 So the last talk this morning will be by Barney Bramham from Borum and he will speak about Poincaré's Last Geometric Theorem, a 21st century proof. Okay thank you very much and it's an honor to be here this week especially to be able to actually give a talk so I thank the organizers very much for that. So as many speakers have said already the last day or so Poincaré spent a lot of time thinking about motions of planets especially three and so back in 1912, a hundred years ago, I suppose he'd been thinking about them for close to 30 years and I would like to talk about an interesting question it was very influential called that he asked shortly before he died and this question which he came to from considering the motion of planets is just a question about maps on a shape like this so we have a two-dimensional surface it looks like a plate with a smaller plate a sort of hole removed from the middle and instead of this motion which was just planets moving under the laws of gravity we now have a law on our disk on our sorry on our annulus which says that if I have a point like a then this law tells me that there's some other point I should move to such as B and no two points go to the same one and this is defined for all points and so on so this whole thing now is called a transformation and he was particularly interested in configurations of planets that after a certain amount of time come back to their original position because then these are ones that you can then understand you know trivially for all time and then maybe once nearby well anyway so he's particularly interested in fixed points of this transformation so before I won't really explain exactly how this this annulus map is arrived at but let me just say for a sort of simpler a similar example I believe he also thought about and and Birkov also is the following situation so you have a surface maybe looks like the surface of an egg sphere like this one and you have a point a particle I've made it look like a planet and it's attached to this and it's there's no friction it's free to move around but it won't fall off maybe there's a magnet or something and if you give it a nudge in any direction then there's a sort of a natural path that will follow and then this is called a geodesic but it will keep going forever let's say and he's interested in the analogous question in this situation for that he asked about for celestial mechanics would be is there a curve that is closes up so this is a difficult question and I'd like to just quickly explain an idea that you may use to approach this and it goes like this you say well supposing I can find one such close curve so I have my my egg and I managed to find one path then I say maybe I examine the other parts that start off from this one and I see where they go and maybe if I move around a lot I find one that closes up so I take a point and I take a direction I don't care about the length of this direction and I nudge my my my particle and it goes along and I see what the natural path is and if I'm lucky maybe it comes back and intersects the red curve let's assume it does then on the second time I intersect it I have a well-defined vector arrow based at this red curve pointing the same side of the red curve as the first vector so in some sense I now have a law which says if I start off with a vector like this one then I can produce another one and I sort of forget about what exactly goes on on the blue curve in between this and then if you think about it abstractly the space of all such vectors we don't care about the length of it well there's sort of a two-dimensional family of them I can move tangent to the curve and I only to the red curve and I only care about vectors pointing to one side and then there's 180 degrees as I sweep out from top to bottom and I could also change the base point as I go around a circle so if you think about it for a minute you actually have something where the natural notion of closeness between these vectors gives you something that looks like a closed annulus so this this space I mentioned a minute ago and now this rule that says I start off with a vector a and I get a vector b now turns into a rule on the annulus so I now if I'm given a point on the annulus such as a this corresponds to a vector on the left hand side I follow the blue path around it gives me another vector and that corresponds to another point on the annulus so now I have a law that says I do move from a to b on the annulus so now we have two points of view of the same same situation and Pongka Ray did analogously for celestial mechanics and for three bodies and he was interested in are there paths which close up and in this case this would correspond to a point on the annulus which is fixed by the transformation so he's particularly interested in these now what's common to both of these approaches is that what's going on is that there's some three-dimensional space with some arrows on it a vector field and you're interested in the motion if you start at a point and you follow these arrows where you do you ever come back to yourself and in all of these he's picked he's cleverly picked a two dimensional sort of slice of this which will call a surface of section which has the nice property that if you start off on a point on this surface and follow the arrows you come back to another point on the surface and this gives you a map on the surface and in fact in both those cases you end up with something that is in some sense an annulus and I just wanted to mention this now because later on in my talk will show that well these are quite hard to find in general there's actually ends up a way using a partial differential equation that allows you to in some sense naturally arrive at surfaces of section if you're lucky and this was where I'll come to so I just wanted to draw a link between what was going on a hundred years ago and what happens a long time later so back to the story so Pong Gray now has a map of the annulus he's interested does this have fixed points because this would correspond to some interesting properties of the of the planets he's studying now he asked himself what properties of this map do I know now and there are two two interesting properties so one is that if you instead of just considering where a single point goes you take a set of points you consider what happens under the transformation well the set of course gets distorted like this cat it turns out it'll have the same area and this is somehow a property or comes from the nature of the laws of motion and then the