 Okay, well thank you very much. It's really a great pleasure to be here today and to be able to see again so many old and good friends. As you can imagine this is not an easy lecture to give, so I think I decided to sort of approach it the only way I can actually approach it, which is to sort of tell you a little bit of my own sort of interactions with Ricardo. And while I do that I try to sort of convey a little bit about him as a person and as a mathematician. That's really the only thing that I can actually do. So let me begin first by telling you a little bit about him. Maybe most of you know all of this, but we might as well run through some of this very, very briefly. So he was born in 48 here in Montevideo. He began studies in engineering at this very same institution. This was the case of many Uruguayan mathematicians. I guess at the time there was no formal studies just in mathematics and the natural path if you wanted to do math is you will come to the School of Engineering with an eye of doing sort of engineering and then maybe switch it off. And I guess many of us did that, I did that myself and then I switched into math. I guess that was the case of Ricardo too and many here in this audience I think. So he completed his PhD under the guidance of Jacob Palace in 1973. He made spectacular contributions to dynamical systems. I think what Enrique said in his lecture is very much true. A lot of his ideas are very much alive and sort of going all over the place. So I think Enrique has told you about only a sort of fraction of the volume of his work. I'm going to tell you in due time about an even smaller fraction somehow the one that I had more contact with. He was invited twice to the ICM which is a sort of remarkable honor. Okay, and he passed away in Montevideo as well in 1995. Just a little bit before that meeting that has been mentioned on serial locations during this conference. That was also a memorable meeting for me. Not just because it took place at the bank, as a lot of people remember somehow, but because it was in some sense a very special meeting. It was a very special moment. So somehow we really, we sort of put a huge effort to have to make this happen. Manier had 11 PhD students. Some of those names in that list have people that have been particularly important to me. Obviously, there's the case of my older brother Miguel, who unfortunately is not here with us today because there is another conference honoring Ricardo at this very same time in Rio. But there you can find names like Kevin Enrich, who I hope he was around a little bit earlier. He was there, yes. Álvaro Robella, these are people that are actually taught many of us courses in mathematics. They are exceptional lecturers. And also, I mean of course there you can find Gonzalo. That has been a sort of huge influence for me. Right, so I'll do maybe a sort of quick trip down memory lane. So I'll tell you a little bit the first time that I actually met Ricardo. So the first thing to say is that I do not claim to have known him well at all. So I really got to know him towards the end of his life. But I got some glimpses into his personality and him as a mathematician too. So the first time that we met him was in the summer of 1986. So there was a big group of us. I mean, we were about at least ten, maybe, who's abbreviated remembers this better than me, at least ten that we were all essentially shipped from here to IMPA somehow. So this was towards the end of a very difficult period for Uruguay. The return of many mathematicians that were abroad happened maybe one or two years before that. And they came and they found a group of young people, I guess, who were very keen to learn mathematics. And we were encouraged and essentially told you should all go to IMPA and take some courses there. That would be a very good thing for you. And indeed it was. This was a sort of amazing thing. I think it was ten year ago I was going to IMPA just at the same time during the summer of 1986. So among them were Beatrice, Alba Ruella, and my brother Miguel. So how did we actually make Ricardo? So Ricardo was a difficult catch. I mean, he wouldn't sort of show up. He wouldn't be at IMPA all the time. He was sort of well known that if he wanted to see him, he had to be after two basically, two in the afternoon. So we managed to catch him in the cafeteria somehow. And then, of course, he was sort of curious. I mean, he wanted to know who was coming from Uruguay. So sort of take these courses. So I went there with my brother, and hello Ricardo, come from Uruguay, blah, blah, blah. And then, you know, he seemed very friendly. And at the time we were taking a course on manifolds, a basic course on manifolds. This was being given by El Cidio Cilín's neto. And we didn't have really much to talk with him, right? I mean, this was around the time he has proved the stability conjecture. We were sort of little people. He was a big celebrity in some sense. So the only thing we could think of is why don't we ask him, you know, this very difficult problem that we cannot solve from the course, and we did. And he actually dispatched the problem more or less immediately after a coffee, a coke, and a cigarette. And then we sort of went away. So that was