 Thank you, Philip, for the introduction and thanks to the organizers for the opportunity to speak here. I'm very pleased to be here. So as Philip said, I will speak on unlikely intersections and distinguished categories. And the work I present today is a joint work in progress with Fabrizio Baruero. And that now is pretty much finished and should appear on the archive soon. So roughly I want to speak about three things and what are unlikely intersections. So especially for people who are not familiar with this field and then what are distinguished categories which is something that Fabrizio and I are proposing to introduce. And then thirdly, what can we prove about unlikely intersections using distinguished categories. So here's a quote about unlikely intersections by Lady Bracknell from the importance of being earnest to lose one parent might be regarded as a misfortune to lose both looks like carelessness. I would love to be the person who came up with this joke but it was Galbini amini. But so this is what you should keep in mind so we have something unlikely that maybe happens once but we don't expect to happen it twice or we don't expect it to happen too many times. So then I want to start with this theorem which is a very special case of a result by Bumbieri-Musser and Zanier from 1999, which in this very special case just says that there are most finitely complex numbers tau such that tau, tau plus one and tau plus three satisfy two independent multiplicative relations. So what does this mean, there exists linearly independent vectors ABCDEF with integer entries such that tau to DA times tau plus one to the B times tau plus three to the C equals one, and they're the same for DE and F. So if we just impose one such relation, then we can solve it for tau and we find a tau that satisfies it and by varying the relation we find many tows. But then the theorem tells us in all but finitely many cases, these tau will not satisfy a second relation that is independent from the first. So yeah I chose this example because here we can easily find some tau for which there are two relations so if we take tau equal to one we get one two and four and then we with this choice of ABCDEF. Yeah we see that these relations hold so there might be or in this case there are finitely many exceptions, yeah but only finitely many of them. So this this special case is an instance of a more general theorem that they proved. So if we take any curve C inside the nth power of the of the complex multiplicative group so that's just the complex numbers without zero to the N. In the example n was equal to three and we take any curve in there that is defined over the field of algebraic numbers q bar and whose coordinate functions are multiplicatively independent modular constant functions. And then the curve contains at most finitely many points that satisfy two independent multiplicative relations. And this was then later shown by more and in combination with with a further result of Bumbini remastered zanier that this stays true. Also for curves that are not necessarily defined over q bar and whose coordinate functions are just multiplicatively independent and not multiplicatively independent modular constant functions. So some non trivial product might be cons might be constant but not equal to one. And this condition that the coordinate functions should be multiplicatively independent it's quite clear that this is necessary, because if this isn't satisfied then we have one non trivial relation that holds on the whole curve. So if we get one relation for free and then we can again impose a second one, and by varying that we find infinitely many points satisfying to independent relations. So, so the guiding philosophy here is that a curve has dimension one. And if we impose two independent multiplicative relations this defines a sub variety of co dimension two. And so, in the absence of any other information, we expect the intersection to have dimension one minus two which is negative which means we expect the intersection to be empty. And then of course we can vary the two relations that we impose. And for each pair of relations we expect an empty intersection but of course in some cases it could happen that that the intersection is not empty. But still, we expect the union of overall these probably empty intersections to be finite. And unless there is a reason for it and not to be finite in the form of a multiplicative relation that holds identically on the curve. Okay, so so before I continue. Here is, I want to introduce some settings so fields are always of characteristic zero and K is our ground field which is algebraically closed of characteristic zero you can think of the complex numbers or the algebraic numbers. And varieties will always be reduced irreducible and defined over K unless stated otherwise, and sub varieties will always be closed. So I want to basically restate the theorem. I already stated but in with a slightly different terminology that is then also well suited to to generalization. And for this I want to speak about a special and weekly special sub varieties. And so here I let G be a connected commutative algebraic group. And so I can introduce this in this generality so for example, the case we had before a power of the multiplicative group of C star, but G could also be an elliptic curve or a power of an elliptic curve, and or an abelian variety. Or also the additive group GA or a power of that. And, and then a special sub variety of G is just a component of an algebraic subgroup of G, while a weekly special sub variety of G is a translate of a connected algebraic subgroup of G by an arbitrary