 All right, well, thank you very much. Thank you to the organizers for the invitation, which took me a long time to finally follow up on, but thank you. And thank you, everyone, for coming. So hopefully the talk will have some new and interesting things for everyone, even though I'm really revisiting some very classical mathematics. So this is a project. This was just a really fun project. This came out of the special semester on illustrating mathematics at ICERM. So in fall 2019, the whole semester was focused on illustrating mathematics. And we all got to be in residence. So I got to be a semester organizer actually here in the upper right here are some of the other semester organizers in a cubicle formation here. And I met Steve Trell, who's at the lower right here. He's currently a postdoc at Stanford. And Edmund Harris, who's an mathematician and artist at the University of Arkansas. And the whole program was this sort of free flowing experimental exploratory mood. And so what we were interested in trying was to draw pictures of algebraic numbers. So my question to start with for everyone is what would it look like if you plotted all the quadratics in the complex plane? And of course, the answer to this really, I suppose, is it would look like that because they are dense. But then, of course, you want to ask the question in a somewhat more nuanced way. So let's imagine that we're going to draw a dot for every quadratic or rational in the complex plane. But I can size them. So some of them can come to the fore and some of them can fade into the background. And you can try lots of different sizes. We tried and I'll show you lots of interesting sizings. But one of the first ones that we tried is to highlight the simplest, most important quadratics in some sense is the discriminant. And this is the picture that you get. So in this picture here, this is the upper half plane. So here we have the real line down here. And this is I right here. And I'm sizing them by the inverse discriminant. I'm also scaling them by the hyperbolic metric because you can see that they crowd up as you head towards the real line here. And so I'm shrinking the dots as we head that direction. And so you can see when you do this, you get a ton of structure. And so one of the things I wanna do is explain to you what's happening in this picture if you can't guess right off. There's a nice fairly simple answer to this. And yeah, so looking at the picture, you probably pick out various features. I think for a number theory crowd, you're probably noticing the tessellation of the upper half plane by fundamental regions of SL2Z in this picture, right? We've got two of these showing up. And in fact, all of these lines, so the picture appears to be completely full of lines. All of these lines are geodesics. They're semicircles centered on the real line. Okay, so what's going on here? Well, each of these lines is sort of densely filled with dots like this. And it looks something like this if you were to just pick one of the lines. But this is actually a picture of rational numbers where I've sized them by inverse denominator. Actually, I think this might be inverse denominator squared here. And so that just means that the integers will be the biggest and then denominator two, these ones will live in between and then we've got the third and two thirds and so on in between those. Hi, Kate, just a question from Andrew Granville. I wonder, could you be precise about what the numbers are? Which numbers? The numbers that are being graphed in this picture? I believe so. Yeah. So these are all the quadratic irrationals in the complex plane. So I'm drawing the upper half plane. So anything which is a root- I don't know what you mean. Are you talking about A plus B root D over C or what are you talking about? Yeah, the roots of quadratic polynomials with coefficients in Q. Okay, thank you. I'm not sure what the alternative was. Sorry, but hopefully that answers it. Yeah, okay. So yeah, so algebraic numbers of degree two. Okay, so here it's just rationals. And I'm sizing them by the inverse of the denominator. And you see this, again, the sort of fractal structure fading in and subsequent intervals like this and so on. And so if we can explain this, then we can guess what the other picture is really showing. So what's happening here is that the rationals are really P1 of Z, right? So instead of thinking of P over Q, think of the vector P comma Q. So if I put the origin down here, so this is a picture of Z squared, the lattice Z squared, then as I look out from the origin, I'm gonna see all of the lattice points. And I've marked in darker gray the ones that are primitive, meaning no common factors in the entries. So that you actually see these dark gray ones and the light gray ones are hidden behind because they're multiples of those. And so from your perspective at the origin here, if you're looking out, you will sort of see a copy of P1 of Z. And the thing to do is to imagine this is sort of like a perspective drawing, right? You're the artist sitting at this position here and here's all of the lattice you're looking at. That's the scene. And you put a screen between you and the scene and copy the picture onto the screen, right? And if you take the closer dots to have larger dots on your screen, right? Because they appear larger because they're closer to you, then you will get a picture that looks something like this, okay? So you could do this in various