second property is that on the two different boundary circles the points are moving on one circle to say counterclockwise and on say the inner circle clockwise or the reverse but in nevertheless in opposite strictly opposite directions and then this is often referred to as a twist so given these assumptions he oh yeah so so if I now took a set like this this cat and applied this I would get a cat of the same area but now it gets distorted in this fashion and then by playing around with many examples he asked if maybe such a map must automatically forget about where it comes from in mechanics just with these two properties I should also say orientation preserving and so on does this map always have a thick two fixed points so that is like these two red points where the transformation doesn't move them if you remove either of these assumptions you can clearly see that this this doesn't have to be true if you ignore area preserving it's easy to find a counter example and if you don't have a twist at the same you could for example rotate in the same direction around so so this is the question he asked and I believe he managed to answer it in a number of cases and I like to explain a beautiful idea that sort of almost works because then something I'll show later there's some sort of resemblance to this and the idea is roughly as follows said supposing I take so first of all I use the fact that this is twisting the inner boundary one direction in the outer boundary another direction so supposing I take a straight line that goes from the inner circle to the outer circle and then I start walking along this line from the inner boundary so if I start on the inner boundary clearly the transformation moves me to the right and if I as I go further up it moves me to the left so the image maybe looks like this of course it could be a lot more complicated than this come back to that nevertheless there must be some point by continuity as I move up that doesn't move either to the to the right or the left of the line and therefore stays on the line so there must be some point like this red one that moves to some other point on the line such as this green one it could be of course closer to the center or the same point so okay so now I could play the same game with another radio line so let me take this one and again I start from from close to the center and move outwards and there must be some point that moves neither to the right nor the left such as this red one and it goes to another point and I can do the same with another line maybe this time the red point moves inwards and then more lines and more lines and if I do this for all of them maybe I end up with a picture like this where I have two circles that are enclosing the central boundary so I have these two circles the red one being taken to the to the green one and in this example they intersect they are of course for intersection points here and of course we think about it what what's special about these intersection points is these are points which stay on the same radio line and go from the red curve to the green curve so assuming they only intersect once on that radio line these have to be fixed points so in this case we were lucky we even got four fixed points he's conjecturing maybe two now why should we have any of these intersection points so if we can show two intersection points we're done so to summarize we've constructed a red circle and a green circle going around this central boundary and we would like to show the intersect in at least two points and then we've proven there are two fixed points so so far we've only used one of the assumptions that was a twist we haven't used area preserving and if we don't use it something's wrong because the statement is clearly false if you don't use this so this comes in now so supposing by bad luck we end up with the green circle and the red circle not intersecting and they must look probably something like this because they have to go around the curve the boundary but then of course this means that the area enclosed inside the red circle is getting mapped to the to the region of much larger area inside the green circle so this is a problem because this map had to be area preserving and so of course we think about it two regions two circles which enclose two regions which have the same area have to intersect in at least two points because somehow either they sit exactly on each other one of them kind of has to come in and then come out again so you end up with two to these intersection points and then these two fixed points so that's basically the argument and it's a beautiful argument because one problem with with it is right at the beginning is that there need not be these unique points that move neither to the right or the left and you could then end up with so actually this picture I drew of course is actually not not what could happen because the regions inside are not going to have the same area but I only realize this just I didn't have time to change this but nevertheless you can make up configurations that clearly you're not going to have to two intersection points was less so a year later nevertheless bookoff actually found a correct argument or argument that really works and it's funny but humans, mathematicians, scientists I guess everyone you know when a question is answered they're really satisfied where you just want to know does the answer to the question suggest new interesting possibilities and things like this and I don't know actually how bookoff's proof was was used in other directions but certainly what Poincare's ideas I believe well I know inspired many people later and in particular Arnold in the 60s he made a conjecture there's a number of conjectures that generalized this question about the annulus and so he managed to make a statement that would include the included spaces of higher than two dimensions and not just a shape like an annulus and so on personally if I had been around at the time I would have put my money on it being a two-dimensional thing I mean I'm clearly wrong but that's what I would have been my gut instinct from the idea that we just looked at it seems somehow that the twist and the air preserving are somehow accounting for the two of the dimensions and so I would have been very surprised that this holds in higher dimensions but Arnold actually finds a statement that turns out to be to be true so now his question is a question in symplectic geometry so it's a geometry of the equations of motion and there's a long story here and I'm not going to say