 Thank you, Philip, Alina, Mike, for the invitation. No, it's certainly I'm happy to be here and to see all these names here. Can't really see anybody, but anyways, thank you. I wanna talk about a joint project with Mertel Mavracki, something we've been working on for the last year or so and we're still working on aspects of this project. And I'm particularly excited right now about the picture that you see in front of you. Many of you will recognize the Mandelbrot set in this image, but what's especially interesting to me is all the structure of these rings that you see that go around or really that all crash into the Mandelbrot set and the way they're overlapping and this set, I should say, this is not a talk in complex dynamics. I'm hoping to get to describing what this picture is at the end of the talk, but I wanted to start with it so you can see what's coming. There's actually some number theory hidden in this picture. In fact, I should say there's some number theory hidden throughout my talk, but it's going to be very hidden. I'm not really going to be talking about much arithmetic anything, even though I put the word arithmetic into my title, because in some sense, everything I'm saying is connected to the algebra and the number theory of the problems that I will describe, but at the same time, my goal is to tell you a little bit about the complex dynamics and the tools from complex analysis that we've been using to address the problems. So with that, actually I'm going to begin at the very beginning. I decided to turn this into a slightly more historical presentation of the subject, if possible. Well, not a complete history, so. All right, so let me start at the beginning. Whoops, I'm on the wrong slide, there we go. Okay, so in complex dynamics and the study of dynamics of a function of one complex variable, the basic thing that we do in the field is we break our space into two parts. We have what we call a fact two set and the Julia set, basic decomposition of either the complex plane or the Riemann sphere in this case, if you have a rational function. Roughly the fact two set is some large open subset of your sphere, which is where the dynamics are stable and the Julia set is what we call the unstable set. It's complement. And these names are attribute to fact two and Julia who really initiated the study of the iteration of this type of function. And so I want to share with you, just go back in time. Don't worry too much about these definitions. I won't actually need these definitions at all for this talk. I want to go back in time and tell you a little about where this came from. This text that you see in front of you is the English translation of prize announcement. It was announced in 1915 and the prize closed in 1918. The submissions had to be done within this period of time. And what was it? It was this, the researchers were asked to investigate the iteration of maps. So dynamics had essentially existed for let's say roughly 15 or 20 years in the sense that with the work of Poincaré, the subject of dynamical systems from a geometric point of view, from a qualitative point of view was really initiated in the work of Poincaré that had taken place around 1900, before really starting before 1900, just before 1900. And as this prize mentions, I'm not gonna read the text, but the prize is saying, well, let's study the iteration of maps, for example, an F which depends rationally on your input variable and turn it from a local question, the study of local dynamics to some kind of global point of view. In this case by global, I simply mean on the entire space as opposed to just a neighborhood of a point, not global and in arithmetic sense. Okay, so the prize was announced, it was closed in 1918. And it's what led to the work of Fatou and Julia whose names I said were used, are now used for this important decomposition. And now I should say there were only three submissions for this prize. And it's not these three people, I'm mentioning these three because it's their work that was very important and played a very important role in what came later. In fact, of these three, only Gaston Julia submitted a manuscript, though Fatou had been actively working at the time and produced something very, very similar at the time. And so Julia won the prize, Fatou did not submit his work. But both of them were building on a theory of the theory of normal families in holomorphic fashions that had been developed by Montel. So Julia won the prize, but what I wanna talk about actually is not his work or Fatou's work directly, but the work of the runner-up. So there was a second prize offered to Samuel Lattes who most people don't know about. And he died in 1918. So in fact, he had already passed away before the prize was given. And I think it was that his family received the prize in his name. And so I wanna say a little bit about what Lattes did. It wasn't considered especially interesting, I think to the others at the time, but it turns out to be quite interesting for the story I wanna tell today. So let me say a little bit about Lattes' work. What he was studying was functional equations that is composition relationships, relations between analytic or maromorphic functions. And especially this particular functional equation where P here represents the virus stress P function, or doubly periodic maromorphic function on the complex plane. So I'm viewing it here as now a map down to the Riemann sphere. And L is a linear transformation. And if L preserves the lattice of periods, then you get a quotient map F. So this L will descend to some map on the Riemann sphere. And Lattes was interested in looking at exactly this relation, which F can arise in this way. So remember, of course, the holomorphic maps from the Riemann sphere to itself are just the rational