different ways, right? Like so you could put yourself at the origin and put a unit circle around and project onto there. But you'll always get approximately the same kind of picture out of this. So the idea being that the dots which are closer to the origin are larger. Okay, so this is the picture of the rationals. And so this looking similar should have a similar explanation and it does. So the thing, the lattice that you're looking at in this case is actually the lattice of the coefficients of the quadratic polynomials that these are the roots of. So we'll make that precise. So the idea is I'm gonna take the space of coefficients. So it's projectivization of the three-dimensional real space here. And I'm interested just in the ones that have complex roots, non-real roots. So I'm gonna take the discriminant to be negative. So I'm gonna take the B squared minus four AC to be negative here, okay? So I'm thinking of each of these vectors as representing a quadratic polynomial. And then inside that where I'm allowing real coefficients I could take the lattice which is those with integer coefficients. And that's gonna give me all of my algebraic numbers of degree two that I'm calling the quadratics. So this is a lattice, a three-dimensional lattice and this is a projectivization of that. And again, I'm gonna think of the things that you're near the origin in some sense as being the arithmetically simple ones, right? So to make this process precise this condition here that the discriminant be negative is actually if you look at the locus in the space of B squared minus four AC, it's a double cone. It's actually like a light cone. So this considered as a quadratic form is a form of signature two one. So this is really what I'm talking about is a Minkowski space, which I'm endowing with that quadratic form from the discriminant. And then the light cone cuts out an interior part and an exterior part. And the interior part is the stuff that has complex conjugate roots and the exterior part is the part that has real roots. So then what I wanna do is projectivize. So I'm going to choose a disc at height one up in the cone and I'm gonna project all of those lattice points onto that disc. And this might be familiar because this is the Klein model of the hyperbolic plane. So if you want to ask what are the geodesics, they're the straight lines across this disc in the Klein model. And that corresponds to taking a plane in the Minkowski space, which intersects the light cone and then intersecting it further with this disc. So I'm gonna project the lattice down onto that disc and then that's the Klein model of hyperbolic space. And then what I wanna do is I want to take for each of those, this is the coefficient space, right? So each of these represents a quadratic polynomial. I want to take its root in the upper half plane. And so I can show you a picture. Here's a top down look at that Klein model. And then if I take its root in the upper half plane, then I'm moving to this model of the hyperbolic plane. So basically moving from the coefficients to the roots is actually just changing, it's just an isometry of two different models of the hyperbolic plane, one being the Klein model from the Minkowski space and the other being the upper half plane model. Okay, so what's happening is that's actually a quadratic formula. So what we find is that the quadratic formula can be thought of just as the hyperbolic isometry between the Klein model and the upper half plane model. Okay, so that's how we move from coefficients to roots. And that's why what we see in the picture is all of these geodesics is because they were lines, they were planes in the Minkowski space, which captured a full two-dimensional collection of quadratic polynomials. Okay, so what we can do with this observation is go back to the story of diphantene approximation. So in the rationales, sorry, in the reals approximating, doing diphantene approximation of real numbers by rational numbers. This picture is sort of an illustration of the way that the rational numbers sit in the real line in some sense, okay? So if I draw, here I'm being explicit, these are disks of radius. This is just a constant out front to size them sort of for visual appeal, but proportional to one over two squared. Then you notice visually that they sort of repel one another. So the larger disks can't be very, two large disks can't be particularly close to one another. And that's actually a very simple observation in the rationales, right? It's just that if you have two rational numbers and they're not actually the same number, then there's a minimum distance between them, which is governed by their denominators, okay? We could grow the disks. Now here in this picture, what I've done is I've grown the disks so that they're still centered at the same place and they're still proportional to one over two squared but I changed that constant. So it's literally one over two squared now. And now they overlap. So here's an integer at this point right here. And then this disk is out here like this. And the way I've drawn it now, each of the disks has a little bit of translucent whiteness to it. So as they overlap, they build up whiteness. So we're moving from black to white through the different grays, okay? And so then what you see in this picture is if you pick a point on the real line somewhere you can ask, am I covered by finitely many or infinitely many of these disks? And of course that's Dirichlet's