 Thank you very much, Philip, and thanks to all the organizers, of course, and especially to Philip and Alina for helping me very much with the virtual modality that I am not practical. So let me start. So I will survey in this talk over some results which are actually known since a few years, but I will, perhaps they will not so well known to all of the audience, and I will present also some applications which seem newer. So the context is of elliptic families. And probably the best known is the Legendre, of course, that you can see the equation. And we have the parameter lambda. This parameter lambda varies over what we call the base. The base B now is the projective line minus three points. And if lambda is inside this space, then the equation defines, in fact, an elliptic curve because the polynomial on the right has no multiple roots. And I write the elliptic curve in a fine form, but let us think of its project completion in P2. So we always think that there is a further point at infinity. So we have a family for varying lambda, but an alternative viewpoint is that we can think of lambda as a variable and also see the equation as defining an elliptic curve over the field, for instance, of rational functions of lambda with the coefficient complex coefficients or coefficients in some other field. Then another important objects in my talk are sections. What is a section? It is a function from the base to the family with the property that the property is written at the bottom of the page for a section. The value at a point of the base must lie in the elliptic curve corresponding to that point. So there is a drawing here. And the drawing is not quite precise. In fact, the point B is not thought as lying in LB, but rather the curve LB corresponds to B. And you can see the image of the section. It is a curve in the space of all elliptic curves. And let me give two examples. One is the zero section. So each elliptic curve has a group low, and we can take an origin in each elliptic curve. The origin I chose it as usual, this point. And the zero section associates to each point on the base the origin. There is a slight abuse of language because this origin, in fact, depends on lambda because it lies in the elliptic curve lying above lambda. But as a point in P2, it is always the same. And then let me give another example, what I call a master's section because David Masser proposed this as a simple example of section to pose some of his problems which motivated the work which, in part, I am going to describe. So this master's section is obtained by fixing the x coordinate in the elliptic equation x equals 2, and then y has this value. We may recover it from the equation. And of course, there is a choice of the sign. So if we want to be precise, we have to think as the section, as defined, not over the original base, but on a quadratic cover of it. So where the sign is well defined. But otherwise, we may forget about this precision, but OK. So for these sections, we consider the torsion set. It is the set of points of the base where the value of the section is a torsion point in the respective elliptic curve. The elliptic curve has a group low. So we have the torsion points. They are the points such that some positive multiple is the origin is 0. And we have this set of points. So let us start by looking at the so-called trivial case is when the section is itself torsion. So it is torsion identically on considering lambda as a variable. This may happen. And an example is, for instance, written here. If on the Legendre curve, you take the point with the x coordinate lambda, the y coordinate will be 0, and this has torsion of order 2 identically. So in this case, since the section is identically torsion, of course, every value of the section will be torsion actually of the same order. And we have that the torsion set, t sigma, is equal to the whole base. And now an issue, let me pause just a moment on the issue, how to check whether a torsion is whether a section is torsion or not. In the case of the example, this is trivial, but we might have a complicated section given by complicated functions, and it may not be immediately clear how to check this. And for this, it is effective, however. There are several methods, a method is from Galois theory, because the Galois group arising from torsion sections are quite large. So by looking at the Galois group of the functions, algebraic functions which appear, we may detect whether a section is or not a torsion. Then we have a height theory. Torsion sections have zero canonical height in the functional sense. And there are other methods, still other methods. One method is due to Manin by looking at differential equations. And another criterion, let me mention it, because it is often very practical, it comes from reduction theory. And you may read it, if the section is torsion, then the field of definitions of the algebraic functions, which appear as the coordinates of the section, they generate a field which is ungrammified over the ground field C of lambda, ungrammified outside the points of bad reduction, 0, 1 and infinity. And this criterion, you may see the example, master section is not torsion, we may immediately see this from this criterion, because the master section, you see it is, it has this square root of 2 times 2 minus lambda. So the minimal field of definition is ramified over 2, over lambda equal to, and this goes out of the bad reduction, so it is, it cannot be torsion. And let me also recall this result, which is not well known as I would have expected. For the Legendre family, this criterion is also a sufficient condition, which means that if we have a section ungrammified outside 0, 1 and