-order expansion in terms of the action With coefficients that show up in the expansion of these guys all of the expected parametric resurgence expressions are completely explicit they Work out as we expect them On the strong coupling side even though everything is completely convergent When you go to this strong cut this tough limit they become divergent expansions But again, they have the correct large-order low-order Relations you can calculate the liang zeros since the expression for the partition function is just a determinant of Bessel functions at finite n at finite n you can just find the complex zeros It's not difficult The expectation is that they form some sort of region like this when you zoom in on the transition point Which is where it's pinching the real axis here at one. This is t and so You can look now at the complex zeros of the panel of a to Hastings-McLeod solution Which form like this and so this wedge here is just that wedge there zoomed in At large n okay, so you can make all of these ideas extremely explicit in this case because you have total control over Finite n as well as infinite n So particularly beautiful Okay, I could go on and on and on about different examples of phase transitions that we do understand But I want to switch to a slightly different tack just in the last 20 minutes Actually make a bit of a reality check here and this is recent and ongoing work with a video cost in Which is that doing these calculations is really difficult and I want to talk now about a really difficult problem Not one of these nice integral problems where you know a lot of structure Just think of some really difficult problem such as QCD Yang Mills Hubbard model something That we're not going to be able to solve completely Can we use resurgence to still do something useful? so In physics this means can we extrapolate from finite information to something? realistic in Mathematic that so that may sound to the mathematicians like a kind of boring problem But I want to convince you at the end that there's a lot of very beautiful mathematics here involving all of these guys and today, I only have time to show you some explicit numerical If you like experiments, but I think they'll convince you that something interesting is going on and in current and ongoing work. We have actual theorems That can tell you that if you give me n terms of an expansion of something We can now tell you what degree of precision you can hope to get in this region of the parameter space Real complex wherever okay, so these are very strong results That don't exist at the moment and they do the route to these results explicitly uses resurgence but Let me not state theorems. Let me show you some Experiments so again, it's motivated by this sort of stuff Imagine you only had a little bit of information here. What could you possibly learn about the middle? Okay, so resurgence suggests that some sort of local analysis like this is supposed to encode global information so the question is how much global information and The reason you might think that this could be successful is that if you work in the Burrell plane and if what you're calculating is actually a resurgent function, which we Usually expect in physics context that these set of natural problems Then it tells you that in the Burrell plane. There should be some orderly structure Okay You don't in advance. You don't know how orderly but it's not going to be totally random and a mess Okay, so now can you develop extrapolation techniques that take advantage of that? And I'm going to illustrate it with Penna V1 because I think everybody in the room is familiar with Penna V So you can relate to it But also because Penna V1 is interesting for physics So that's another reason but the main reason is that we can do Extreme precision tests because we know everything about Penna V1 analytically Okay, so let's start here's Penna V1 You make a formal expansion at large x and You see already from the balance of this that there's a square root of x over 6 I'm going to choose the minus sign for a special reason and then it's a simple matter to generate these coefficients And this is what the first few of them look like Okay, and this expansion explicitly develops the tritron quay solution to P1 and I'm picking the tritron quay because in some sense The tritron quay is the most difficult because it's the most finely tuned You can also do this for tron quay and the general solution, but that's less interesting. This is the biggest challenge here Okay, so the question is if I take How many co if I take some number of these coefficients and these coefficients are just developed out here at x equals infinity and How much do those coefficients know about the rest of the complex x plane? Okay, so a video jokes that one coefficient is not enough a thousand of coefficients is too many It's somewhere in between So how many do we need? So again as Riccardo pointed out the other day There's this inherent five-fold symmetry of the equation, but I'm just going to start with data That was developed out here, and I'm going to ask Do those coefficients know about the global structure of this solution? And how much do they know? Okay, and of course there are these phase transition stokes and anti-stokes lines that are the most interesting points So if you look at the plots on the digital library Unfortunately, they use the opposite convention of x so it's flipped, but it's as if we're trying to come in on this separatrix And then go off into the rest of the complex plane, you know, it's exponentially sensitive to the boundary conditions So let's start first of all no big surprise the coefficients are factorally divergent Fine, so we make a Burrell transform fine If we had this is convergent has radius convergence one in my normalization So if we knew this function if we really knew what that function was we could just write this integral representation of the original solution Okay, but we don't know this function. We only know something about that function So if I only have a certain number of coefficients, what can I do in the Burrell plane? Which gives me a well-behaved? Burrell of P which somehow captures global information. That's the challenge So I truncate this expansion because I can only calculate some certain number of these coefficients So the first obvious step to do is to make a party approximate for this. Okay, it's a finite order polynomial You make it into a pad a