my sort of first encounter with Ricardo. So that was actually a memorable summer at INPA. A few months later, he came to Montevideo and gave an unforgettable lecture here at the email. So once again, I guess, I mean, my... Probably he wasn't coming that often before 85, essentially. It was essentially when Jorge Levovitz returned to the country that we started to see more of Ricardo, essentially. So he came and he gave this incredible lecture on geodesic flows. I really don't know why he picked that topic. He could have talked about the stability conjecture. He had lots of possible topics to talk about, but he decided to talk about geodesic flows. This was a couple of years after his work with Freire on Manifolds Without Conjugate Points. A kind of well-known term telling you that if you have a geodesic flow on a manifold without conjugate points, then the topological entropy coincides with the volume entropy that was defined by François earlier in the lecture in the meeting today. And he had just around that time, a few years earlier, given a new proof of a very famous term by Klingenberg that said that if you have an anosov geodesic flow, then you didn't have conjugate points. So conjugate points, as briefly as if you go to the universal cover, two points are joined by a unique geodesic. So if you have a manifold with negative curvature, this is obvious. But if you just assume an anosov, this is actually nontrivial. It's a nontrivial fact. So the proof by Klingenberg was a proof using the most theory of the loop space Ricardo gave a completely new proof and he had this sort of way of looking at the geodesic flows that we were fascinated with. At least I was. My first contact with the geodesic flow was essentially through Jorge Levovitz. Jorge had his own way of looking at things, but somehow seeing Ricardo lecture was unforgettable. I mean, it was not just the quality of what he was talking about, it was the quality of the delivery. I mean, you saw just a glimpse a little bit in the video how good he was at sort of explaining things. So his lectures were really sort of impeccable. So I guess those who have visited the institute here of Mathematics know very well of the informal gatherings on the corridor. So the seminar will always start late. I mean, you could not expect that it will start on time. It will have sort of, you know, it will start at time x, it will never start at time x. It will be sort of at least 30 minutes later. So here's a, so I went yesterday and took a picture of the corridor because I think I have this sort of wonderful memories of it. It looks a little different, I mean, in some sense it looks a little different today than what it did maybe 20 plus years ago. It looks a lot cleaner to begin with. But the seminar would take place in that room and then people will come out and sort of gather there, you know, classes of people and there will be conversations about more or less anything. The conversations will sort of move from sort of politics to football to mathematics. When Ricardo gave this seminar, he sort of came, sort of sat there on the corridor and we started chatting. So this was actually my first memory of a sort of kind of proper conversation with him in which I got a glimpse of his refined and lethal sense of humor. So his sense of humor was something that you had to be very careful about. And it's very difficult for me not to talk about Ricardo without mentioning Jorge. So I think when I sort of exchanged emails with Enrique a few days ago, he sent me some slides that contained that picture. I haven't seen that picture before. I don't know if any of you had seen it. I see Martin nodding. I've never seen a picture before of Ricardo and Jorge together. And I think this is a great one. Okay, so as I said, I never really got to know Ricardo that well. So I left in 1987 for my PhD and my further interactions with him were essentially in the sort of period 91, 94 after I finished my PhD. And I was sort of more or less like many people after finished the PhD without really knowing what to do and looking for a direction, looking for a bit of a future. And somehow things happened in such a way that I was sort of able to sort of interact with him. But this was also a particular difficult moment for him in which he was already suffering very much with his disease. During this period, I mean we sort of visited him with Miguel in his flat here in Montevideo. We had some sort of wonderful discussions. And I think one thing that I really wanted to say was that there was some sort of incredible moments, even during his weakest moments, you could see actually him switching the topic of the conversation to something that would be of interest to us. I mean that was sort of quite amazing. Okay, so I'd like to tell you a little bit then now about my own personal mathematical journey next to the ideas of Ricardo a little bit. Okay, so I would be touching upon his work on Lagrangian systems. So I will tell you a little bit about that. So in this slide I will just simply mention the three main papers that he published on that. So the first two were actually published in non-linearity. One actually, I mean the preprint was ready for many years but it was published one year after he passed