functions. So which rational functions arise as quotients of linear maps in this fashion. By the way, so the study of these functional equations was something considered very important and goes back some number of years. He certainly wasn't the first to look at relations of this sort. And soon after the work of Lattes, I just wanna mention that Joseph Ritt had done a very careful and deep analysis also of these functional equations of this sort. And specifically in the case where all of the maps are polynomials, polynomials of one complex variable, Ritt completed a complete classification of these, what we now call a decompositions of polynomials. When can you take one polynomial and decompose it in say two different ways? So what are all the polynomials of this sort that satisfy this kind of equation? So that's now complete. And I just wanted to mention that if you do the same thing for rational functions where F, G, H and Q are rational functions of one variable, we do not yet have a complete description of what the factors, the composition factors can be. Although we know a whole lot more now that we did at the time and that's a whole nother story and an interesting history as well as very interesting recent research that I don't have time to talk about. But I just wanna mention that that is something that people are still trying to understand is how, what is the decomposition theory in the case of rational functions? Anyhow, going back to Lattes. So again, so what was Lattes looking at? So Lattes is looking at the rational functions down here on the Riemann sphere, which are quotients of a linear transformation. Of course, we know, especially with this audience we know that a linear map which is preserving a lattice on the plane then factors through the quotient by this lattice in particular the corresponding elliptic curve if lambda represents the lattice of periods for your virus draws P function. And so, I mean, of course this depends on the, I'm writing it as if it were just one function but obviously, we know it depends on the lattice. So here we have now of course our elliptic curve and now we can think of this Lattes map as really as I've drawn here the quotient of an endomorphism on an elliptic curve which of course is linear from the complex plane but we can think of it as a map on the elliptic curve. Okay, oh, and by the way and some of you know this very well but others if you've never thought about this kind of relation I wanna emphasize that these rational functions that you get are very explicit. You can write down formulas for these rational functions F if you have the data of your lattice and let's say you choose for example the linear transformation which is multiplication by two on the complex plane which we know will induce an endomorphism of any elliptic curve multiplication by two. And so here suppose we take the square lattice so we get the square torus as the quotient then if we choose coordinates on the Riemann sphere we fix coordinates and whichever coordinates you choose you can actually write down a formula. So for this particular example this is a formula for this rational function. Notice it's a right, this is multiplication by two. So it's topologically degree four and the corresponding quotient here is a degree four rational function. Of course it'll always be the square whatever you're multiplying by. That's where in dimension two here. Okay, so just to show you Lattes original paper this is the paper from 1918. This is just one screenshot of a digital image of page 28. Anyways, that was published in the Académie des Sciences this journal and so I just wanna point out that this example that I've chosen happens to be exactly the example that Lattes was studying was presenting as an interesting example from a dynamical point of view. Notice here P is our Weierstrass P function. The image point is the quotient of the multiplication by two. This is the square lattice. And amusingly there is a typographical error here. This numerator should be a z squared plus one and not a z squared minus one. So I'm not sure if that's an artifact of the digital image that I have but probably not. It seems to be in a typo in the original paper. In any case, this was exactly the example that Lattes was studying. And one of the things he observed was that as a dynamical system as a dynamical system on the Riemann sphere the periodic points are dense in the Riemann sphere. And now let me come to that. So let me go back to this picture. So again, I have my what's now called a Lattes map. So let me now define it officially. So we'll say that a rational function on the Riemann sphere is Lattes named after Samuel Lattes if it's the quotient of an endomorphism of an elliptic curve in exactly this fashion. And notice by the way, of course, this quotient if you think of it as a quotient from the elliptic curve to the Riemann sphere this is simply the gluing of a point with its additive inverse. Because of course here where I just identifying Z and negative Z in the coordinates on the complex plane. Okay, so what did I wanna show you? Oh yes, about the periodic points. So this is very basic. If you take just reminding you the definition of a torsion point and in dynamics the notion of a point with finite orbit something's pre-periodic if under iteration the point eventually cycles. So it has a finite orbit. The word orbit just represents the sequence of points as you apply your map repeatedly. So, and then the basic observation is that a point upstairs in the elliptic curve is a torsion point if and only if it has a finite orbit under multiplication by two if you multiply by two repeatedly and it's a torsion point it can only take finitely many values on your elliptic curve. So that the orbit has to be finite and vice versa if the orbit is finite then