much about this except how it relates to what I want to say in the next part of my talk so in the 80s Gromov then introduced a new tool to this this geometry of the equations of motion and he showed that solutions to certain elliptic partial differential equation called pseudomorphic curves in the context of a symplectic space have wonderful properties now then there's this somehow was crucial to answering Arnold's question so just before Gromov Ali Ashberg answered the Arnold's conjecture for all two-dimensional surfaces to which it would apply and then Conley and Zaynder managed to do all dimensions for this particular class so Torai or even dimensional Torai so and then Floor sort of combining Conley Zaynder's ideas and Gromov's managed to do a substantial chunk of all of all the spaces and then there are many other people who also contributed to this I mentioned just a few so important so relevant for the rest of my talk is that in the 90s Hoffa-Wazowski and Zaynder discovered that in a certain framework these these these these pseudomorphic curves actually naturally give you a surface of section if you for certain Hamiltonian certain certain three-dimensional spaces with a sort of Hamiltonian vector field on now in general you it's not at all clear that even today that you actually have a surface of section coming from this is not known for every contact three manifold for example but they showed that in many cases sort of how to construct them and this will be important for what I want to say in a minute and in fact using this technology a pseudomorphic curves and surface of section to come from these turns out you can go sort of historically backwards and return and actually prove Pankare's question again and this is what I like to explain because it has a particularly a geometric feel to it and and in some sense once you if you assume that you have the existence of these solutions to this equation and so on then you end up with a picture that's not dissimilar to the one I showed ten minutes ago so so to explain this I'd like to so so just recall what the statement is the statement was you have a transformation of the surface and annulus it rotates the boundaries in opposite directions and it preserves area so let me reformulate this as a statement for disk maps just just for convenience so now I've sort of essentially shrunk the inner boundary to a point and I'm saying okay I now have a map on the disk it preserves area it fixes the origin and it sends the points on the boundary in some sense to the opposite direction to the points infinitesimally close or the linearization at the at the origin and then now your conclusion would like to be two fixed points besides the origin of course so now to apply these pseudomorphic curve techniques I'd like to re-embed kind of go backwards and re-embed the disk in a three-dimensional space with a flow but now pick the flow to be kind of convenient and the space to be topologically simple so I'm going to make it look like a Torah a solid Torah so a donor so I have a three-dimensional space here and visiting putting this disk map in here and now for the moment forget that this is a twist map just keep it area preserving so we put it in here and now we reconstruct some sort of vector field on the three-dimensional space so that if I started a point on the disk call it P and I follow the vector field I go in a curve and I come back to some other point I'll call it tfp here on the on the disk and it's the same map so fixed points of the map on the disk of course correspond to periodical bits of this flow here that go once around so in this situation it turns out that let me first open this up this is kind of more convenient for the next pictures so I'm just slicing it here and opening it up turns out that you can actually fill up the whole space by surfaces of section now of course they're all they're originally it was filled by surface of section that look like you'll these disc slices are also surface of section but here's a ones that are different they're not topologically discs these are now annulises each of them so what do we have here so roughly we have a filling of the space by surfaces I've written a foliation of course it's really a singular the foliation and this consists of a typical leaf looks like this so it's topologically an annulus because the boundary here should be identified with the one here it won't in general I just drew in my picture I made it look exactly flat like this of course it's going to wiggle around a lot and then the rest of the leaves looking similar they're all looking like annually and the relation to the dynamics is that the vector field that we have on the interior of each of these surfaces is strictly is never tangent to it so it's points always in some direction that's the red arrows they in this case they point upwards and on the boundary they are tangent so a small caveat is that they don't have to be tangent to the boundary of these curves that lie in the boundary of the three manifold but sister so in particular the boundaries of these surfaces if they exist have to be on periodic orbits of this flow on the solid torus so correspond to fixed points of the original map and these surfaces come from these are not pseudomorphic curves these are in an odd dimensional space but these are the projection of pseudomorphic curves in a four-dimensional space down to this picture here so now to explain supposing we actually have a map that's a twist map so this thing exists generically I've a generic air preserving map then you can actually get this of course it still doesn't immediately imply necessarily a lot of periodic orbits because it could be a much simpler looking affiliation a priori so let me take a slice like this and say well we said that the flow is transverse to the leaves in the three-dimensional picture so lying a little bit intuitively at least in the two-dimensional slice the flow is somehow transverse to the leaves of course the leaves could be moving around as we as we walk around the solid torus but approximately this is this is the case so you end up with a picture like this these big bold dots are now corresponding to fixed points of my map and these other leaves you have the flow transverse to them here so so now I'd like to apply this to the situation that we actually have a twist so now we say supposing we have a map with it as I said that you're moving infinitesimally in a different direction around the origin which is a fixed point to on the boundary and now we say okay supposing we have