 All right, so for the last talk of this morning, we have Thomas Grimm is gonna speak about the timeness in March theory and physics. Thank you very much. Well, thanks a lot. First of all, thanks to the organizers for putting together this very nice workshop and inviting me here. So I'm a physicist. I will also give some sort of the physics style talk but I will try to make some contact with recent mathematical development and hopefully help a little bit to further make a pitch between the two disciplines. The work is based on, or this talk is based on a couple of papers. Let me just highlight a small number here. One set of papers is very recent which is this Michael Douglas and my host of Low and Schlechter. And then I will, because I want to make this contact with mathematics, I will also mention this work with Benjamin Bucker, Christian Schell and Jakob Zimmermann, which is a specific mathematical theory. Let me begin and let me start with some motivation. Well, my motivation starts in mathematics and I essentially just broadly state the following fact that there is a much recent activity in mapping out the same part of mathematics. And in fact, this pain geometry which I will briefly introduce has now found its way into various parts of mathematics from algebraic geometry, arithmetic geometry, number theory and so on. So it's a very active field and this might come a little bit as a surprise because originally this concept of painness which I will introduce actually comes from mathematical logic and more precisely from model theory and it essentially just to kind of get a rough understanding for the introductory part is essentially states that when you do geometry or when we ask certain questions you can draw your sets and your functions which you want to use to describe the system. You can just draw it from a space specific structure and these structures are called omenomy. And I will motivate a little bit why these have been introduced in logic later on. Now that this has found its way this kind of pain geometry or these omenomal structures is indicated here that this has found its way into what theory this happened rather recently. And I give you a rather biased and incomplete list of papers which I know reasonably well and what this entails to is, for example it was shown that the period map or the period integrals they're actually drawn from such a specific pain set and become such a specific omenomal. Then there are multiple axonal type conjectures for hot structures. They are quite a deeper statements that maybe the kind of the most involved statement is kind of the last year's proof of the geometric country and both of these periods. What I will talk about is actually how this omenomal structures or pain geometry how they are used in establishing a finiteness result for a self-proclaimed project. Now why, how did I get into this field? Actually it's one of these exciting things which was mentioned before that there is a story which came out of physics which turned into a concrete mathematical result namely it arose from a physics conjecture. So in physics we have the problem in the string theory setting to solve a certain problem which you might call the fifth force problems and this is solved as Arthur described by fluxes. These fluxes are something like generalized electromagnetic fields in some compact spaces but from a mathematical point of view there are nothing else than integral classes in on your mind. And the settings which people considered for quite some dark time is the following. They considered a 12 dimensional background in a theory called F theory but that's not important. And the 12 dimensional background consisted of a Cala Vial four fold so a real eight dimensional space. So it has a very non-trivial middle chromology and a four dimensional space which is supposed to be our universe. And then in order to solve this problem one introduces this integral class and this integral class has to satisfy a constraint namely that it's wedge product that itself has to be an integer. And in fact this integer cannot be too large to have specific constraints but it has to be some fixed. Now if you solve the equations of motion the physical equations then this cannot just be any integral class but it has to be what you might want to call a self dual integral class. Namely it under the watch star operation has to map to itself. Now of course this gives a constraint on what is the complex structure of the space in order that such a condition is set. Now what is the conjecture which led to the theorem namely in almost 20 years ago now it was suggested that the number of solutions to string theory not necessary only to this setting is a certain bound of the vacuum energy KK scale compactification while in all of these things are not important in the following but the statement was that this is finite. There should be just finitely many solutions. So now if we apply this to the setting then it means that we should have finitely many solutions to these equations. And this is a very, very hard problem. And so now let's turn this into a mathematics problem. So it's a very hard problem but it's sufficiently concrete that you can translate it into what you do. Now let's formulate the same problem in what you do. Now let's consider M2P a smooth complex algebraic variety and you can think about this as the modular space of