you extract a relation and so the same thing actually holds for the quotient map. So a point downstairs will be finite orbit or pre-periodic if and only if the point upstairs is torsion. And this was an observation of La Tess. So I just to have a little picture here of course we know that these torsion points being the rational multiples of those of those lattice points these are dense inside your elliptic curve. So they project to a dense set of points inside your Riemann sphere. And so, oops, went too far. Where's the statement? Oh, there it is, okay. And so in particular the pre-periodic points of the map are dense in the Riemann sphere. So this was an example using modern language. This was an example of a map where the Julia set is the entire Riemann sphere what's called the chaotic set. Now, from a dynamical point of view the dynamics here is very simple. It's a linear map. So we fully understand the dynamics on the elliptic curve and therefore also on the quotient. But from a complex analytic point of view that it's considered unstable or chaotic because of the fact that all of these repelling cycles are, sorry, all of these periodic cycles are repelling. We have multiplication by two here which is bigger than one. So points are being pushed away from each other. So that's what gives rise to a sort of chaotic behavior. And so that's why the Julia set or the unstable set is everything under iteration. If you start with a small perturbation of your point and you start iterating points get pushed away from each other. So I think my understanding is that La Tess was interested in these examples because, ah, look, here's this map defined on the entire Riemann sphere and it looks like it has nice behavior but actually all points are unstable. Small perturbations lead to very different behavior in the long run. But Fatou and Julia were not particularly interested in this construction and it was more or less forgotten about from a dynamical point of view as far as I can tell as far as I understand for years. And so, but it turns out from a modern point of view this class of examples of rational functions this is just a very small collection of rational functions among all of these dynamical systems they actually play an interesting role and an important role in complex dynamics. I mean, there are always exceptions to in complex dynamics when we're studying rational functions in general the La Tess maps are always an exception to almost all theorems and almost all statements and almost all proofs. They kind of play a special role and they have to be excluded for various reasons. But sometimes we really wanna focus on them. So I wanna give you an example of a problem that I had been thinking about for a number of years actually, many of you have seen me talk about problems of this sort over the last few years. So let me state an example of one of these problems that's turns out to be directly related to studying La Tess maps. Okay, so this problem question has nothing a priori to do with dynamical systems. It's a question that I learned first appeared in an article of Bogomolov and Schinkel from 2007. It was revisited 10 years later in a follow-up actually, this is the publication year and this is a pre-print year. So maybe it was in any case more than 10 years later revisited by Bogomolov together with Fu and Schinkel. And what was the question? So the question is, suppose you're given two elliptic curves, two elliptic curves, each of which has its own collection of torsion points. So let's say I have the torsion points over here on this elliptic curve, it's a dense set and the torsion points over here on this elliptic curve, some dense set. And then they were asking the question when you project them down to P1 via the same projection I was talking about before, just the natural two to one projection where you glue a point to its inverse. What is the overlap of these torsion images? What is the overlap? How big can it be? How big can this set be? Just as a set, just counting the number of points. Now I've very roughly said some kind of vague motivation. This is a very imprecise, perhaps funny thing to say even a funny question to look at. But roughly, I mean, the idea is you have this elliptic curve, it has some geometry. I'm working over the complex numbers, let me remind you, working over the complex numbers. But one could ask the question of if you're given an elliptic curve, let's say you forget about the structure, which is the origin. What is the order of the given torsion points? And you just look at the geometry of those torsion points. So to what extent does that geometry of the whole set or of a certain subset determine the elliptic curve? How many of these points do you need to really pin down the elliptic curve if you forget the torsion order? So I wrote that roughly as how does the geometry of E and its torsion points determine the arithmetic by which I mean, say, what field is it defined over? Well, how much information do you really need to pin down the elliptic curve? Is how I like to think of it. Now, how many points do you need or how large is this overlap? First of all, I just want to remind everybody this is a finite intersection. There are a number of ways to see this finiteness, but the quickest way to see it, although I'm not actually giving you the proof, but it turns out one can see this as an immediate corollary of a big theorem, namely of Reino's theorem from the early 1980s, proving the Manian Mumford conjecture. I'm not going to say how this argument goes, but what is the statement? The statement is that, I wrote equality. Equality is a funny thing to say. So if this intersection were infinite, then it means that actually the elliptic curves are isomorphic and the projections are the same projection via this isomorphism, the