theorem. So alpha is irrational. If and only if there are infinitely many rational numbers that are good approximations in this sense. So alpha lies within the disk centered at P over Q of radius one over Q squared. So in some sense here, this picture of course is always going to be imperfect because it's an approximation but in the ideal picture, all of the irrationals here are bright white because they're covered by infinitely many of these disks. Okay. And then of course is Roth's theorem. So this is asking about what happens if you slightly increase the size of the disks. Sorry, if you slightly decrease the size of the disks. So if we increase this exponent here, we're asking for a tighter approximation. And then the algebraic numbers, then they end up having only finitely many approximations so lying under only finite many disks. Okay. So we can ask the same, we can revisit those classical sorts of questions. Now in this picture, and it seems like there should be sort of a analogous statement. If you try to draw this picture, by the way, by expanding the distance that they overlap, you don't get a nice picture. So it's easier to scale them down so that they look like repelling stellar bodies or something instead of overlapping disks when you're in this dimension and just, there isn't enough space somehow. The information in this picture is somehow actually carried in the extra dimension not on the line itself. So all right. So then the question that we'd be asking is, how well can we approximate a complex number by quadratics by like the set of algebraic numbers of degree at most two. Now we need the correct notion of arithmetic complexity for these numbers. So before we were approximating by rationals and we were measuring the quality of the approximation in terms of the denominator. But really what we're measuring in terms of this is the height of that rational number, which should be the maximum of the size of its numerator and denominator. So we wanna generalize that because if we're gonna approximate by a quadratic then we want to know the arithmetic complexity of the quadratic in order to control the quality of the approximation. So these P and Q are actually the coefficients of the minimal polynomial for that rational number. So what we do is we look at the minimal polynomial for the quadratic. And I'm gonna scale my minimal polynomials so that the coefficients are all integers but don't share a common factor. So not monic but that's what I'll call the minimal polynomial. And then you could generalize this notion. So there's two classical ways to generalize it as pretty much, I mean, I'm telling you many things that you've already know in fact in this talk right now. So the naive height here would be just to take the max of all of those coefficients or you could do the V-height. And for that you have to pick a field K which contains your algebraic number alpha and then do this product over the different places for that field K. And if you normalize by this exponent then you can choose any K which contains alpha. Okay, so now these are essentially in some way equivalent, right? They should be telling you the same sort of information and that can be made precise by this relationship. So if D is actually the degree of alpha over Q then you can relate these two different notions. But in the picture that I was drawing I was not actually using these. I was using the discriminant and this is really not the same thing. The discriminant is not somehow equivalent to this these other notions of height. So, but what happens if we look at the picture with say naive height or you'll get a very similar picture if you use the Bay height, it looks like this. So what's happening is that as you head out from I here. So for example, you go from X to X plus one the height will actually increase and so the picture will fade off. And in some sense that's maybe not what you want because the arrangement of the dots here in this region is the same as over here. And if you wanted to get a stronger approximation statement you could actually sort of translate over and figure out what's going on and then translate back. So there's something to be said for using the discriminant because it's actually literally periodic with respect to SL2Z. Okay, so there's actually so let's be slightly more explicit, right? The SL2Z acts as Mobius transformations on the upper half plane. So on the roots, the quadratic points. And in fact, then there's a representation on the coefficient space as well. So there's some representation of PSL2Z and PSL3Z which is acting on the coefficients which is just composing on the inside with your Mobius transformation. And so when you draw it with the discriminant it's actually periodic with respect to SL2Z. Okay, and I think I said these things. Okay, yeah, and I'll just point out that there is this relationship at least between discriminant and height but you don't really expect inequality in the other direction. Okay, so, all right. So let's go to the question of approximating complex numbers by algebraic numbers of some bounded degree. So we want to approximate by degree at most D. So Coaxima defines something which is basically equivalent to what I have on the screen here, which is to ask the question, as we ask to approximate alpha by algebraic beta of degree at most D, we could ask for different sort of qualities of approximation. So this K is changing. And at some point you reach this cusp where it switches