 OK, thank you very much. Yeah, so I will try to present some little bit, some kind of perspective on resurgence through algebraic geometry. So I recall that if you have series sum over n, which is divergent, but coefficients grow like n factorial, then you make Laplace transform, take sum over n factorial z to power n, z to goes to 0, and it will be germ of analytic function near 0. And the property of resurgence has endless analytic continuation. And in the talk of Jean-Nacal, we heard some about otarchy and the isographic form, and now the main guess. This thing means that you have a polarized exponential hodge structure of infinite rank. Yeah, so kind of like infinite dimensional break geometry. And it looks that the words have these meanings. So what is, and this exponential hodge structure of infinite rank, it's something, if you make Laplace transform, we'll get a variation of hodge structure on a fine line, maybe with infinitely many points. It will be usual variation of hodge structures of polarized pure hodge structures on C minus, in general, infinite set, maybe everywhere dense, some countable set. And you get representation of fundamental group of these two, some J infinity over integers, which will be integrality of this desolations, coefficients roughly. Yeah, so what is this exponential hodge structure of infinite rank? Let's say a finite rank, because for infinite rank, we don't have yet rigorous theory. Sir? This is pure hodge structure. Pure, yeah. Or mixed, maybe, yeah, maybe, yeah. Right, yeah. But let's recall what are usual hodge structures. Let's start with pure hodge structure. It's a vector space over C, this hodge filtration, n of vh. And lattice, so you multiply not compatible with this iteration. And you have a property, such that if you consider real in evolution, so lattice gives you real structure on H. And now you can see the f bar and f will be opposite filtrations up to shift. So you get decomposition of H into sum of HPQ. Let's say P plus Q is equal to n. And yeah, that's a usual hodge structure, which appears in the co-molge of, let's say, complex projective variety. And then there's something called polarisation, which come from the ample bundle. You get non-degenerate baleenar form on integral lattice, which is symmetric or skew-symmetric, depending on parity of weight. And such that it induces baleenar forms on, it induces a pseudo-hermitian form HPQ for HPQ. Namely, you can see the pairing of alpha and alpha bar for alpha belonging to HPQ. And this has signature minus 1 to power p. So it will be either positive or negative, depending on the things. And if you take sum of this appropriate sum, you get hermitian form. So you get Hilbert's space structure, minus 1p of this form on HPQ. It will be positive definite to get Hilbert's space structure. OK, that's a usual hodge structure, which appears on co-molge of projective varieties. And exponential hodge structures, it's some notion which we introduced about 10 years ago with Kacarkov and Pantyev. It's that in mere symmetry of Landau-Gisbert models, it's related to exponential integrals. These are red-to-user integrals, and the definition of exponential hodge structures is the following. You get a holomorphic vector bundle. On C, this coordinate, we denote it U. But it's the same as Planck constant with a small parameter zeta in this one or that in this situation. So we get this holomorphic bundle. So we get just a family of spaces depending on U. And then on U-plane, we have special point U equal to 0. And we get a connection on this bundle. Holomorphic connection has second order pole. The second order pole that U equal to 0. And in particular, you will have certain monodromes. And we assume that flat sections of the bundle, long rays, with sections on rays, grow like exponent minus some constant times U. In principle, they can have more different singularities. It will be a simple exponential type singularities. Then what we get on each array, we get filtration by order of growth. We get this Deline-Mangrange filtration. And what is assumed that we have certain lattice. So the extra data is given to have a lattice, hUz, sitting in hU, for U non-equal to 0, which is covalently constant with respect to filtration, with vector connection. And all terms of filtration are also integer subspaces. I define it over oz. So this will be analog of hot structure. Actually, the usual hot structure, it's a particular case when this is not the second order pole, but first order pole in the monodromes plus minus 1. In all terms of Deline-Mangrange filtration, I define it over z. So this analog of hot structure and polarization is given by the following data. You get for U non-equal to 0, you get non-degenerate pairing. It opposite sides of filtration, which is covalently constant and also compatible with integer lattice on hUz is integer valued. And then there is a way to formulate sign of polarization, which eventually gives you some Hilbert space structure on hU. And also, from this sequence, it can deduce certain filtrations on this hU without z, labeled not only by integers, but maybe some fractional numbers. So that's its abstract story and where it appears. Appears in exponential intervals. So your h is finite