 First, I want to say that if you are confused or you think I'm going too fast, you want to ask a question, just feel free to unmute yourself and interrupt me. And then, as suggested by Benach, I will put the notes of my talk in this Zoom chat. So let me put it again so that you can download the PDF notes of my talk and in the middle of the lecture, if you forget something that I said before, you can look at the notes, go back and look at the notes. I think it's helpful. So let's start. The title of my mini course is the Frobenius Structure Conjecture for Logo-Kalabiya Varietis and the reference is my paper with Sean Keogh on archive. So here is the plan of this mini course. First, I will talk about the statement of the Frobenius Structure theorem and applications to cluster algebra and applications to the modular space of logo-kalabiya pairs. And in the second lecture, I will talk about skeletal curves, which is a key notion in the theory. In the third lecture, I will talk about deformation invariance, tail conditions, gluing, associativity, and convexity. It's lots of properties about curve counting. And in the last lecture, I will talk about scattering diagram and comparison with the work of gross hacking keel on cluster algebra and then more details about application to the study of modular spaces of kalabiya pairs. Okay, so let's start the first session. Someone asks that he does not see the link of the lecture note. I can put it again. Anyway, it's in the chat, you can just download. Otherwise, if you cannot download, I will also put it on my website afterwards. So let's start the first session statement of the main theorem and applications. So the motivation comes from mirror symmetry, which is a conjectural duality between kalabiya manifolds. So roughly, mirror symmetry says that for any given any kalabiya variety X, there is a mirror kalabiya variety X check. And by the mirror symmetry philosophy, one may build the mirror kalabiya variety X check by counting curves in X. So the Frobenius structure conjecture of gross hacking keel is a precise yet simple formulation of this mirror symmetry philosophy for log kalabiya varieties, which boils down to intricate relations of counts of rational curves. And the first question is what curves do we count. So let me describe the setup first for the Frobenius structure conjecture. Here is the setup. We fix K any field of characteristic zero. For example, you can take K to be complex numbers. But our theory is algebraic. We do not use the analytic structure on C. And then we have you an affine smooth log kalabiya variety over K. So here by log kalabiya, I mean it means that all log pluricononical bundles are trivialized by the tensor power of some volume form. I will give some examples of log kalabiya varieties later. And then we denote by K of you the field of rational functions on our log kalabiya variety. And we denote And then we introduce a set SKUZ. The set is called integer points in the essential skeleton of you. For the moment, it's just the words. I will explain later the mathematical meaning of essential skeleton. But for the moment, it's just terminology. And we have an explicit description of this set SKUZ as follows. So it's just a zero disjoint union of M new where M is a positive natural number and the new is a divisory evaluation on the field of rational functions KU where our volume form omega has a pole. So divisory evaluation. In other words, it's just given by some divisor on some compactification of you. So now, let's fix a normal crossing compactification you inside the Y with compliment we denote by D D is y minus you. So this is the picture we have a Y projective normal variety and our log kalabiya variety you lives inside and the D is divisor at infinity. So this is our setup and a base field of characteristic zero some log kalabiya variety you and then we consider for the moment just this explicit set, which is just multiples of some divisory evaluation. And then we also have some compactification. And the next, we want to count rational curves in you. So what are the curves that we want to count. So we want to define some counts of rational curves. We are given. First, we are given an end couple P one to P and denoted by this board P where each PJ is just a point in SKU Z. In other words, just some multiple of some divisor. And we also, we are also given a curve class beta inside the NE YZ cone of effective curves, which lives in the in the space of one cycles, modular numerical equivalence. So you can just think of it as a homology class. Here we write a cone of effective curves and in terms of cycles just so that our description is purely algebraic. And the next, since each point in this SKU Z has this explicit description in terms of just some multiple of some divisor. So we write PJ equals MJ new J for all non zero PJ. And we assume that each new J is just given by some component in the. This is always possible if we make a big enough blow up. And now the counts that we want to define is the following. So we define the count eta P beta to be the number of closed rational curves in Y of class beta that intersect the divisor DJ with order MJ for every non zero PJ. Here is the picture. So that's all we want to count. Just this kind of red curves. In Y of given class beta whose intersection with the boundary D is given by our end couple. Yeah, you mean intersect this order with some kind of tangency order. Yeah, and then say, yes, and then say order MJ. And there's no conditions for components. Yeah. For PI equals to zero, they are somehow living in the interior of the curve. There are no conditions. They have to, they should not go into the boundary. So. Yeah, so we just define the count to be such red curves. And you may ask why the count is well defined. So, let me also give a precise mathematical formulation. But heuristically, it's just that we counted the set of such curves. So the precise mathematical formulation is the following. We consider that H bold P beta be the space of maps from P1 with N plus one mark points. So we put P1 to PN, small P1 to PN, and we put also an extra mark point as to why we consider the such space of maps, satisfying the following conditions. First, for every non zero PJ, we ask F PJ to meet the in open part of the component DJ with order MJ. As Maxime said, tangency order MJ. And also, we ask that there are no intersections as no other intersections with the boundary. And then we ask that the class F push forward of fundamental class of P1 is beta. And then we can prove a simple lemma saying that if we consider the map five from our modular space of such maps to M zero N plus one times you where M zero N plus one is just the modular space of N plus one points in P1. So the first factor of our map five is just the taking domain of this taking domain. And the second factor of our map five is take evaluation of the marked point S. So, we have a simple lemma saying that this natural map five is finite a doll over as a risky dance open subset of the target. And this can be proved using the deformation theory of curves. Now, using this lemma, we can precisely define our accounts. So now we can precisely define our accounts of rational curves, eat up all the P beta to be simply degree of the finite a data map above. Let me recapitulate that in order to state this conjecture of gross hacking kill. We have to define some counts of rational curves. And heuristically we just account to this kind of red curves, but mathematically precisely it's also very easy to count, because we can set up this modular space and we show that the map is just finite a dollar and then we just take a degree of this finite a dollar map, and this is sorry to interrupt. Do you understand that you assume that there's no other intersection points of curve is a boundary. Yes, because he didn't. Ah, yes, it's written. Sorry. Yes. Yes. So, yeah. Tony, can I ask a question? So I saw the formula you gave. Sorry, let me put out. So, okay, so you assume that. Oh, so you assume that each PJ can be expressed as the linear combination of the VG. So the VG is actually so all the pages are actually the valuations on the on the function field of the you. And