between having infinitely many good approximations in that sense to finally many good approximations. And so that cusp, that point where this changes is this exponent case of D. So if we set D to be one, then we're asking about approximating by rationals. And so Dirichlet's theorem says that if you're rational then it's at most two and if you're irrational it's at least two. And Roth's theorem says that it is two for algebraic numbers, that's where it swaps if you combine actually Dirichlet and Roth's, right? And Sprinjuk says that for almost all alpha, this is the generic behavior. Two is the place where things should switch over. And of course you can make numbers which are much better approximately. You can construct things that are counter examples or that aren't in this set almost all. Okay, so what about a higher degree? So Sprinjuk has a result that says what these should be in general. So for almost all alpha, KD of alpha should be D plus one in the reels, so if alpha is real. But then if you're in the complex numbers and not in the reels, so you're off the real line, then it's D plus one over two. So this is kind of interesting because it says that this is a smaller exponent. So it gets hard to approximate at sort of worse, approximation level. So they're actually harder to approximate in the complex plane. And you might pick that up from the picture, maybe let's go back to the picture here in the sense that these seem to cluster really strongly on the real line. Okay, so this just tells us basically that in the complex numbers we expect a different sort of behavior in the reels. And okay, so let's go revisit the sort of classical statements that we've talked about already for the rationals in the situation just of quadratics. So we said that distinct rational sort of repel in some sense. And if you think about the proof behind this, this is really, if you think about what you're doing when you take the difference of these two things, right? The number is AD minus BC. So in that coefficient space in some sense, you are pairing vectors, they're integer vectors. So the pairing has to be an integer. If it's not zero, it's at least one. So that's one way of viewing that proof. And there is a generalization of this for algebraic numbers. So if you have distinct algebraic numbers, alpha and beta, and you know D, which is the degree of a field which contains both of them. So you take any field that contains both of them, then that'll give you a sort of repulsion statement in terms of the height. And this isn't too difficult, but we're using the V-height here. So we're using the triangle inequality in absolute values and stuff. And the proof doesn't look sort of the same in flavor as it does in the rational case, really. So, yeah, go for it. Can you go back to slides? I'm just curious, yeah, right there. So there's D plus one versus D plus one over two. Does this have anything to do with like, or could this be reinterpreted as H, the complexity H for when you have two embeddings into the complexes? Like totally real versus, like there's a square in there. You know what I'm saying? Like the norm. Square. Except a square. And so maybe these are somehow, one could try to interpret as being the same. It's just that H is what changes. You know what I mean? We can talk about it later. I was just curious if that was... Yeah, I think I might have an idea what you mean, but I'm not sure. I mean, yeah, I'll have to think a little bit more about it. I mean, I haven't thought about it that way at all, no. Thanks. Yeah, okay. But yeah, thanks for interrupting with questions and stuff. The talk, if anything, is probably short to some. Okay, so let's go back to the repulsion statement. So we were just talking about the rational and algebraic numbers, sort of repulsion statements. And so you can make a similar statement if you wanna talk about quadratics, but from the perspective of the picture. So this is a slight variation on the theme. You take the hyperbolic distance is in some sense more natural from the perspective of the picture. And then what you do is you look back at the coefficient space. And the coefficient space is endowed with the bilinear pairing from the quadratic form which is the discriminant. So you have a pairing in there. And if you take two different elements of the lattice, then the pairing is an integer. And if it's not zero, if we're all inside the light comb it's not gonna be zero unless they're the same. So that pairing is at least one then. And that leads to this statement, which says really this is the hyperbolic distance as calculated here in terms of coming from the Mikowski space. And so really, the art coach of one is zero. So this is really the quantity just like you would have in the classical situation. And so you've got one over your discriminant. Okay, so it looks pretty similar. And it's interesting that if they're from the same field then they repel just a little bit better. It comes out, naturally comes out of this. And I think that's actually visible here too because you're taking to the degree of the fields containing both of them. Okay, and so now let's go back to the Dirichlet and Roth type question. So Buzho and Averetze have a paper from 2009 in which they have a lot of wonderful results about approximation of numbers in the complex plane by algebraic numbers. Yeah, was there a question? No. Okay, so suppose we take an algebraic number alpha which is in the complex plane, but not