 Thank you so much. Welcome. So this is supposed to be a really elementary talk. So anytime, stop me. I don't have a problem, but shout, because I'll try to remember to look at you, but I may not always. So wave your hands, and I'll stop and answer a question anytime. So what do we mean by a numerative geometry? We have a geometric problem, and instead of finding the solutions explicitly, we just want to know how many solutions the problem has. First of all, this is obviously interesting, even if you want to solve the problem explicitly, because you know how many you should look for, or in particular if the number is 0, you know you shouldn't bother. But as we will see later, it also has a use. So the idea is count solutions to a geometric problem. So let me start with a very, very simple example, which will actually play a role later in this talk. So if I give you two points, and I ask how many lines contain p and q? Well, in this case, it's pretty obvious that the answer is 1. There is one line through two points. This, as I said, I wanted to start elementary, so this is extremely elementary. But notice that what I didn't tell you is the context in which the question is asked. So these questions I could ask in R2 or in Rn, or I could ask it over any field, really, if I make sense of what lines are. For instance, if I work in Cn, then I can consider complex lines, and then the answer is still 1. The other remark, which is pretty simple, but in some sense pretty trivial, that the answer is 1 if p and q are two different points. So my problem has some parameters. In this case, the points p and q. And the number of solutions doesn't depend on the parameters so long as they don't become too special. So so long as p and q are what is called in general position, which just means they are different in this case. So let me give you another example, this example 1. And example 2, this was a problem. I told you I draw it in the plane because the blackboard is two dimensional, but you can view it as a problem in any dimension. But my next problem will be a problem in space. So let's work in R3 or C3. We'll go back soon to as to why I'm always interested in the complex numbers. You are given two points and a line. And you ask yourself many lines. The property satisfy, sorry, I didn't mean that. I meant two lines, L1. Satisfy the property that is in R and L1 and L2 intersect R. So this means that R is incident to both L1 and L2. You see, this in some sense is, again, a completely elementary problem. You could explain it to school kids, really. All you need to do is know what it means to be a line. But it's already a little bit less evident than this one. And I will leave you as an exercise to show that the answer is still one. Although in this case, again, if the data are in general position, if P, L1, and L2 are general. And by general, I don't want, I will specify later what I mean by general. But basically, it means that the ones which are bad are very special. Like in this case, the special fact was that P was equal to Q. For instance, here, of course, if you take L1 is equal to L2, then there will be infinitely many solutions. So that is clearly a case I don't want to look at. And I suggest you do the exercise. It's kind of nice. Basically, you take a plane which contains P and L1 and a plane which contains P and L2, and you intersect them. And if this intersection is a line, then you have to prove that it is the only line. Well, so far, all the computation problems I've given you have answered one. So the question is, can I have interesting problems where the answer isn't one? And the answer is, of course, yes. We are again in R3. And I have now four lines. And I ask myself, how many lines are satisfied intersected? So the first question is, why is this even a good question? It looks like I'm kind of random. Why did I give four lines, not 3, not 5? Well, the first question is, of course, I have to choose reasonably. Like here, if I had given only one point, I would have infinitely many solutions. And if I had given three points, if I take three points at random, there will be no line through one of them. So the first thing you notice when you do these kind of problems is you have to figure out how many conditions it makes sense to give. You give too little. You get infinitely many solutions. You give too strong conditions. You get nothing. You want to find a good mean. So why is four a good number? Why do I expect four to be right? And the answer is that, in some sense, the set of all lines in R3 is what is called the four-dimensional manifold. Or informally speaking, it depends on four parameters. This is kind of when you give a line in the plane. This is two parameters because the line is determined by the intersection with this axis and the intersection with this axis. So it's one parameter here and one parameter here. And you can do the same in space, but now in space you do the same, but you do it with planes. So you just take a point, you take your line, and you intersect it with this plane. And with this plane, you have two parameters, two coordinates on this plane and two coordinates on this plane. And then the condition of intersecting a given line. So if you take two random lines in space, they don't intersect because space is big and lines are tiny. But the condition of intersecting a given line has co-dimension one. And the condition intersecting has co-dimension one. It imposes exactly one condition. Again, this can be viewed as an exercise in linear algebra. If you know how to model in linear algebra real geometry problems. And so the idea is if you have four variables to get a finite number of results you want to put four equations. So what is the answer? And the answer in this case is two. So if