away. And then the third one which is the one that I will really talk about the most was a paper that he actually never completed. So he wrote the paper and he wrote a series of theorems and wonderful conjectures too, I think. And the proofs were actually given quite a few years later by Gonzalo, by Delgado, another of his former PhD students, and by Renato Triaga, which I guess he must be also attending the other conference in Rio at the moment. So the paper was actually published twice. So it was published as a conference proceedings, I think edited by Francois here in the audience as well. Going back to that conference in 1995 and then it was reprinted a couple of years later in the bulletin of the Brazilian mass society with the follow-up of the proofs by Gonzalo, Renato and Jorge Delgado. So it's really on that sort of third paper that I would like to tell you a little bit about. Some of the ideas. So the first thing to say just before I go to the next slide is that again there is a sort of very, I mean in some sense a very kind of clear connection with the School of Engineering too. So I mean, so back when I was taking my courses at the School of Engineering, there were these incredible courses on mechanics. And there was Mechanics 1, Mechanics 2, and these were actually fantastic, particularly the course on Lagrangian mechanics. I did have conversations about these courses with Ricardo too, who found it actually very difficult. I mean there was a pretty tough somehow. And you could say that in some sense the work that he's done, you know, you could sort of trace a very continuous line from those courses immediately into his work on Lagrangian systems. We'll try to make a case for that in a minute. So I will mostly focus on this third paper and this is really an attempt to understand the dynamics on the various energy levels using variational methods. So I guess something that we are all very familiar with at some point, we must, you know, and I will go through that very quickly. The solutions of the Lagrange equations come essentially from a variation principle. And I think to sort of get into the theory, I will tell you right away the class of Lagrangians that Ricardo was discussing. So we'll not do it in the most general form. So for those of you that went to Gonzalo's lecture yesterday, there is a more general framework for this. You can take a convex, superlinear Lagrangian, blah, blah, blah. But I think, you know, the theory is already, and in some sense it's already a theory about that class of Lagrangians which in some sense are the most classical Lagrangians from classical mechanics. So you put kinetic energy and the kinetic energy comes in the form of a Riemannian metric. You have a potential, u of x, but then you also have a magnetic potential that comes on the way of a one-form theta. So in the velocities, so if you look about the, if you fix a point, so x is a point on the manifold and v is just a tangent vector. If you fix a point x, this is just a polynomial of degree 2 in the velocities where for high v's, the metric dominates, but for small v's, the one-form starts sort of getting into the way and produces some effects in the magnetic field, and this will come up in a minute. So that system, as you know, I guess I learned here, the School of Engineering has a first integral of motion which is the energy, which is just the kinetic energy plus the potential. That's the first integral of the system. So I wrote it in the most classical way because I could not resist switching to q, q dot, because I guess a few days before coming to the lecture, Martin told me this is a lecture potentially for a general audience. So I thought if I change the xv to q, q dot, I would be able to get away with it. And then I changed the theta to a because that's a physically relevant way of writing it for the magnetic potential. So that's the Lagrangian. So the action of a curve that actually Gonzalo told us about yesterday is just the integral of the Lagrangian. That's the action of any given curve. And if you look at the critical points of that, you know, you do gamma s and you differentiate, you respect two s and you just differentiate the Euler equations. The Euler Lagrange equations sort of pop up. Wonderful second order ODEs. Somehow they produce a dynamical system on the tangent bundle of the manifold. So the phase space now is positions and velocities. It's a space of all q, q dots. And the function E, which is defined on TM, is now a first integral. This is actually if you're working on a bounded space on a compact space, then this flow is defined for all times t and becomes a complete flow. And this is really the dynamical object of study. Now there's a sort of bit of geometry coming up that will be relevant in a couple of minutes. That is, again, very, very classical. So perhaps for this one, I might as well draw a little picture. So the geometry of those energy levels depends on the value, on the amount of energy you have, unsurprisingly. So if your energy is sort of very high, then the energy levels will project into the entire thing. But if you don't have enough energy, the energy levels might not project over anything, over everything, and there's this sort of threshold value, E0, which