point. And then it's easy to check that if we intersect to special sub varieties, any component of the intersection will again be special and the same for weekly special. And therefore, if we have a sub variety V of G, we can define it's a special closure be in angular brackets to be the smallest special sub variety containing V. And similarly we can define the weekly special closure to be the smallest weekly special sub variety containing V. Okay, and just to give a concrete example if we're again in a power of the multiplicative group, then special sub varieties are just components of sub varieties defined by equations of this form. And then special points are torsion points so so points whose coordinates are all roots of unity and weekly special sub varieties and our components of sub varieties defined by equations. So, from x1 to the a1 product up to xn to the an but now just equal to some constant element y of the field, not necessarily one. Okay, so, and having introduced this this terminology, and it's now a good point to go a bit back in history. I speak about this theorem of Laurent, which is the, the toric version of the man in Mumford conjecture, which says that if we have a sub variety V of the power of the multiplicative group it contains as a risky then set of torsion points if and only if it is special. So, so one direction here is quite easy if you have a component of an algebraic subgroup it will contain as a risky then set of torsion points and the non trivial direction is the other one. Or equivalently, and we can also say that a sub variety of a power of the multiplicative group contains at most finitely many maximal special sub varieties. So, in the sub variety you might have, for example, infinitely many torsion points, but then the theorem tells you that the sub variety must contain a special sub variety of positive dimension. And if we go to maximal special sub varieties there, there are most finitely many of them. So, in particular, what, what does this mean for curves and non special curve contains at most finitely many special points. So, so this statement is somewhat similar to the statement before that if no relation holds identically on the curve then there are most finitely many points that satisfy two independent relations. And soon we will see a statement that unifies both these statements. Okay, so here's, here's again the theorem from before now in a bit more general context so, and so I, this is attributed to various people because this is actually not a theorem but more a scheme of a theorem I'll say something about that in a minute. So here the the ambient variety G is a semi abelian variety so, for example a power of the multiplicative group, or an abelian variety, or, I mean the most general case would be an extension of an abelian variety by a power of the multiplicative group. And we have a curve C in there that satisfies a certain hypothesis hype see I'll say in a minute what what this is, and then the set Sigma of points on the curve, and such that the dimension of the special closure of the point is at most the dimension of G minus two. This set is finite. So as I said this is a scheme of a theorem, and so actually many theorems combined into one. And this table here shows some instances where we know that this theorem is true. The case I already spoke about is if G is a power of the multiplicative group. And the base field is the complex numbers and the hypothesis on the curve is that the special closure of the curve should be equal to G. So this just means there's no non trivial multiplicative relation between the coordinate functions of the curve. And just the theorem I stated before, and as I said this is known, thanks to work of moran and the Bumbieri master and that's on the air. And then this theorem was also proved for certain special abelian varieties a over Q bar. So for example, and powers of elliptic curves or a billion varieties with complex multiplication. And sometimes under the stronger hypothesis that the weekly special closure of the curve should be equal to G. So so which in the GM case would say that no non trivial product of the coordinate functions is constant. And so with the weaker hypothesis that just the special closure of C is equal to G. And so this was done by by various peoples and in the first decade of of this millennium. In 2016, and how beggar and Pila and prove this theorem for an arbitrary abelian variety a over Q bar with the weaker but the necessary hypothesis. And in a pre print that is accepted for publication but but has not appeared yet for Britzy on the eye, then extended this to to abelian varieties overseas. And in recent work in progress, Barreiro, Kine and Schmidt, and even treated an arbitrary semi abelian variety but, but now again only over over Q bar. Yeah, so so these are the instances where where this theorem is is known. So, before I continue and give now a general conjecture that somehow unifies the money month for statement and the statement in this theorem I have to introduce yet more terminology but but this will be the last reformulation so and for a sub variety V of G, I can define its defect to be the difference in dimension between the special closure of V and V. And then if I have a chain of sub varieties w inside V inside G, and I call w optimal for V in G. And if there is no way of making w bigger, while still remaining inside V, without increasing the defect. So, any sub variety you that's in V and strictly larger than w must have defect larger than the defect of w. And then we can also define the weak defect and being weakly optimal just by replacing here the special closure by the