one of these almost diagonal pad days because there's this sort of odd power here That's not important. This is completely algorithmic You can just press a button on math Mathematica or maybe it'll generate the party for you zero work and then the zeros of of P and Q Tell you some information some global information about this function B of P so for example you can look at the polls and You see these pictures similar to what we've seen already in the conference These polls are trying to represent a cut whatever that means. Okay, since you're in Pad a world there can only be polls You're working with polynomials There are some results in the literature about how Pad a tries to represent cuts Looking at the density of these the distribution of these polls. You can actually learn information about What type of cut it is? That's actually extremely difficult. It turns out there's a much easier way to do it But at least looking at this we it looks like branch points from plus or minus I Now this step already is fantastic If you take the Raw, this is with n equal 10 10 terms if you take the raw Padae From out to infinity down towards zero. This is what you would get if you apply a Padae to it This is what you would get and this dashed line here is The exact answer starting at zero. So you go all the way down to 0.2 0.25 This is from 10 terms where you press the button once it's not difficult. Okay, so this is fantastic, but we can do much better The much better is a magic step that physicists knew about in various contexts long ago But I claimed that the reason it works is because of resurgence so you make a conformal map to bring these The this doubly cut plane here into this unit disc and The reason this is important is because actually this is not just one cut This is actually many cuts if you believe that it's a resurgent problem But there are and they're layered on top of one another and you can't see that here You can't see it easily So these repeated cuts here Should map will map under this conformal map to here here here okay So now you just do something also algorithmic and trivial you map the finite Burrell transform into the conformal disc you re-expand and repad day totally algorithmic and Now you look for the poles there and this is what you get The leading one the sub leading one the next one the next one So you've suddenly discovered the resurgent structure just by making a conformal map okay Also Extremely simple step easy to implement and it has this feature that it separates these poles Into the separate resurgent branch points Okay, there is the poles outside of this disc Our higher Riemann surface contributions because of course this thing must be convergent inside the disc So there can't be poles inside the disc So there's also resurgent information buried in them, but I'm not going to talk about it today I just want to make the point that this conformal map Reveals the resurgent structure in an instant so we can now take advantage of that by working with these conformal map Part days and we find that we get now much much much higher precision and at the end of the talk I'll explain why where that's coming from But I stress that physicists already knew this Well, they knew that making this conformal map was somehow better But I don't think it was appreciated why and how much better They're exactly these Yeah So if you map back you see in the in the original p-plane you see now the resurgent structure so just a simple test you can now use that version of the Burrell transform and You get the solid blue line Compared to what you get if use if you use Padé Burrell and you see you get Better precision down to lower values of X on the real axis, okay But now let's do some more precision tests Let's look for resurgence so we know that the leading singularity is a plus or minus I and The coefficient and we know that it's a square root branch cut we can test that in many many ways So you can ask what's the coefficient? And that coefficient should be the stokes constant and for Penelope 1 we know the stokes constant So here I can plot When n equals 10 right 10 coefficients only The approach to this critical point, so this is the Burrell plane P so I'm going to approach this critical point here and This dotted line is the stokes constant For Penelope 1 and the red line is if you used the Padé Burrell and you see it's approaching fairly well You know this is point nine six here But it breaks down as you approach the the branch point Whereas the black line is the con formally mapped guy and it approaches the stokes constant perfectly Okay So I can get five digits of precision with ten input terms approaching this leading singularity Okay, whereas with Padé Burrell. You can't even get one digit of precision from that I can now do Right so along this direction where you do your Burrell integral if x is real and positive Everything's fine, but if you want to rotate x in the complex plane You have to be able to rotate P in the complex plane and the most interesting Difficult point is going to be as you approach this These lines of cuts Okay, so now I'm going to skim along this cut And if we do it with the Burrell Padé Burrell as you approach the first singularity It's diverging as it should but it's not diverging. Well It's missing as you get very close and then when you go beyond that first one you just get garbage Okay, and there's no sign whatsoever of the further singularities Now with the conformally mapped guy I Can skim perfectly smoothly along this cut You see the first one you see the second one if you zoom in here you see the third one So let me zoom in on the second one. So this would be the second singularity here there's a jump there and Resurgence tells you that the coefficient of that jump the size of that jump should be a half the Stokes-Stomson squared Which is this which it is Okay So you can even do high precision probing of not just the leading singularity, but the sub leading singularity, etc And that's why you can get such incredible precision with this guy Which you can't get with this guy Okay in some parts of the complex x-plane Okay, so we can also zoom in from infinity down to some point and repadé. That's another option and In order to do that you need very high precision values for the function and its first derivative here But that's exactly what this method gives you gives you almost absurd precision so now let's do some diagnostics, let's test this and