so those VG, I'm not for sure. So those VG are just given by like a localizing the sort of like the real functions are localizing the structure shift at the boundary strata the general points of the boundary strata. The new J supposed to be a Greek letter. It's just the same as some divisor at infinity for some compactification. Oh, I see. I see. So it's not a localization of something or. Okay, okay. It's just a divisor at infinity. I said the divisory evaluation. It's just the given divisory evaluation. In other words, it's just given by some divisor at infinity. So it's just a divisor. Oh, I see. I see. Okay, thank you very much. Yeah, we it's divisory evaluation of the field of rational functions on our variety, which is the same as some divisor. Yeah, yeah, thanks. No problem. So, yeah, so then the count is very simple, just a degree of a finite data map. And since it's just a degree of finite data map. We remarked that these counts are sometimes called naive counts. In the sense that we make no use of the virtual fundamental classes. So now we have defined our accounts, and we see that our accounts depend on two parameters, both the P and the beta. And there is a natural question. If we vary our parameter, both the P, which is n tuple of points, just multiples of divisory evaluations, and curve class, if we vary our parameters, we get infinitely many numbers. Ita P beta. So, natural question is what is the relation between them. In order to answer this question, let's assemble the numbers, ita P beta into generating series as follows. So first, we want to assemble all the curve class beta, and we just assemble like this, we consider R to be this meaning the monoid ring of this any YZ integer points in the cone of curves. In other words, positive integer linear combinations of effective curve classes. So we let R be the monoid ring of this over Z, which is nothing but just direct sum of a lot of copies of Z. And with basis in NEYZ. So we denote, it's just a notation, we denote the basis vector to be Z to the power beta. So it's just the most straightforward way to put all possible beta together. And the next, we want to assemble all the P together, and we consider a to be the free R module with basis inside SKUZ. So, we define a to be this free R module, which is simply just copies, a lot of copies of our parameterized a lot of copies of our parameterized by points in SKUZ. And we denote the basis vector by C type P. So here, Z to the beta and the C type P, they are just a notation, they are just notations. So, all what we do are just put all the possibilities of beta and all the possibilities of P together, nothing else. Then, we want to put all these numbers eta P beta together, and we define the following notion called Frobenius Perry. So, we let this multi-linear map from some product of a to R be the R multi-linear map, which sends the basis vectors, the basis elements, theta P1, theta Pn into just some over our count eta P beta over all possible choices of a curve class. So, we define this Frobenius Perry just to put all possibilities of P and beta together. Is it some finite automatically? It's a good question. So, first, I said that there is nothing mysterious here. All what we do is the most straightforward way to put all possibilities of parameters together. And the second, as Maxim remarked, the sum is actually a finite sum, which follows from the affinities of you. So, now, with these preparations, we can state our main theorem called Frobenius structure theorem. So, we assume that U contains an open split algebraic torus. I will remark on this assumption a bit later. And then, the following hold. First, the R multi-linear map we just defined from and product of a to R is degenerate. So, this Frobenius Perry assembling together all the counts, it's a non-degenerate Perry. Non-degenerate means this and the usual notion of non-degenerate for Perry. And the second, there exists a unique finitely generated commutative associative R algebra structure on A such that, so for the moment, A is just a free R module. And the theorem says that on this R module structure, there exists a unique finitely generated commutative associative R algebra structure such that, first, this base vector theta 0 is 1 in the algebra. And second, if we take the pairing of n elements A1 to An, then this is equal to the trace of the product multiplying A1 to An using this algebra structure. Here by trace, we mean taking the coefficient of theta 0 inside this product. So this is the second statement. And third statement. So now, let's consider Td, some distributed torus with co-character group generated by the irreducible components of D, and then we have a natural equivalent action of Td on this family. So, we start with A just as some R module. And in the second statement, we equip A with some algebra structure. So now, we are able to take a spec of this A, which maps to spec of R, because A is R algebra. And we denote spec of A by V. So, from the motivations of mirror symmetry, we call A the mirror algebra, and curly V is a mirror family. They are just words for the moment. And statement 3 says that we have some torus generated by irreducible components of D, and it acts equivalently on this mirror family, just spec A over spec R. Probably it doesn't make much sense, just a single statement as this, but I will explain later that this equivalent action is very useful. So, we have this family and recall that our R is defined over integers. Our base spec R is defined over integers. It was just a monoid ring over these curve classes. In other words, positive integer combinations of effective curve classes. Tony, small question. Is this monoid finitely generated or not? It's not necessarily finitely generated. Not necessarily. But later, when we really need it to be finitely generated, we will do something. For the moment, it's not. Yeah, so we have this spec R, and it's defined over Z, and now let's base change to Q. So, yeah, although this spec R is bad, it's not finitely generated, but this family is good, as we will see, it's relatively good. So, spec R is defined over Z, and now let us base change to Q. And we define VQ to be spec AQ to spec RQ. Just base change this map to Q. Then statement four says that this family over Q is a flat family of a fine varieties of same dimension as U. Each fiber of this family is a Gorenstein semi log canonical and K trivial, meaning that the canonical class is trivial. And moreover, the generic fiber of this family is log canonical and log K trivial. So here, if you are not familiar with the words Gorenstein semi log canonical or log canonical, they are just conditions on singularities. So, somehow, it just says that we get a flat family of a fine varieties of the same dimension as U with some manageable singularities. It's unreasonable to expect everything to be smooth. So, but we have a pretty good control on the singularities, and the generic fiber is a log Kala B override. Tony, yeah, so it's not smooth, generic fibers also not smooth in general. In general, it's possible that none of the fibers are smooth. Yeah, but can you apply your original construction to also varieties with some kind of singularities like those which you obtained because you assume you start with smooths for writing, then to some non smooth varieties. Yes. So, in our paper, we only consider the smooth case, but it's possible also to extend to singular case. And for that, one has to be as more technical at the many places, but theoretically, it is possible, possible also to start with a singular and obtain singular. Maybe after I explain more details in the construction, you will see better. Like, we can actually just remove the smoothness assumption. So may I ask, yes. So you, what you mean by the, well, what's about the singularity class. You allow low terminal singularity. Oh, for the moment, we just assume our log Kala be out to be smooth. But you said that you can extend to some money singular. Yeah, but it's, it's technical and we didn't really. So of course, we have the singularities has to be co dimension at least two. And yeah, then we guess log canonical should be good enough, but we haven't yet checked