 First of all, I'd like to thank all the organizers, especially Ezequiel and Martín, for the invitation to be here for me. It's a great honor. It's a big pleasure. It's my first time in Uruguay and it's a great privilege to be here. So, my main interest in research is dynamical systems coming from physics and geometry and many of these systems are Hamiltonian in nature. They are perhaps better said, they are examples of Hamiltonian systems and among many topics that interest me, perhaps the one that interests me most is the existence of global surfaces of section. So, let me explain a bit what that is and then, I guess just from the definition to be clear, what are they good for? And actually, these gadgets are, these objects are deeply connected to the birth of many fields in mathematics, especially a simplistic topology, dynamical systems. I'll try to explain as best as I can and please interrupt me if anything is not clear, okay? I'm sorry for speaking in English because my Spanish is not very good. So, I have about 45-50 minutes. So, what's a global surface section? So, let's say we're interested in a flow on a three-manifold. Let's say you have a vector field and this vector field generates a flow. So, then one is interested in studying the dynamical properties of these systems, okay? Right, so if you take a point P, then the system evolves in time and then at a given time T, the point P is evolved to a point phi T of P. So, for each T you have a transformation of the phase space. These phases is the name you give for the space, the three-manifold where this system is defined, all right? And a global surface of section is an embedded compact orientable surface inside the phase space, the three-manifold, you know, with the following properties. So, the boundary of the surface consists of periodic orbits. The flow, the vector field generating the flow, let's say the flow, so the vector field generating the flow is transverse to the interior. Of course, it's not transverse to the boundary because the boundary, the boundary is tangent to the boundary, the periodic orbits. And three, so for every point P, not in the boundary, there are positive times such that if I float P by some positive time T plus I end up in S and if I float P by minus some positive time, or let's say by some negative time, you end up in S, right? So, this means that you have this embedded surface, have some periodic orbits here, and then the flow, so for every point you're going to flow and eventually you're going to come back to the surface itself or if you go negative time, you're going to hit the surface again. So, every single trajectory, which is not one of these periodic orbits, will hit this surface infinitely many often in the future in the past, which means that you can define a first return time for every point. So, if you consider points in the surface out of the boundary, you have a first return time and with this first return time you can also define a first return map. And then this two-dimensional morphism encodes the whole dynamics of the three-dimensional flow. So, you can reduce this third of the dynamics of the three-dimensional flow to the dynamics of the return map. And then this is a big achievement if you find an object like this because two-dimensional dynamics is much more well developed than the three-dimensional dynamics. So, I have much more, so, reduces the study of the flow on M to the study of the return map back to the interior. So, you have given a point P, let's say P pointing the surface not in the boundary, then you have a first return time and you have a first return map. So, this different morphism side captures all the dynamics. Let me draw a bunch of examples, a few examples at least. So, of course you can consider suspension flows. If you have, of course, if you have a diffeomorphism of a surface, let's say an open surface, you can cook up some... If the diffeomorphism is not so well, not so badly behaved near the boundary of the surface, you can consider the mapping torus, you can consider the flow which gives the suspension of the diffeomorphism and if the diffeomorphism is not so bad near the boundary you can sort of close it up and get a global surface of section, some three-dimensional manifold, but that example was artificially constructed. So, let me now discuss some examples which are not so artificially constructed. So, as I said before, my interest is in Hamiltonian dynamics. Just to recall, if you have some Hamiltonian, let's say in standard 2n-dimensional phase space, what do I mean by that? I mean just a function on an open set of R2n, then you can consider the Hamilton's equations. Let's say, even by this set of differential equations and it's a feature, so this is a special kind of flow and it's a feature of these types of flow that the flow preserves the levels of the function H. So, if you start considering functions on R4 or open sets of R4, the level sets will be giving you examples of three manifolds with equipped flows. So, one particular example is, let's say you consider two uncoupled harmonic oscillators of the same