We can test penalty one to very high precision because we know everything So let's go from infinity down to zero It's a central connection problem There's no closed solution for this in penalty one and it's one of the standard test cases that numerical analysis people do and The very best numerical analysis people get 14 digits of precision Okay So with this method starting with 50 terms at infinity we get 64 digits of precision Okay, so you can calculate y y prime y double prime at zero and it's zero to 65 digits Okay Starting with just 50 terms if I start with 10 terms I get you know 20-something digits of precision And the very best numerical analysis people get 14. So why is this happening? Well now I can go beyond This is the x-plane so coming from infinity I just went down to zero, but I can keep going I can go to negative x and we see the poles on the negative x-axis just Just plot so now we can go into the complex plane and just wander through the complex plane And we know that as we cross these stokes lines and anti-stokes lines There are transitions exponential things are going to appear and disappear and come in So how much of this global information is encoded in this input data which just came from here? Okay, so again recall this five-fold structure. We're going to go this this and eventually into here so there's connection formulas from the work of Kitaya from this beautiful paper of Stavros and company That relates the behavior of the function say along this line to the behavior of the function along this line It's a very precise relation that this difference here behaves like this so here's the real part of this difference and I Don't know which is which the blue one is This and the red one is this at large x so you see they agree very well Okay This is just with I think this plot was with 50 terms Now we can look at connecting stokes line so we can connect this stokes line with this stokes line and The difference between these two with a particular phase is exponentially decaying with a pre-factor depending on the stokes constant again Perfect agreement at large x Remember this is coming from a finite amount of information here And we're talking now about the most difficult directions in the complex x-plane And with just a certain number of terms here we can map out all of this complicated And analytic continuation properties of this pen of a solution so It really is true That these this expansion is encoding in some clever way this global information So we can now go into the pole region so there's this conjecture of to Brevin So the old work of but through was that at large distances the poles Look like the poles of the vice-truss functions suitably scaled For the treat on K these poles only appear in this 2 pi over 5 wedge For the other ones they can leak outside depending on the stokes on the trend series parameter But for the treat on K. They're only in this way So here they are from our Extrapolated solution the blue dots that we're starting with 10 terms the red dots are starting with 50 terms Okay, they form this beautiful lattice. You can use old work of a video Using this transesentatics where you resum the instant times and we can use this to make a prediction for the Location of these poles as you cross into the treat on K region You just insert the stokes constant for C and this is the agreement I plot it for the first line of poles So this is also very much like what Riccardo was showing as you go into this region There's these trans series out here turn into things that look better described by Meromorphic functions with poles and the locations of these Poles are perfectly captured by this extrapolation The pain of air equation tells you that the expansion round any pole looks like this There's a second order pole with residue one there's one more constant the other boundary condition parameter So we can just test our solution and expand it around this first pole And see how well it satisfies this structure and the answer is every coefficient there matches to 30 digits in the Okay, so in fact, there's no real sense to describing what happens in pen of a 1 at x equals 0 because x equals 0 is not special But at the first pole it is special And so now we have high precision values for the location of the first pole and for the first constant about the first pole Which has a spectral interpretation in terms of the cubic oscillator Okay, so I'm over time already. So let me quickly flash two other examples That you when can study so here's a problem from quantum field theory the cusp anomalous dimension So you can start at strong coupling, which is a divergent expansion You can start at weak coupling which is a convergent expansion and these black dots are what you get from applying this Mapping and you see it interpolates perfectly well between the two limits Since so gay is here. I'm going to show a few pictures related to his Beautiful work on complex transomance theory. So one of the examples in their paper If you do Padé Borel, you generate this cut like thing If you do this and it's you know, you have to do some fancy stuff That takes advantage of the special structure of the problem to recognize that this is actually a sequence of poles If you just to make a conformal map of this it again breaks up into the images of these Poles so you see the resurgence structure immediately simply by making the conformal map You can then plot along that axis along this and The let's say I'm plotting the real part of that here and the blue lines is the conformally mapped guy and the red line is What if you would get if you use Padé Borel and you see for the first singularity? They're pretty much the same But the Padé Borel degrades as you go to the higher singularities Which that means that if you wanted to do some analytic continuation in the physical variable you would need a well-behaved Borel transform near these singularities and what this is showing is that the conformally mapped Borel transform is much better near the singularities Okay, so let me conclude because I'm over. I'm not going to bother saying some of this Let me just say a few words about Why this is happening? so Padé is well known to involve three term recursion relations as You change the order of the Padé as soon as you hear the word three term recursion relations You should think of orthogonal polynomials Soon as you think of orthogonal polynomials and making the polynomials larger You should