all the details. And also, that's not our first priority for the moment to extend to singular case, because actually, in the singular case, we can just make blow up and assume smooth. Then we will lose the defineness. So somehow these are, these, they are related to these questions. So another question is the, what do you mean by low Kala be out. So do you, do you have, have some natural computation. Yes, this is a very good question. So actually, we also have a natural compactification of this family V. Let's denote, say the compactification is X. And then by log Kala be out, I just mean that K X plus the boundary divisor is trivial. And the Paris LC. The pair is generic. Yeah, the generic fiber is log canonical as a pair as a pair also. Yeah, for the interior and also as a pair. Okay, thank you. Yeah, no problem. Oh, sorry, Tony, can I ask a question. Yeah. So you said that each generic fiber is a log Kala be out. So which means that for each generic fiber, you actually choose the compactification. But so this complication is consistent for the whole like family or it just each fiber will have its own like a special compactification. The specification is for the whole family over spec R. And you will see that I will mention later today that we can not only compactify fiber wise, but we can even compactify the base. And finally, we can make everything to be compact. And that's why I mentioned in the beginning that we can apply our theory to the study of the modular space of log Kala be out pairs. So we obtain compactification of modular space. Oh, okay. Thank you very much. Thank you. So, yes. So this is the first statement of the main serum. The first is that non degeneracy second is the existence of algebra structure. So that is the equivariant the tourist action and the force is the singularities of the mirror family. So let me make some remarks. Remark, we can remove the dependence of our mirror algebra a on the compactification. Why, by setting all curve classes to zero. So, somehow, if we ignore curve classes, then the compactification doesn't really matter, or what matters is the you. So we set all curve classes to zero by taking tensor product. We denote this by a you and we take a tensor product with Z over R where our maps to the sense every Z to the beta to one. In other words, sending all beta to zero. So the compactification doesn't really matter. We can ignore it if we forget a curve classes. And second remark, the assumption that you contains a tourist always holds in dimension two, but not always in dimension three. And this assumption plays two roles in our theory. First, it allows a degeneration of our mirror family to a torque variety. And this is crucial for the proof of one, which is non degeneracy and for the proof of four, which is the study of singularities. And the second role it plays is that it greatly simplifies the enumerative part of the theory. So, in fact, the enumerative part of the theory. One can have a more sophisticated enumerative theory in order not to use this tourist assumption. But it's more technical and in fact that many geometric ideas of the enumerative theory are already present and are much easier to illustrate in this simplified situation. Sorry, Tony, one question. It's because this conditions really looks out of blueses containing contours. Does it have also hold for mirror family varieties? In some cases, it also holds. Like we have to assume some not always, but in the final case, I think it always holds. And also, as I told you, it's because if contains local labial contains the torus and contains infinitely many torus by mutations. Yes. So we will see the mutations later. So, but we still, although we used this assumption, and we conjecture that the serum should probably still hold without the assumption. Then I want to mention that the original conjecture of growth hacking keel is a variant of this serum, which was stated via log of written invariance instead of naive counts of rational curves. So the statement is more complicated involving log of written invariance and which does not have the torus assumption. And growth and the seabird they are working on the mirror construction problem in greater generality using their theory of punctured log of curves. And it is not evident whether their mirror construction will coincide with ours using no Archimedean curves. Recently, there is a preprint by Mark and Julia, which showed the comparison in some special cases, but in general, it's not clear to how to compare log invariance with no Archimedean invariance. Yeah. So that's the main serum. And now I would like to talk about structure constants of the mirror algebra. Sorry, may I ask one more question? Yes. So when you define this naive count, so you already assume the rationality or rationality of what? Of, I mean, your variety. No, it's not necessary. For the naive count, we just, for any log calabiol, for any affine, smooth log calabiol. I see. Yeah. So, yeah. So, let's recall that the second statement of our theorem says that there exists a unique finitely generated commutative associative algebra structure. And it's stated as existence, but we have a concrete construction of the multiplication on this algebra. So, let me now describe what is construction of structure constants of this mirror algebra. That is, we want to answer the question that how are products defined in the mirror algebra. Recall, the mirror algebra A as a module is just a free our module with basis parameterized by points in this SKUZ, which are just multiples of divisory evaluation. So, in order to define products, it's enough to define what the product does on basis elements. So, given P1 to Pn in SKUZ, which is just multiples of some divisor, we write the product in the mirror algebra A as the following. So, we take a product of these basis elements, theta P1 to theta Pn, and it will be an element in A. So, since A is free our module with basis in SKUZ, so we can write this element as sum over all q in SKUZ of this basis vector theta q with sum coefficient. And this coefficient is in R, and we recall that R is this is a monoid ring over curve classes. So, we can again write this as sum of all possible curve classes over all possible curve classes of this basis vector z to the gamma and with sum coefficient, and we denote this coefficient by chi P1 to Pn q gamma, and this chi is just called the structure constants. So, this formula for multiplication is just unfolding the definition of A as R module, just write it in terms of basis, and then the coefficient is called the structure constants. So, here is the idea for the definition of structure constants. Inspired by Kant-Savage homological mirror symmetry, we would like to define the structure constants chi as counts of holomorphic disks. Unfortunately, disks do not make sense in algebraic geometry. So, we have to go to analytic geometry. Sorry, sorry, I like to interrupt. Yes. A question is that, what is the inspiration behind this count chi as a kind of count of holomorphic disks using Kant-Savage homological mirror symmetry means can you keep the idea? I did not understand it clearly. Yeah, because homological mirror symmetry says some equivalence of categories between Foucaille category and it's like Foucaille category and the derived category of coherent shifts, then one can try to think how do we represent functions on the coherent side, because here these elements, they are like functions on the Lore-Kalabi-Avoretic. So, how do we see these kind of products from the picture of the symplectic geometry in the Foucaille category side, and then we will see that they should somehow be determined by counts of holomorphic disks. Which side is counting, I understand this side is counting, the sky is counting some Foucaille category. This is the holomorphic side. These theta's are on which side, in these coherent shift sides or in the Foucaille category? So, if you want to think in terms of mirror symmetry, here where we count is the A side, and what we obtain, this theta, they are on the B side. Okay. But this is just heuristic idea from homological mirror symmetry and it suggests to count some holomorphic disks. Sorry to interrupt, I think