period. So, let's say the round three sphere in R4 is an energy level of this Hamiltonian and every single periodic orbit is periodic, and the orbits of this flow are the so-called hop-fibration. So, every orbit, so the flow is looking like that. Let's say you start in some point, let me identify, let me write it, that's complex coordinates. So, the flow is really looking like this formula. So, you see that every single trajectory is periodic, the period is pi, and a global surface of section here is just given by, if you choose any complex subspace, any complex plane which doesn't go through the origin, let's say choose a complex plane here, and if you project it radially onto the sphere, just like that, you're going to end up with a disk whose boundary, of course this picture is bad because it's not a two-dimensional situation, but you're going to end up with a disk whose boundary is one of the hop circles, and every single trajectory is going to hit this, I mean it's going to be a global surface of section just like that, disc-like global surface of section. Of course the dynamic here is very simple, the return time, the return time is just constant function equal to pi, return map is just the identity map, there's nothing much going on, that's an example, maybe the easiest one. So, here's another example, if you consider, let's say if you consider a positively curved sphere of evolution inside R3, to have some symmetry axis here, it's not very symmetric, so we have some convex graph like that, and you rotate it around the an axis, and you consider the geodesic flow on the surface, and you consider the geodesic flow on the unit sphere bundle. So the phase space is now the unit sphere bundle, which is just SO3 or RP3, the flow is still integrable, I mean still an integral Hamiltonian system just like this one, and the integral is the Cleho integral, and using the Cleho integral you find that, so when there's sphere here is getting fatter, you end up with a closed geodesic, which is this equator, and you can consider the Birch of Anulus, over this simple geodesic, which is this equator, which is the set of vectors in the unit sphere bundle, so a set of unit vectors pointing, say, to the northern hemisphere. So at every point here you have this bouquet of unit vectors pointing north, and then so this is embedded anulus inside the unit sphere bundle, whose boundary consists of two closed geodesics, this simple geodesic going that way, and this other one going that way, the flow is transverse, and because of this integrability you know that every trajectory is going to go down, and then have to go up again. So trajectories are going to flip north and southern hemisphere infinitely many often, you get a return map to this anulus, right? That's an example of the Birch of Anulus, that's another example of global surface action. Both these families of examples, both this example or this family of examples, are examples of so-called integrable Hamiltonian systems, and the word integrable really means that integrable, I guess maybe a century ago, and I guess still today is used as a synonym of understood or understandable, which means we understand everything, right? We understand what's going on. So it's not so surprising since we understand everything, it's not so exciting to find something which is supposed to help us understanding everything, so it's backwards engineering, not so exciting. Sorry? Indentity has to be transverse to the flow, because the boundary consists of periodic orbits, right? So it's not transverse. So the first, I guess, maybe one of the, not the oldest, but one of the oldest theorems about the existence of global surface action, for situations where you're really considering systems which are very, very far from being integrable, is a theorem of Birkhoff, which is saying the following. So let me draw a picture, it's going to look a lot like that, but it's not to be thought of as the same picture. Let us consider any sphere equipped to sphere, equipped with a metric with positive curvature, right? So let's say here, possibly curved sphere. Then you consider the geodesic flow. Let's say you find a simple closed geodesic, which means a geodesic which doesn't self-intersect, like an embedded loop like that, almost the same picture. And then you consider exactly the same annulus. Choose a hemisphere, and on top of each point of this global surface, of this geodesic, you put this bouquet of unit vectors pointing to one of the hemispheres. And then you have an annulus, let's say this geodesic's a-gamma. You have an annulus embedded in the unit sphere bundle. So the theorem of Birkhoff says that, let's say the curvature is plotted everywhere, and gamma is a simple closed geodesic. Then a-gamma is a global surface section. So really talking about any possibly curved geodesic flow next to. So this is very, very large class of systems, very, very far from being integrable. So it's a quite useful, I mean it's quite strong a useful statement, right? And in fact you can prove a lot of stuff using this theorem. But the proof is not, the proof is too geometric to shed