infinity, then it is necessarily torsion. This was brought by Chioda already in the 70s, using elliptic, theory of elliptic surfaces. But recently, with Pietro Corpavier, we gave a proof depending on modular function. Anyway, let us forget about this now, and let us suppose that sigma is not torsion identical. And then we ask, what can be said about the torsion set? The torsion set, you see it is the set of points where the value of the section is torsion. Now we suppose that the section is not identically torsion, and we ask when it becomes torsion. So this is somewhat when the structure of the section is different, becomes different on specializing. And we may say a few things about this set. And the first observation there is that it is small in a sense. If everything, for instance, is defined over the field of algebraic numbers, then a result of silver mandate proved already in the 80s. And this set of torsion values, say, is a set of algebraic points of bounded height. So this is quite a strong result. The set shows that the set is very sparse. For instance, it contains only finitely many points of bounded degree over q, only finitely many rational points, but also finitely many quadratic irrationals, and so on. And before this result of silver mandate, there was some work of Neron who used this to construct elliptic curves of a larger rank. But that was somewhat weaker, not really weaker in the logical sense, but the result of silver mandate is very, very strong and efficient for applications. And let me mention another result very interesting about the set is sigma was proved by the Marco and Mavracki around 2015. They proved that in a suitable sense that I will not explain it for time reasons, it didn't contribute. So let me just mention that this Galois refers to the set of points on the base. So it has nothing to do with the Galois groups of the torsion points. We are looking at the points where the section becomes torsion. And they are algebraic points with a certain Galois group. But this Galois group is very difficult to understand. And this is a result above this Galois group. And the set is Galois equidistributed. So the conjugates go somewhat everywhere on the base with respect to certain measure. So this shows that the result of silver mandate shows that T sigma is small in a sense. But we may also look from the other viewpoint and say that T sigma is not too small. For instance, we may expect that it is infinite. Why? Because we are looking at the points where the section becomes torsion. But we do not fix the order of torsion. So we allow any possible torsion order. And so for each order, we can expect to find some point where the section has just this order. And by taking the union, we may expect that the set is infinite. And indeed it is. Although this is less trivial to prove, at least to my knowledge, than what one could perhaps expect. One proof, for instance, can be given depending on Siegel's theorem on integral points over function fields. So this is not difficult to like the number field case, but still it is not completely trivial. So this gives as an ingredient to prove that T sigma is infinite. But a much more precise result can be proved either by the method of a Mavracki and Markov but also using the so-called Betty map of a section. I won't define the Betty map. However, just to mention this Betty map, we look, since every elliptic curve is holomorphically equivalent to a complex torus, one looks at the values of the section, not in the elliptic curve model in the algebraic model, but in the torus. And using the periods, one may define this Betty map. So it is defined in terms of elliptic logarithm. And it has been studied recently by, it was studied implicitly introduced by Manin, if not before. And recently we have studied it with Ivan Dre and Pietro Corvaya, but also independently Philippe Habegger and Giang Gau have studied it with applications and Claire Walsang also for other applications to algebraic geometry. So using this Betty map of the section, one proves that the set of torsion values is dense in the complex topology. So this shows that it is not too small. It is small in a sense, but not too small. And if we look at the real values, we can look at the real points on the base. For instance, for master section, we can look at the real values of lambda where the section takes torsion values. Still the Betty map implies that this is dense, again also in the reals. And with Brian Lawrence, we have proved, however, that it is never dense in the periodic points in the algebraic closure. So there is a marked difference of behavior from the alchemydian viewpoint and long alchemydian. Okay, let us now consider a second section. So our first section will be the master section. And let me choose this as a second section. So you can see that this lies in the Legendre elliptic curve. Again, we have a base. We have to take another quadratic cover to be well defined. And we may ask, a master asked, long ago, before 2010, he asked about the set of values of lambda where both sections become torsion. And so he expected that since for each section, the set of torsion values of lambda is sparse, then if we take the intersection, then it is even more sparse and probably finite. And we proved this in 2010, around 2010. So for instance, for these two sections, the former section and this one, the intersection of the two sets is finite. And I should remark that here we allow lambda to be any complex number. So it will be automatically algebraic if it lies in the set. But we do not put any restriction on the degree or we forget about any such restriction. So we proved this finiteness. And let me say that the finiteness requires two