there will be some possible confusion. You have your variety, you and have compactification, yeah? Yes. And before you said that you can always arrange compactification, so it goes to individual devices, not intersectional devices. Yes. But now, but when you make generating series here, you would consider numerical effective classes and some fixed compactification. Fixed compactification. Yes. So, now Karof will go to intersections of devices, which you never mentioned. But when we make this generating series, for example, when we make, where is the generator? Yeah. When we make this generating series, this y is fixed compactification. So, in the complication, more complicated pictures? No. This y is fixed compactification. But when we count this eta, we make some arbitrary blow up. Yeah. And this is just some over all possible curve classes on the blow up, which projects to bed. Oh, yeah. It's just temporary, one can make temporary blow ups. Okay. Is it clear? Okay. Yeah. So, but this is just a way of, of like a technical detail. So, one can also use, if one doesn't like a temporary blow up, then one can use the log Goromov written invariance by growth and the seabird and the work with with with the curves going into corners. So either one make blow up or one use log invariance. Yeah. So, yes. So by homological mirror symmetry, one can construct the count this structure constants by counting some holomorphic discs. But unfortunately, the discs do not make sense in algebraic geometry. So let's go to analytic geometry. Oh, sorry, Tony, to interrupt you. Yeah. Oh, even though I definitely agree that even though it doesn't really make sense for accounting of the homomorphic discs, but is it possible to define those structure constants by using the tropical tropical counting stuff. Yeah, I mean, we will see how tropical curves come later in the series, but this account is not a purely tropical count. It's actually the multiplicities of the count cannot be seen always from the tropical picture. It's like a combined tropical and analytic. Oh, somehow multiplicities, like when you count the tropical curves, for example, you apply the Mikaukin multiplicity formula. Because we just not just account combinatorial objects. We also need to count with multiplicity. And in the Torah case, one apply Mikaukin multiplicity formula, but here we are not in the Torah case. And the multiplicity does not one can not just read it out from the combinatorics. Sorry to just also say it's inspired by mirror symmetry. Do you mean some there are some major some Lagrangian varieties somewhere or corresponding to this city basis elements. We have clear picture. Oh, you mean how to compare with the symplectic cohomology. Yes, this is also possible. I think as long as the symplectic cohomology are regularly defined. It's also possible to compare with like you mean. Yes, yes, how should the cohomology of the focaya category. So, sorry, but then this Lagrangian, but these theaters are in the coherency coherence site, not in the on the ship site, how the Lagrangians in these theaters because just previously Mark you made this this this site structures constant coming from the focaya means Lagrangian thing but how this theater is related to this Lagrangian. Yeah, theta is on the B side and the case the counts are on the A side. That's what how mirror symmetry works. Yes, yes, but the coherence ships this you're counting the core and ships are not on the Lagrangians are coming from the A side is it wrong or am I wrong. Yeah, they are on the A side coherence ships on the B side. Here, theta are like a sections of the coherent ships. And so what is the relation with this Lagrangian that makes him say that maybe it's too complicated, I think it's good. Yes, so the thing I want to emphasize is that this is a lot of very powerful heuristic picture of a homological mirror symmetry, which suggests a lot of this kind of mysterious constructions we make. But what is good about the the story here is that we will just make everything algebraic and so it stays purely in algebraic geometry and and everything we can have a rigorous foundation. Yeah, so. Yes, so we want to count a holomorphic discs, but it doesn't make sense in algebraic geometry. So we go to analytic geometry. And of course, the first choice one try is complex analytic geometry. But it doesn't work well, because if we count discs in complex analytic geometry, as suggested by the homological mirror symmetry conjecture, we obtain complicated curve the infinity structures. And it's not a clear how to get well defined accounts from these structures. So I was the solution is to use the knowledge media analytic geometry. And then we can actually have a very simple and direct definition of the structure constants by counting knowledge media and curves. So explain this in the second how to count this knowledge media and curves and but, however, studying the properties of such counts requires more work, which will be explained in later lectures. So just the first explain like what what is this simple direct definition of structure constants by counting knowledge media discs. So, yeah, so to count knowledge media and curves first, we have to go to knowledge media geometry. So we do better some have some knowledge media field. And it's very simple. We have our base field, okay. And we just equipped equip our base field, okay, with the trivial absolute value, which is, which sends any nonzero element to one, and which sends zero to zero. For example, even if we start with the field of complex numbers, we just ignore the usual complex norm complex norm, and we replace it with the trivial episode value. And then all of a sudden this K becomes a knowledge media field. So that's how we apply knowledge media geometry here. So this K is a knowledge media field, and our log Calabi or variety is defined over K, we can apply their coverage and the notification to our K variety you, and we obtain your own. And this and the notification, it's a K, a K analytic space in the sense of Vladimir, but it's, and this contract, this construction has many analogous properties to complex and the notification. And for your information, if you have never seen this knowledge media and notification, it's quite easy to describe the underlying set of the K analytic space. So, the underlying set consists just of pairs cosine new, where cosine is a scheme theoretic point of view, and the new is an episode value on the residual field at the cosine extending the episode value on K, our base field, which is in the trivial value, the case, the extending condition is the vacuous. So new is just an episode value on the residual field. And we see from this description that the set SKUZ, which we define explicitly as multiples of divisorial valuations. They, it just lives inside the UN, because they are just the episode values on on the residual field of the generic point. They are valuations on the field of rational functions, which is just the residual field of the generic point. So they, it lives inside. And by assumption, our U contains a torus. We denote our torus by TM and being the core character lattice. Then, since this only depends on the generic point. And we have SKUZ is the same as the SK of the torus. And one can compute easily that this SK of the torus is just the core character lattice. Yeah, so that's all about compact, all about any notification. And now we are in purely non-archimedean world. And recall that our goal is to define the structure constant chi p1 to pn q gamma by counting non-archimedean analytic disks. Again, let me first also give the heuristic picture. So heuristically, what are we counting? Heuristically, what kind of non-archimedean disks we are counting? So here is the picture. Yeah, so heuristically, we define this structure constant chi p1 to pn q gamma as the number of this kind of red disks. So it's the number of disks delta, this red disk delta in yon. Such that first, we ask such that first, we ask that the disk intersects every dj with order mj. And the second, we ask the boundary of the disk to go to the point you go to the point q. Remember, I explained that this SKUZ just lives in the identification