you have this problem and everything is general, you will have two lines. So how do we get a result like this? So there is a classical method which was the degeneration method. So the idea, I told you that you want to avoid two special cases. You always want things to be general. And the degeneration method goes against this. It says, okay, let's make things a bit special, but still so that there's only finally many results. And then the results we get in this way will be the right one. So you can, you assume that L1 and L2 are in the same plane, two dimensional plane. And also R3, L4 that call it the P1 is a big two dimensional FI space. And this is also, and also that they intersect, that L1 intersected L2 is X. This is not so special because once they are in the same plane, most of the time they intersect. And L3 intersected L4 is Y, another point. And then the two solutions are, one is the line through and Y. So we are back to the original picture. This is one solution. And the other solution is the intersection of these two planes. But the question when you apply this degeneration method is you have to justify that what you're doing still that the result is robust. Is doesn't change by taking this very special configuration that this result which you have found under these special conditions will hold in general. So the problem is to justify this. And in order to do that, and this is something that I want to introduce. So the general approach to this kind of problems is to start, for in this case, I told you we start with all lines. And I've told you it's a four dimensional method manifold and informally I've said this depends on form parameter. But in fact, what you do is you really study, you define G1 R3, let me say G13, set of lines in space. And this is a call it G. This is a four dimensional manifold. And you find a compactification. What do I mean by compactification? Well, this is not compact. It's not compact. Let me explain to you when you do lines in the plane. What is a line in the plane? Well, it has an equation of this form with the A, B cannot be zero zero because it has to be a first degree equation. And of course, it's not unique. So I have to take the set of triples A, B, C such that A, B is not zero zero. And then I have to ask myself when two such equations define the same line. And the answer is if they are multiple of each other. So I have to divide by the relationship that if I multiply the equation, I define the same line. Two equations define the same line if and only if they're multiple of each other. So if we take this quotient, while this quotient, these are the set of lines in the plane. Instead of explaining you why the lines in the space are not compact, I will explain it in the plane because I can do so explicitly. And of course, this is contained inside the set of all A, B, C which are not zero zero zero, they're modular, the same equivalence relation. And this is what is called the projective plane. It is compact. So what you have is you have something compact and this thing here, how much bigger is it? The lines in the plane are a projective plane minus one point. So what is missing? The set of zero zero C, such that C is different from zero, modular this equivalence, but modular this equivalence, this is just one point. So what happens is you have taken something compact and you have taken away a point and what you get is basically never compact. It's like you take a circle, this is compact, you take away a point. This is no longer compact because you can have a sequence which had the limit before where the missing point was and now this sequence has no limit anymore. And so this tells you that this doesn't work. And so that the first thing is that you see when you work in this context it is natural to work projective space and to use projective geometry methods. For instance, in this case, this missing point has an interpretation. This can be seen as the lines in a projective plane. And this missing point is what is called line at infinity. So there is first of all, even if you are only looking at this very simple things you are led to working with projective geometry. And it turns out that this has a compactification which is actually, this is not the real name. This is the compactification is the one which is called like this, it's called grass manian variety. And variety is just the word we use for manifolds when we want to insist that this manifold is defined by polynomial equations instead of arbitrary equations. And this is actually compact. And it corresponds to lines in projective space. What you do is you study intersection theory. So step one is find a compact, what is usually called modular space. The word modular goes back, was invented by Riemann. And it essentially corresponds to this notion that as he said, he proved that complex Riemann surfaces of genus G have 3G minus 3 modular. So it depends on 3G minus 3 parameters. But in current mathematics, modular space just means you start with a collection of geometric objects like lines in space, lines in the plane, all conics. And you take the set of these things and say it has yet another natural geometric structure which it is worth investigating in its own right. So why do I want it compact? Because now it makes sense to do intersection. Let me call this x to make it clear that it's something we have to find. Intersection theory on x. And again, this is something that can be developed in general that if you have a variety, you have what is called a homology ring which basically keeps track of how sub varieties intersect. This is just a ring. It's usually given by generators and relations. So it's a very simple object, although not always easy to compute. And the point of having something compact is