this Calabriao manifold. Calabriao for example, but the theorem will be general. It doesn't have to be Calabriao on the sphere. And now in order to smooth it you might have to resolve some similarities but there are no theorems that we can always. Now let's consider the Hodge bundle over this base, right over this modular space and your Hodge bundle will be just some abstract PQ splitting, right? Which follows a variation of Hodge structure. So the fibers of this bundle actually split like this and I will look at even weight Hodge structure. Then I have to tell you what is the analog of this Hodge star operated? Well, in Hodge theory that is the so-called vial operator and the vial operator just acts on the element of the PQ by the following action named the I to the P minus. If you want, that's the definition of the vial operator on such a Hodge filtration and Hodge splitting and geometric concepts really turns into the Hodge star and the model. Okay. No, no, no, the right one. I think the right one. Are you mean the pronunciation or? Oh, sorry, yeah, yeah, I pronounce when I put German there. Anyway, could not stop there. So furthermore, the polarisation, the introducer polarisation on this Hodge structure and this Hodge polarisation is just some inner product, which for example, if you're in a geometric setting, you can just think of it as the veg between two points. That's how we would see it from geometry. And now what is the theorem? The theorem states that for any integer L bigger than zero, the locus of self to integral classes, namely of all the classes on, of all the elements x, v of the Hodge bundle, which satisfy the following conditions. First of all, the v is quantized. So that's what Arthur would call, physicists would call a flux. So it's really an integral class. And this integral class actually satisfies the self duality relation under the well-known scale of horizon. And furthermore, if you impose the condition that q, the product of this integral class with itself is actually equal to L, then the resulting set of solutions to this, like this space as L, actually is a set from one of these which you pick out of one of these all minimal structures, namely the set. Now, why is this interesting? It essentially states from general theorems that S has only finitely many connected components and therefore it proves exactly this conjecture, which was around for 20 years. Now, this is quite exciting. I will come back to it a little bit later, but after working together with these mathematicians, and of course, I would have never been able to prove such a thing without these brilliant math collaborators. You said I didn't prove that conjecture, but you mean it proves that the finiteness of these classes, but not of the old. No, it proves this conjecture in this specific setting, if you fix the typology. If you assume that you have finitely many contact-alive-out-conflict, it really proves the conjecture in the old setting, but otherwise, it just proves it in this. Yes, in order to prove kind of these more general, but also note that the theorem is much more general. It doesn't even have to come from geometry. It can be some abstract variation of structure. It doesn't have to be Calabria, or Calabria-conditioned, like, no, no, no. But, sorry, maybe a two-fold question, but which, so in these conjectures of calculus and company, they have these conditions you mentioned, KK scale and so on. They are satisfied. They are satisfied automatically in this setting. So there is no condition, like this theorem does not. That's what we discussed before. So the hard part is, you take these physical conjectures, apply it to some setting where you can get rid of the physical conditions, where they are automatically satisfied in some sense, and then prove it generally. And what about the uniformity in, let's say, some kind of moduli or something like that? Yeah, I will come back. Okay. Yes, still in the beginning. So how do you seem to be satisfied in your theorem? You know that when you have the self-dual vacuum, that the vacuum energy is actually zero. So the bound on the vacuum, these are really solutions which have zero vacuum. So the bound is satisfied. And... Exactly, these are actually zero solutions. So that fits all. Anyway, so this is how I entered the field of tangometry which was absolutely crucial, as I will point out in this in a few words later on. And it turns out that it's such a nice story which actually goes beyond, so to say, what businesses have used so far. And so I did through my knowledge about theories coming from string theory. And what I noticed is actually that the sort of tenderness property seems to be everywhere in this effective theory. And so what I suggested essentially is to think about this in the following way, that in fact that this tenderness is a property of effective theories which come out of string theory. Now this theorem essentially is an evidence for that because I told you that this set of solutions is a tame set. So it's part of the specific structure. And in fact, we know from physics that around each of these solutions there's an effective theory and then it would be a tame theory. So it's very strong evidence but this statement is much, much broader and of course there's a lot to be checked with. Now, it got even more exciting when we realized that it's not just in string theory