dimensional. Yeah, here it will be finite dimensional. And what I want to say is that at least some past intervals one can see is infinite dimensional examples of this. Appears in exponential intervals, namely, imagine that x, the following situation. Suppose x is a final algebraic variety. One can think about just coordinate space. And suppose we have some polynomial, some function. I'll just give you the simplest possible example. Assume that this polynomial has isolated points, isolated critical points. And there's some condition. I forgot what is the name. It says that something this polynomial has no critical points at infinity. Theme, yeah. Suppose you have this polynomial. What it will be is this h of u. We can see the space of algebraic n forms on x. And more doubt by, let's say, the quotient space by the image d minus u d plus multiplication by ds, applied to n minus 1 forms. It will be finite dimensional space. And the dimension of this h u will be number of critical points. So we get a vector bundle. Now I want to say what is the lattice. For unit not equal to 0, the lattice, it will become all of the following set. We consider x. And consider pre-image of the domain when real part divide 1 over us is also function. And we take pre-image of the domain when the real part is very, very big. And take homology with integer k fusions. So the dual space, or dual lattice, can be sorted certain cycles over which real part of s divided by u goes to plus infinity. And now what we can do if you have a pairing between h of u and such cycles, we can integrate over such cycle of the function to integrate s minus u and times some algebraic n form representing class in this homology. So this integral will be convergent. And this integral gives you a family of lattices. It gives you, it will say it's covalently constant. It gives you a certain formula for this connection. It has second order pole. But the main story is that this guy has definition when u equal to 0 as well. And that's like analog of hodge filtration. What is this pairing? I want to describe the pairing on kind of, the pairing is also completely natural in this situation. I will describe pairing on level of homology. I'm going to go to dual, get pairing on level of homology. Namely, let's assume the target of u is generic. Then we get the basis of h u z dual. Consisting of what's called left shift symbols. Namely, what you do? You have critical values of our polynomial. Now we draw a straight path along which this SO u will minus some constants. It will be real and goes to plus infinity. So we get a straight path. And over the straight path, in ambient manifold, we draw a family of, nearby, we have kind of vanishing cycles, some small sphere, s minus 1 dimensional sphere. And we get family of, and then one continuously extends this family of s minus 1 dimensional sphere, we get a copy of Rn embedded in our space. And the bending depends on this critical point. So there are critical, x alpha will be critical points. And then you get s of x alpha. And for each critical point, we get this left shift symbols. It gets convergent integral. So you get this convergent integral. And then we change direction. Argument of your u goes to minus u. You just get left shift, dual cycle going to the opposite direction. Acclaim it is, and it's easy to see that it's kind of currently constant. And you get duality between, it's just boundary pairing between homogenous fiber u and fiber minus u. OK, yeah, so that's kind of scientific way to formulate this integral over left shift symbols. And here's the whole story. One can reformulate in kind of more elementary ways, namely for each alpha and in generic argument of u. So we can write what is the integral over left shift symbol. It will be integral, we choose again some expression. We integrate over the left shift symbol, starting in point x alpha. We integrate some n form, which you write something like polynomial function of x multiplied by some reference form, which you write dnx here. And f will be some polynomial function. So we get this integral to calculate. And then when u goes to 0 along the ray, we have a symmetric expansion to the exponent of my critical value, S of x alpha divided by u. Then we multiply by square root of 2 pi h bar to the dimension over 2, or 2 pi u, sorry, to dimension over 2. Then we divide by determinant of second derivative point x alpha for quadratic approximation theory. And then start with series starting from f of u x alpha plus you get certain series in formal power series in u. There will be value of this function, this point, and then we get extra correction. And this series has factorial growth. And maybe this series is denoted by, it's a series which has factorial growth, but it's actually equal to certain number, which I kind of denote by square root of determinant. Yeah, there you are. It's also here, right? And this I denote by something like g alpha of u. It's now its actual function, which has a asymptotic expansion. I remove all irrelevant terms. So oops, I didn't read it. You get just a bunch of functions, g alpha of u, alpha for 1 to number of critical points, which have defined outside of some stocks raise, direction raise, but this function have the same asymptotic