of you. So in particular, the point q, the q becomes a point in the identification of you. So we just ask the boundary of the disk to go to you to go to q. And in fact, in non-archimedean geometry, the boundary is a single point. Here I drew a heuristic archimedean picture, but the non-archimedean picture, in the non-archimedean picture, the boundary is a point. It's always easier to think in terms of the archimedean picture. Next condition is that we ask that if we compute derivative of this disk at the boundary, then we want the derivative also to be equal to q. Here q is sort of as a direction. And this notion of derivative, we can make sense of this notion of derivative using the theory of skeletal curves that I will explain later. And the last condition is that What is the meaning of derivative? What derivative? It's too late to explain. We have a map from disk into the variety. So it's derivative of this map. This disk lives inside the variety. So here we have a derivative. And the precise sense can only be understood using the theory of skeletal curves. Because here when we make derivative, it's not clear where to go. And the final condition is we ask the curve class of this disk in the limiting sense to be gamma. So this is the heuristic picture for the counts for structure constants. And maybe I think it's a good place to make a break. Five minutes break. May I ask you a question? So you saw the in the second condition said so the boundary of the data maps to the queue, which is point in the Berkeley and identification. But in your picture is not really mapped to cure. It's just the boundary of the contents to cure something. Yeah, because this is Archimedean picture, then the boundary is s one. No, I say, I say, but in the knowledge in medium picture, I think later I also have some knowledge in medium picture. You will see that the boundary really maps to cure. Oh, really? Yeah. Okay, okay, I see. Okay, okay, so make small break to 35. 11. 35. 37. 37. So I just the resume. Yeah. Yes. So, yeah, so we want to define structure constants, Kai, by counting our Archimedean discs and heuristically, we just count this kind of red discs. But there is a trouble is that, unlike the situation for counting closed curves above, the space of all such discs satisfying these conditions is in fact the infinite dimensional. And the question is, how can we extract a finite accounting number from this infinite dimensional space. And the idea is just to add one more condition. So, these four conditions are conditions that somehow suggested by homological mirror symmetry heuristics, and now we want to add, but they are not sufficient. They, the space, we want to add sufficiently many conditions to have a finite dimensional space. And now we want to add one more condition. In order to reduce to finite dimension by imposing a regularity condition on the boundary of our disk delta called the Toric tail condition. So we ask that we ask that by analytic continuation. Since we are in analytic geometry, we can do analytic continuation at the boundary. We ask that our disk extends to a closed rational curve. Being why are satisfying the following two conditions. First, if we write Z to be the opposite of Q, and then Z is just some multiple of some divisor we denote by DZ. Then we ask the tail T to hit DZ, the divisor DZ with order MZ. Yeah, and the second, we ask that the punctured tail, which is the tail T minus this hitting point Z, we ask the punctured tail to lie inside our torus. And this is one more extra condition we impose somehow about the boundary of our disk. And then this Toric condition together with all the previous heuristic conditions implies that we obtain a finite accounts of no Archimedean disks. So that's heuristic picture. And in fact, the precise mathematical formulation is not so complicated. And I can also give it fairly quickly. So let me do it here. What is the precise mathematical formulation in order to realize this heuristic idea. In the first step, we have to figure out the class of the added tail. Since we, I mean, we only specify the class of the disk, and now we have an extra tail, we need to figure out the class of the tail and the claim is that the class of the tail is just equal to the class of the closure of any general translation of the one parameter subgroup in the torus given by the point Q. And this is easy to see from tropical picture. So first step, we figure out the class. And the second step, whenever we want to count something with a better set up the modular space for these extended curves. So, so recall that now we have an extra divisor dZ, Z was the opposite of Q. And we add this extra Z into our tuple. We used to have an N tuple P1 to PN. Now we added one more Z. And we also have a beta extended curve class. Then as in the definition of counts for the Frobenius pairing, we consider the modular space H PZ beta. Where we marked, we mark label with where we label all the marked points. Now we have N plus two marked points. We label all the marked points as P1 to PN and ZS. So we are just setting up the modular space for this extended curve. And we use almost the same modular space as in the definition of the Frobenius pairing. And we recall that this modular space we used in the definition of Frobenius pairing consists just of rational curves in Y of class beta whose intersection numbers with the boundary D are given by our tuple PZ. Just the modular space of closed rational curves. And the recall that we also have a natural map taking domain of our rational curve with marked points. Taking domain so we go to the modular space of P1 with N plus two marked points. And we also take evaluation of S of the extra mark point. That's the second step. And the third step. So now we want to define the count as degree of something. So we'd better pick something. So we should pick first some point in this first factor. And we pick the right modulus in the first factor. And we let mu be the divisory evaluation given by the divisor in the natural delin one for the compactification parameterizing nodal curves. Which has a node separating the first and marked points with the last two marked points. So that gives a point mu. And then we let q tilde be the pair consists of mu in the first factor and q in the second factor. And recall by deformation theory that our map of phi is finite a tau over a Zariski open of the target. And therefore this if we take but this point it's very generic point. So it lives in any Zariski open. So if we take a pre image of this point it's a finite. So more precisely it's finite over the residual field at this q tilde. So somehow we want to take this degree of this finite analytic space as our count. But if we think for a second. We realize that not all curves in this pre image are extensions of disks by Toric tails, because they can have a wild stuff. And we want to discard those wild curves. Then for the last step. We impose the Toric tail condition. And so here is the Toric tail condition. So we define a map F from rational curve with n plus two marked points to why on is said to satisfy the Toric tail condition if the following holds. Um, so now I have drew the non-archimede and picture of this curve. And you'll see that the non-archimede and picture of curve is a tree. And we have a many marked points P1 P2 P3 P4 P5 ZS. Then we let the gamma be the convex hole of all the marked points, which will be the red part union the blue part. And since it's just a tree, since the C is the rational, it's just a tree, we have a canonical retraction from the tree to gamma. And then the tail is simply the pre-image of the retraction of the interval connecting S and Z. So this part is a tail. And the Toric tail condition is to ask that the punctured tail lives inside the torus FT minus Z. This blue part together with all the green small pieces is our tail. And we ask this part minus Z to live in our torus. In other words, the punctured tail lies in the torus. So finally, we let F be the subset of this finite set satisfying the Toric tail condition. And now we are ready for the precise definition of a structure constant. So we just define our structure constant to be a chi to be the