is the same as the maximum of the potential. And if you have energy more than E0, then you will project over everything. And if you don't, you will not. So I don't know if your money for LEM is just some sort of, if you like, some surface of hydrogenous. So remember sort of TM, you're looking at positions and velocities. So if your energy is less than E0, it could very well be that the projection of this is some sort of subset of M that doesn't cover everything. So the particles will sort of go, but then will reach that sort of boundary with velocities here, essentially, and then go back in. I will show you one very dramatic example of this, where the particles, you don't have enough energy and then the particles do not hit the entire configuration space. Okay, so let's keep that particular special value in mind because there will be now several values, special values coming up, associated with Riccardo's paper. Okay, so here's the dramatic example. This is my favorite pendulum. I don't know if you ever have came across this one. So this is called the Botafumero. It's a pendulum. Okay, so you have to be considerate here and sort of imagine that that rope is not bending and it's a sort of rigid rope, but let's assume that for the moment. Have you ever come across this pendulum? Okay, so this is, the story is very interesting. So this is in Galicia. The sort of idea is that you want to burn some incense and you want to distribute the incense all along the cathedral. So you have many approaches for this. You can burn many small little things of incense. Or you would have this amazing idea that the Galicians had, which is to sort of create this huge container with incense, hang it as a pendulum from the middle of this incredible cathedral and then make it swing in this sort of very dramatic fashion. So then the whole church and the audience will be engulfed in this incense cloud. Okay, so this is amazing, right? So the system of pulleys is from 1604 and they managed to push this thing to velocities reaching, you know, almost 68 kilometers per hour inside this very valuable cathedral. So that's a pendulum. Now fortunately that pendulum never reaches the entire phase space. This is a spherical pendulum. The entire phase space will have been the entire two sphere and the Galician engineers have designed that so that that will not happen. So people will be safe. So that's an example where you don't have enough energy fortunately to sort of hit the entire phase space. You know, one idea that they haven't implemented yet if you really want to make it like the sort of Lagrangian system that I had before. So this would be purely kinetic energy plus potential. But you can imagine the sort of engineering putting some charge on that and then putting huge magnets on the walls of the cathedral and then it would be a perfect system fitting the theory as I wrote it before. Okay. So that fortunately has energy less than it does. So the cathedrals and the visitors are supposed to be fairly safe in principle. Okay. So how did Ricardo go into this Lagrangian system? So I think part of the story begins actually with a paper by John Mather from 1991 who wrote a foundational paper on the subject. So there's a paper and there's a title. And I think that paper had a very strong influence on Ricardo actually. So he actually came across that paper way before the paper was published and kind of decided to work on this somehow. I mean there may be other reasons that I don't know about but I know that this paper was a huge influence on actually Ricardo doing some work on these Lagrangian systems. So John Mather was trying to sort of essentially generalize his theory of twist maps on cylinders to higher dimensions and his model for high dimensional twist maps were exactly these Lagrangian systems except and it has to be said that in Mather's case he was looking at time dependent periodic Lagrangians. So they were not autonomous systems as the ones that had been describing but they had a dependence on time. So the main idea in that paper by John Mather is essentially to replace curves by measures and then rather than trying to minimize over curves you try to really minimize over measures basically. And then you would hope that the minimizing measures will be special and they will be giving you information about the dynamics of your system. This is essentially the main idea. So the action of a measure was also defined by Gonzalo but I remind you briefly you have a well probability measure on the tangent bundle you just integrate with respect to the measure mu. Now in my view Ricardo managed to bring this theory of John Mather to a completely new level with a completely new set of insights particularly in the autonomous case. I think he sort of picked up quite a few things that hadn't been picked up by John Mather mainly because I think Mather was actually interested in different things. I mean my understanding of this is correct. Mather always had an eye towards provisional diffusion and always saw this theory as a mechanism for construction. They saw it sort of going off to infinity basically in the time-dependent case. So this was an exciting time since more or less independently Albert Fatih was actually developing his weekend