omnipresent. Actually in quantum theory and just in general quantum systems, for example, when you talk about Feynman's amplitude, then it is actually, they also turn out to be tame function. So there is quite a set of objects in physics which are satisfying these criteria. Okay, now having kind of given you some motivation where the excitement came from, now I should tell you more precisely what is this tame geometry or tameness but it's the theory of omninimal structure. It is a generalized kinetism principle. And if you want, you could say it's sort of a finiteness of geometric complexity. As there is a nice introductory book by Fundandy's and there have been quite some activity in recent years. I mentioned here the Fields Institute Program in Germany last week, what's mentioned there will be actually a whole program in terms of omninimal structure. Now, what does it do? This principle, it avoids wild functions and sets as the name suggests. So there are no sets with infinitely many disconnected components. So the indentals and the lattices are out. If you do a tame geometry, you're not allowed to talk about the integer. It's a constraint, quite a strong constraint. And there are no complicated functions. Well, a complicated time, but they are not allowed to be too wild. For example, this function here, which is like infinitely many different times differentiable, but it has like this infinitely many accumulation points of zeros in the center. This is not a tame function. And why? Because you could use the sign to define the integers by looking at the zero. Now, the interesting part, as I already mentioned, is the motivation came from logic. And there it actually came from trying to avoid difficult question about undecidability, like Goethe's theorem of undecidability. This, as you probably all know, is about the integer. So if you kind of get rid of the integers, then you have better chance to be decided. But you want to focus on another part of property rather than the undecidability. Now, why is it maybe most interesting to this workshop? It's the following. It actually realizes the growth and the extreme of having a mathematical framework for Jormann. The kind of diversity idea that the topology used in mathematics is designed by mathematicians doing analysis, and it's much too wild and much less restrictive. In fact, he proposed this sort of a tame topology, and exactly this is what came out of his O-minimum structure and logic. So now what is the definition of such an O-minimum structure? It actually collects subsets of RN. So it's something really, really basic. For each N, it collects subsets of RN, which are closed under finite unions, finite intersections, complements, and products. And these all correspond to logical symbols. That would be or, and end, and so forth. And it has to be closed under projections. So if I give you a higher dimensional set and you project it down, it should still be part of this collection of stuff. Linear projections should be allowed. In logic, this would be the existential quantity. So there exists an X with the following. Now, in order to make it richer than algebraic geometry, you already include the real polynomials in these sets. So of course, otherwise the zero set satisfies all of these conditions, but now you include also the real polynomials, already the rich class of sets and functions. Now, what is now this O-minimality property, this tamedness property? It's actually remarkably simple. It's essentially, it's just a statement that the only allowed subsets of the real line are finite unions of points of integral. These intervals can be infinitely long, but only finite unions. So the integers would be an infinite union of points. So it's not part of, sorry, so that tames the structure. And this is kind of was introduced as a very powerful property by Fann and Ries, who incidentally is an U-tank. You have a nice. In any case, the sets in the O-minimal structure, S are then called the tamed sets, functions of, called tamed function if they are graphed with a tamed set. And then you can define tamed manifolds, tamed bundles, and a whole tamed geometry. And the point is that actually most of things which geometries have done over the years can already be defined in this because you never actually allowed these wild things to happen. The tricky part is there are many known O-minimal structures, but they are very difficult to construct. Essentially it's a statement which functions do you allow such that they don't break these axioms. So the simplest one are just the algebraic sets, semi-algebraic sets. You just take the zeroes of real polynomials and you put them into the structure. Then if you complete it with these projections and so on with these axioms, you get inequalities into these statements. And these are called the semi-algebraic sets and they form such an O-minimal structure. So this is kind of the most basic thing. It's a non-trivial theorem that is working that you really get this semi-algebraic set, but it gets non-trivial once you want to add more functions to the set for construction. So what happens if I take polynomials and add new functions at to generate the sets? So which functions am I allowed to use? And I just use an arbitrarily complicated function or do they violate these axes? And that's very, very complicated. And in fact, the logical perspective is the