projections are the same projection down to P1. So I'm abbreviating that as equality. So otherwise, the intersection is actually finite. So you can really count it. So you can really ask this question, how large is this overlap? How big can it be? All right. So we don't know how large it can be. That's still essentially an open question, although I will say something about it in a moment. We don't know how large this intersection can be, but one thing we do know is that there's a uniform bound on the size of this intersection. And so explicitly the statement is, there's a uniform bound. So I'm working with pairs of elliptic curves to find over C and these projections, where I just glue a point to its additive inverse. So I project down to P1. But of course the projection, I'm not fixing a priori my coordinates on P1. So I could post-compose a projection with a mervious transformation so I can move my projection. So there's really for each elliptic curve, there's a three-dimensional family, three complex dimensional family of projections down to this sphere. So the statement is, over all pairs of elliptic curves and all choices of projections, there is a uniform bound on the size of this intersection, the number of points in common down here. Unless of course the pair is the same pair, as I said before, in which case they're what I'm calling equal. The elliptic curves are isomorphic and the projections were the same choice of projection. Up to the symmetries, of course. So anyway, so this was explicitly stated, this is proved in this form by P1O, just announced this year. But I wanna mention that this, there's a lot that goes into this story, it can approve of this statement. Well, I mentioned that the finiteness alone can be deduced from Ray Nose theorem on Mani Mumford conjecture. And in fact, this uniform version can also be deduced by the uniform versions in the Mani Mumford uniform theorems, I should say now versions of Mani Mumford. So I've just written the statement here that there are a number of different ways to see now this uniform bound. So I've said some other works that I wanna mention all from this last handful of years. I'm also mentioning some of my own work that was not treating this all possible pairs, but we were looking at a particular subfamily, a certain complex family of such pairs and we were able to find a uniform bound. But one thing I should say, these different, so there are in fact several proofs that this uniform bound exists, they all hold over C now, so that was sort of what we were excited about to get this bound that doesn't depend on the fields over which E1 and E2 are defined, but they are very special to this setting. So this group thinking about elliptic curves or a billion varieties, I mean, they are fairly special to this setting. Oh, I know what I wanted to say next. This does not answer the question, of course. This does not answer the question, how large is this overlap? And so very importantly, I want to mention what the current record holder is and some of you have seen this, that foo together with Stoll in a preprint, I guess from last year, I think from last year, produce an explicit example where the intersection is has size 34, not just at least 34, but exactly 34. So they're able to prove for a particular pair of elliptic curves with particular projections that you can get exactly 34 points. And so I think this is fantastic. And I understand from them that this really should be, this is the record holder for now, but maybe it even wins, that it might very well be the maximum. So it's possible that we can take this M to be, well, I wrote less than so, let's say 35. In any case, I don't know what the optimal M is. And the proofs here do not give us a particularly nice bound or maybe no bound at all as far as I'm aware. So one could still ask to what extent can we really determine is this truly the winner? What kind of bound can we get? The proof that I had given with Holly and Hasi some years ago, we did not make this particular argument effective, but in a similar setting, in a related setting we did, and the bound that we ended up with was huge astronomical. I mean, it was on the order of 10 to the power 100, but not even for this setting, it was some other setting. So we don't actually know what kind of bounds we can get explicitly. So I find that an interesting question as well. Anyways, lattes maps, what does this have to do with what I was talking about? How am I doing on time, by the way? I see 11, 27. Okay, lattes maps. All right, so remember what I said before, this elementary observation that lattes had made in his 1918 paper, these torsion images on the Riemann sphere are precisely the sets of pre-periodic points, points with finite orbit for some dynamical system, in particular these maps that I'm calling lattes maps. So we know that there's some dynamical interpretation to this problem. And so it's very natural to ask, it's very natural to just ask, well, why restrict to these types of maps? Why not ask the same question for any pair of maps on the Riemann sphere? So for example, one could ask, given a pair of rational functions, morphisms from P1 to P1, again, working over the field of complex numbers here, how large can this intersection be between the sets of pre-periodic points? And I have to add, in addition, is this gonna give us any new information anyways? I mean, it may be nice to generalize, but perhaps also a proof, a dynamical proof might give us some extra information about the group setting, this case of looking at pairs of elliptic curves. So I've written that here. Would a general proof provide any new insight into the group setting? Not clear. Not clear. Okay, what was I gonna say? Oh, I guess I should observe. I mean, as in the case of pairs of elliptic curves, there