some light onto the general problem of finding this, okay, strong statement by Birkhoff. So here's another strong statement, actually much more strong than this one. Maybe I should write it in bigger space. Every convex energy level inside, I mean when I say convex I really mean smooth, compact, strictly convex. Inside R4 with standard symplatic structure. If you don't know what a symplatic structure is, it doesn't really matter. I'm just talking about the standard Hamiltonian equations you write for a flow, right? So just some nice strictly convex hyper-surface inside R4. And then you consider any Hamiltonian which realizes this hyper-surface as a regular energy level, anyone. So every such guy carries a disk-like global surface of section. So the situation is a bit like this one, right? You find an embedded disk which is a global surface section. And here the statement is saying that this is not only true for this particular integrable system. This is true for every convex energy level. In particular it covers, I mean, such energy levels they double cover the geodesic flow here, so this theorem somehow encoded there. Yeah, but I will sweep the word dynamically below the carpet because I don't have time to explain. But yeah, there is a generalization of the notion of being convex in the symplatic dynamics world and things get more complicated but I'll skip to the simpler. Which means, I mean, there is no picture, right? Which just means that inside this three-dimensional space which is just a copy of S3, you find this disk bounded by a periodic orbit and then you have a return map. So really find these guys. For instance, as a corollary you find two or infinitely many periodic orbits because the return map, if you take advantage of the fact that this is a Hamiltonian system, the return map will preserve a finite area form on this disk so by Brouwer's translation theorem you find a fixed point and then if you remove this fixed point you end up with a map on an open annulus by a very strong result of John Frank's. If you find another periodic point you'll find infinitely many periodic points. It's a really strong statement. All right. But I want to go back to the origins of this notion. I mean, I want to really try to go back in history and find where exactly this idea was born and it was born exactly from Poincaré's studies of the restricted three-body problem where he was trying to do exactly the jump from studying differential equations from a very hard analytic point of view trying to solve everything explicitly with pages and pages of power series. He was trying to jump from this point of view to a point of view where he studied the dynamics qualitatively. He tried to understand how the orbits behave and so on. He was really trying to find objects that would help him to understand the flow more qualitatively than trying to solve everything explicitly. So let me describe a bit the problem. So Planer, Sechler, three-body problem. So the three-body problem is just what you think it is. It's just three masses moving according to Newton's law of gravitation. So have mass one, mass two, mass three, and then they attract each other according to Newton's law of gravitation. He wanted to study the dynamics of these three bodies. So I have three adjectives that I have to explain. Planer, Sechler, and restricted. Planer means that... that's what I mean. It just means what you think it is. Everything, the movement takes place in the plane. It is a plane where the movement takes place. All right. Restricted means the following. You sort of... Right. Let's say the positions here are z1, z2, and z3. So how do the equations look like? z1.th equals to m2, z2 minus z1 plus m3, z3 minus z1, and so on, right? And finally... Okay. Something's wrong here. z1 minus z2, z3 minus 1, z1, z3, and 2, 3. Okay. So restricted means the following. You see, the equation for the third body doesn't have m3. So you can... If you want, you can forget about... Put m3 equal to zero here and here, this means that the first two bodies are going to move as a solution of the two-body problem, and this follows whatever it follows, right? So the situation is... That's what restricted means, and circular now means the following, that if you have a two-body problem, then the relative position solves Kepler's problem, and then the circular here means that follows a circular solution, okay? So these two guys are relatively moving as a circle, and this is moving according to this differential equation. Is this clear? Are there any questions about the three-body problem? So first thing you can do is you can put... So this should be thought of as some massless satellite, some massless particle, which doesn't influence the movement of these two particles. And then we can talk about the center of mass of these two particles, and by choosing the inertial coordinate system, you can put the center of mass, let's say the given point, say the origin, and Z1 and Z2 are going to be moving around circles, moving in circles around the center of mass with the same angular speed. So you can consider a rotating frame where