that when you go down to zero dimensional things, they have a well-defined degree. And the other thing that you always want to do, so want to always work with the complex numbers or an algebraically closed field. So why do I insist on this? Well, this is the only case in which this example isn't particularly significant because linearly you can't see this. But of course we all know that there are geometric problems where over the real numbers, we don't have a good solutions. Like how many points a circle lie? And of course the answer is, I don't know because the answer could be two or the answer could be zero. So this seems like a bad problem. And the answer is this is a bad problem because we are working over the real numbers. If you work over the complex numbers, the equation of a circle and of a line, this system has two solutions. Or one, again, I have to, of course there is this possibility, this one I can actually draw because, and you see that is actually an interesting point because this will tell me that if you remember here, I had this approach of the generation, of instead of addressing the complicated general problem I can't understand, I try to look at the somewhat more special case which I understand. And you see here, having a tangent line looks very special. And indeed, so why is there this one? And the answer is that this one counts for two. In fact, a classical approach to this would be to say that here the intersection is really two points but the second is infinitely near to the first. This is the language of algebraic geometry from I would say about a hundred years ago, a hundred to a hundred and fifty. Nowadays, what we say is that the intersection is a scheme of length two, yes length two, which means it has a richer structure than just the set theoretic structure and this is instead mathematics from the 60s, 1960s. So about 50 years old. So you can see we get closer and closer to the present time. So once you have this, so this setup, so what about I told you that always, what does this always mean? You can say, look you're ridiculous, we are interested in real numbers, we live in the real world. And the answer is, at least let me qualify, always work with C or an algebraically closed field first. So typically if you want to count solutions over the real, your, the standard approach is you first work over the complex numbers and then you'll find a way to figure out how many of the complex solutions you have found are actually real. And this is a quite, it's a real algebraic geometry is a thing, people studied these and there are all kinds of interesting results of which I'm not going to talk about because I have only one hour. So, so far I want to focus in this talk about counting curves because this has been a subject of much study in the last 30 years. Not that people didn't do it before, mind you. Let me give you an example of an old result. So early 1900, if you take S in three space, say in C, C3, well, I will let me just be honest. It's always projected space. Let me just be a cubic surface. So what do I mean? Surface just means it has dimension two and the cubic means it's given by a degree three polynomial. IE there exists a homogeneous polynomial in three variables of degree three such that S is the zero locals. So remember projective space is something you do by taking vectors, modules, scalars. So in general, it doesn't make sense to compute the value of a polynomial on a vector if then you are free to multiply the vector by a scalar. But if the polynomial is homogeneous, then multiply by a scalar will just multiply the result by a power of the scalar, in this case, the power three. And so if it's zero, it stays zero. So Z of F just means the set where F is zero. And the answer is, and so I'm still haven't asked you a question, so what is the question? The question is how many lines are in S? And the answer is actually 27. And this is a very beautiful and very classical results. There are, it works only over the complex numbers, the questions of under which assumptions, how many of these 27 complex lines are real is an interesting debating question. They have a lot of interesting geometry. And again, here I have worked with cubic surface, so something of degree three. And the point is that if the degree is at most two, you get infinity lines. And if the degree is at least four, and F is general, then you get. So again, it's the same phenomenon. You know, you have to be just to get a finite number of results, you have to be just right. So far, I told you I want to discuss counting curves, but so far I've only counted lines. Can I count something which are not lines? And the answer is yes. For instance, let me count, tell you how you count something a bit more general than lines, which are rational curves. So what is a rational curve? Well, informally, it's so, well, a curve, if you do, in the same infinity word, a curve is an image of the real, of an interval inside space. And rational means that it's parameterized by rational functions. So you have a C inside PN, a curve. And I will not have a precise definition because this will be a consequence of, is a rational curve. Even only if there exists an F from P1, PN, map, given by FF0, FN, these are FI are homogenous polynomials of degree D. So this is a rational curve, and this is a rational curve. D. So this is a rational curve of degree D, such that C is F of P1, and for general, for X in C minus a finite set, F minus one of X is one point. So let me give you some examples of rational curves. Well, first of all, so why are they called rational? Because when I work with projective space, it's very hard to see projective space. So what I typically do inside here, I just take CN. So I just take that one coordinate is