expansion, which doesn't depend on direction. Formal expansion is the same. Stokes, stocks, formal expansion doesn't depend on the sector. So you get just a bunch of formal power series, which you can calculate on computer using formal expansion at this point. But you get actual functions as well. When you divide it, you get a function. Yeah, you get actual functions, yeah. And this function satisfies kind of jump property. Jump, if you go through some ray where argument ray, when it says it argument of u is equal to argument of difference between two critical values, yeah? It's a self of 1 minus self of 2, say? When you get this ray, then what we see? It's kind of standard picture. What happens if you, on the one side of the ray, if you on one side of the ray, and then we get rotation, we get slightly different homologic classes. And you see that one homologic class doesn't change. So one function stays analytic. g alpha 2 is analytic. So jump from the ray of g alpha 2 is 0. And jump of g alpha 1 is a certain multiple integer of some integer in alpha 1, alpha 2 multiplied by g alpha 1, alpha 2. But because of this factor, we multiply by exponentially small term. Take exponent s of minus s of x alpha 2 minus s of s alpha 1 divided by h bar, which is very, very small. Then it has no exponentially small. So it has trivial expansion. Sorry? It's from u. Oh, yeah, sorry. Yeah, because I kind of make notation. I have in mind this h bar and u. This was the same, eventually. Maxime, what is offered? Delta, jump, a kind of jump of a half function defined on two sides on the right. Consider difference. I think, yeah. So it's equivalent to, yeah. So it's kind of in completely elementary mathematical terms. All this hot structure, besides this positivity process of polarizations, it's very elementary object. The polarization itself, it's kind of tricky stuff. It's called TT star equations. And we will not talk about this thing at all. OK, yeah. So we got this completely clear picture in finite dimensions. Now go to infinite dimensions. Yeah, suppose I have a certain remaining manifold, maybe two points of this remaining manifold. And I assume that manifold can be complicitized to some complex manifold. You can have algebraic manifold. And metric will be algebraic tensor on this manifold. Also get this big complex manifold. Now, m is an example you can try to see, like, m is a sphere. Or something you can write explicitly, formulas. Sphere, or ellipsoid, or hyperbolic space. Maybe flat space, even, yeah. Or something with a very nice concrete formula. Now this then have infinite dimensional complex manifold, x, which will be the different space of path connecting these two points, m0 and 1. Maps phi from 0, 1 to mc, m4 is m0, m4 1 is m1. It's infinite dimensional complex manifold. Plus it contains kind of half infinite dimensional things past sitting in real path, in real locals, boundary condition. And now the Feynman integral, which you can see that the action functional will be 1 half of integral from 0 to 1 d phi, let's say dt squared dt. So you get this functional integral. And now when we integrate over xr, expand of s of phi divided, let's say by u, but now maybe I denote it by Planck constant and defies what the expression means in path integral. So it's analog of this situation of finite dimensional exponential integral. And all this should have some kind of critical points, form of power series expansion, and then actual functions and all this geometry. Let me explain you kind of basic trick here. So we integrate how to calculate this integral. When h bar is equal to 1, so I get this usual linear measure, it's a Brownian motion on your manifold. And you start from point m0 and it's point m1 and time 1. And that should be equal by the value of heat kernel, exponent minus 1 half of Laplacian on x real. And you use these connotations. So consider heat kernel for time 1 between points 0 and 1, m0, m1. And what to do for general h bar? The idea is it's very simple. So you multiply, I think, divided by 1 over h bar. Let's keep on half. So you get this thing. And you see that in time, this thing is homogeneous of degree minus 1. So one says that its putted time bar is equal to ht. So tau bar will belongs to an interval of 0 h bar. We will identify 10 points. And then it will be equal to just equal df over dt bar. So you see that this integral is equal to the heat kernel for small time to h bar. So you get this nice formula. So it will be heat kernel for small time. So the prediction is the following. If consider heat kernel for small time and divide by this leading term, you get some asymptotic expansion, then should be resurgence series. This should be resurgence series. And now, so it should have some exponential small term. So we should divide by exponent minus, I think, 1 over h bar times something like square of length distance between those points, which will be leading growth of expansion. So it should be kind of like series in h bar. And this should be resurgence. That's actually a completely open question. For general sinfinities, we remain in mind that we really don't know. This guy is resurgence. And I think it's true only for some special algebraic varieties. So things close to them. So