weekly special closure. And so so how does this relate to the previous statement so if we have a curve inside G, who special closure is equal to G, and then the defect of the curve is by by plugging into the definition is just the dimension of G minus one. So a point on the curve will be optimal for the curve in G, and if and only if its defect is strictly smaller than the defect of the curve and the defect of a point is just the dimension of the special closure of the point. So the optimal points here are just the points whose special closure has a dimension at most the dimension of G minus two so we get the set of optimal points in this, in this case is precisely this set Sigma that we had in the theorem before, and where the theorem said it was finite. Okay, so I hope that and this is sufficiently convincing and to you to make this conjectural generalization of the theorem and due to a silver and and in this formulation due to have a grand Pila, which basically says if we have any sub variety V inside G. It contains at most finitely many optimal sub varieties. And here I want to introduce this statement, Z P G M D M which says that any sub variety of dimension at most M contains at most finitely many optimal sub varieties of defect at most. And just because later, and I want to speak about the result where that says if we know this for some M and D, we also know it in a more general context for the same M and D. And so it's it's not necessary to prove it for all M and D. There's also a different form of this conjecture due to pink, where maybe the unlikely intersections aspect is a bit more clear. And this says if we have a sub variety V of G, and by G superscript K, we denote the union of all special sub varieties of G of co dimension at least K. Then if the special closure of V is equal to G. And then the intersection of V with all special sub varieties of co dimension at least the dimension of V plus one is not so risky dense in V. So you have something of dimension V and you intersect it with things of co dimension larger than the dimension of V. And you expect the intersection, not to be so risky dense in V, unless there is a reason for it in that V is contained in the proper special sub variety of G. So it's, it's fairly elementary to see that the upper formulation of the conjecture implies the lower one. And one thing that came out of the work of Fabrizio myself is that in this context of semi a billion varieties also the other direction holds so these two conjectures are actually equivalent. So, so this includes one in one fourth since the optimal sub varieties of defect zero of V. And these are precisely the maximal special sub varieties of V. So, as the sub varieties of defect zero are the special sub varieties and being optimal just means not being contained in a larger special sub variety contained in V. So, this statement ZPG M zero for any M this is precisely one in one fourth for G proven by Renault in the Abelian case, and then hindering the in the most general case. And then I can't go into details but the, the, the silver pink conjecture also implies the more the land conjecture. This is proven of course by faultings and, and, and hindering. And so in particular to give a bit down to earth example. It implies that if we have a polynomial equation in two variables like for instance this one that defines a curve of genius at least to this will have the most finitely many solutions in in rational numbers X and Y. And the silver pink conjecture. So somehow, many important conjectures and results in number theory are summarized by it. Okay, so, so up to now we were in the world of commutative algebraic groups or semi a billion varieties. I want to introduce another world, the modular world where all these concepts that I've introduced so far, and also exist and the corresponding conjectures can be made. Now, Gabriel, the question in the chat room for you. Please just ask away. Yeah, I'm sorry. Gabriel, could you just comment a bit on the choice of this genius to curve. Oh, and, and what, how come what are these numbers with. Well, the, the birth years of, of three people are in there. And I think all these people are are now in, in this seminar, as speakers or as parts of the audience. Who was born in 130. Only three people. No, I think also there's nobody here who was born in the year and one. Okay, somebody was born then. Nobody who's here now. Well, anyway. Okay. Yes. So, yeah, so there's there's this whole other modular world of Shimura varieties or mixed Shimura varieties if this frightens you then I can say that it also frightens me and and some examples will will soon be forthcoming. But the important thing is that that all these these concepts I've introduced up to now also exist in that context and and and also the version of the silver pink conjecture. And what you, you might be familiar with in that case is again the D equal to zero case that corresponds to one in Montfort, and which is the Andre art conjecture, which for Shimura varieties of a billion type is known by by work of Zimmerman. Yeah, so some examples so typical examples here are our modular spaces of a billion varieties and and families of a billion varieties. For example, the the affine line a one, which is the the modular space of elliptic curves so think of of the J invariant and or a G, which is the modular space of principally polarized a billion varieties of dimension G, and which which is a of dimension G times G plus one over two. And or to also give a family of a billion varieties. For instance, the Legendre family of elliptic curves that is defined by by this equation y squared z is is x x minus