non-zero, like the first one, and then the map is just given by F1 over F0, FN over F0, and these are rational functions, which are homogenous of degree zero. So let me give you examples where lines, you have that T goes to TAT plus B, for instance. This is a line, you are all familiar with this, but also any degree two curve. For instance, if you take the circle, the usual circle, X squared plus Y squared is equal to one. Of course, you can parameterize it by cos T sin T, but this is not what we are interested in, because these are not polynomials. So the idea is we are always working in the language of algebraic geometry. So this means pretend all the special functions you learned don't exist, pretend only the four operations are there, polynomials, rational functions, it makes taking derivatives very, very easy. So how do you parameterize a circle if you aren't allowed to use trigonometric functions? Well, I hope you are familiar with this. You can parameterize it in this way. It's a parametrization of the circle where T is the tangent for halves, or if you want is the tangent of theta. So this is rational of degree two. And so in fact, any rational, any degree two curve in P two is rational. It's pretty easy. The idea is you take, you fix a point. How do you get this one? What about if I decide that not only I am not allowed to use trigonometric functions in my final form, but I just want to pretend I've never heard about trigonometric functions. How do I get this? And the answer is I fix this point and then I take a line and I choose Q in such a way that it goes through this point here. And then I take P of T, that's second point of intersection and I get this formula. So this means I take X squared plus Y squared equal one and then I substitute this, I get a second degree equation in X. I know one solution and I look for the other. And the same method fix a point and consider the lines going through it works for any conic, any degree two curve. On the other hand, most cubics, most degree three curves aren't rational. This is actually a non-trivial fact. And those that are what is called, there are two kinds of rational cubics. One is done like this and one is done like this. So for instance, this one is parameter is given by Y squared is equal to X cubed. And of course this only makes sense if X is, that's I'm drawing always the real part and the parameterization is given by X is equal to T squared and Y is equal to three cubed. Or if you want to give it in homogeneous coordinate, this is S cubed, ST squared, T cubed. And you can try and figure out how to get something looking like this. So you can ask yourself, so in some sense what does it mean that a curve is rational? It means that you, if you view it as one equation in two variables, it means you have a very explicit formula for all the solutions. So another way of saying it, f x y equals zero is rational. I can parameterize all solutions with rational functions. In particular, since they are rational, it means I can find solutions over some fields of the reals. I can find easily, I can compute solutions over Q. And if you take this formula and you apply it over Q and you work a little bit, you get a formula for all ketangoric triples. All triples of integers that can be the sides of a triangle with a, if you look at all triples M and M such that all of these are all integers, it turns out that these have a parametrization and the parametrization is essentially this one. So you can find the three integers that are the sides of a rectangular, of a triangle with an angle of 90 degrees, even only if they are of this form with some factor of two that I have to check, but it's something like that. So it means it's not just about finding the solutions over the complex numbers, over the reals numbers, any field you are interested in, you know how to find the solutions. So these are very good. So you could ask yourself, just very simply, I told you that degree one and two, everything is rational and I kind of explained how it works. I told you without explaining that most degree three curves are intracional and the way you prove it is you can again count parameters and you find that it's just not enough rational polynomial to parametrize everything. For instance, the degree three curves have nine parameters and the rational ones have eight parameters and it only gets worse from there. So in general, degree D rational curves have D minus one parameters. And I can leave you as an exercise to find how many parameters have degree D polynomials in two variables, but let me tell you it's quadratic in D. And so it certainly grows larger than three D minus one, which is only linear. But once you have this parameter space, you have a very natural question. Namely, let me call and D to be the set of C in P two such that C is rational degree D curve and C contains P one, P three D minus one where P one, P three D minus one are fixed points. Why do I do this? Because saying that a curve, a polynomial contains a point means imposing a zero. So it's a linear equation on the coefficients. So if you want an alternative is to say we are looking into the linear space of all degree D curves and we are counting the degree, the intersection with linear spaces until we get to dimension zero of the locus of rational curves. So this is a very classical problem and it is classical, which means early 1900 that this is a well defined, this makes sense. So that there is such a finite number which is achieved by taking general points in the sense we discussed until now. And in fact, there is a very, very pretty argument involving just topology over the complex numbers which tells you that, so let me compute the first ones. Well, N one, these are the lines to two points. So it's one, it's