one of these fillings by surface of section I just described and let me just draw the transverse slice of these fillings it's easier to draw so supposing that this this this this foliation actually contains this particular fixed point at the origin then I mean this is this is not rigorous but I'm saying supposing locally around there it the leaves are meeting in this fashion here this just coming in all directions and supposing on the boundary the leaves are coming out like this so we'd like to conclude somehow that the structure of this foliation is rich enough that we can conclude two more fixed points right so worst possible scenario is that it's just boring and the leaves do this they just go from the end to the out well then of course that means the flow has to be transverse to the leaves on the interior so in some sense the flow is either moving counterclockwise or clockwise whichever but then of course we're assuming that there's a twist so the flow is transverse in different directions near the boundary and near to the interior so we have a problem so this can't happen so clearly all the leaves that start off at the origin cannot connect to the boundary so there has to be something extra in the foliation that prevents this something getting in the way so we would like to conclude two fixed points so let me try and assume that we can get away with this was just one so maybe something like this we have one and we have a leaf a curve that closes around and then we haven't filled in the rest of the picture but somehow imagine that the rest of the curves are then prevented from going to the boundary by this other ring in the middle okay then I'm still not done because I've only got one fixed point besides the origin but then if we think about it of course this leaf is also a surface of section in the solid three and the solid torus so again the flow has to move strictly outwards or strictly inwards we don't know which let's say it's outwards and that would mean that the whole region enclosed by this circle would end up when I come back again in a region with strictly greater area so we have a problem so there has to be a little bit more in there and that has to be at least one more fixed point so there's now sort of two windows one of which the vectors can move out one on which the flow can move inwards and so we get our two fixed points and you can make that reverse this is what the rest of the foliation could look like I mean of course there could well be many more fixed points but that is how it could look like the rest of the leaves now of course this actually only works if you have one of these foliations and so it works which you do have generically so if you took a dense set there's a dense set in the smooth topology on the space of air-preserving dipimorphism from the disk and you can find set all this up and you can obtain this and you can get the two fixed points now Poincare's statement is not closed under limits right if you see you can't conclude from the truth if the statements true for a generic map that it's true for for all air-preserving maps because the two fixed points could collide nevertheless if you buy this argument and you keep track of this obstacle in the middle these two red curves then you can take a limit with those and then you can get any air-preserving dipimorphism and you get again still at least two fixed points so this is really using Gromov's compactness there so before then we really only use sort of topological properties of these sections of the vectors of treaty transverse eventually to do that take a limit then you really need there's a analysis behind that and actually use Gromov compactness to get the full story so everything I just said more or less did not I mean apart from when I actually tried to prove there are two fixed points the whole existence of these surfaces of section filling up the space that does not require there to be a twist at all so you could hope maybe if you can construct these maybe you can apply these to other situations and it turns out there are there are a great many of these foliations by surface of section as I said if you take a typical three manifold with a suit let's say a contact three manifold you don't know there even exists a single surface of section coming from a pseudomorphic curve but in the particular this particularly a nice case where the flow is really two-dimensional because of the way we constructed it and you can find them and in fact you can pick any homology class on the boundary and you can make sure your leaves meet that homology class you can pick any single one period of orbit and ensure that that is a period of orbit in the boundary of one of your curves as an asymptotic orbit and you can also do this for sort of coverings of these these mapping Torah and so on so there's there's infinitely many of these that are not just trivially related so there's a pro a lot of information here so we construct if we if you have this then we just observe that it's very easy to go from there to say well if I happen to have a twist at the origin then there's also two fixed points but what about in general so there are a number of things you can do and I just mentioned one that just just because it relates to the notion of a twist so you might ask so okay if I have an error preserving map on the desk let's say generalize our notion of a twist map to be not just that there's the fixed point is at the origin and there's a little twist in some direction compared to the boundary but there are some fixed points somewhere on the interior such that infinitesimally you have this rotation in a different direction to the boundary and then of course if you took a typical map at random it would have a twist because this is you could just give a little perturbation to anything if it wasn't and you would have a twist or at least at least for some iterate so how about if you don't allow yourself to take these perturbations and you just say i'm just given any smooth error preserving this map is there any way to say which ones have a twist in some sense and which ones don't so clearly so what are the ones that clearly don't well clearly the identity map doesn't have a twist and and more generally any map which has a finite iterate is the identity clearly doesn't have a twist another another another class of maps of the disk error preserving maps of the disk that clearly don't have a twist of those that have only one fixed point and no other periodic points because if that one fixed point was a twist on craze theorem would