instead of seeing them moving like that, you make them stay still. This is a known inertial choice of coordinates, so you're breaking... you're not choosing a system of coordinates again, which are inertial anymore. But anyway, you can do it. Let's say... Let's put Z1 here, standing still. Z2 here, standing still. Let's say the mass of Z2 is bigger, so the center of mass is a bit closer here. And then the satellite does something. And then it's moving somewhere. Let's say this is the x-axis. This is the y-axis. All right. And then it turns out... that's a miracle now, that it turns out if you write down, if you really write down the equations for the relative position of the satellite with respect to the heavier body, you're going to end up with a Hamiltonian system. You're going to end up with a Hamiltonian system. So how does it work? I don't have time to explain everything, but it's going to be a Hamiltonian defined on an open set of R4, right? And it turns out that this Hamiltonian has five critical... actually four critical levels, five critical points, and Poincaré was studying the situation below the lowest critical level. So below the lowest critical level, the satellite doesn't have too much energy to get closer to both primaries. Let me call primaries these heavy guys. Satellite doesn't go close to this heavy guy or that heavy guy at the same time. I mean, they have to choose. So there are three so-called heel regions. They look like that. So one heel region is what's inside here, the other heel region is what's inside here, and the third heel region is what's outside here. One unbounded one. So satellite is moving here, or here, or outside. So let's say we are talking about our moon. Let's say there is more mass here than here. Let's say we simplify the study of the movement of our moon as just considering the Sun, the Earth, and the moon is the satellite, and nothing else makes this approximation. So this is Earth, this is Sun, and the moon is here doing something. Almost periodic. Of course, the moon also doesn't move on a plane, but it's almost the same plane anyway. All right. But Poincare was considered a situation here near the heavy primary, and if you make the limit, as all the mass is concentrated here, what's really happening is that this heel region gets smaller and smaller, this point gets closer to the center of mass, this is converted to a certain disc, and then you see that one can... So the Hamiltonian that you get is the so-called rotating Kepler problem, and you can really find solutions which are, he called, direct and retrograde. So there is one periodic solution where the satellite does this, and there is one periodic solution where the satellite does this. And that's already a bit of an insight from Poincare from the point of view of the studies because, you know, how did he find these solutions? He considered the problem, which is not to be considered, but he considered the problem where all mass is concentrated here, you study what's happening there, you find some periodic solutions, and then you try to continue it as you distribute a little bit of the mass here. So he was inventing the continuation method just to find these orbits. Already something special here. Not only he did this, but he found that, you know, you can regularize collisions with the heavy primary. And if you regularize collisions with the heavy primary, what you get is that you leave the movement of the satellite here to an energy level inside R4. It's a nice, compact energy level inside R4, which is defilmorphic to a three-sphere, and inside there, you see that these two orbits, they bound, they form a hop flink. They form a hop flink, and they're the boundary of an annulus, which is a global surface section. And because he found a global surface section which is an annulus, Poincaré was able to think about what's now a day called the Poincaré-Birkel theorem. He wanted to study conditions on this return map on the annulus, which would allow him to find infinitely many periodic orbits. So in one stroke, he invented the continuation method. He used regularization. He introduced the notion of global surface section. He invented the Poincaré-Birkel theorem to find infinitely many periodic orbits of the three-body problem. And as we know, the Poincaré-Birkel theorem is just a special case of what's now a day known as the Arno conjectures. That was the seat also, floor feuding, symplatic field feuding, things like Augustine's going to talk about later. So, I mean, he really devastated a lot of stuff here just with one movement. So the message I wanted to send is that Poincaré found this annulus-like global surface section. Okay, so let me just draw a picture. So it's very hard to draw pictures. The energy level after regularization is going to be a three-sphere again, so think about R3 of some point, very far away. And then you have these two orbits, the retrograde and the direct ones, are forming this hopflink. And then you can try to picture an annulus which has boundary in these guys, an embedded annulus. If you look at Chancy-Nez entering Wikipedia, you'll