length of F. Since F is a zero dimensional scheme. It's just the length or degree. In other words, it's just the cardinality after passing to algebraic closure. So if you are lost with this precise definition, it's not a problem. All I want to say about this precise definition is that it's actually not so complicated to define this number. Although proving properties is a bit more complicated, but just to give the precise algebra geometric definition. It's quite simple. And then we have the following theorem, which says that the structure constants defined as above as length of this F. They are independent of the choice of torus. And the furthermore, for our multiplication rule, which we wrote like this, the sums. And a priori, when we write the sum, it's the sum over infinite set. And a priori, this multiplication rule is just a formal multiplication rule. It doesn't give algebra structure. But we prove that these two sums are actually finite sums. They give the finitely generated commutative associative algebra structure on the mirror algebra a in the Frobenius structure zero. So then I have a remark. So as I said, the precise definition of structure constants is rather self contained and straight forward, but the proof of their properties as in the above theorem requires more machineries to be set up, especially the deformation invariance and the theory of spectral curves. And before we plunge into the technologies inside. Let us first illustrate the two applications of the Frobenius structure theorem in research areas beyond the mirror symmetry and the enumerative geometry. So now I will talk about applications. And starting from next lecture, I'll talk about more, I'll talk more about the technologies inside this enumerative theory. There are no questions. I will continue. So the first application I would like to talk about is to cluster algebras. And we make a comparison with the work of a gross hacking keel condensate bits on cluster algebras. Now here is our theorem. So let X curly X be a focal contour of skew symmetric X cluster variety. Which is, I do not recall the definition, but roughly, it's defined from some combinatorial data by gluing a lot of tourists. And we assume that the algebra of global functions is finally generated. And the, if we take spec of the algebra of global functions, we get smooth variety. And we also assume that the canonical map from curly X to you is an open immersion. So it's something which we can very hard to check. Yes, it's general. If you have general weaver and how to check that you is smooth. Yes, from combinatorial data, it's difficult to check. Yes. But as I said in the beginning, it's possible to relax our smoothness assumption. But here I want to state this theorem more as an illustration of like the idea for the moment. So, example, yeah, these conditions, it's not easy to see from combinatorics. But if we showed that some nice variety has a cluster structure, then they are good. So, for example, double Bruja cells in... No, the French citizen should say Brua cells, not Bruja. Brua, okay. The cells are in semi-simple complex groups. They are examples of such focacontra of skew symmetric X cluster varieties. And now we can apply our Frobenius structure theorem to this cluster variety, and we obtain our mirror algebra. So, here, we don't specify any compactification for this log-calabi-yaw variety. And as I remarked above, it's okay because as long as we set all curve classes to zero, then our mirror algebra AU is independent of any compactification. And the next... Let's denote by mirror X, the combinatorially defined mirror algebra of Gorotha-Hackin-Kio-Kontasevitz, then the following hold. So, first, we prove a comparison theorem saying that the combinatorially defined structure constants in the mirror algebra, mirror X... I have a question on the notation, and you have a Kar-Karlik-Kai on focontra of skew symmetric on X, and then what is Y here? I mean, I'm just... Ah, Y is any compactification of U. And X is here? What is X here? Then what is... Clearly X. No, no, the skew symmetric, sorry. This X cluster variety is just a terminology. Okay. This X doesn't mean some variety, it's just X cluster is a word. Okay, okay, thanks. No problem. No problem. So, we prove a comparison theorem saying that the combinatorially defined structure constants on the mirror algebra in GHKK are equal to our geometrically defined structure constants on our mirror algebra AU. So, this gives an isomorphism between our AU and their mirror algebra. And the second, the mirror algebras are independent of choice of cluster structure. In other words, they are canonically determined by the variety AU. So, that just follows from the first, because our mirror algebra is independent of choice of the torus inside. It's associated canonically to the variety, while it's not clear from the combinatorial construction. So, let me mention some consequences of this comparison theorem. First, let's recall that our naive counts are always non-negative integers, because it's just defined as a degree or length of some scheme. So, they are obviously non-negative integers, and this gives a much more conceptual proof of the positivity of the structure constants. And also, of the coefficients of the scattering diagrams, which in turn implies the positivity in the Laurent phenomenon for cluster algebras. So, I will explain probably in the last lecture some ideas in the proof of the comparison, which we have to use some wall crossing structure or scattering diagrams. And we compare not only the structure constants, but also the scattering diagram. And this comparison, since our numbers by definition are just non-negative integers, and this implies the positivity result for cluster algebras. Whose original proof, I think now it has several different proofs. And here we give a geometric proof. And the second consequence is that we obtain a proof of GHKK broken line convexity conjecture, which is some conjecture about convexity property of broken lines in the paper of GHKK, which is not easy to see from the combinatorial construction, but it follows quite easily from the geometry of non-accommodian analytical curves. And third consequence of the comparison theorem is that we can remove some technical assumption in GHKK's paper for the construction of the mirror algebra. So the technical assumption is called EGM assumption, which is short for enough global monomials. And in GHKK's paper, the mirror algebra structure on what they call the canonical algebra. The algebra structure can only be defined under this EGM assumption, but that's no longer necessary from the geometric approach. Is it realistic improvement to other examples when you don't have EGM but still have mirror algebra? Do you have an example where it works? Actually, I don't really, I mean, you mean, are there examples where EGM is not satisfied? Yeah. Yeah, I don't know. But how do you prove that EGM is always satisfied? Yeah. I don't know. EGM is not a simple condition. So fourth consequence is that the comparison theorem shows that the mirror algebra, as I said, just the second statement, the mirror algebra is independent of choice of cluster structure, which was also conjectured in the paper of GHKK. And it can be shown that it's possible for a variety to have different cluster structures. Like, we can have two different ways of writing it as the union of infinite union of Torah. And the final, finally, I want to mention a consequence for the comparison theorem is that it gives a geometric explanation of a change of scattering diagram under mutation. There is some complicated formula for change of scattering diagram, which can be seen from counting this non-comedian curves. So that's all I want to say for comparison with for application to cluster algebra. Oh, this is second, yeah, Tony. Yes. Both in cluster algebras, you can have, can construct non-community of algebras, yeah, like Q deformations. Q, so your question is how to make a quantum deformation of our accounts. Yeah. And also the suppositivity for quantum deformations. For quantum deformation. But this, I don't know for the moment, we still have to think about how to make it. So in that two