theorem. So there was a lot of somehow going on at the time actually Ricardo was working on this paper piece. So the paper of Ricardo has this great idea about these threshold values. These critical values which nowadays they are being named in honor of Ricardo they are called the Manier critical values and I will tell you about a few of them again they were defined in the lecture by Gonzalo earlier but I will remind you of the definition. It's a very simple idea somehow based sort of on the following fact. The Lagrangians that I wrote down have this very special form. There's a kinetic energy plus something linear plus a potential. So for each x-fix you can imagine these things they're just sort of parabolas basically. You can think of them. So they will have a minimum at every point. So if you pick your K large enough the L plus K will be positive. So if you take your K very large the L plus K will be positive and then the action of any closed curve will be positive. So then the idea of this critical value is really very simple. You say well let's try to decrease this value of K until somehow I kind of manage to find somewhere a closed curve that has negative action. Or you can go the other way. The way this is defined I sort of say it's defined by the infimum but you can also do it the other way around coming from the bottom. So you can just introduce in the paper this critical value. And it turns out that this critical value had all sorts of connections with methods theory. So one of the first theorems that is actually stated in that paper of Ricardo is a characterization of this very simple threshold value at C of L in terms of this sort of sort of invariant measures or the action of these measures. So he proved that another way by which you can recapture this critical value is as minus the minimum of the action of the Lagrangian running over all the invariant measures of the earlier Lagrangian law. And you can actually check it's easy to check from the definition that this critical value C of L is greater than or equal than this value E naught that I defined before. That has the property that the energy there was projected over everything. It's again very easy to check. Now how does this connect with actually methods theory? Well, I can explain that sort of quickly. The theory has cages, naturally. What are these cages? Well, if you pick a close one form you can consider a new Lagrangian which is L plus omega or L minus omega. This would be exactly the same kind, right? If you just simply change in the one form by adding another close form. But of course if you think a little bit about if you have a close form you don't change the actual magnetic field. You're just changing the potential but not the magnetic field. So if you don't change the magnetic field the particle, the Lagrange flow is just the same, yeah? You have a slightly different Lagrangian but the orbits would be exactly the same. Even better, when you change the Lagrangian by a close one form it really only depends if you're looking in particular at these sort of actions of invariant measures and so on all the critical value it really only depends on the class of the form. Because if you were to change if you were to change the close one form by another let's say df when you integrate that df along a close two of it you get zero, yeah? So what is really relevant of a close one form is just the class of the form not the actual close one form. So this immediately gives somehow a function on co-homology on the first-term co-homology group. So if you define a function alpha of omega as this sort of critical value of L minus omega this gives you now a perfectly defined function from h1 into R. And it turns out that the previous term actually tells you that this is exactly Mathes' alpha function that plays a key function that appears in the paper by Mathes. Mathes' paper is about all this sort of alpha and beta functions of this one in co-homology and other in homology but somehow this definition of the critical value was capturing kind of immediately this Mathes' alpha function. So that was Ricardo's first theorem. There were many theorems in that paper but somehow also if you have that function on the co-homology classes and the theory has all these cages one thing that you can actually show which is, you know, not so hard to believe is that some of the sort of convexity of the Lagrangian is inherited by this alpha function and this function alpha becomes also convex and super linear growing faster than linear so therefore it has a unique minimum. So what Ricardo did in his paper is he said, oh well, this is a special thing somehow. I mean this critical value somehow doesn't depend on the gauges and you're sort of picking up one really now distinguished value associated to the system somehow. So he defined the strict critical value and C0L precisely says the minimum of this critical values of L minus omega. And this is very special. It has turned out to have connections with symplectic topology so there's a sort of wonderful kind of universe associated with this critical value which Manier was immediately sensing. I mean he was picked up various other things but I guess probably at that time he didn't picked up actually the depth