z x minus lambda z, which for varying lambda not equal to zero or one. The parameterizes a varying elliptic curve in the in the project of play. And special points here are just torsion points on fibers with complex multiplication, or in the case where you just have a modular space they are just points corresponding to to see a million varieties. And I hope this and gives a bit of a picture of this modular world. And okay so so that's part one what what are unlikely intersections now what are distinguished categories so so this is something we propose to unify both these worlds that I've introduced up to now into one context. So if unified proofs of certain statements and also prove new results. Okay, so. So we have a category C with objects V morphisms M and the covariant functor F to the category of varieties over K. And the functor is a is a technical nuisance and just think of the inclusion functor so as he is a subcategory of the category of varieties this will not always be true but but then. For simplicity thing like this, then we call this category distinguished if it satisfies for axioms so first of all it should have a direct products and that are mapped by F to direct products. And second, and it. It almost has a fibered products in a sense. So, and if we have two morphisms. Phi from X to Z and the style from Y to Z. And then after applying the functor F, we can look at the, and the fibered product inside the direct product. And then axiom tells us that we can cover this fibered product by the images of finitely many distinguished morphisms. So there there are morphisms. Phi one up to Phi N from some X I into the direct product X times Y, such that the the fibered product is covered by the images of the F of Phi I. And three, just says that the category has has a final object that is mapped to a single point by the functor. And axiom for says, if we have any morphism Phi then first of all its image should be closed. Then it should factorize in a certain way. So as you see in this diagram, so we have a finite surjective cover W of X and after passing to this cover. The morphism should factor into first a morphism with irreducible fibers of constant dimension and then a morphism with finite fibers. So it's good here to think about algebraic groups. And so if you have any homomorphism between algebraic groups you can first quotient out the identity component of the kernel. And then you're left with a morphism with finite fibers. Of course, these properties are properties of the images of these morphisms under the the functor. Okay, so, so some examples. So, and, okay, there's a completely trivial example where the objects are just powers of a fixed variety. And axiom are just of the form we take a couple x one up to x n, and we map it to x i one up to x I am for some choice of indices, and maybe with repetition. And the function is just the inclusion function. So so everything that comes later kind of also makes sense in this trivial example but but all the statements are of course trivial. Then an interesting example are connected a commutative algebraic groups so so these are the objects and the morphisms are homomorphisms of algebraic groups composed with translations by torsion points. And the functor is again the inclusion functor. And so so if you think this through if we have to algebraic group to connected commutative algebraic groups their product will again be a connected commutative algebraic group. And for the fiber product the point just is that the the fiber product is an algebraic subgroup. And so we can cover it by translates of its identity component by torsion points. And then the final object is of course just the trivial group and then this a four is also satisfied by essentially what I said before. And also we know that the image of a homomorphism of algebraic groups is always closed. So we can also take full sub subcategories of this category, and where we restrict the class of objects for example to be just the abelian varieties or just the semi abelian varieties just powers of the multiplicative group, or just powers of the additive group. And then you might have expected that the third example is is in the modular world so connected mixed Shimura varieties. So, and these are the reason why we need this functor. So I don't want to go into the technical details here, but but basically so objects here are our triples of this form px plus gamma, and px plus is a connected mixed Shimura data man gamma is a is a congruent subgroup. And then we can look at certain morphisms between such triples, and the functor associates to such a triple a certain variety over q bar. And here the functor is needed so we somehow have to keep track of this data associated to the variety so it, it could happen that for for different triples. So that so or it does happen that different non isomorphic triples are just mapped to a point by the functor, but but we somehow need this additional information about where this point came from. Okay, and then and then here also, and we can just look at connected Shimura varieties or just that connected mixed Shimura varieties of Cougar type. So, so what is the, the upshot of this so. So the elements of the I will call distinguished varieties and the elements of m distinguished morphisms and from now on I, I just will often tacitly identified them with their images under the functor this is a fairly harmless. And then I can define in this setting what a special sub right is, it's just the image of a distinguished morphism. And I can also define weekly special to be a component of the image of the pre image of a point and for a pair of distinguished