dimensional case, Pierre, he has some work about quantum counts. But I discussed with him. First, it's not easy how to make it work for curves with boundaries. However, it's not clear how to generalize to higher dimensions. But this is a very interesting question. So that's all I want to say for application to cluster algebra. If you are not familiar with cluster algebra, then you can ignore this and now I will explain applications to the modular space of long collabial pairs. So now we forget about cluster algebra and we go back to algebraic geometry. So we have the following conjecture in my ongoing work with Paul and Sean concerning the modular space of collabial pairs. And the conjecture that any connected component to queue of the modular space of triples, X, E, which is some of E1 to EN, and the theta, where X is a connected smooth projective complex variety. And E is a divisor inside the anti-canonical class. It's a normal crossing divisor with a zero stratum. Zero stratum just means I have many intersections of pieces, so finally I get a point. We assume that every piece is a smooth and the last piece theta inside the X is an ample divisor, not containing any zero stratum of E. So we consider any connected component of modular space of such triples and we conjecture this connected component to be unirrational. So we have a more precise form of the conjecture above for the compactified modular space. And I state first this version because in order to describe the conjecture for the compactified modular space, we need to recall a bit more notions from birational geometry. So let's now recall something from birational geometry. First, I recall the notion of a pair in birational geometry. A pair X delta consists of a reduced pure dimensional variety X and an effective Q divisor delta, none of whose irreducible components is contained in the singularity of X. For example, our YD is such a pair, but we also allow X to be non normal here. Then I recall the notion of KSBA stable pair. It is a proper semi log canonical pair X delta such that KX plus delta is ample. Here semi log canonical is a condition on singularity generalizing nodal singularities to higher dimensions. And then we recall that a log sorry a collabial pair is a proper semi log canonical pair X delta such that KX plus delta is rationally Q rationally equivalent to zero. So finally, a polarized log collabial pair is a triple Xe theta where Xe is a collabial pair and the theta is an ample divisor not containing any semi log canonical center of Xe. In other words, if we add to E a sufficiently small multiple of H, we put E and H together, then we obtain a stable pair. What is H? Oh, theta. It should be theta. I'm sorry. I forgot to draw the circle. Yes. So, yeah, let me so I just recalled the four notions from birational geometry first notion of pair just a variety with the divisor and the second notion of a stable pair. So we ask that the pair has manageable singularities. And stability condition is the same as usual stability condition for curves. We just ask K plus the divisor to be ample. And then we have collabial pair, which instead of asking it to be ample we ask it to be trivial. And finally, we have polarized the collabial pair. It's collabial pair together with a polarization, theta. And we ask the whole thing to have manageable singularity, which has two equivalent formulation, either we ask this polarization theta not to contain any bad center of the pair collabial pair, or in other words, we ask that if we add a small multiple of the polarization to E, then we get a stable pair. This should be theta. Maybe I can draw. But my draw button is blocked by zoom. I cannot click the draw button. I just want to add a circle. Yes. So now that's for sufficiently small epsilon. If we take X, we take the we take epsilon multiple of theta. This should be theta epsilon multiple of theta to E, and we consider this pair. And our modular space Q of triples X E theta immerses into SP, the modular space of stable pairs. And this SP modular space is is a generalization of the deline month for the stack mg and bar of stable curves. And we let now since our modular space Q of triples immerses into SP, we denote the Q bar, the closure of Q in SP. So this SP, SP is of infinite type because we can vary many things, but it's proven by Alexi, Hayekan, McKernan, and the shoe that connected the components of this modular space SP are proper. And it's also proven later that they are actually projective. So now we can state a precise form of our conjecture that Q bar admits a finite cover by a complete toric variety. So the weak form is we conjecture Q to be unirrational. And the process form is that there is a natural compactification of Q. And if we use the right notion of singularities, if we add to Q the right degenerate, good degenerations, SLC degenerations, then we get canonical compactification, and this compactified modular space. So we conjected to admit a finite cover by some proper complete toric variety. And with Paul and Sean, we verify our conjecture in the two dimensional case. So the theorem is that this conjecture holds when X is a del petro surface, E is anti-canonical cycle of minus one curves, and the theta lives in the anti-canonical linear system. So the proof of the theorem is based on the synthetic construction of the family of triples, X E theta, we study modular space of triples, and we have a construction of this modular space of triples over the toric variety T. And this family of triples is an extension and the compactification of the affine mirror family we discussed above, and which works at the same level of generality as in the Frobenius structure theorem. So the rough idea for constructing the family over T is to compactify not only fiber-wise but also on the base of the mirror family in the Frobenius structure theorem. So let me give a bit more details. Excuse me? Yes. Did you claim that the combinatorial, well, the dual complex of E remains the same in this torus T? Yes. I see. Yes. Okay. Yeah, thank you for the question. So, the idea is that we have to start with YD, the mirror kalabia pair. That also explains why the two-dimensional case is easier because it's self-mirror in the del petro case. In general, we have to apply the mirror construction twice. And we assume that the interior Y minus D is a smooth affine containing a Zariski open torus. And we also assume D to be normal crossing and the Y funnel. And let K to Y be the canonical bundle. And we denote by K to K bar a contraction of the zero section. Yes. So now the zero section is just Y. So we just contract Y inside the case. And we noted that we have identification of pika groups for K, for K over K bar, and also for K, for Y. And since what we contract is just Y, then we also have identification of a cone of curves of the relative K over K bar and Y, and identification of the napkin. And then we construct a complete fan sec K over K bar, denoted by sec K over K bar, supported on the pika vector space K over K bar, called the secondary fan for YD. So it is a generalization of the classical Gelfand Kapronov-Zelovinsky secondary fan for reflexive polytops. That's why we give it the name secondary fan. And by construction, it is a coarsening of the more event for K. So I will omit the construction for this talk. And it's some coarsening of the more event. And the more event is a fan supported on this pika vector space, which is defined using more equivalence relation for divisors. So just to recapitulate, I'm just explaining the idea for the construction of this universal family using the mirror symmetry machine. And we start with some mirror funnel, YD. We take some canonical bundle and the contract zero section. And from this one can we construct some complete fan called the secondary fan, because it generalizes the classical secondary fan. And also, and by construction, it uses some more sphere. And next, we conjecture that the toric variety associated to this secondary fan here should be the base toric variety for our family of triples, Xe theta. Because we recall that the conjecture says that the modular space, compactified modular space has a cover by some toric variety, and