following. So it means essentially that you formulate a statement about your setting using your standard symbols in your ring over R and then your set of functions. And then you can formulate any for the logical statement. So why is it that there is a lot of excitement because it's made non-trivial to add these functions. And in particular, it was considered a big breakthrough to show that the exponential function, the real exponential function is such a tame function. Now, the user question when I give the talk is like, well, look at the, it looked extremely tame, right? It looked as if how could it be that there is a non-trivial theorem there, but it actually took very long to prove this one day challenging. And the fact is that you have to not only just stare at the function that it looked very tame, you actually have to look at every higher dimensional set generated by this function and make sure that you never by any projection or intersection violate these axes. And that can be very challenging. No, no, it can be an infinite set, but it has to be connected. So in the real line, an infinite integral is fine. And of course the exponential in the two-dimensional set. So that was a big breakthrough to the field and from then on, kind of took up some steam and now it kind of goes into all these different mathematics. So now the other interesting part is that you can add restricted analytic function, take an analytic function restricted to a smaller interval and you can show it still is, can define a minimal structure. In fact, you can also combine both. Then you can combine the restricted analytic function and the exponential function. And if you add all of these functions together, it's still an all minimal structure. And that's exactly the all minimal structure which appeared in the theorem in which I mentioned in the very beginning. Now there are much more fancy construction like the Parthian extension where you can include solution to first order differential equation. So tameness goes well with first order differential equation. Unfortunately, not so well with second order differential equation and physics many of these places are second order. So it's special once you can add. Now, just to point out that this is something which is still ongoing only last year, it was shown that there exists an all minimal structure where the gamma function and the zeta function are defined. So that's, so it's by no means something that has ended in the nineties but it's still an active field done by the nineties. So just to give you a feeling, so that's how a tame function look like every tame function from the real line to the real line, you can always partition the domain into finitely many intervals. And on each interval, the function will be monotonic and continue. And so in particular, any such function can have only finitely many minima and maxima. I think you can differentiate and it has to have a tail tame to infinity. So there will always be a last interval where the function gets boring because it cannot wiggle around there. Then it's like for the exponential function that's what we have to do. The other functions are out. And that's kind of the remarkable thing that the sine function on the real line is not a tame function. That's what I mentioned before, you could also use it to define. Now, why is this all exciting? It's exciting because the things which appear in Hodge theory are actually tame functions. So the teamness of the period map was shown in this 2018 paper and essentially it proceeds in a couple of steps. The first thing which you show is that the veil operator is actually defineable. So it's a tame function in this R and X. So why do you need the X? So like maybe just to point this out. The point is that you know that this is not an algebraic function, right? Because it's all the bigger books equation. It has infinitely many exponential corrections. So you need the exponential to define the function. So it cannot just be some purely analytical thing which is respected. So you need the exponential. And it has to be analytic because we know that on a small disk near the, any point there is a whole analytic expansion of the theory. And for the veil operator, you can show that using the input in the orbit theorem. That's the key part here. And essentially you see it as a map from your base space, from your modular space into the orthogonal group of your pairing, of this edge product divided by the orthogonal group of the pairing evaluated for some fixed veil operator. Yeah, yeah, that's why it's a difficult theorem. First of all, the matrix equation by certain matrix. Certain matrix. Well, the point is that once you want to show this, you're not showing it using bigger books equation. So this is really an independent theorem which you show using as a public watch theory. It really, of course, these two things are equivalent. We are Gauss-Mannouin and so on, but you don't show it using the differential equation. So in other words, there are special differential equations appearing in watch theory. And they are still for certain things. Yeah, but complex exponentials are not so good. It's a complex experiment of design. So that's the non-trivial part. Yeah. I will show you. Yeah. No, this is C, it's the vial of the plane. No, I interpret it as the set of all watch operators are elements of the group G, which is the orthogonal