morphisms so we are five from y to X side from y to Z and and we fix a close point in Z. And take a component of phi of psi inverse of Z. And in all the examples these definitions coincide with the usual ones so for example for algebraic groups you can go back and check that this gives exactly the same as as the definition I gave earlier. And then we can actually coincide with pinks definition of these concepts in the case of mixed Shimura varieties where we we kind of got the idea from. And then once we have a special and weekly special sub varieties defined, and I will tell you in a minute that, for example, components of the intersection of special sub varieties are special. So then we can again define the special closure the weekly special closure at the defect and the weak defect just as we did for sub varieties of connected commutative algebraic groups. And then similarly we can also define what it means to be optimal or what it means to be weekly optimal. So in particular we can formulate and a silver pink statement like the one we had before. Okay, so now why these axioms maybe they seem somewhat random. So, I don't have a justification for a one about the existence of direct products this just a part from that it seemed natural to to demand this and that it's satisfied in all the examples. But I'm concerning the other axioms. So first of all a for tells us that the image of a distinguished morphism is closed. So from this we get that special sub varieties are closed. This is certainly something we'd, we'd want. And then special sub varieties are weekly special because of a three so a three says there's always a morphism that maps everything to a point. And so if I have the image of an extinguished morphism it's also the image of the pre image of a point just just by taking the morphism that maps everything to a point. And then with a two in the presence of a one actually is equivalent to say that irreducible components of intersections of special sub varieties are special. So, so that's why there is this axiom about fiber products. And then with a two and a four we can also prove some other natural things like components of images or pre images of special sub varieties are special and the same for weekly special and components of intersections of weekly special sub varieties are weekly special so many, many, many natural properties that you'd want to hold. And the factorization in a for so that you can factorize this with these three morphisms 515253 with their special properties this is is crucial in order to be able to prove anything non trivial. So I guess if you if you drop that then you could just take all projective varieties and all morphisms between them. And of course then not much could be proven, but imposing this this certain kind of factorization actually and rigidifies the situation somewhat. Okay, so so now for the third part what can we we prove about unlikely intersections using distinguished categories so I already gave you some basic statements that we can prove. And also, if if somebody is familiar with the so called effect condition this is also something we can prove with a one to a four. Yeah, but what I want to speak about now is a bit different. And so, and I told you that we generalize this theorem of hope a grand pila from Cuba or to see. And so we realize that some steps in this process of extending the base field ever quite formal and could maybe also work in in much bigger generality. I don't want to speak about now but but unfortunately I have to first introduce a fifth axiom that that you see here. And that informally says that weekly optimal sub varieties come in finitely many families. So if X is a distinguished variety and V is a sub variety then there exists a finite set of pairs phi and psi of distinguished morphisms, such that for for any sub variety w inside we that is weekly optimal for V. So we have a pair phi psi in this set, and the point small set in Z psi, such that. Okay, phi should have finite fibers and the weekly special closure of this w is a new reducible component of phi of sign verse of said. So, so going back to to GM to the end this just says if you have a sub right TV inside GM to the end. And you have a weekly optimal sub variety there off you look at the weekly special closure. And this will be a translate of one of finitely many sub Torah of GM to the end. So, so this is also sometimes called a structure theorem. And then since I, I talked about extending the base field so if you have a distinguished category over K, and you have a bigger algebraically closed field L containing K. And then we can just compose the functor with the base change functor and we get a distinguished category over L so all these properties are preserved if we just base change everything. So you could imagine that we go from Q bar to see, for example. And then we call a category, a distinguished category very distinguished if if every base change with respect to an extension of finite transcendence degree satisfies a five satisfies this this weak finance property. So, so yeah this this might seem somewhat artificial but actually there's many examples that satisfy this let's say for for K of finite transcendence degree over Q bar. For example, a billion varieties satisfy this powers of the multiplicative group as semi a billion varieties connected she more varieties and also connected mixed she more varieties of Kuga time so for example the deluge under a family of elliptic curves, satisfies this. And so you can see here the names and associated to these. In these cases, and non examples here are just any connected commutative algebraic group and also