now we make it precise that this toric variety should be the toric variety associated to our secondary fan. And now recall, we have the affine mirror family curly V defined as a spec of our mirror algebra to spec R. R was just the monoid ring of monoid ring generated by curve classes. So we recall our affine mirror family from the Frobenius structure theorem, this mirror family, V over spec R. And note that the dual of the cone of curves and EY is the nef cone, nef Y. So if we take spec R, it's just the toric variety associated to the nef cone. So now it's finitely generated here. No, it's a funnel. Yeah, for funnel. Yes, for funnel, it's finitely generated. So, yes. So now, we can compactify and extend. So theorem, the affine mirror variety V to the toric variety associated to the nef cone from the Frobenius structure theorem compactifies and extends to a mirror family to a family of Calabiol pairs, curly X curly E to this complete the toric variety. The toric variety associated to the secondary fan. And moreover, we have an ample divisor theta in the total space such that the generic fiber X together with the E plus some sufficiently small multiple of theta is a stable pair for sufficiently small epsilon. So the theorem just says that in the our final context, we can not only fiber wise compactify, but we can also compactify the base. And we obtain a proper a family of Calabiol pairs over this complete base, which is toric variety associated to the secondary fan. And furthermore, there is a canonical theta divisor, given by vanishing of some some of theta functions, such that if we can take a generic fiber, then this becomes a stable pair. In other words, X E theta becomes a polarized Calabiol pair. And we conjecture this stability to hold. We conjecture this stability to hold not just for generic fiber, but for all fibers, but currently we can only prove it in dimension to the stability for all fibers. So here is the precise. Yes. So do you have chemical choice of data as a divisor? Yes, it's a canonical seat. Yeah, depending on Depending on the data we in the setup. Yes. So here is the precise statement in the two dimensional case. We assume why to be smooth their petal surface. Then the family of pairs X E over the toric variety of the secondary fan is a flat family of Calabiol pairs. Meaning it has semi log canonical singularities and has a trivial canonical class. And the second, this boundary is a trivial family of a cycle of rational curves. So the boundary is everywhere and it's trivial. And the third, for every fiber over the structural torus of the base toric variety. X is a del petal surface with at worst deval singularities. That and the E, the boundary E is an anti canonical cycle of rational curves. So we can compute the number of kx degree minus one, because here we have a singularities. So it's difficult to talk about the self intersection. We talk about intersection with the canonical class. And the furthermore, we can compute that the self intersection number of the anti canonical class Is equal to the number of irreducible components of the So it's interesting to compute the self intersection number because by definition this is called the degree of a del petal surface. And then for sufficiently small epsilon. If we add an epsilon multiple of our divisor theta to the boundary E, we get a family of stable pairs. Finally, we show that the induced map from the toric variety associated to the secondary fan to the modular space of stable pairs is a finite map. So that's a detailed description in the two dimensional case. And we see that over. Our base is a toric variety and the fibers, they are SLC. Some fibers are not normal. But we but over the structure torus of the base, the fibers are pretty nice. They are normal. And they are actually del petal surface with the canonical singularities. Finally, I state a corollary of the theorem. If we start with a smooth complex del petal surface Y and the D in Y, an anti canonical cycle of minus one curves. Then the generic fiber of the mirror family is a smooth. And the one fiber will be the original ID. And if we consider the image of the finite map from this toric variety associated to the secondary fan to SP, the modular space of stable pairs. Then we show that the image is just the closure Q bar that we talked before mentioned before for the modular space Q of triple X E theta for the original Y D. So that verifies the precise form of our conjecture in the two dimensional del petal surface case. But in higher dimensions. It's still very difficult. We needed the full mirror symmetry machine to work in full generality in higher dimensions for all possible singularities in order to consider the question for the modular space of Calabria pairs. So that's all I want to explain today. And for the next lecture, I will start by recalling by explaining some basic notions of essential skeleton. Following the work of Tim King, which generalizes condescendence of a man and many other authors. And then I will talk about the skeletal curves in the next lecture. Thank you very much for your attention. Okay, thank you. Yeah, if there are few, you have maybe few minutes for questions. Thanks. Yeah, I don't understand that there are many things to do, for example, you have to remove open torus condition. Yeah, that's the first. That's also about for the for being with a structure conjecture. And a finest assumption, we must keep. Otherwise, we don't have a finiteness of sums, we only have a formal formal mirror families. You have to prove that mirror to mirror is original variety. Yes, so this application to the modular space is really requires everything to be figured out in the mirror symmetry world. May I ask, so for the fifth claim in the so yes, just this induced map, I mean, yes, the map from the toilet to the module. Yes, you may need to be dominant. Just as it cannot be dominant because SP is so big, it contains everything. But it dominates. Yeah, but here, in the collidery, we further prove that it dominates a connected component. So, in general, like there's a, well, some possibility to live when this XE is fixed, but to take a look right. A CTA can always move even in our example, CTA is allowed to move. So why are X and E fixed. X and E can also move. But I mean, is there some direction where X and E fixed while theta only theta moves. That's a sub space. Yeah, but it's not on tree. But if only theta moves, it's just a projective space. It's linear system. Open part of the open part. Yes. Yes. And we need a CTA in this question because without a CTA, the modular space doesn't compactify. The nice compactification is only possible if we put a CTA. It's like in the story for the Dalin bound for the stack of curves. We only have the modular space of the stable curves. Otherwise, we cannot compactify. So the CTA is like auxiliary polarization in order to have nice compactification. And stratification induced by the, so this map respect the stratification. You have a stratification which can be defined by the combinatorial structure of the Xe. And also you have Toric structure. So it's, so you have Toric stratification. Yeah, here we, you mean we have, it's a Toric variety. So we have Toric strat. Yeah. On the other hand, you can consider the stratification by thinking of combinatorial structure of Xe, the dual complex. But the dual complex is fixed. Oh. Oh, you mean in the compactification, in the compactification. In the compactification, yes. Yeah, in the compactification, the dual complex will be subdivided. Yeah. Yeah, that's what I meant. Yes. So the two stratification coincides? It's not clear. We have a fairly explicit description. But they do not coincide, even in the two dimensional case from the explicit description. Yeah. Because I think that even if we go to some boundary strata, the dual complex may not change, may not necessarily change. Okay. Thank you. Okay. I think it's not really time to stop us. Thank you again, Tony, for the beautiful lecture. Thank you very much. Continue. I close the meeting now. I will put the slides also to my website. Yeah. I think it's also through the webpage for the seminar at HACM. Yes. So see you again on Thursday.