the problem I started with and two, these are all conics because they are all rational through five points. This is also one, this is a very classical result. It goes back all the way to the Greeks because D equal two, so 3D minus one is five and this is lines. So N three is rational cubics, eight points and here the answer is 12. And I will not give the proof but let me say that there is an easy, once you know enough about topology, topological proof. It is about using Euler characteristic and the fact that the non-rational cubics contributes zero to the Euler characteristic because topologically they are tori and so they have Euler characteristic zero. Well, and so this, anyway, let me tell you this is a simple argument and as I said, this is something we understand at least for a hundred years. So when was this problem solved? And the answer is this was solved around 94. And this is Duke to Konsevich and he gave a recursive formula for N D which means in particular that to determine all it's enough to know N one. Precisely the easy, easy, easy example we started with. So how come you have an open problem for a hundred years? How does somebody break through a problem which has been sitting around for a hundred years? And why am I talking to you about a result which is 20 years ago? So the point is that this result wasn't, you know, you could say, okay, this kind of solves the problem, but it wasn't the end of anything. Rather it was the beginning. So this was the beginning, the beginning 20 years and ongoing of counting curves. So what are the ingredients? What changed in the hundred years between the question and the answer? And what changed is that the physics came to play a role. What changed is that string theory started so what does string theory say? Of course I'm not a physicist, so if you're a physicist cover your ears. String theory says you shouldn't think of particles as points, but you should think of them as very tiny circles. And so before we are doing when we compute how particles interact, you use Feynman diagrams. We have two particles. They bang into, the time goes on like here. They bang into each other. They become one particle and then at some point maybe this particle will split again. And we don't see this, but we use this to understand the interaction. Of course the actual interaction takes place in a huge expensive accelerator. And now string theory, you do the same, but with little circles and what you get is a tube. And you can already see one big difference that here, you see here in some sense this is the point where they get together. You have these two circles that touch, merge in a big circle and then they split again. This one is singular, there are special points. This one is not. And these circles are very tiny. So topologically this is just a sphere with four punctures. And of course what we know from Feynman diagram is that we have to consider even more complicated diagrams like this, all of them. And it turns out that counting these, well these don't look like curves, but remember these are over the complex numbers. So these are actually Riemann surfaces are complex curves. So physics, there was a series, it was a key idea of Witten that came up with an idea, an ansatz on how to count curves. And this in turn was based on previous work of Gromov from the differential geometry side who had discovered. So let me go all the way back. One of the things that makes this problem so difficult is to find, I told you, you always want to compactify. For lines in space, the correct compactification is very easy, just take line in projective space. Here the problem is the recent and obvious compactification that you can work with. And the idea was that Gromov came up with a suitable compactification in the concept of something he called pseudo holomorphic curves which are basically, it's like holomorphic if your complex structure is what is called the normal complex structure. And then there was, and the underlying language of course was that there was a compactification of moduli of Riemann surfaces with punctures. And this has a natural compactification which is called the Linn-Mannford in case there are no punctures and then there is also a version. So this is like 1968, approximately. So what happened is written is a physicist that that time he was in Princeton. I think he still is but this is irrelevant and the Linn also was there. And so the two of them started talking to each other about this. And we found a way, they found, we all found the other thing is there was a huge effort of the mathematical community to get, so there were physicists computing numbers. For instance, at some point physicists did a similar computation. I told you you count lines on a quartic surface. Physics started counting curves, rational curves on a quintic three fold. And also mathematician counted them and got different numbers. And it turned out that the mathematician had mis-programmed their computer. But of course our number, the mathematicians, not mine, but our mathematicians number was right. So it was correctly proven. It was wrong because the program was wrong. But at least it was. So what has happened since then? So where does this subject go? What has been done in the last 20 years? Well, I told you, we started having, notice that these are ideas from physics. But usually you think of a physicist using mathematics like Einstein wants to develop general relativity and finds out that people have already invented the Romanian geometry and all he has to do is to change a sign and he can ask for helping doing that. But what has happened in this case is that ideas from physics generated new definitions and new solutions to all problems in mathematics. So this led to the definition of