of this critical value but what he did pick up and this he sort of mentioned immediately is that all these measures that were popping up in Mathis paper all these measures have to have high energy somehow. I mean so I remember actually Ricardo saying you know yeah and this you know what comes out of this is that all these measures by Mathis have to have energy, high energy energy at least is zero or higher something that was actually picked up by Diaz Carnero early on somehow that this Mathis measures had kind of special property related to the energy in the autonomous case. So then Ricardo had a question sort of tailored particularly for my brother and myself. So in one of the sort of visits that we made to his house he told us well he has a question for you too. So he said so I think this is very likely to be true. So if you have one of these energy levels in a Lagrangian system that is hyperbolic, there is an Osso in the sense that Enrique was telling us a little bit earlier it should have high energy. And I'll tell you in a minute why he was actually making that that sort of conjecture. I mean so he asked that question so we went away sort of thinking I mean this is great right? I mean this is a new problem straight from the master particularly tailored for us we got pretty excited somehow we got something that kind of nice that we could work on. So just before I tell you what it's so in some sense somehow the rest of the talk is a resolution to that question of Ricardo I think. So before I get into that I have to tell you a little bit about that there are actually many critical values. So this is a critical value C of L then you can change the L by L minus omega because of this cage then there is the minimum of that but then there are other things that you can do which are completely natural. So with Miguel we started looking at coverings and we said well why not? I mean this is a completely natural thing to do if you take just a covering of your manifold you can lift a Lagrangian and then if you look at the definition of a critical value you can define that the critical value on that covering. Completely natural thing to do that value would be defined and then the natural question was how these values were changing with the coverings so that these critical values were insensitive on the finite covers but they were going down as you were going up in covers. So you can naturally associate now two more critical values in terms of coverings because you know if you have a manifold you have all kinds of coverings of course but there are two distinguished ones there is a universal cover that we all know I guess is a kind of pretty well known covering which is the sort of abelian cover essentially given topologically by the kernel of the Hurewitz map from pi1 into h1. So naturally since the universe covers everything that's the smallest of the critical value but there was this other one from the abelian cover. And all these critical values are somehow always greater or equal than the critical values that the botafumero never reaches. It's important to keep in mind. So the resolution to Riccardo's question came in several stages unfortunately this was a couple of years after Riccardo passed away so with Miguel we first proved that this strict critical value that Riccardo had defined was quite naturally the same as the critical value of the abelian cover. But we managed to produce examples that somehow were giving a negative answer to Riccardo's question to produce an example of an anose of energy level with energy in between the critical value of the universal cover and the critical value of the abelian cover. And then one or two years after we somehow actually caught the right theorem. So if the energy level is an anose, then actually there is something going on exactly as Riccardo suggested. He just didn't quite give the right critical value. But I mean this is a really kind of minor thing in a way. So this is the universal there's a critical value of the universal covering, the one that has to that works for hyperbolicity. And then with Albert Fatih so this he kind of immediately realized that there was a kind of you know group theoretic reason for this to happen. This an anose of energy levels happens in services of hygienists in which the of course the fundamental group is not amenable. So somehow this sort of gap between the universal cover and the abelian cover only happens when the pi one is not amenable. A lot of this theory was actually being done on tori and of course on tori these two critical values are always the same. So that was the sort of the answer to the question. So now I'd like to tell you a little bit about why somehow Ricardo asked a question in the first place and this goes back to the theorem of Klingember and his sort of new proof of Klingember's theorem. So the key ingredients somehow to to produce this sort of phenomenon that the energy of an anose of energy of an anose of level has to be sufficiently high comes is at the heart of all this is this Klingember's theorem somehow. And it's something that I would tell you in a minute that which again I picked up from Ricardo and I have never forgotten somehow. This is like the essential thing that one has to look at. So the