in the reals. Then the question is what is this exponent of Koch's mom? And it breaks into cases a little bit. So if D is even and the degree of alpha is bigger than D. So D is the degree of the stuff you're trying to approximate use as approximations, right? So we're trying to approximate alpha with things of degree at most D. So if the degree of alpha is the bigger then there's two possibilities here. We saw before this was the generic one but it could actually be something different. And then in the other case here, maybe, so this includes if D is odd, then again it depends on whether the degree of alpha or D is the larger but then you're in this situation. So that's to say that they're able to compute this exponent pretty much all the time. Sometimes you can't decide the orbit almost always. Now let's go to the D equals two case which we've been talking about. So we're trying to approximate alpha which is something of say larger algebraic degree an algebraic number of larger degree by algebraic numbers of degree two. Then the answer is either two or three halves and so there's two distinct types of behavior that might arise and what it depends on is whether one alpha, alpha bar and alpha plus alpha bar so that's complex kind to get there are Q linearly dependent or not. Okay, so in the geometry you can interpret this algebraic condition that distinguishes the cases as a geometric one. So they are Q linearly dependent if and only if they actually lie on, let's go back to the picture for a sec. If they actually lie on one of these algebraic, one of these what I'm calling rational geodesics, okay. So these rational geodesics remember they correspond to planes in the coefficient space which capture a two dimensional sub lattice of the Z cubed lattice. And so I'll call those rational geodesics and if you're on what the statement says if you're on a rational geodesic then the exponent is larger, you are better approximable. So here's a picture of that happening. So in this picture in gray we have the original quadratics plotted and then in the two different colors of blue we have cortex and the cortex are colored light and dark according to whether they have two pairs of complex conjugate roots or just one pair of complex conjugate roots I think. I don't remember which is which though. Anyway, and so if you zoom in and see what's happening if I pick some cortex that I'm interested in approximating if you're not on the geodesic then you can only be approximated by quadratics when they fly by along the nearby geodesics, right? Because all the quadratics are lying on these geodesics and so they can only sort of fly by near you. But if you lie on one of those geodesics yourself then within that geodesic you have a whole two dimensional family of stuff that has a chance of approximating here. And so that explains the difference in the exponent. All right, so if you want to go at this naively from the point of view of the pictures we've been drawing then you might ask about again hyperbolic distance and in terms of the discriminant. And so then you can recover a sort of quadratic version of Dirichlet's theorem. So what this is saying is suppose we've got some complex number, which is not itself a quadratic irrational. So it's not a self a quadratic number. Then suppose it's lying on a rational geodesic. Then there are infinitely many quadratics beta that approximate to within this quality. So here the exponent is two on the discriminant. Okay, so it looks a lot like Dirichlet's theorem. And this constant up here can depend on alpha in this case. Now, if we don't assume that we're lying on a rational geodesic then we get a weaker result that we can only get to an exponent of three halves. So the proof of this is actually the same as Dirichlet's classic proof of Dirichlet's theorem. So here I'm just at the top, this is restating the first part of what's on this slide. And the idea is to take, if you take the number that you're interested in approximating alpha, it could be transcendental or something, that's fine. You can still talk about it and it's complex conjugate, right? So you can still write down a quadratic polynomial that satisfies it just won't have rational coefficients. So the coefficients can be given in terms of alpha plus alpha bar and alpha times alpha bar. So it has a location in the coefficient space. And so we work in the coefficient space because that's where we have this nice discrete lattice to work with. So we find this point in the coefficient space that's associated to alpha. And then you look modulo the lattice. So you're looking really modulo Z, Z, Z in the projectivization. And you look at the multiples of that point in the coefficient space. And by the original principle, eventually two of those multiples will be in some little subdivision of that cube. So it's just like a higher dimensional version of Dirichlet's original proof. And so that means that they're close, that their difference is close to a lattice element. So that gives you an element of the lattice which should be a good approximation to that point. And then you go via these linear forms. So that tells you that these linear forms should be small. And then you move to the discriminant pairing if you wanna have a statement about the discriminant. And so that means that you have two elements of the coefficient space that you're pairing in that Minkowski pairing from the discriminant. And so that'll be small. And then you carry it back