find a very nice picture of these annulus. This is exactly the picture you get if you consider Biko's theorem, right? I mean, the unisphere bond is Rp3. Of course, it's doubly covered by S3. And then these two geodesics going... You have just one simple geodesic, but you can consider two-pedic orbits, one going each way along the geodesic. If you take this link, if you go to the universal covering, which is S3, you find a hopflink as well. Same thing. And Biko's annulus is isotopic to the annulus of Poincaré. All right. Now, is there anything... I mean, of course, it didn't explain what I'm just talking about. Poincaré and... But if there is anything that I should explain a bit more, if there is anything that I said, which is not so clear, please stop me. So I started 35 or 25? 15 minutes, okay. All right. So let me discuss... I mean, as you know that... I mean, you may or may not know that celestial mechanics was, in the turn of the century, from 19th century to 20th century, one of the, let's say, stronger sources of insight for many things in mathematics. And until today, many problems from that period are still open and they're still very hard to solve. One stupid problem, stupid between quotes, is the general existence problem of a direct orbit, right? So even below, for energy levels, below the lowest critical value of this Hamiltonian. I mean, the Hamiltonian that you see here, I can write it down for you. I mean, I'm not going to write it down because I have to define all these variables. But the Hamiltonian that you see here is certainly unbounded from below and above. So what you see is that you go from below, you don't see any critical values. Eventually you pass a first critical value. It's exactly the moment where the satellite has a bit of energy and can go from, can go near both primaries if given the right initial condition. Anyway. But you can consider this picture below the first critical value and you consider the situation where the satellite moves inside one of these two bounded regions, right? So there's, let's say this one. And if you don't insist in fixing that one of the primaries has more mass than the other, just consider any mass ratio. Then it doesn't matter which one you look. Right? So let's consider any mass ratio. Any mass ratio between the primaries. And let's consider the movement of the satellite for this subcritical energy level inside one of the bounded regions. And so if you consider the case of this sun, earth, moon, our moon, we would be talking about the sun very heavy here. Let's say the earth here. And you see the moon, our moon here. And our moon is so-called direct. What does it mean? It means that it moves around the earth in the same sense as the earth moves around the sun. That's what it means to be direct. But there are also solutions which are retrograde, which means that the moon would do like that. I mean, it would move around the earth in the opposite sense as the earth moves around the sun. Some moons in the Solar System are direct. Some are retrograde. I don't know which ones, but I know there are. I know that our moon is direct. And still an open problem today to find one direct periodic orbit for arbitrary mass ratio. Poincaré found direct orbits near the heavy primary when the mass is almost all concentrated in the heavy primary. But if you want to let the mass ratio be anything, it's still an open problem to find direct orbits. Finding retrograde orbits was already solved by Birkhoff. He invented the so-called Birkhoff shooting method, and he was able to find... So, I mean, if you want to find some orbit which it does like this, you'll find. Just one simple circuit, no self-intersections, you'll find these orbits. Or here, any mass ratio doesn't matter. So the strategy of Birkhoff to try to find... So why is it that there are... It's much easier to find retrograde orbits. And then maybe this is a bit connected to what Augustine is going to talk about, is... There is a morse... This is like a modern explanation. People knew that from a century ago. But there is some kind of morse feeding for the action functional, whose critical points govern the periodic orbits. And the retrograde ones are the ones which are giving certain robust energy levels which capture the topology of the contact structure, whatever that means. And the retrograde ones, are very, very long and very, very unstable, very, very high index properties. The direct. The direct, sorry, the direct. So if you think, for example, as... Let's say you think of two uncoupled harmonic oscillators, but now we don't think of them as being harmonic oscillators with the same period. Right? So there will be one periodic... So you have two periodic orbits. Let's say that the periods are incommensurable, so they have only two periodic orbits. One periodic orbit is you freeze one oscillator and the other is oscillating, and the other periodic orbits do the opposite, right? You freeze one oscillator and the other is... Or this one increases also. Okay. One of these periodic orbits is going to be... It will have low action and will have nice index properties. But