great theorem of Ricardo is the following. So it says that if a Jurassic flow admits a continuous environmental arrangement of bundle then the sub bundle is transversal to the vertical sub bundle. So let me just explain a little bit about this. So if you have an anose of Jurassic flow you have the stable and unstable bundles so the Jurassic flow is very easy to check that the unstable and unstable bundles are actually Lagrangian. So in other words this sort of symplectic form in phase space has to vanish on this sub space. And it's very easy to see because this two form is invariant so you just pick a couple of vectors in this stable subspace you let T go to infinity and that goes to zero. So the form has to be zero on any pair of vectors. So they have to be Lagrangian. So and in the anose of case of course they are just continuous so what Ricardo did was he sort of realized that somehow the only thing that actually mattered there in the closed case was not the hyperbolicity but just having this continuous invariant Lagrangian sub bundles and then the key property in clean and best theorem was the transversality of these sub bundles with the vertical sub bundle. The vertical sub bundle is just the kernel of the projection map. So it's essentially you stand on a point and it's just a fiber over the point. So the key geometrical property that a an anose of geodesic flow has is that this is stable and then stable bundles have to be transversed to the vertical. And I remember Ricardo clearly saying you need to forget about no conjugate points. This is really the property everything comes out of this. So once you prove this transversality of the stable bundle with vertical then the rest comes from student comparison ODE standard thing. This is the key property. And he gave a wonderful proof of this using the Maslow index. So he gave a kind of a very cool proof using the Maslow index and the Maslow cycle. Now his proof used the fact that the geodesic flow was reversible. So time reversible means that if you know if you go along the orbit and then if you reverse time you can come back also along an orbit. So this will happen for any classical system of the form kinetic energy plus potential but if you put a magnetic field the reversibility gets destroyed. Because the magnetic field twists kind of always in one direction. So you go if you go one way and then if you change time you come in a different orbit. So it's not reversible. So Ricardo wanted to know I mean and this was sort of with the clock going back a little bit in time at sort of 1991 wanted to know well I mean there had to be a cling ember theorem for this more general class of Hamiltonians where my proof using the reversibilities replaced by something else. And this was another question that he posed earlier to Miguel and myself and when I when I came back from my PhD that I came straight back here to Uruguay my first project with Miguel was actually to look into this come up with a new proof of this that didn't use the reversibility. And I think this is perhaps maybe this is a kind key point of the lecture I guess today this was a wonderful moment I will tell you in a few minutes why we came up with this proof this was a proof that we gave it was published in 1984 but somehow we started working on this in 1991 a paper that I'm very happy with and somehow I do have wonderful memories of discussing this paper with Ricardo and Miguel and Ricardo helped us a lot with the writing somehow he kind of liked the sort of idea he liked the project and then you know there we felt his sense of humor as I will try to show you in a meaning his lethal sense of humor more than once he volunteered at some point I had to write the introduction for you and then I'll show you how to do it he did it and it was actually way much better than the ones we wrote and so there's a sort of I did check this with Miguel so at some point in the paper emails going back and forth this was the early days of email and at some point we needed to deal with the muscle of cycle which is not quite a manifold but it behaves as if it were a manifold it has some singularities it's stratified the bad part has clearly mentioned 3 so it almost works as if it were a manifold but you know we needed to get the technology right so at some point you know we needed to give the right reference and we sort of inserted a phrase in the paper you know saying that this is a you know this was a stratified cycle in the sense of with me or something like that and then Ricardo came back with the following comment I thought it was great I did check I did check with Miguel I mean this is yes yes no don't know he said that yes he did so so I went back I read the paper again it didn't look too bad I went to listen to the symphony again the fourth movement it's not pedantic it's majestic actually which I'm not saying that the introduction or the phrase is majestic in any form but again I think Ricardo in some sense at his best I mean his sort of sense of humor was sort of spectacular he was very fond of music too somehow he was really incredibly knowledgeable you could sort of switch the conversation very quickly from mathematics to music he would do that more properly with my older brother who was much more