by the hyperbolic distance and you get this statement. So it really is exactly Dirichlet's theorem. Okay. So then you can ask for, what about algebraic numbers being poorly approximable? So take some alpha, which is algebraic, but is not itself quadratic. Then if it lies on a rational geodesic, then you can show that if you, we just had that two exponent two approximation we expect infinitely many, but if we go to two plus epsilon, there's only finally many. So the approximation, just as in Roth's theorem here, breaks if you go any larger than two. And then if you're on a rational geodesic or not, so this was if you were on the rational geodesic, but now whether you are or not, you can ask for approximations that come not from your own geodesic in any case. So ignore those really good ones that come from being on a geodesic and ask about the rest of them. And you find then a similar statement, but with the exponent three halves. So there's only finally many of those sort of from nearby geodesics approximations. And so the proof of this is basically to do the same thing that we did in the Dirichlet's theorem proof that we just saw. You move to those linear forms that we saw in the last slide right here. And then since we have linear forms, we can apply Schmidt subspace theorem. So neither of these results is really new. They're slightly different in some way than what Bujona-Verze proved because I'm talking about hyperbolic distance, which locally is the same as you put in distance. So that's not very important, but using the discriminant. And so you can kind of recover versions of each of these from each other, right? It's really basically telling you this about the same phenomenon. All right. So that tells you sort of in depth some of what you can do with the quadratics picture. Now I want to go to what else happens in the universe. So for example, cubics. What happens if you draw the same picture for cubics? And if you take a look at this picture, it's wild. It doesn't look anything like the quadratic picture, right? So they're colored by the way by what happens with the real root. So if you have a cubic which has a complex conjugate pair of roots, I'm drawing the complex conjugate pair, the complex conjugate root in the upper half plane and then coloring to show you what the real root is. I'm not drawing those which are actually quadratics. And you can see the quadratic picture as a ghost in here because the cubics actually stay away from the quadratics. In fact, you can prove that the cubics can't lie on the rational geodesics. So there's actually these white lines passing through the entire picture. Now here's a detail of this picture. So it looks crazy, but when you look up close, you can see, well, first of all, well, we called them algebraic number of star scapes, right? But you can see that there's still some of that kind of linear structure happening, right? You still see these lines that look kind of like we were having the rationals and the quadratics. But now you've got like, this looks like a star with solar flares and these look like spiral galaxies and stuff. And then in here, it just looks like a mess. So there's clearly still structure going on, but now it's too busy to understand by looking at this picture. So you could try to follow now the same geometry that we did in the quadratics case. So you could ask, okay, let's try the discriminant again. So let's take a look at now the coefficient space is spaced with four coefficients projectivized again. So we have a copy of RP3. So this is a drawing of RP3. So what this is is you scale everything to length one and you get this ball, but then you still have to identify the surface of the ball antipodally, right? And so we haven't done that in the picture. So these are just like little balls. So this is a picture of the discriminant locus inside there. And then the discriminant locus has a sort of singular locus on it. So the discriminant picks up at zero if you have a double root. And then the singular locus inside it here will pick up when you have a triple root. And the discriminant itself breaks the space into two pieces. So there's this sort of inner piece here where you have three real roots. And then another piece out here where you have two complex conjugate roots in one real root. So this is sort of the geometry corresponding to the Minkowski space, but now it's much more complicated in the Cubics case. So are there any questions and stuff showing up in the chat? I have a question. Yeah. Why are there two, like for each of ABCD, why are there two pictures? Oh, why do I have a top and a bottom? Yes. Yeah, this is just two different views. So this is just rotated. I see. Okay. Okay. Got it. Thank you. Yeah. Sorry. I should have said that. Yeah. Any other questions? Okay. So you can try to run the same story and ask what do these pieces look like? So what I want to look at is because I'm interested in that picture I just showed you of the complex plane. I want to look at this piece that has two complex conjugate roots. So if you're a Cubic and you have two complex conjugate roots, you also have a real root. So I could draw this is the upper half plane here. This is the real line. So I could draw a black dot for my complex upper half plane root and a white dot for my real root. And then to capture the information of the real root from over here at the position of the complex root, I could give a direction. So if I give a tangent vector here, then