group of the pair. Okay, so now the vial operator period map is taking the vial operator and actually modding out the orthogonal group of this infinite line of this pairing on the plane. And now proving that this is the same function is much more difficult. It actually uses the S of two or the theorem. And that's the, so to say, one of the non-trivial steps in the, in the Buckettinger theorem. And once you have this, you can lift the statement that the period map itself, so not the vial operator, but the period map itself is defined in the island. So in contrast to looking at the differential equation, this tameless statement comes really from the Hodge theory statement, from the Hodge theory analysis. So it's proved absolutely. Now, let me remind you of a very famous theory which relates to a talk which was mentioned yesterday by Katani DeLiemann and Kaplan from 95. It actually states that if you have the integer L bigger than zero, that the locus of clinical Hodge classes, namely classes which are of type T and they are in the integral. Yeah. And they have this condition. So this tetcal condition or this constraint from physics, it's already in this paper from 95. If you take this into account, then actually locus of these equations, where these equations are satisfied is algebra. So this is really a remarkable theorem because as we just discussed, the periods are horribly complicated. But as soon as you restrict the periods to be of type T, D, D as the specifying the reforms, actually this locus can be described by a polynomial equation and all these exponential corrections, they actually can't. And that's a very, very non-trivial statement. It seems it indicates that something very special happens. And of course, the very special thing which happens is related to the Hodge concept. No, no, but things are going this way. Actually, we can discuss that. Okay, so why is this a very famous theorem? It also follows from the Hodge conjecture. So you can also assume the Hodge conjecture for a projective Kena manifold and then use this assumption to prove the same theorem. So it's sort of a very strong evidence that the Hodge conjecture is true. And it's very general, so it's not example-based, so that's why it's considered one of the strongholds. And in fact, it covers this finiteness of the special case where this G4, now in physics language for Calabria 4-fold, where the G4 is actually of type 2-2. So if the G4 is of type 2-2, it's actually even more constraining, namely that the locus is algebraic and that's what we also probably would hope comes with that. So the original proof of use is Hodge theory and the Nilfurtian orbit theorem, but what is the nice reason, the thing is that there is an alternative to the use is tameness of the theory. And remarkably, this tameness, the proof using tameness is just two lines long. Of course, you have to use art theorems from the geometry, but essentially the statement is, if you have a complex analytic function, which the periods are known to be, which is also tamed, then it has to be algebraic. So it just follows from the general O minimal chart here. So for Calabria 4-fold, that means that if I satisfy this equation, which Arthur also displayed in slightly different form, so if I have this function here, then this statement here is just that it is of type 2-2. And there are more equations than unknowns. So it's an unlikely intersection theory and these strong statements. Now, the important point to notice is that the relies on polymorphicity, but in physical situation, polymorphicity is most of the time absent. And in fact, in this more general theorem, it is absent. So actually tameness seems to be a property which is preserved away from the holomorphic world. So that's also, yeah, what some physicists like to point out. Well, we can compute holomorphic things. Well, now we have a property which allows us to go beyond the holomorphic world. And in fact, in this theorem, when you look at the self-tool forms, actually you have not considering only a complex locals, but it's actually a real locals and beyond the real ones. We first used, tried to use asymptotic Hodge theory to show it and it turns out to be notoriously difficult. So only for one parameter is possible, but for one parameter. So I give you here the details of the proof, but I have not time to go through. So it's really a clever use of tameness of the period map together with the fact that these lattices were used into finitely many orbits if you impose this step. So now a question to the mathematician. So what is the analog of the Hodge conjecture in this case? So what are the cycles associated to the self-tool class? So that would be very interesting for holography because in holography we often replace like flux by cycles and then we have sort of brain configuration. And this is an open question. So it's, and maybe one of you has a good guess for this. So what is the analog of the algebraic cycle which you have in the Hodge conjecture? What is the analog here for the self-tool? And there should be something. Anyway, I wanted to quickly mention the tetpo conjecture but I want to be brief on this. So a new conjecture which is very concrete is this tetpo conjecture. So I told you that in all of these theorems there is this constraint on the self-intersection of this integral class. And in fact, this arises from consistent