think of Zego asymptotics of orthogonal polynomials And now the idea is that if you apply this mechanism and in particular Some results of Barry Simon and Dominic about Characterizing the asymptotics of these orthogonal polynomials you can apply that in Borel world in in the Borel plane To your representations of the Borel transform and you can get really precise estimates of the errors Okay, so don't work in the physical domain work in the Borel domain where there's more structure and Using these properties of continued fractions, etc That's why you can actually develop estimates of how much precision you can get So that'll be my next talk. So, thank you Yes, two things and if you do conformal mappings, yes, perhaps you could simply Are on the other cases where you can actually perform the integration because now the Laplace Transform is on the on the finite segment on which the series converges and then there are many options. Yeah That is useful But remember we where you go in the Borel variable Depends on where you want to be in the physical variable, right? You rotate in the physical variable that corresponds to some Transformation in the Borel plane. So in certain cases it might be better to work inside the Conformal disc in other cases. It might be better to go back to the people, but you have both options I have a question. So if we look at The tree trunk care of fine level one, you wrote down the first stokes constant That you know can be numerically Approximately very well And and then recognize Is there For that stokes constant Yeah, of course so Yeah, it's square root of That was the recognition No, no, I mean you mean in this method I'll talk about how to numerically compute and then maybe the stokes constant for P1 is known from Isomodromic deformation, but also from asymptotics not from isomodromic Is there a formula for the stokes constant that one can From asymptotics, yes, so if we do did this right you come at it from both directions and you apply Single validness and you get it immediately. This is in his I don't remember which paper but So either asymptotics or osomodromic you get the closed-form stokes constant With this method you can get it to 100 digits And you either recognize it and work with it or you are happy with a hundred digits precision and you just use it Okay So and what prevents you to get all the stokes constant that Ricardo was looking for? If you want them numerically nothing exactly because For getting the stokes constant, there's one stokes constant in this no in the treat on K Give two examples of Magical confirmational summation that you didn't explain how to find it and the reason so the reason yeah Sorry, I ran out of time. So so the okay, so let me tell you the secret so So in this part a transformation so part a turn something into a ratio of polynomials right But you can represent that in more efficient ways You can write as a partial fraction representation in which case you have the poles and the residues turns out That's even better than per day, but you can do something even better which you can convert it to a continued fraction Now it doesn't sound much better because they're the same thing But it turns out to be much more numerically stable and now when you study the large n limit of continued fractions There's something called a terminate. You can actually choose how to terminate right and as soon as you do that you study the large order behavior of the three-term recursion relations that come up from the continued fraction representation You discover the conformal map. It's the limit point of that large n behavior So that's what that's the deep reason of why these conformal maps are the right thing to do I guarantee that these convergence No, but that's what Simon and demonic showed they have an if and only if Condition on the behavior of the continued fraction coefficients that guarantee you that you have a well-defined measure That defines the set of orthogonal polynomials Very beautiful result Yeah, no just some remarks should be answered about long time ago should be some software just Drove by your mouse the past itself The best of the past and then should be formal how to change Conform in bed to disk and then making your snap and try to yeah, so there there are there are I haven't done it yet But there I've tried I've tracked down the material, which is these numerical generations of conformal maps It's very efficient, but it can clearly be applied to this problem Yes Just very beginning to talk to the slush and programs look at an idea of what that's just in this where there are Yeah, no, I don't know where they live. Yeah, sorry. I mean in the matrix models. Yes, but How many coefficients they mentioned that you would the goal is to try to apply this to Qcd and gang mills and stuff like that How many coefficients do we have right now for here? You knew So I have a drawer in my office full of examples and Sergey is going to give me a few more to put in there So you know g-minus two in QED is the sort of most famous calculation in quantum field theory. There are five coefficients Okay There are certain quantities in Qcd where people have calculated 35 Coefficients a typical number is more like 10 or 15 in statistical physics Very often there are calculations that might have 30 terms There are of course some do you have an idea of what the conformal transformation is supposed to be in that case? In general, no, but once you find the branch cut I can tell you that the the behave the dependence on n Goes like e to the minus square root of n times x where x is your large parameter So e to the minus square root So that gives you some idea that as you increase n you're going to get exponential increase But you know we take what we can get I mean I can't tell Qcd people to calculate a hundred coefficients in But the the the message is that something your order of the order 10 Contains useful information if that wasn't true if you needed a thousand then forget it Okay, but we have several examples now that tell you that of the order of 10 you may be able to learn something interesting But then can you try to do the calculation and see how it compares with experiment? Yeah, well This is all very new Yes about your Matthew equation plot may I just also suggest Alternative is to use one over H bar on the exit on the horizontal axis Like to see what it looks like It's like the first part that I Just showing it like an energy spectrum