are cases where this intersection is not finite. I mean, it turns out there's a similar description. We understand completely when this intersection is infinite. It's only infinite, exactly like the pairs of elliptic curve setting. If the two maps are essentially the same, I wrote equality in that case. It's a similar thing here, but in the case of dynamics, you have some extra symmetries that arise. So it could be that F and G share an iterate. It could be that they share an iterate up to some kind of symmetry. We don't have a complete description of when the pre-periodic points for F coincide with the pre-periodic points for G, but we almost do. That's related to the decomposition theory that I had mentioned earlier. But in any case, what I wanna say is, unless F and G are essentially the same from a dynamical point of view, this intersection is finite, just as it was in the case of pairs of elliptic curves. Was there a question? No? Okay. All right. So, oh, right. I was gonna show some pictures, sorry. So I showed these pictures, some Julia sets earlier. So just to put this in context, remember now that these sets of periodic or pre-periodic points are no longer or not necessarily dense in the entire Riemann sphere. Now it's interesting, beautiful fractal structure. Here's the closure of the set of pre-periodic points for some particular polynomial example. In this example, the pre-periodic points are dense in the boundary of this set. And then there's a discrete set of them inside in the interior of the black region. Anyways, you get this nice fractal pictures and in fact, what does it mean to intersect them? Here's the little animation. We just take the two and we overlap them. And then you can see geometrically how these things overlap. So in dynamics, of course, we have a lot more tools to really study the intersection. In the case of elliptic curves, it was just a dense set and a dense set and that's a mess, but here we actually have some nice geometric structure as well, so there's more to study here. The picture that I showed at the beginning of the talk is going to be related to this and I'll come back to that. Okay, so what's the theorem? It turns out, and this is the brand new work that we have almost made available. I have not yet posted, but in this new work with Mertou Mafraki, we have been able to give a uniform bound in the dynamical context for almost all pairs of maps. There are still some exceptions that we're thinking about for almost all pairs. So let me at least read the statement. The statement is that if you fix the degree of the maps, so for each degree, this is the degree of the rational functions, there is a uniform bound on the number of common pre-periodic points, the number of shared pre-periodic points that holds for all pairs in a Zariski open subset of the space of all pairs. So rat D here denotes the set of all rational functions of a given degree D with complex coefficients. I didn't write the C in the statement. I should have written C for this audience for complex rational functions. So it's all pairs except there is some certain sub-variety of possible exceptions which has to include all the pairs of maps that share iterates, pairs of maps that have some symmetries, pairs of maps that have other features that you, or when F equals G, for example, the entire sub-variety where F equals G, there are some obvious things you have to throw out and we haven't yet worked out exactly what to throw out but we're in the process. So in any case, we know actually that, yes, it does extend. It's not just for pairs of elliptic curves or certain examples that I had already taken care of prior in some work with Holly and Hesse that I mentioned before for quadratic polynomials. It turns out this holds more generally. Well, here's what I wanna emphasize. The proofs in the other setting in the case of elliptic curves while the statement is not arithmetic at all. There's no number theory in the statement whatsoever. It was about elliptic curves over the field of complex numbers. Just as this is about rational functions over the field of complex numbers. In fact, the proof is very arithmetic and very much inspired by this recent work. Very much inspired by this recent work. In fact, we wouldn't have been able to do it if this recent work had not been completed. So this was our, it grew out in an effort to understand a lot of these recent developments on these uniform versions of Money and Mumford, uniform version of Mordell conjectures that all came out within the last few years. So this was all an effort to synthesize all the stuff that was happening. And very especially, let me say, what we're actually using. So here's when I say what we're actually doing is combining some complex dynamics. So there is some theory of stability of complex dynamical systems together with the arithmetic. So the proofs of this kind of statement or the ones in the case of elliptic curves we don't know how to prove them without using number theoretic methods or arithmetic geometry tools. And there's no reason to assume we'd have to be able to prove them without arithmetic tools. Although since the statement is in some sense purely geometric, it would be nice to know if it's possible. In any case, we're happy to take advantage of all the tools from arithmetic geometry that exist out there. And so I wanna mention especially the work of Yuan and Zhang that we're actually using in order to prove this statement. And some of the theory that was developed very recently by Gautier and Vigny and a certain important special case of this theorem that was already carried out for one parameter families of pairs that was done by Myrto, my co-author together with Harry Schmidt. And what we're really doing is mimicking what they had already done for one parameter families