let me show this example, which I found extremely striking. This is an example when consider just manifold will be hyperbolic plane. Yes, it's maybe the simplest example, which could be. Because in this case, we have just only one geodesic, between two points. And for some classical formula for the heat kernel, one can express that this m0 exponent minus Laplacian and half Laplacian 1. And you multiply by h bar is equal to the following thing. It's certain constant. I get something. And then you get certain integral. It will be small s, not capital S. But it's related to the critical values of my function. This formula, the L is distance between my two points. No negative real number. So this thing is given by certain integral. And what here goes on? You can see the critical values of c on the space of compressive height pass. Pass in a compressive height hyperbolic plane. And consider what are critical values? The critical values are the following. You get certain points on a parabola, integer points on the parabola. The points on parabola, as you can see, you get a bunch of lefty symbols. And this is my integral of the certain lefty symbols. And in fact, if you look on this integral, you'll get certain finite dimensional. Actually, you say that you identify up to the simple factors in the front. You identify the infinite dimension integral with finite dimension integral, but over the remaining surface, which is infinite genus remaining surface, ramified this countable set of critical values. You can see this function of small s, this one of square root of hyperbolic cosine minus hyperbolic cosine. It's something which is too valued. If it's not on a sphere, it will have a cloud, which I did. Yes, yes. No, and that is the sphere. So on a sphere, if you have two points, you have not only one judizic, but you go around, around, around. So it means that to lengths, you add this length of the periodic judizic, which is purely imaginary. You get some kind of length plus all these points have the following meaning. You consider this length plus 2 pi i n, where n is an integer. One is real, and the other one I'm just adding this here. Yeah, but of imaginary radius i. So the critical value should be. Yeah, all these points is a set of all these points which I consider. So you see you get these integer points on a parabola. And moreover, this example I think is already very interesting because from the answer, you can see, you have a prediction how many, you have prediction how this function jumps when you rotate zeta. So these integer numbers n alpha 1, alpha 2, for each two critical points. And it's a very simple number to calculate for this interval, plus minus 1. And the claim that should be coincide with something very interesting because if you look what is alpha 1, alpha 2 in general situations, some kind of number of certain gradient lines from one critical point to another. And gradient lines will be gradient lines in the space of pass. So it means that you should solve certain pass in the space of pass. It's a map from surface. And it means that gradient lines, it will be number of certain surfaces satisfying certain kind of Cauchy-Riemann equation in compressifying space. Yes, so it's kind of very non-trivial prediction that from this answer, we can see that everything fits together. And we get kind of convergent answer only if you know this number of pass from power series. And this will be given by this picture. So that's the main example, non-trivial example, which I have to tell you. It's about this heat kernel on hyperbolic space. One can ask similar stuff. What in small modification? I can study the state-of-the-sphere. But for example, in this case, it will be case of ellipsoid with non-equal axis in R3. Sorry, MR, yeah. It's ellipsoid, actually of any dimension. Why ellipsoids are nice? For ellipsoids, this geodesic flow is integral. So one can write lengths of geodesics to some set of functions. But also eigenvalues of Laplacian's also kind of known because it belongs to family of commuting operators. For example, one can do kind of really extreme case. When ellipsoid became a flat, it will became like double of disk. Imagine disk glued to itself along the boundary. Then geodesics will be this polygons labeled by pair of integers. So it will be lengths of geodesics and calculate like close things. It will be whatever, q times 2 cosine 2 pi p of q, something like this. And eigenvalues of Laplacian will be because they're decomposed by 0s of some basic functions. So in this case, there will be certain great identities, certain infinite sum, personality combination along this series of basic functions that will be interesting to calculate all data explicitly. In this case, yeah. So here, what was the trick? This action for this free particle was homogeneous in time degree minus 1 because of df over dt squared times dt will be homogeneous of degree minus 1. And then we can make this calculation. One can make kind of similar exercise. It's related to very old work by Andrea Voros. So consider operator, consider another example. Instead of Laplacian, consider secretary plus potential. And the whole story, the main thing is it should be also homogeneous function. Like I said, one can generate