knowledgeable than me in classical music but that was sort of wonderful wonderful to see and there were many sort of exchanges of these sort of nature somehow kind of private ones but they were really very good so the actually just before I go into this this is something else that didn't impress me I mean his command of English was remarkable I mean we actually saw it in the video how well he could explain so he he read huge amounts so he was some sort of anglophile in some sense and his knowledge of the English and sort of American literature was extensive and deep I mean actually quite remarkable so he was he was different really many levels the the transversality of the stable and unstable bundle with B already forces the energy to be bigger than A0 and this was actually in some sense our first observation that we told Ricardo oh if you since we have this new proof in the transversality of the stable and stable bundle this is actually telling you that the energy has to be high enough you cannot have an an awesome energy level that will produce a sort of hills region like that and Ricardo said oh really yes you can't and he he really liked that sort of observation somehow he said oh I thought if you sort of you know put a system there with enough balls bouncing against each other like those sort of systems in mechanics that we saw the school of engineering it had to be an awesome no it can't it can't because the once you hit the boundary the vertical vector field kind of becomes vertical and if you are looking at the transversality if you prove the transversality of the weak stable bundles with the vertical you violate exactly that if you had a something that projects that didn't project over anything so that kind of made not just our day but our years to come in some sense you know the fact that Ricardo found this interesting made us incredibly happy and now you can see why he asked us that question early on so once he came up with these critical values and these critical values were bigger than E0 then he said oh I have a question special for you too so why don't you now try to prove that this is also an energy level has energy even higher somehow which is higher than this sort of threshold given by the critical value story there so this was really made us very happy for a long time it makes me happy even today okay so to get the stronger property that K is bigger than Cu one has to combine the transversality of the weak bundles with a topological argument and somehow use a little bit more and then once you know that these are actually the grangians you go to the universal covering and then the whole proof kind of unfolds and this will somehow what we did later on with Gonzalo we will not do okay and this is enough to give K bigger than Cu just before I finish I think I have just wanted to slide more quickly and I will be over there is still an open problem here which is actually quite interesting we still don't know and this is kind of natural now in the context of Lingon-Basthen whether every energy level K less than Cu have gone to be points right so I think I will finish just with two slides because I think it's quite I feel quite fitting to finish with this for reasons that you will see in a minute so Ricardo had did a lot of work on the on these Lagrangian systems from the generosity point of view and he introduced the right notion of generosity which is perturbing and this is complete I hope it's completely natural by now at this point perturbing just with potentials functions that depend only on the point on the point X because if you do that you will not leave the class and he proved an exceptional theorem in this direction again was published in 86 but it was proved quite early on Gonzalo mentioned this theorem a little bit earlier mentioned this theorem in his lecture and he said that there is a generic set such that if you have a potential in that generic set and then the Lagrangian L plus Phi has actually a unique minimizing measure and in addition that unique minimizing measure is uniquely organic so his view was that somehow if you were allowing for generosity properties you can make all this theory all this sort of math theory very very sharp and you will sort of start kind of getting a very very clear picture of what was going on and in fact he conjectured and this was a wonderful conjecture in that one of those papers that for every Lagrangian you can find a generic set such that that unique minimizing measure is actually supported in a periodic orbit and when I say periodic orbit I mean periodic orbit and sort of a clearing point as in Gonzalo's lecture and again I guess I'm putting the most expensive conjecture there because it's free at this point I guess as Gonzalo said so it was also known from quite a few years ago that the unique measure if it is supported in a periodic orbit then generically the orbit will be hyperbolic I think the Gonzalo and Renato did that with these sort of canal interpretations that Gonzalo explained in his lecture and I think I would like to finish and leave you with this which I think is a very fitting final to see one of Ricardo's PhD students after decades of spectacular work including his conjecture for surfaces in the sea to topology thank you