something which are called the Gromov-Witton, you guess why. Invariance for smooth projective varieties. That they have been computed in a large, large number of cases. The other key idea is that, so these are just counting curves in a very general way. And there's another key idea I have no time to develop but which is extremely fruitful is I told you here that there is that Konsevich gave a recursive formula but I didn't tell you how the recursive formula looked like. And the way you give such a formula is you make a formal power series like this. This is, it's not exactly like this. There are some coefficients I'm hiding but this is a formal power series and the recursive formula is a differential equation that the Z satisfies. So what has been done is both explicit computations in a number of cases like PN for instance. So explicit recursive formulas, differential equations on satisfied by the generating functions. So you take all these invariants, you put them into a huge power series with a lot of variables because you have to take into account all manners of data. Some of these power series in particular are what is called modular forms or quasi-modular forms. So in turn they have all kinds of beautiful behavior. And then when I told you that the physicist bested mathematician in counting Quintic three-folds. So it looks like I'm giving numbers at random. Degree three is two-dimensional, degree five, three-dimensional. But the answer is there is something called Calabi-Yau three-folds, which again play a big role in string theory. And the typical example is a Quintic in P4. And for this we have three different ways of counting curves. So there's the Gromov-Witten, then there's the Donaldson-Thomas, and there's Panderi-Pander-Thomas, and there are translations, relations between them partially conjectural. So there's conjecture relating them, and some of these conjectures are proven. And the idea is why do we have different ways they correspond to different choices of compactifications. So what is the final message I want to give you? You can start with a very simple, very classical, very geometric problem. This, if you can keep a high school student interested, if you have a high school student interested in mathematics, you can explain them what this NDE is in say one hour. They won't know how to compute it, but they can understand it. So it's a very, even if you are interested in very concrete, very classical, a numerative geometry problem, it actually pays to have all these material at your disposal. And the material is on the one side, the notion of modular space and of ways of compactifying modular spaces, because as I told you different, it doesn't, it's not true that it doesn't depend on the compactification. Some case it does, some case it doesn't. So which compactification you choose will impact which number you get. Secondly, you have to have a way to work with these. I was very general saying that sometimes I use the word manifold, sometimes I use the word variety. In fact, most of the time, these modular spaces are actually the lean-mumpford algebraic stacks where the lean-mumpford are the same because after all, this is what their compactification is. The first algebraic stack is the stack of curves of genus G. So even though the notion of modularity of curves of genus G goes back to Riemann, we actually really understand how it works only from the late 60s. And in fact, there are still things we don't know about it, but at least we have the right language. Then we have to be able to do and understand the background in intersection theory. And finally, there are all these computational techniques. Let me just mention another one. I told you these are all invariants. So invariants under what? Well, invariants under changing, moving a little bit your target where you are counting things like counting curves, lines in a cubic surface. It doesn't depend on the precise equation of the cubic surface, the number is still 27. But the other thing you can do since it's a polynomial, you can change the field. So you can go from the complex numbers to characteristic P. So for some reason, if you like characteristic P, you can work over characteristic P or you can do a different kind of limit which is called, it's a non-archidia. It's a different kind of limit, I'm not going to say. There is something which is called tropical geometry which reduces these Gromov-Witten computations to combinatorics, let's say. This is a very, very rough sketch of what tropical geometry is. Which means there is room in this research field for many, many different approaches. You can be somebody who just proves that compactifications exist and behave well. You can be somebody trying to find ways to do intersection theory on them. You can be somebody trying to find an intelligent way to package the data you get from the intersection theory to generating functions and finding differential equations for them. Or you can compare different counting problems. And finally, if you like combinatorics, there is plenty of combinatorics available. So what I'm going to close with is this was just the starting point and we haven't seen the end yet. We still have problems open, we still have work to do. Thank you. Any questions? So why are these formulas always recursive? Is there a geometric explanation for the recursion? Oh, yes, indeed. Well, they're not always, but some are recursive and there is a geometric explanation. There is, I'm not going to give details, but basically there is something which is called WDVB equation and this is written die graph twice. These are two brothers. And so these are physicists, by the way. But the way mathematicians understand it is that the two count curves going through with end points going