it'll tell me which geodesic I have to follow to hit the boundary of the hyperbolic space at that particular real root. Okay. Really what's happening is I can think of that information, the information of the set of a complex conjugate pair of roots and a real root as a point in the upper half plane together with a tangent vector. So this is like a unit tangent bundle on the upper half plane. And then I'm seeing that space two different ways. So I'm seeing it either as the space of coefficients. So these are this projective space, the piece of it with negative discriminant, or I'm seeing it as a sort of symmetric power because I've got a set in here of roots of that polynomial. So basically then the map going from the picture with the coefficients to the roots is some isometry between two different ways of writing down the unit tangent bundle to the hyperbolic plane. Okay. And again, you've got an SL2 action on both sides if you like. So it's clear how that behaves on a unit tangent bundle here. And so it's realized on both sides. Over here it's like pre-composing. And over here it's acting on that point by Mobius transformations. Okay. So now this is the geometry that governs what's happening in the cubic case. So how can we draw that unit tangent bundle? Well, one way is at least topologically, you can break it down into a product like this and make a solid torus. So each slice of the torus is a disc model of hyperbolic space. So each one of these is a copy of the upper half plane as a disc. Okay. So that's all of the complex roots. And then you could decide for your real root. It's somewhere on the real line. You can include infinity. So it's on an S1. And so that's moving this direction around the torus. Okay. So every point in the solid torus corresponds to a choice of a point in the upper half plane and then a point on the real line. Okay. And so then we can draw this in its natural world. It should really be a three-dimensional lot of two-dimensional picture. And so this is software created by David Dumas to visualize Fuxian groups, which I can now show you here live. So this is my solid torus where I filled in all of those points, which are the cubics. Make it a little darker. And you can see that it has the same sort of structure. It's just like a projectivized lattice structure again. And there's a slice you can clearly see right here. But when you smash it down onto just one copy of the plane, then it becomes too messy to understand the structure. So this is something you can play with online. I'll give you the link at the end or whatever. And oh, I should give a plug for David's software. So he made this because he was interested in visualizing Fuxian groups. So here's PSL2Z. And what he's doing is taking basically the orbit of one point in the upper half plane with its tangent vector under the group. And then you see, because this is a picture of the tangent model, you just see the orbit as other points in the tangent model. Now he's put it on the outside. So, you know, you can flip the torus through its surface. So this happens to have things on the outside. Anyway, it's beautiful and fun to play with. And since he had made that, it was easy for us to just upload our dataset to it. You can also take slices of this. So if you take, these are qubits that have a particular form. So the leading coefficient is the same as the linear coefficient. Then you'll get a slice of it and you get these beautiful things. This one looks like a Movia strip when you look at it inside the viewer. And here's some other slices. So now I can just sort of enjoy some pretty pictures. These are depressed qubits. So this is if you take out the X squared term and then you can really see, then we have something of the correct dimension and you can really see the sort of projectivized lattice structure. Did I skip one? Yeah, here's another one. This is where I'm forcing the quadratic term to be the same as the constant term. And you can see this looks, it still has the same structure, but there's definitely something different happening in this picture than in this picture. So this big dot at the center is I. The unit circle is here in both of the pictures. And, but in this one, something crazy is happening at I. And you can answer this with the geometry. Somehow what's happening is that when you project down onto just this one copy of the upper half plane, you have all of these linear features, right? All of these lines in the starscape. And if you happen to line one of those up, so it's a fiber itself. So that whole linear feature maps right down onto the point I, then you'll get this density happening. And so this would seem to say, and you could probably write down something like this, that if you're the point I, you are better approximable by cubics of this form, then you are of cubics of this form, by cubics of this form, right? Okay. So here's a picture. These are the same two pictures where I've just, I've shown some of these, what we just call linear starscapes because I'm just taking a two-dimensional slice of the coefficient lattice, which becomes a one-dimensional feature in the pictures. And so you can just see what some of those look like in detail. And here you can see how they're passing through here and they can actually turn around and pass through again. And, oh, and here I've written down what the fiber is. So actually this collection, this family of cubics right here is a subfamily of this where it always