powers of the additive group. And, yeah, this might not be too surprising since, for instance, the more the land conjecture also becomes falls in in powers of the additive group so, and somehow at this step these these algebraic groups are or sift out. So, probably a general mixed she more varieties satisfy this so so there has been recent work on the archive on on mixed action will that that should probably imply this property in general. Okay, so now I can present this theorem that more or less says if we know silver pink over some field we also know it over any bigger field. So, so kind of if you know it for some varieties defined over Q bar you will also know it for some varieties overseas. Okay, but so let me give the precise statement. So we have non negative integers m and n we have k inside L an extension of algebraic closed fields, and see is a very distinguished category over K over the smaller field, and x is a distinguished variety. So first of all, if we suppose that we know said PX prime MD for every distinguished variety X prime. Then we also know ZP XL MD so so ZP for for the base change of X to L so you still have X in a way but but now you allow some varieties that could be defined over L. And then we have a second statement where we can get away with a bit weaker hypothesis. Namely we just need to know ZP X prime M minus one D and ZP X prime MD minus one for every distinguished variety X prime. And then if we have a sub variety V of XL of dimension at most M, which is not the base change. So suppose me that I have a five minutes left. And so which is not the base change of a sub variety of X so it's not already defined over K. And then this V will contain at most finitely many optimal sub varieties of defect at most so this is the ZP XL MD statement, but only for sub varieties that are not already defined over the smaller field. And then we can get away with with a bit weaker hypothesis. So to state it because it's too technical but actually we don't need to know it for every X prime, but somehow only for those X prime that arise if we start with X and we apply axiom a five, and maybe apply it again and again. And in each step we somehow get more distinguished varieties turning up. And we only need to know it for those that arise in this process so in the case of a billion varieties we would for example only need to know it for M all quotients of the ability and variety X, which which is of course much less restrictive than needing to know it for every variety. Okay. Okay so so that's a fairly abstract so in the last four minutes I want to speak about some unconditional applications of this. So so this is a statement of the form something implies something else and I want to give some examples where we know the first something. And remind you this calligraphic a to is the modular space of principally polarized abelian surfaces over Q bar. And this is a connected she more a variety of dimension three for which the under art conjecture holds, which is all we need to know to be able to apply our theorem. And also calligraphic E as I said is the Legendre family of elliptic curves over Q bar, and it to the G denotes the the g fold fibered power of the Legendre family. And this is a is the connected makes two more variety of cougar type of dimension g plus one. There is again a scheme of a theorem so so X is a distinguished variety over Q bar C is a curve inside X defined over K. And satisfying that the special closure is equal to X K and the hypothesis hype see. So the sigma of points on the curve whose special closure has dimension, most the dimension of X minus two is finite so so the same as I had in the beginning. And now again a table will appear and with some instances where we can prove this. So the the first instance is in in a two X is a two. And then the hypothesis is that the curve is not already defined over Q bar so the curve should not be the base change of a curve over Q bar. So in this case we we apply the the second part of our theorem where we only need the weaker hypothesis, and here that the Q bar case is actually still open, although crystal and and Martin or showed some special cases. And then if we are in a in a hybrid power of the Legendre family we need no additional hypothesis on the curve. In this case that the Q bar case and follows from combining results of all these people mentioned here. And then we can apply our theorem to deduce the same over any algebraically closed field of characteristics zero. And the third instance would be, if we're in a semi a billion variety G over Q bar, and where the Q bar cases is due to Barrow era Q and Schmidt in in work in progress. And just to give a concrete example of our result in in a two. So I let h t denote the hyperlip the curve of genius to defined by this equation why squared equals x x minus one x minus theta one t up to x minus theta three T for some polynomials, he taught I with complex coefficients. And they T be it's Jacobian and I suppose that the generic endomorphism ring of this Jacobian is said, and that this defines a curve in in a to that cannot be defined over Q bar. And in this case what the theorem says is that there are most finitely many specializations at zero in C of T, such that one of these three possibilities holds. So J T zero is simple and its endomorphism ring is a Z module of rank for or J T zero is isogenous to a square of an elliptic curve or J T zero is isogenous to a product of an elliptic curve and the CM elliptic curve. So this is just spelled out what and what silver pink for a curve in a two means. And with this, I'd like to end my talk and thank you all for your attention.