somewhere, you have a modular space of maps, of data which are F from C to X, a degree D curve. And then you have P1, PN in C. This is your modular space that you are studying. Pretend this is P1 and this is PN. And if you want your curve to go through so and so many points, you parametrize them first on the curve. And this goes, this is what is called M bar GN, XD for historical reason. And this goes to M bar GN, where you forget M and you go to C, X1, XN. And then you stabilize, but let me pretend I don't have to. And so when you look at the special case with four points, if you take, this means genus zero, it means the curve is P1. And you have four marked points. Now if you have four points, three points on P1, you can have that the first three are zero one, infinity and the fourth is T. So this would be M04 without a bar. It means just the four points on P1. This is like P1 minus three points. So this is not compact, how you compactify. And the answer is you do this grom of bubbling. So you add two points on one side. And the way you get the WDVV is you compare integrals over two different special points of M bar zero four, where you have four special points and you compare what happens if you put one and two together and three and four together. And what happens when you put one and three together and two and four together? So this is, it is in some sense by itself, it isn't really a recursive formula. It is a functional equation. And then of course, once you have a functional equation, and you have a formal power series, this functional equation will yield a recursive formula. But basically it is the geometry of the modular space. For the ingenious one, there is also another relation, the Getzler-Pander relation. And there are more complicated relation, these various auto constraints that come from subtle properties of the geometry. So you shouldn't view them as recursive as much as functional equation, differential equation systems of differential equations which are satisfied by your solution. It just happens that if you have differential equation of formal power series, then this leads to recursive formulas. And the differential equations by themselves have, you know, have, of course, the part, I don't know of the story, but have this big connection with integrable systems. And that's what, you know, there's a whole thing I didn't say. This is Frobenius-Manipulse. It's something defined by my colleague, Boris Dubrovin. And so there is a lot of geometry that I have hidden from you, which yields differential equations. And then differential equations from formal power series means recursive formulas. Just one more. Can you invert, if you have, you start off with an object, you look at a modular space associated with it, compactify it in some way, if it is not compact, do intersection theory on it. Is that solving some enumerative problem? Yeah, that is always a good question. So that you get, that is another problem you have. That you want to see, after you have done all your computation, what its enumerative significance is, if add. For instance, another example I would have liked to talk about, if I had time which I didn't, is you can count curves on KT surfaces, which aren't Gromov-Witten invariance, because I told you Gromov-Witten doesn't care if you move the variety a bit. And if you take a KT surface and move it a bit, it has no curves. So Gromov-Witten invariance are zero, but there is a Gromov-Witten-like count. And the count existed, like two or three or five years before people could actually prove that this count was counting actual curves on an actual surface. So you have somehow, again, it's like it's a huge problem and everybody looks at the corner. So even if you have the intersection theory done, you still have the question, is this number geometrically significant? And that is typically a different question addressed very often by a different person. Thank you. Any other question or comment? No, it's just a request. Can we see the Conservatives formula if it's not too... Well, you could if I was the kind of person that can write a formula. I suggest you Google it. It's much better than taking the chart. I have it actually written, but I don't trust my ability because it has a couple of factorials in it and they can never get them right. Sorry. Is it a question by a different person or by the same person? So this differential equation of Conservatives, was it first predicted by the physicists or... Yes. Okay. All these equations, the Vida... It's basically... Yeah, it was all predict... Not all, but this particular group of equations was predicted by the physicists. So Conservatives' contribution is to actually prove that they satisfied it? Yes. Well, no, actually in this case, this particular one is really new. So the general idea of having such equations is due to Whitton, but this particular recursive formula is due to Conci average and of course you could say that the whole proof is based also on work by Conci average money and bed and money because what he proved is that if you have such and such invariance, so what they did is they took the viewpoint of the physicists and formalized them into properties of the invariance and then he showed, okay, if the properties can be proven, then you get as... If that was really the smart thing, this idea of this double the generation, this is really something the physicist... The formula is the physicist formula, but this way of doing it and this way of applying it are due to Conci average and then Conci average and money together formulating the axiomatic approach and bed and money proved the axiomatic approach to genus zero, which was enough for this specific problem. Any further comment, question? Okay, let's thank Barbara one more time.