coupling to gravity. But if you have a non-compact space you don't have to impose. And now the tetpo conjecture in the weakest form essentially says that if I fix such a self-intersection L and I make the dimension of my space larger than this L actually the dimension of the Hodge locus should be bigger than zero. So that's a very weak statement of the tetpo conjecture. And this comes from examples and so on. And it's a remarkable statement and it could be in principle proved using, you could imagine that there is a proof using these techniques but it's much harder than this finite aspect because it's very important to do. It's not enough to use asymptotic Hodge theory for this you really need more typical information. You need something about a quantized structure. But certainly it's an interesting thing to consider. So why do we believe that this is not completely absurd? We actually proved in all asymptotic regimes using asymptotic Hodge theory. At least if your periods are given by S and two orbits you can see that this statement is very, very reasonable. Essentially because everything splits up into blocks and if you want to reduce the dimension of the Hodge locus you have to have an integral class in the block for some rational. Which condition, this condition here? I just pick an L, just pick it, there's no condition. I just, no, no, no, there's no condition that just comes out. This is not a condition. This condition comes that it's an asymptotic statement that the dimension of the base of the modular space has to be sufficiently large compared to the number which comes out. So this is the non-trivial condition. Yeah, but you can just pick it. No, there's not a capital. It's just given by the set intersection. No, you pick up, I think it's all fine, but you pick a flux and you compute the number which comes out or you just pick a number L and then you look at your money fold and you want to be sure that all the fluxes that you consider satisfy this condition for given L. No, no, no, this is not L. This is L despite the definition now. That's just a definition of L. Ah, you're disturbed by the word condition here. Yeah, but you have to fix an L. Right, but you don't need to, you just pick an L and then you consider all the set of satisfy this condition and then your string theories, you know that certain Ls are particularly nice. In any case, so that's the mathematical statement, the weakest mathematical statement which you can make and I believe, and then you could try to prove this. And I'm optimistic that at least for the Hodge locals using this theory of unlike the intersections, probably you are able to prove this eventually if it is true. Ah, so because I wanted to make the weakest statement there, I believe it's true. So what I don't know, so what I, okay, I don't know, but I believe that this number here depends on the Hodge structure on the money fold you consider. So roughly you have an estimate of this number here, what is kind of the biggest SL2 representation which you have in the other part, in the limit of the Hodge structure. And for Kalaviyao 4-folds, they are very small. That's why these numbers are very small. But if I have a Kalaviyao 1,000, then these numbers will be much bigger. Because it will be related to the representation theory. I know that there are many non-trivial discussions about this in the master's picture, but this question has not about those. Yeah, I don't see, we can discuss this after. I don't think that there's, I think these maps are so nice that this has a delta prime dimension theory. There's so much known about the Hodge loafer that it has a delta prime dimension theory, but who can discuss that? I agree. That's exactly the challenge. The challenge is to take the positive physics literature and translate it into something mathematical with the positive equation. So my second question is, can one of you please proof this conjecture for the Hodge locus or for the self-dual locus? So I think Arthur would say for the self-dual locus, it's certainly not true. It's a feeling, but I agree that it's much more challenging. Okay, so I'm already over time, right? Okay, so maybe it's fine. So now let me turn to quantum field theories. And essentially in quantum field theories, we look at a very different physical question and you will see in a moment how this is related to what I just discussed. Namely, we consider scattering amplitude, of course, in our real functions. They are very complicated functions which appear in a physical setting and they give you some profitability of how particles scatter. They are in physics defined over all possible processes and so on, it's horribly complicated. But in any case, clever physicists have developed this expansion into Feynman integrals. And the nice thing about this is that these Feynman integrals, each one of them is a proper integral which we can compute. They are very complicated integrals. Yeah, yeah, yeah, sure, sure, yeah. Sure, sure, yeah, yeah, yeah. Oh, no, I don't worry. I will come back to this. Okay, so what is the theorem which we now proved? Now, there is a hot star here. Why? Because that's a physics theorem. The difference between the mathematics theorem which is true and the physics theorem which is true if you, yeah, don't open your eyes to writing. So they are good arguments that this is true. So what is for any renormalizable