kinds of checking so that P and Q are not identically torsion, which is an accessory condition to obtain, of course, finiteness. But we have also to check that P and Q are not linearly dependent because if we had a linear dependence holding identically, then it is easy to see that being torsion for one section forces the other to be torsion as well. And then we would obtain an infinite set. So the finiteness requires some assumptions on the sections, but otherwise it holds. And indeed, there are generalizations to several sections or even to a billion pencils. So not only in elliptic curves, we may generalize with similar finiteness conclusion. Let me only recall two results that I will use later. If we wanted the simultaneous torsion, like in the example that I gave, then if the sections do not map in a subgroup family, which amounts to be linearly independent in our case, then we have finiteness. This was proved jointly with Massa. And Baroero and Capuano a few years ago treated arbitrary families, and they improved the more general result because they did not require the families to be torsion, but just to lie in subgroup families of co-dimension two or more. So this is milder. If we work in an ambient of high dimensions, this is a milder condition than requiring torsion. And still if the co-dimension is two and they intersect with the union of subgroup families, then they obtain finiteness under the appropriate assumptions that are technical and I don't want to recall them in precision, but I hope this is sufficient to give an idea of the result. Let me say that Gioca, Sia and Tucker had done some special cases of this result before, so went beyond torsion, but not as generally as the Baroero and Capuano. Let me give now some applications. So recall that for the Massa section, I have rewritten it, the set of real values of lambda, real values B of lambda for which P of B is torsion is dense. Of course, I have to assume that P of B is real, so I require that B is real number less than two. And there are an infinity, this is dense in the half line lambda less than two. This may be probed with a betting method. Now suppose I change the problem apparently by a little. And I ask for the purely imaginary torsion values instead of the real torsion values. So and now we obtain finiteness. There are only somewhat surprisingly, perhaps not so surprisingly, but the formulation may seem similar. We prescribe here real and we obtain dense and here we prescribe purely imaginary and we obtain only finitely many. And the reason at bottom is that the set of purely imaginary does not form a subgroup. But anyway, let me state in precision, so there are only finitely many T in a real T such that P of I, T is torsion, where of course in the corresponding elliptic curve, the one in the Legendre family with the parameter lambda equal to I, T. Let me sketch the proof, which is very simple. And with the trick that we used also elsewhere with David Masse. So suppose that P of I, T is torsion inside the corresponding elliptic curve. Then we conjugate and we obtain that the conjugate point is torsion on another elliptic curve, the conjugate elliptic curve. And the conjugate now is obtained just by changing sign. So now we have two torsion sections. Now in two families, the family L lambda and L minus lambda. So it is, we are in the situation of the former issue with two torsion sections, but now the elliptic families are not the same. And in fact, it may be checked that they are not isogenous, these two families. So the product family, since they are not isogenous, has maybe proved quite easily, but let us forget about this issue, it can be proved. And then this implies that the abelian family consisting of the fiber product over the lambda line has no non-trivial subgroup family. And so this means that the assumption of independence of the two sections is automatically satisfied. You remember that in the theorem that we had before this one, if the sections, the first one, we needed for finalness, we needed to know that the sections do not map in the subgroup family. And here this is automatic because the curves are not isogenous. So we applied just the previous result and we obtained finalness of the purely imaginary torsion values, contrary to the real torsion values. And the one I may generalize, this has never written down, but I believe one may generalize to real curves defined in the complex plane by algebraic equations relating real and imaginary parts. Now let me go to the main applications, which concerns elliptical billiards. The issue of the elliptical billiards goes back to long ago. I owe it to Peter Sarnak, reference of the astronomer Jesuit priest Boskovich, who studied elliptical billiards already in the 18th century. So it has been studied also later, mainly by Ponsale, Jacobi and many others. And until recently, this topic has attracted interest. Let me say a few things. So the elliptical billiard consists of a table bounded by an ellipse. And we may think to play a billiard game. And with the usual reflection law, so the trajectory hits the border, the ellipse C, and becomes reflected by the rule that the two segments form equal angles with the tangent at the point where it hits the boundary. And we obtain this billiard trajectory. It is a very interesting theorem capable of a Euclidean proof, elementary proof by geometry, but somewhat subtle proof that if we start a billiard trajectory, then all segments in the trajectory are tangent to the same to a same conic, which is confocal with the original ellipse. It has the same fochi, and it may be an ellipse or a hyperbola. In the picture, you see it is an ellipse, the caustic, it is called the caustic. So