the other one will be very long and will have very bad index properties. And that refers to the difference between direct and retrograde here. So Bikov taught it to be a good idea to prove the following. Bikov taught that if you regularize... So you always find... He proved that you can always find the retrograde orbit. You can always find one. Let's say things are moving like that, so the retrograde looks like this. So if you lift this orbit... If you do this regularization procedure, you end up with some flow on S3, and this orbit is going to be one of the components of this link. I mean, you don't know the link yet, but let's say you find one of the components by regularizing and looking at what happens to the retrograde orbit. So Bikov conjectured that it might be possible that the lift of the retrograde orbit will always bound a disk-like global surface section. He conjectured that. So if you... And then, if this is the case, you will find a fixed point of the return map by applying Brouwer's translation theorem, and the fixed point could be a good candidate for the direct orbit. So this could be a strategy to find the direct orbit. Two very hard steps. First, find a global surface section. Deciding whether these retrograde orbits are going to bound a global surface section or not. And then deciding if you find some fixed point which is going to be a direct orbit. So... But the conjecture was exactly this. Regularized collisions. Look at the lift of the retrograde orbit and try to prove that it bounds a global surface section. That's Bikov's conjecture. That's Bikov's conjecture too today. Anyway, I'm not sure if I'm going to go into any of my results in this talk. I have some results on this problem, but it's very, very mild partial results so I don't bother telling you about them. Conjecture is wide open. I don't think there is anything that I know how to do that sheds any light onto the general case. But I do want to tell you a bit about situations where you... I mean, it's very hard to decide whether you have global surface section or not. There is a very general theory developed by... In the second half of the 20th century started with the work of Saul Schwartzman of asymptotic cycles. David Fried and then Denis Sullivan was all summarized recently by the AT&G's. He called these Schwartzman-Fried-Sullivan-Fury. So there is a very general theory that tries to attack this problem of existence of global surface section any flow in three dimensions. That's great. One drawback is that this is too hard of a problem. I mean, you end up with theorems which have very, very strong consequences but also require very, very strong hypotheses which are hard to check. I don't want to go too much into detail. I just want to say that in some situations you do find... orbits say... let's say one of these components, let's say you find two... you find the whole hop flink but it's hard to decide whether they bound the global surface section or not. And then, but still, it might be the case that the Poincaré-Bicot theorem... it's a theorem... it might be that the mechanism behind it is not, let's say, purely coming from dynamical systems. It might be the case that there is some sort of Morse theory behind this. And it is the case. If you had a corpus you would find by looking at global surface section even in cases where you don't have a global surface section. So, I mean, you can try to picture Poincaré's annulus here, which is a bit harder but you can also try to picture the falling annulus. Just remove this fixed point, right? And you have this punctured disc which is an annulus. It's an easier annulus to picture. Okay, so the last five minutes I just want to describe a situation where you can find... you can overcome the problem of deciding whether global surface sections exist or not and to obtain conclusions. So, let's say you have this... you have some energy level which is a copy of S3. And let's say for some reason, like in many situations in celestial mechanics, you find a hop-flink of periodic orbits. Let's say you find one. Then, if you have a periodic orbit or a flow, you can always do the following. You can consider the linearized dynamics transverse along the periodic orbit transverse to the flow, right? Since you have only two dimensions in the transverse of the orbit, basically you can cook up defilmorphism of the circle, right? Because you take a ray, look at a linearized flow and you see how much this ray rotated and you come back to the starting points at the infinitesimal level. This is a circle defilmorphism which has a rotation number. That's called this number, the transverse rotation number. Of course, to have it as a real number you have to sort of trivialize the whole linearized dynamics along the orbit which means you have to choose coordinates somehow because the rotation number is only defined up to an integer anyway. But if you look, let's say this guy here has transverse rotation number theta zero and this guy here has transverse rotation number theta one. If you look at this annulus map which in this case, let's say let's say it extends