quantum field theory with finitely many particles and intersection of finite, all finite-loop amplitudes. So that's about the asymptotic series. I cut at finite-loop, obtained functions of the masses, external momentum and a couple of them. And so essentially, whenever you compute one of these complicated integrals, it actually is the same function. So there is some hidden finiteness property in all perturbative quantum field theory amplitudes. And it's a very non-trivial theorem because it essentially states that there is a non-trivial or can be non-trivial interrelations between algebraic relations between Feynman amplitudes and symmetries of the system. Now, where does it come from? I don't have a lot of time. Essentially, I just state the basic punchline. It's essentially coming from the idea that Feynman integrals can be written as very generally as relative periods of some associated geometry. Many here in the audience have worked on this. So there will be talks in the later part of the workshop. And the remarkable fact that they have been proved to be tamed functions also now translate into the fact that all the Feynman amplitudes. Now, we can come to Albert's question. So we know this is just an asymptotic series. We know that if I sum them all up, I get infinity, which is not an analytic function, but it's some trans-series. So what about timeness in a non-detervative quantum field theory? So a non-detervative quantum field theory actually are interested in computing correlation functions with a complete action, like with a complete path integral. And this is very, very complicated. But what I wanted to do is I wanted to just say some physicists computed this for me and it's a very complicated function. It goes a lot in ingenuity in computing it, but I want to look at the final result as a function of the parameters, for example, of the masses and of the positions on space time and want to ask the question, is this a tamed function? Now, let's look at the simple example and then we will come to Albert's question. The simple example is when you do that in zero dimensions and you look at five to the fourth area, then you can compute it and actually what you find is the best function. And the best function is not a theory. So in fact, is this tamed? Well, it does not have an analytic expansion around the coupling. So it's precisely this feature which was mentioned. It's just a trans-series expansion as a resurgent phenomenon and so on. So it's definitely not an R and X. However, the remarkable thing is there is a other O-minimal structure, a bigger O-minimal structure which accommodates all these functions or many of these functions which have these resurgence phenomena and these are so-called trussian structures and you can show actually that this is part of the trussian structure. And this is rather remarkable because only last year O-minimality has been sharpened and there has been some sort of new notion of O-minimality then which also captures some sort of complexity, the degree of the phenomena which was asked before which is really encoded in the structure itself. And this non-detervative amplitude is actually part of the structure. And we are now going to bring this into physics section. Now, there are multiple challenges in mathematics that may skip this. The question I have to you is what about exponential periods? What is the developed theory? Is there a developed theory of exponential period? Because exponential periods are the thing which you need. I see my patient smiling here. I don't know if it's good or bad. So if there are developed theory of exponential periods which can be used to show that exponential periods are definable in this Parfian O-minimal structure. And with this, I want to end because I'm running out of time. What is the, so to say, the idea now of this whole program is we want to actually map out the tame parts of physics. And we made this very concrete entity had led to multiple new conjectures which I have no doubt will eventually have some impact on mathematics. We essentially stated our precise ideas about conformal field theories, which part of conformal field theory should be came. And as I mentioned already before, we stated more or less precisely because here the objects are not as well defined as an conformal field theory, what we expect from effective theories coming out of quantum gravity. And with this, I'd like to thank you very much. So perhaps, okay. So I was just wondering, until you mentioned you're saying that that tameness works, you get something that is less of a function that it's tame? Yeah. Is that true if you do matrix problems? I don't know. So that, so this has just kind of completely, it's completely exactly, that's what you should know. So it has entered physics only recently, right? Through these efforts. And now you can look at all sorts of interesting questions. For example, a very interesting question which I also can pose to you is, is the topological string partition function which we now understand to better extent, also non-deteriorative, is this a tame function? At each loop order, you can ask, is are these tame functions, are matrix models tame function? We are now looking at conformal field theories in two dimensions, better mean of written models and asked, are our conjectures about conformal field theories in two dimensions true and are all the