many variables. I just start with one variable. So we have this homogeneous function. And when we calculate trace of exponent minus lambda h bar, how to calculate it? Again, by Feynman-Catz formula, it will be integral of the space of pass this periodic boundary condition to r1, this coordinate x. I have a separator. And I multiply by exponent minus action. We denote by s lambda of phi d phi. s lambda of phi is just integral 0 to 1 and half d phi by dt squared phi to the power 2n is dt. It's a discrete spectrum. It's a discrete spectrum because the spectrum is discrete because its potential goes to plus infinity. It has a val. Sorry, you're right. And now I try to rewrite as the integral from 0 to 1. I want to rescale time and rescale everything, time and phi to make its integral from 0 to 1. Namely, I take tau lambda is lambda 1 minus t. And then when rate phi lambda is equal to, if you make a rescaling, then s lambda will be integral of 0 to 1 when I go to deal the variables. Multiply by lambda to a certain power, which is you get just homogeneity game. Nothing tricky here. And then the conclusion will be the following. Then if you take consider trace function, whatever zeta goes to, trace exponent minus zeta h to power n over n plus 1, should admit an endless analytic continuation. So that's what's under the word predicted. But as far as it's not on the level of pass interval. So it's this simple homogeneity game with pass integrals. So there are two basic examples when it's very, very concrete. So it was a case of free particle and also with homogenous potential. There is still another example. So suppose we get, use the same name, but now one can write another pass integral very badly defined. Suppose I have a complex manifold like a tangent model homogenearily holomorphic simply-active manifold, which contains two complex Lagrangian sub-manifolds and 0, L1. And what I want to write, I want to integrate. The space xc will be no real space at all here. It will be just my infinite dimensional complex manifold. It will be space of pass. f of 0 belongs to L0, f of 1 belongs to L1, and n is back to tangent bundle. And as an action, I choose mc, mc, sorry, mc. I have the same infinite dimensional space. As an action functional, which I consider here, it will be, is defined only up to a constant and only locally. So I just have to write what is the differential of the things. It's, it's will be homomorphic one form, which will be integral of two form of, of my pass. When I have one parameter family of pass, I get surface and integrate to get the defined thing. Yeah, so, so roughly think it's like in mechanics, you write something like these things plus boundary terms. First, it's called first-order formalism. And plus boundary terms. Yeah, yeah, in principle, one can include this random walk. I think certain terms, like you add certain function, it depends on three parameters. So it will be function on cotangent bundle m times some time variable to c, that's function h. You can add this such term. And if the thing is quadratic in p, this is, we'll know it's equivalent to pass integral one can integrate over p variable and get exactly random walk. But now I ignore these things completely. So it seems it's completely geometric. I have this integral of one form. So the question, what is it? And what we'll have here? Yeah, in general, it's pretty unclear. How interpreted what is the interval? Yeah, it's, and what research property of what we should expect from this interval? But at least it looks at some, for certain kind of class of example, this could be the following. Suppose L0, get some L0, city can cotangent bundle. Here I draw fibers of my cotangent bundle. And it will be like this, this is L0. And L1 will be cotangent far by a certain point. Then I get finitely many intersection points. And kind of the guess is if L0 kind of corresponds like spectral curve, again, it's long story how to identify parameters to algebraic bundle with a connection. Then this asymptotic series, which you should get here, should be like WKB series for formal solutions of the models. We try it like, you write L0, it's graph of differential, graph of D, maybe called F0, some multivalent function. And then it will be, you write a solution as you remove this main term to get asymptotic series in each bar. So that's something which with Yan we kind of part of general product we have with Yan on wall crossing. And what is going on? Here's some kind of really interesting effect here right to the following story. On the space of pass, we don't have really one valued function to get functions defined up to constant. We have a closed one form, and that means the following. We can, in the whole integral, we can go to universal cover. And then we go down, we can twist this arbitrary rank one local system on space of pass. So we get certain torus, C star 2, of the space of pass. By which you can twist your integral. Yeah, for example, the same station can be in finite dimension of the story. You can twist your variable connection by rank one local system, consider again commulge of pair and so on. So you get things depending on a point on a torus. And what happens in infinite dimension? There are certain directions of Planck