to the boundary let's say you remove this point, you blow it up, right? You put an infinitesimal circle there. What you're going to be seeing here is the rotation number in this boundary component with annulus is going to be theta one, let's say that way. And the rotation number here in this boundary component is going to be one over theta zero just because this is a transverse rotation number. So you want to return to the disc that is a one over factor. One over is inverse. Okay. So by the Poincare Bico theorem if let's say theta one different than one over theta zero you have infinitely many periodic points or, and periodic points means really infinitely many periodic orbits of the flow. Okay. Let's say there is one. Let's pretend let's pretend there is one. Then you would find annulus and it's not hard to check that this would be the rotation numbers on this boundary component. So we pretended that there was a disc here global surface section. What happens if there is no global surface section? So you might want to try to prove a theorem where this condition which makes sense even if there is no disc like global surface section would imply the existence of infinitely many periodic points. That's the would be good to prove this right? So this is exactly one theorem that okay, maybe I state the result which means it, so let me state it like that if you have in R4, if you have a star shaped energy level let's say star shaped with respect to the origin so it's a defilmorphic to S3, some nice energy level let's say inside this energy level you find a hot link of periodic orbits right? Let's say so gamma one gamma zero gamma one inside energy level forming so periodic orbits forming a hot link so but you don't have a disc you don't have a disc like global surface section but you can look at their transverse rotation numbers so the situation basically says that there is a slightly fancier way of doing this which is the following so consider vector looking like let's say let's say in the plane here let's say the vector this vector here is theta zero one and this vector here is one theta one right? second coordinate equal to one first coordinate equal to one so if these guys are not collinear which in that case means exactly that so you find an open wedge here which will contain infinitely many points of the integer lattice right? each point in the integer lattice let's say P and Q the theorem says it will correspond to a periodic orbit which links P times with gamma zero and Q times with theta one which would be exactly that the orbit would you would find from applying Poincare Bico theorem to this annulus map that's the statement of the theorem which means that you don't have in this situation you don't have to get the global surface section get something just from the flow and then one corollary is half of a corollary is this statement provides half of a new proof of the existence of infinitely many periodic closed geodesics on a Riemannian two-sphere I know my time is so I have this theorem celebrated by a theorem by John Franks and Victor Bangert every Riemannian closed geodesic flow on S2 carries infinitely many periodic closed geodesics so there is a way to prove this using this theorem plus a lot of stuff from Nancy Hingstone so Hingstone plus our theorem implies the theorem so that's the proof let me just finish by describing this proof in one minute so the space of embedded loops in S2 carries a three-dimensional homology class let's say embedded loops in S2 model are short ones this guy carries a three-dimensional homology class which is just like that for every pair of antipodal points in S2 you have this family of long short loops right so for every point in S2 you have this one-dimensional but you have another RP2 parameter so you have a three-dimensional homology class so there is a curve shortening flow from Grayson's work in the 80s saying that you can flow down you can shorten there is a flow that will shorten these loops right and it will preserve the property of being embedded so you have this three-dimensional class model of short ones this will because it's a non-trivial homology class this will cook up a special closed-jordesic which is going to be simple that's one important step the key point is that so this is a three-dimensional class this means that the Morse index plus the nullity of this closed-jordesic is going to be bigger equal to 3 okay so as I said before a simple closed-jordesic on S2 gives you two periodic orbits on the unit sphere boundary the jodesic going in both directions so let's leave everything to S3 this is going to leave to a Hopf link right and there are two possibilities either these vectors coincide or not so in the case they coincide they form a very special critical point of energy functional and there is a very very deep theorem by Nancy Hingston saying that in this case there will be infinitely many periodic orbits that case we cannot handle but if they don't coincide then this theorem handles this case and then that would be the steps of this a new proof of this theorem sorry for the extra time let's stop here