correlation functions tame function? So now you have this huge part of physics literature which produces non-trivial functions. And you can ask if they are part of this tame, of the tame physics. Once it's part of the tame physics, you can use this strong theorem from mathematics to show very general properties about algebraic relations among amplitude and thought. So that's my vision for the future. Two comments. One is that exponential periods have a fairly well-developed framework. I mean, as you know, GKZ periods in general are exponential periods. And the ones that are periods of variations in odd structure are just some resonant case. But there's been an exponential hodge theory developed by various people who goes back to the work of Savan a while earlier and Peter Nielsen of your raison had a monograph on the subject. So probably talk to them as a test. I got it. The second thing would be that I don't see a reason off the top of my head for why integer classes and h40 plus h22 plus h04 should have any cycle interpretation any more than right on the tractor points should have a cycle theoretic interpretation for a lot of you have three points, right? If that doesn't make them not special in some way, minimal sense, like suitable for the math. I mean, it's not a good thing to take office, right? So, I mean, specialness in the hodge theory says it's a rather narrow notion. Yeah. So, yeah. It would be nice to have a definite answer to this. So your feeling is that there's no- Yeah, but it would be nice to have a- I think so. In the light of these holography papers which attack the scenario which are presented as KKL two scenarios which is very, very popular, big story around it. It would, there is some suggestion of some sort of tool interpretation of the cycle but I don't know if this is by any means complete. So, I don't think that in any case it would be nice to exclude then it would be nice to exclude that there is such. I only want to make a silly physicist's comment. So, if the universe is compact or the sitter which is politely then the entropy is finite so everything is tame. So, tameness is really deep in the heart of physics. Yeah. Thank you. Even more silly physicist's question. So, what is completely now say a condition on our consistent theory to become the quantum gravity? Just before you said, it's just a property of ordinary quantum filters without gravity. Now here you have this picture with quantum gravity. So, what goes wrong if you sort of what happened to find a theory which is not in any way, like very nicely speaking, a classical dynamic system is a chaotic, a fractal behavior is very awful. So, what was wrong if you tried to cover things like this to quantum gravity? Yeah. So, I gave here a few examples of systems which are not tame and we also do not expect to be coupled to random gravity. So, one example is something like you have a scalar potential which forces you on to an infinite spiral in its vacuum, right? So, this goes again, many of the conjectures we had about the nature of quantum gravity in quantum language. And in particular, the distance conjecture would be severely challenged. So, I give you some compact region in space and of course in any compact region of space I can draw an infinite spiral. And if such an infinite spiral comes from a scalar potential constraint theory, I would violate the distance conjecture. If I lower the energy, I'm in this infinite spiral and it's just ordinary region in space. But this is not a tame, that's not a tame function. This is of course a very on a very abstract, elevated level, you know. And actually people try to construct models of this type and they never work. But what I mean, for example, they typically swamp land conditions boil down to decay properties of black holes, you know. Typically it's always the same story that some kind of entropy balance or whatever some things are violated. So what does a concrete play say for such an example, you know? Where would quantum gravity go wrong? There would be a contradiction of... Well, I mean, yeah, this is unfortunately an example where the distance conjecture is also equally badly motivated or like whole procedure. So it's not a good example to make my point, but I have another example here. For example, functions which have discrete infinite order symmetries, they can also never be taken, right? Because the sine function has an infinite order symmetry just to have infinitely many minimized, we want. And this symmetry is believed to be absent in the theory of quantum gravity. We have like whole arguments for it. And so it ties in nicely with this conjecture. Of course, it doesn't follow for many of these conjectures, it's really a new principle. Yes, sorry, another point occurred to me. So what about the famous series of ADS-5 and the S5 vacuum? Is any flux going to infinity? How do you exclude that from our considerations? Well, I mean, this is not part of this talk. So you have to put a cutoff at the end here. I mean, I understand that this is logical, but this seems to be a very by hand physicist thing to do to put a cutoff. I mean, there seems to be, as a series, there seems to be a perfectly well-defined set of these series connected by domain walls, which are not, we don't form a same set. We can talk about this in the frame. All right, so we thank the speaker again. And I'll see you at lunch.