to go to higher dimensional families. And the input that we needed to go to the higher dimensional families really comes from these articles. So we're really indebted to all of this recent work that had happened essentially all during this last couple of years. And by the way, I should say this method does give actually a new proof or at least new ingredients in the proofs of the case of elliptic curves. And in the case of elliptic curves, we know exactly what the exceptional variety is there. I mean, it's only when E1 is equal to E2. So you really get a complete statement. So we also get some new insight I'd like to say and maybe I'm gonna tell you about that if I have time some new insight into the original problem that I had been thinking about the case of pairs of elliptic curves. But before I do that, one consequence or special case I'd like to share, which we found amusing which did not seem to follow at least not immediately from the uniform money, Mumford type results that were already out there. So just for entertainment, let me share this corollary, this consequence. Suppose you look at the set of roots of unity in the complex plane. And then we look at all elliptic curves overseas projected down to the Riemann sphere via the same projection. By the way, I should again remind you, so before I make the statement, this projection is just, for example, view your elliptic curve in Weierstrass form and then project to the X coordinate. So that's what this projection is. But as I said, you can also post-compose by a member's transformation on the Riemann sphere. So you can take the image of those torsion points and then move them around by automorphisms. In any case, overall the entire four dimensional family of elliptic curves plus projections, there is a uniform bound on the intersection of the torsion points with the roots of unity. For each individual elliptic curve, we know that this intersection is finite, but the statement is that no matter how, which elliptic curve you take and no matter how you move them around those torsion points around on the Riemann sphere, this intersection is uniformly bounded. You can't make it bigger than something. Now, again, I don't know how big this can be. There could be some interesting particular elliptic curves where this intersection is huge, but I would say typically, it's probably not any bigger than the number four. In fact, one thing that we prove from these methods is that actually there's a Zariski dense set of pairs, elliptic curve plus projection, for which that intersection is at least four. And I don't mean like in some families, it's really there's some collection of points where you see that the intersection becomes four, which is some Zariski dense set of these elliptic curves equipped with these projections. But so the method gives you this kind of statement, but it doesn't quite give you what we'd like to know for maybe generally speaking, maybe for a general choice of elliptic curve plus projection, the bound is actually four, but we don't yet know. So that is still an open question. Anyways, one can do similar things over here. We know that generally, I mean, that is to say, there's a Zariski dense set of pairs for which this intersection is as large as four times the degree minus one, but we don't know what an optimal general bound is, and we certainly don't know what an optimal actual bound is. Although there's some record holders there too. And so I'd like to mention that later, some work of Doyle and Hyde about what are the record holders for this type of intersection. And so hopefully I'll be able to get to that at the end of my time, but I probably won't. Okay, in any case, this is a special case. And here again, there are no exceptions. Let me emphasize, this is not a general result. I'm not cutting out some sub-variety. This is all elliptic curves equipped with their projections. There's always a finite intersection and there's a uniform bound. That's what the statement is. Okay. All right, so what's the theory? So this audience, hold on a second, go back for one second. This audience may or may not be familiar already with the, wait, where was the, oh here, with this work about uniform on a Mumford and all of these recent preprints that happened and the work of Kuna and Dimitrov and Gau and Habaker and Yuan and Zhang and there are a lot of authors that go into the story and some of you are already familiar with that. I'm not going to talk about that work at all. I'm just going to take all of that stuff as kind of a black box. And what I want to teach you in 10 minutes is the other ingredient, the complex analysis and the stability input, which is exactly what gave just a little bit of extra input into the case of elliptic curves as well and the case of lattes snaps, specifically. All right, so what is this theory of stability here? This is sort of a description just by picture. I won't prove any theorems or really state anything in detail, but what we were doing, so what Mirto and I are doing is we were building on what I'm going to call a classical bifurcation theory. In the theory of complex dynamics, there's a notion of stability and a notion of bifurcation for families of dynamical systems on the remote sphere. And what we were developing with some analog of this that works for our setting, what notion of stability is the right thing to look at when you're dealing with a pair of maps and not just one map. And so we're calling this pairwise bifurcation theory just for lack of a better name. But so what is the actual theory? So in complex dynamics, one thing that we learn is the following. So let's just imagine you have now, okay, a holomorphic family of maps on the remote sphere. So your parameter here, so I'm going to assume it's algebraics. So let's assume that my parameter