has I as a root, but you can have any rational number on the real line as a root. And so that all ends up mapping right down to there. So these are, so these pictures are really fun to play with. So we, we sort of got this far. And now I'm just gonna show you a few pictures and a few questions. So here are some cortex. And so the natural question is to ask what happens in, what happens to the geometric story in higher dimensions? I mean, the discriminant was great for the quadratics case because of its nice tight relationship with PSL2Z and because it gives a pairing on the Mikowski space, which was a convenient thing to work with, but you don't have as nice a story as the, as the degree increases. And in fact, even in the cubic case, we had that the piece of RP3 that had two conjugate roots and one real root was at least one SL2R, PSL2R orbit was just one orbit. But actually when you go to higher, it'll break into more orbits and the geometry gets much more complicated. Another thing that you asked from these pictures, I think just motivated by the pictures is that you can see these linear star scapes, they seem to repel one another somehow, except where they cross. And so you might actually ask, you might be able to phrase some sorts of questions that ask about how well you were approximated by the geodesics. So instead of by a one-dimensional thing by a two-dimensional thing or something like that. Well, depending on you're in the coefficient space, the root space, or whatever. Yeah, so, and you can see very special things are always happening around the unit circle. Here's a picture, which goes right up to quintix, but again, it's a slice. And you see these interesting sort of denser regions. There's something interesting happening in the geometry there. And again, that should have some repercussions of some kind in terms of approximation, perhaps. Here is a picture, a quartix, but where the coloring now is according to what's happening with the real root. So this is where you have two kinds. So gray is where you have no real roots. So you have two pairs of complex conjugate roots. And then I think how we colored this, which is a little suspect, is we took the first root reported by Sage and said whether it was negative or positive and that was red or blue. So yeah, and so maybe there are better ways, this is an illustration question, there are better ways to capture what's happening with more of the roots once you're in higher degree and of course you have more roots to tell something about and you have more dimensions and it becomes more and more difficult. Another question that I think is interesting is the, in the rational picture where you have the stars, the rational stars skip where you just have the dots in one line. That's the, those are showing the fairy fractions in some way, right? Like because the big dots build the littler ones in between them by medians and so on. And that actually generates the continued fraction algorithm for approximating by rationals. And so this picture, it makes me wonder if you can write down a continued fraction algorithm which would approximate complex numbers by quadratic irrationals. So by quadratic numbers from all different fields by some sort of fairy subdivision type thing. Okay, and so yeah, and I wanted to show you show this picture, we didn't make this picture. This is not from our project. This is Gabriel Dorseman Hopkins and Candy Shoe. And I should say a little something about the paper. We wrote a paper out of this project. And because this project had, you know this some experimental visual flavor and it was really sort of revisiting some classical things with new illustration. We wrote this paper which was meant, it's a bit of an experiment but it was meant to provide something of interest to people from all different levels. So not just number theorists or mathematicians doing research but also to students and just anybody who's interested in the images. So we spent the time to explain a little bit about projective space, for example, and things like that. So as an experiment, of course it succeeds in some ways and fails in others but we really did try to make it so that it could be used, for example for undergraduate research. And so that's what Gabe did. So Candy Shoe did an honors thesis with him in which they took a look at drawing some of these pictures. So this is a picture of Quintex. It's done with a different method. So instead of dots, it's just sort of like clouds. It's done pixel by pixel. So this is all of the Quintex in this sort of glowing seafoam approach. And then what they decided to do was to size the dots by some other features. So on the, this is Quintex. On the left is just Quintex sized by the usual discriminant or height or something like that. And on the right, they've made certain of those ones brighter. They've made the ones that have smaller Galois group brighter. So what's interesting is you immediately do see that those lie on the same rational geodesics you're seeing some of those appear rightly in the picture. And you can actually convince yourself that this has to happen. If you have a quartic that lies on one of those rational geodesics, it can't have the full S4 Galois group. And so, but you know, they just decided to draw pictures for the sake of the experimentation. And then you see this visually happening before you have even thought to ask the question. All right, so I think I will stop there. Thank you very much.