constant for which you change parameterization space. You kind of apply nonlinear automorphism of the torus. So you get things which kind of very close to kind of a Calvaronian picture. You have some nonlinear changes along the race. You get its kind of new infinite dimensional effect. There are kind of new walls, new rays. Said that we're going to race. You can apply diff morphism, analytic diff morphism of torus, of some domain of torus depending on small parameter. Yeah, so you get more involved picture with some kind of nonlinear changes as well. Before, we had just vector bundle with basis and transformational grace. But now you have vector bundle on torus and parameterization. Is it not sure this is a vector bundle of rings? Which is sort of small? Yes, yes. Yeah, but eventually it should give again the same resurgent properties of the whole story. So yeah, so I didn't had a time. Yeah, that's essentially all one-dimensional examples. And in principle, and what is completely untouched, one can try to make, I guess, imitate at least the question for two-dimensional theories, like there's the Minovitan or three-dimensional with Sharon Simons. Yeah, so again, one get actual values of integrals from physics, and then there's the same picture predicts you resurgent structure. Thank you. I see, yeah. It's a question. So you consider this free particle on hyperbolic space. You can think of it on circles, here, or? Any varieties here. So for the general case, the general group, there is a formula called deus thermotekman, which essentially we write this as a sample of geodesics on this space. But as far as I remember. No, no, this is certain sub-deus thermotekman for any remaining manifold. If you can see the trace of exponent minus ET of Laplacian, it has singularities on a major axis equal to lengths of geodesics, yeah. But for example, if I just found the particle on a circle, again, I can write a sum over geodesics. Yes, yes. There are different fluctuations around them. Yes, yes, you got the exact formula, yeah. It's exact formula. Yes, exactly. Like, you got c to function, yes. I see, OK, so there is no asymptotic expansion. Yes, it's trivial, yes. In this case, it's trivial, yeah. And in these general cases where this deus thermotekman applies, it's also the same structure, right? No, but the conjecture, it's for more or less general remaining manifold with algebraic metric. Then we get very complicated expansions. Yes, if it's not, if it's not homogeneous, yeah. Can you describe the whole structure of infinite rank in terms of the section function or related to homology? Yeah, no, there's a kind of abstract one can write. There's convulsion of parabola. What is it? But you see this number of this gradient lines integer numbers in alpha 1 over 2 gives you the center of the day, kind of step-by-step construction of this infinite. It should be space generated by all geodesics, yeah. I think it should be pure in this case, yeah. More questions? Yeah, I would like to understand everything, obviously. But how far are we, or you, from effective computation on the finite path integral for, say, a non-homogeneous problem? Yeah, for general potential. I think it's for general potential, yeah. In principle, it looks that if it's not homogeneous, this gives you kind of few steps procedure, which is completely mechanical, one can calculate critical points in complex domain. Then one should count how many gradient lines. It's a number of solutions of certain, like pseudo-homomorphic disks somewhere, or some integer numbers which one can calculate. And then you get form power series which you do at each point. And then after that, it should produce actual numbers. Can you replace it by so many numbers? Yeah, because it should, yeah, the story is the following. This integers and alpha, and those jump formals for J, which depends only on critical values and integer numbers. So it gives you a way to glue certain homomorphic vector bundle with sterilized infinity structured zero. And then there's form power series after you solve the problem. Convergent series, yeah. Yes, yes, yes, it's completely general procedure. That's from this perspective, it will be. It's effective, yeah, it will get to actual number. Yeah, it will get it, convergent, thank you. Is it clear that if you write for 420, if you write the formal WKB solution, such as in the classical, or is the same as you get a feminine integral, diagonal exception? So is it clear? No, no, no, no, no, no, no, no, no, no, it's not clear. A comment on your question? Yes. So from the feminine integral point of view, that's very difficult. But if you do Bender-Buhl type analysis, you can actually computerize. Yeah, you can, yeah, because you substitute the things to your expansion, you get to. Yeah, but it's purely computational. It's purely computational things, yeah, it's a computer will collect your first, I think at the end of the negate. I think even the determinational part is not completely clear though. Right. Yeah, but no, but there's also, no, there are many things here. For example, here, that's dimension, yeah? It's with zeta function regularization then. This will be zeta prime of zero, yeah, yeah, so it's in field theory, all terms make sense, yeah.