is living in some Riemann surface and let's say a quasi-projective complex algebraic curve, X. So this is my parameter lambda and then I have a family. So in other words, the coefficients of these rational functions you can view is rational or meromorphic functions on some underlying compact curve. And then there are finitely many points that you have to throw out because they wouldn't be some well-defined dynamical system. Okay, so I have a holomorphic family of maps, which I'm going to view as one map on a surface. So I have my base curve X and then I have my product space X cross the Riemann sphere X cross P one. And so each individual map is acting on one of these fibers. Now, for any fixed map, you have lots and lots of periodic points, infinitely many periodic points in fact, and most of them are repelling points. The derivative near those points is, of the return map is bigger than one. So points are getting pushed away. And those repelling points, you can follow in a little neighborhood. So each individual point, you can parameterize, you can follow holomorphically in a little neighborhood. So that's, whoops, getting ahead of myself. That's what these little black lines are supposed to be, is maybe some collection of repelling periodic points that can be parameterized. And they fit together in certain regions of space, they might fit together into some foliation or lamination of some subset of this surface, this complex surface. Now stability, dynamical stability refers to, when you have a family like this, refers to maps for which, well, which lie in this open region where those periodic points really look like this. They can be followed holomorphically in a family and they're not colliding with each other. They're not colliding with each other. So wait, stability equals an open region. This is very bad English. I mean just to say that stability is the concept for which the periodic points are not colliding. This is the standard notion of stability. And so what I was drawing here is maybe you have another periodic point for this family F, which over some parameters is colliding with your other periodic points. And that's necessarily inducing bifurcations in a dynamical sense. Because collisions, collisions of periodic points force repelling behavior to transition to attracting behavior. So collisions actually force a transition of cycle type. And so this is a bad thing from a stability point of view. So there's a real bifurcation taking place when some periodic point that you're following collides with another one. By the way, this is also the typical thing you expect to see. So for example, in the famous family of quadratic polynomials, it's exactly the boundary of the Mandelbrot set, which is the set of parameters where these collisions happen. That is to say, well, the closure of the set of parameters where these collisions happen. So if you're following all the periodic points, these algebraic curves of periodic points that live in the surface in your family, and you see where they fail, where they fail to have some nice structure, now they're not filling out the whole space. Because not every point is periodic. I mean, some of the points are periodic. The closure of these repelling ones is precisely the Julia set. So the closure here is just filling out some fractal piece of a vertical slice. But this boundary here corresponds to the closure of the set where you have these collisions. OK, so it turns out, and one of the things we learned in complex dynamics, a very important theorem that was proved by McMullen in the 1980s, is that these collisions essentially always happen. You cannot avoid these collisions in the case where you're dealing with an algebraic family, so parameterized by some algebraic curve in this way. In other words, the theorem is that if you're given a family, and if it happens to be stable over the entire x, so you really have this nice family of algebraic curves that fit together into something that looks like a foliation, it can only happen if the family is either trivial or a family of lattesmaps, namely if the family came from a family of elliptic curves. So I already introduced lattesmaps, so you already know what they are. In other words, coming from just deforming the complex structure on your elliptic curve and taking the corresponding family of maps, those are called lattesmaps. So the only time you can have this stability is if it's a family of lattesmaps, unless it's just one map, it's constant. The isotrivial setting would be just one map, and then, of course, there are no collisions, because the map is not changing. So the lattesmaps are exceptional already in this way. This is maybe the only theorem in the state where you see their exceptionality, they're exceptional to almost all statements. So otherwise, for any other family, you expect to have bifurcations like you have here in this Mandelbrot set. So one of the ingredients that Mirito and I are using in our study, well, I haven't told you what pairwise bifurcation theory is yet, but one of the ingredients that Mirito and I are using is a sort of extension of this type of result. So I just want to mention a theorem that I had proved some years ago, which was building on some work of Dujardin and Fav, which was the following, which was rather than looking at the motion of all the periodic points, let's just look at one single, say, algebraic curve inside this surface, that's this p. So I look at one algebraic curve inside this surface and a family of maps f. And if this curve avoids collisions with the periodic points, all the periodic points, or all but finitely many, but let's say all the periodic points of f, then that curve must have been, say, either a periodic point itself or the family is trivial. Okay, so one more time. If you have some