the trajectory is described by drawing successive tangents to this caustic. We may also write equations, for instance, you can see the equations for the caustic for the ellipse, and let us sort of frame this picture. We may view the thing as follows that the tangents to the caustic are points are identified by points in the so-called dual conic. The dual conic is a conic which parameterizes the tangents to the original conic. This is well known a projective geometry going back to centuries ago. And the billiard shot, we may view as a billiard shot as a pair consisting of a point on the boundary of the ellipse and the line, and the line is tangent to the caustic. So we may view the billiard shot once we know the caustic as a pair consisting of a point on the line, so lying in the product ellipse times the dual of the caustic. And the point lies on the line. And for a given caustic, these pairs form a curve of genus 1. And so the Jacobian is an elliptic curve. And we have the billiard map associated to such a pair, so to a point on this curve, it associates another point on this curve. It is an automorphism of the curve of genus 1, once that we have the caustic. Of course, corresponding to a given point, we may have several caustics depending on the shot, on the direction where we hit the ball. And the theorem, famous theorem of Ponsolet and the Jacobi gave a proof in terms of elliptic curves that beta is a translation in the Jacobian. So it depends only on the caustic. And so the billiard map is like doing a translation in this elliptic curve. And it follows that the trajectory is periodic, if and only if the translation is torsion. This was a part of a theorem of Ponsolet, for the so-called games of Ponsolet, anyway. So let us play some billiard games, possible billiard games. So we take the first point on the border, the point P0 with coordinates minus 1 and 0. We chose a slope, psi, and corresponding to the slope, we have a caustic. The rule is very easy to find, and this is written here. The parameter s square of the caustic is related to the slope by the equation that you see. And it turns out that this elliptic family has the Legendre parameter given by s square divided by c square, where c defines the ellipse. You may take a real number less than 1. And we may prove easily that, so we have a section, because the billiard map gives a section to each slope. We have a point, we have a family of elliptic curves, depending on each caustic defines an elliptic curve. So we have a family. And for each element of the family, we have a billiard translation because of the theorem of Ponsolet Jacobi. And so we have a section, and this section is not torsion. This is trivial in a way, because it just says that not all billiard trajectories are periodic. So it is very easy to prove by continuity. And we may actually prove that there are billiard trajectories of any given period. It is, we may use a variational principle. If we take inside the ellipse a polygon of n, we choose an integer n, positive integer n. And we take a polygon of n sides and maximal length. And it is easy to see by the principle of Fermat, here on Fermat, this is a billiard trajectory. And so by compactness, we find one, and by the theorem of Ponsolet, we may assume that the vertex is in P0, because the periodicity does not depend on the vertex. And there are trajectories periodic of any given order, which is quite intuitive. A non-periodic trajectory will be dense in the shaded region, which also is rather intuitive and can be proved easily by this description. Here in the picture, you see a hyperbolic caustic. And the slopes of the periodic trajectories correspond to the torsion set, because the trajectory is periodic only if the section given by the billiard map assumes a torsion value. So this provides the link with the previous context. Let me do an example with two players. So we have two players playing in a billiard elliptical billiard. And suppose that they start at the same point, P0, and the play. So the first player chooses a slope, for instance. And the second player uses the same slope, but increasing the angle by a fixed amount, alpha. So one can ask how often will both trajectories be periodic? And the answer is only finitely many times. So let me remark that for usual rectangular billiards, this is usually false for a dense set of angles. And actually, one may go further and prove that if we have a billiard defined by a parallelogram, then the finiteness holds if and only if the parallelogram defines a lattice with no complex multiplication in C. So also for the usual billiards, the question makes sense. But the finiteness does not always hold. So let me give a sketch of the proof that we have finiteness of this situation, where both trajectories are periodic, where we require that the directions form a fixed angle. Now we have two sections. We have two billiard maps corresponding to the two players. And both sections individually are not torsion. So we are asking now where, when both sections will assume torsion values simultaneously, like in the problem posed by Masser. And again, the results that we proved gives finiteness unless the sections are dependent. But again, to show that the sections now are not dependent, it suffices to show that the two families are not isogenous. And again, this is not difficult to matter. One uses, for instance, the fact that if two elliptic curves are isogenous, then their J invariance satisfy a modular equation and the modular equations are monic. So each of the J values is integral over the ring generated by the other one. And so there is a verification that I will skip. And it is possible to use this criterion with modular equation to