tell you there were many more periodic points and we're just assuming those not and these do exist for example irrational rotations but they're also more interesting egodic examples and these maps clearly don't have a twist so there's two classes of maps here that clearly don't have a twist those that are roots of unity and those that have a single periodic point and it turns out all the others do have a twist if you allow yourself to take an iterate so the reason for this i won't explain but it comes down to the fact that we observed in the in the proof earlier that we found in our proof of Poincare's question we found that these two extra fixed points came from the fact that we had a foliation here which had this structure on the right hand side so on the left hand side all the leaves are kind of tree-like you could uh they don't have this sort of topology that you have on the right hand side we actually have a closed a closed chain of leaves so a twist kind of implies this picture and it turns out that actually the converse is also true if you have this picture you also have a twist so you can use this the cycles are equivalent to twists to prove the statement i just said which is that every smooth orientation preserving error preserving map of the disk is one of the following either has precisely one periodic point or it's a root of unity or there's some iterate that has an interior fixed point about which the motion is infinitesimally in a different direction to the boundary of course to me that rigorous you have to speak of covering maps and so on but intuitively that's clear so uh and then uh there are other things one can do and uh with with with this uh this structure that's there and i won't talk about those now um but uh i'll well uh stop there yeah thank you the questions remarks does this proof work for continuous map or do you assume smoothness uh i smooth smoothness yes okay yes yes yes right yeah yeah thank you yes so uh patrice la calva is and very interestingly in it from not using supermorphic curves at all comes up with um a picture very similar to this so he shows that there's a single or one-dimensional foliation if you have only a continuous error preserving or measure preserving map on a sphere or a disk and um uh this doesn't have a he doesn't associate it with a mapping torus instead he has a picture like this and he shows that given uh uh given a point and then the point that the transformation takes it to you can find an arc that is then connects these two that is transverse to every leaf in his single foliation and with this he was able to prove very strong results for continuous error preserving maps and he came through this not through considering pds but by a generalization of the brah uh plane trans translation theorem so that's very interesting he ends up with uh with these singular with these one-dimensional foliations with this transverse property from this point of view yeah thanks yeah just as there is some time maybe you could take five minutes to explain where where where the foliation comes from from this holomorphic disk is it possible so um right so so um so the foliations come from maybe i go back a second to this picture of the mapping torus so so here we are so here so really if you take a uh maybe this picture so here we have a three-dimensional space if you cross with this with the real line you have an uncompact four-dimensional space and you can equip this with a with an almost complex structure that is invariant in the art in this direction in the art coordinate in this four-dimensional space so now um where these leaves come from in here is these are the projection of surfaces that live in this four dimensional space you just project down into the three-dimensional space here and these surfaces if you can show that they are invaded upstairs and have the property that when you shift them up and down they don't intersect themselves then this corresponds downstairs into them having this transversality property there's also an energy condition that has to be the finite energy condition has to be satisfied and in some senses is very like uh i mean these curves are very much like negative gradient flows i mean so it's sort of i mean these come historically from the variational principle for which these periodical bits come from so in that in hindsight there's there's another natural links even though it's such a difficult you know amazing concept to have come up with and these come from that so then where do the leaves in the four dimensional space come from well these unlike the dynamics are actually very stable under perturbation so you can start off with a flow on the solid torus that is not the one you're interested in but as simple as possible for example in an irrational rotation and then you find by hand a foliation by uh by the by by these holomorphic curves and then you can homotope all the data so that means the vector field and there's the background symplectic structure from this model situation to the situation you're interested in and when you do this you can carry over all the information all the all these leaves that you started off with and at some point sort of in some sense at the last minute everything can kind of break up and you started off with a picture that's actually rather simple there's just a single binding orbit going through the center and just leaves going around it's really an an open book at that point but in the final picture you end up with is much richer and you end up with potentially a whole collection of these period of orbits and that's roughly where this comes from and this was a technique discovered by Hoffer-Vazosky and Zehnder like i said in the the 90s and they did this for the situation where instead of having a solid torus you have a three-dimensional sphere and you have a certain kind of this is sort of like an energy surface of a certain kind of Hamiltonian on an ambient symplectic manifold and they were constructed a filling of the three sphere by surfaces transverse to the vector field they were interested and they got marvelous results out of this in that situation where the flow is genuinely three-dimensional unlike here they could really only construct one foliation here because you really have something that's coming from a two-dimensional situation you can actually construct infinitely many and there's multiplayer around it so perhaps you can also in for the three sphere but that's presumably harder and so is that helpful maybe more questions remarks so if not we thank the speaker again