algebraic curve in this surface and I have this family of maps, and if this curve does not collide with the periodic points of f, then it must have been that it was a periodic point or eventually periodic for the entire family, or it means that the whole family has to be trivial. There are no other options. And so this is really building on this McMullen theorem, the proof is sort of mimicking what McMullen had done when he was classifying stable families. And I just want to mention that there's a very special case of this, that this audience, at least some of this audience is already familiar with. So I want to mention in the case of La Tes maps, where I'm thinking of a family of elliptic curves or I could really think of an elliptic surface over a base curve, there's the foliation of those torsion points that, I mean, the torsion points fit together to give you a foliation of the total surface and what's called the Betty foliation. And in that case, we know that the only leaves of the foliation that are actually algebraic curves that close up to give you quasi-projective algebraic curves over your punctured base are exactly the torsion leaves. So the only algebraic leaves in that foliation are the torsion points themselves. And that's, could actually be viewed really as a sort of special case of this phenomenon. So it's sort of extending this idea to non La Tes type maps to say, if you have something that could be a leaf of a foliation, then it had to have been the analog of torsion. It had to have been pre-periodic for the whole family unless the family is trivial. So I see that it's 1150 and so this is the end of the talk. So I want to just now bring this back to what we were doing. So how is this related to what I was doing? I haven't told you what pairwise bifurcation theory is. So just to say that what we were doing is using this idea with this complex analytic proof to compare two different maps. Sorry, let me skip ahead here. To compare two different maps. So one map, one family of maps is stable if its own periodic points do not collide. But a pair of families is stable if the periodic points of one do not collide with the periodic points of the other. So for one family, they might have internal collisions but I don't want collisions like you see here. The green family should not collide with the black and red family. So stability for the pair means they stay away from each other. And there's a description that one can give. One can completely characterize like in the case of McMullen's theorem. Building on some earlier work of Levin and Pshatitsky and Baker and Aramanko and some work from the 1980s and 90s in complex dynamics one can actually completely describe when a pair of maps is stable with respect to each other, with respect to each other. And this is what we're actually building on to talk about what we needed for the uniform bound. In other words, the following. I'm gonna show you one slide that gives you a sketch of how the proof goes with zero information. I apologize, there's in some sense, zero information on this slide or maybe too much information on this slide. Let's just take as a giant black box all of the number theory, all of the arithmetic geometry I'm just gonna pretend that this entire audience already knows all the stuff that I do not know so well the intersection theory of a delically metrized line bundles. Here's what we actually do. In order to run, to apply some of the theorems from this work of Yuan and Zhang and prior to that, there's a work of Cune, Gotie and Vinyi, Dimitrov, Gao and Hamburger. There's always a hypothesis of something called non-degenerate. And it always boils down to when does this particular object you're interested in satisfy this hypothesis? So I guess this notion of non-degeneracy I understand originated and maybe some, maybe Philip can say in a work of Philip and then has been studied also by Xi'an Gao and others and sort of developing what should this non-degenerate mean and then how do we characterize it? There's dynamical analogs. And so what we actually do is what we're showing is a particular measure what we're calling is a pairwise bifurcation measure is non-zero. Therefore, we feed it into this machine. It implies a certain non-degeneracy and the output is height inequalities and arithmetic equity distribution. This is very formal. And then from that point on we can mimic some work that Mirito had already done together with Harry Schmidt to get the uniform balance we want. And now I've gone three minutes over time and so I should finish by coming back to this picture to tell you what it actually was. This was an illustration of that measure that object that we need to prove non-degeneracy in one very special case, one parameter family where you take one of the maps to be constant just the squaring function. And the other family of maps is the famous quadratic family. And this set which unlike the usual quadratic polynomials which bifurcates only at the boundary of the Mandelbrot set when you do pairwise bifurcations there are bifurcations on the entire Mandelbrot set including the interior. So the bifurcations are everything in black together with these rings that go around and crash into the Mandelbrot set. And this is the set which is exactly where these pairwise these two families of maps are bifurcating sort of with respect to each other. This one doesn't bifurcate at all. This one only bifurcates by itself on the boundary of the Mandelbrot set but as a pair this beautiful image is illustrating the set. And I didn't see this image I had been wanting to have a picture of it and finally last week two graduate students Caroline Davis and Alex Capriamba at a conference in complex dynamics helped me produce this image and so I'm grateful to them. And I think that's where I should stop.