show that the families are not isogenous. It suffices to compute the lambda and the J invariant once we know the lambda to compare the two J invariants. And looking at the poles, one concludes that there cannot be integrality of each one over the ring generated by the other. Let me give a second example of billion with a hole. So we have a hole and two balls, A and B. Our task is to hit B with the first ball so that B goes eventually to the hole. And we may hit B directly or we may first hit the border and then go to B and so on. So there are infinitely many ways in which we can hit B with A. And we ask in how many of these ways then B will go to the hole. So to discover all the possibilities that send B eventually into the hole, L, A of our elliptic family corresponding to the choice of the slope and the winning slope must obey two conditions. First, we want that for some integer M, after M rebounds, we reach the ball B. And then the second condition is that for another integer, the shot from B goes to the hole. So we have two conditions. And we remark that we think of the game as being such that when A hits B, then B will continue in the same direction, the same trajectory of A. So this is like asking that the trajectory from A goes eventually to H, but passes first through B. And now we may also use the Jacobi proof of Ponsonnier theorem, which the previous correspondence with elliptic families. And we may translate into two conditions among three sections. So we have two sections associated to B and the hole. And we have this system. So this system describes two linear dependence conditions on the sections. And now we again can deduce fineness by using, for instance, the result by Barroero and Capuano. But we have to verify that the three sections are linearly independent. This may be done with some more difficulty than in the previous cases that I illustrated. And this may be done so we recover this result that is stated. There are only finitely many slopes, sending B eventually into the hole. And there is an exception when A and B are the two fochi of the ellipse. And in this case, every slope sends A to B. And for an infinity of slopes, B goes into H. So this is a true exception. The fineness theorem, let me remark, it could be proved even with weaker results than Barroero and Capuano due to G, O, C and T. So let me, the last few minutes, give an alternative description and formulate a general problem to describe the billiard, elliptical billiard. We may also work with the surface obtained by the square of the ellipse, C times C. So the pairs of points on the ellipse. You see in the picture, we have a pair of points. If they are distinct, they define a line. Also, if they are equal, they define a tangent. And so we may think of a pair of point as a billiard trajectory from the first point to the second. And in this picture, the billiard map becomes a rational automorphism of this surface. It turns out that this automorphism is not well-defined everywhere. But this can be made irregular after blowing up. And then we have curves, L, A and L, B, consisting of the pairs, such that the trajectory contains a given point A or B. So we have to each point in the interior of the ellipse defines a curve in this surface. And the problem before can be reformulated as follows. So given, and also in the more general form, given a surface and three curves on the surface and an endomorphism of the surface, let us call it beta, which is in practice in the example is a map. So the problem is to describe the set of points in the first curve such that the orbit under iteration of this endomorphism falls eventually into both curves with a different stages. So after M iterations, it falls into the second curve. And after N iteration, it falls into the third. This problem, so the case of the billiard that we have seen is a special case of this problem. And this problem is also in the spirit of the dynamical model long conjecture studied by authors like Gioca, Tucker, Xheng, Shovu Xheng, and others. So we can put the previous billiard problem in a more familiar frame. So the above billiard game is an example when we obtain finiteness, finiteness of the points on the first curve. And the proof method relies in the results of Baroeiro Capuano and my joint results with Maser. And so these results in turn involve counting methods developed with PILA. But we have another example of exactly the same situation where the methods are entirely different. This situation is the following. We take the, as the surface, the projective plane. And we may take the simple case of three lines in the projective plane. And beta by beta, the endomorphism, we take simply a linear automorphism of the projective plane. Then the powers, the iteration of this linear endomorphism are expressed by a matrix whose entries are linear recurrence sequences. And eventually, the problem turns into an equation written here, AMBN equals CNDM, where these sequences are linear recurrences. So we find the problem now amounts to an equation amount for linear recurrences and in two variables. So, and this may be dealt with also, may be dealt to obtain under suitable assumption finiteness. But may be dealt with using very different methods. This time, we have to use the Schmitt's subspace theorem and the corollaries by Evertsch, Schliekewein, Wanderporten, obtained from the Schmitt's theorem to the linear recurrences. So I conclude here that it is somewhat surprising that the two examples formulated in a very similar way and also with simple objects, surfaces, and lines here and certain curves in the other example may be treated, both may be treated by, but with entirely different methods. So I would stop here.